Mathematical Problems in Emerging Manufacturing ... - Hindawi

Mathematical Problems in Engineering

Mathematical Problems in Emerging Manufacturing Systems Management

Guest Editors Taho Yang Mu-Chen Chen Felix TS Chan Chiwoon Cho and Vikas Kumar

Mathematical Problems in EmergingManufacturing Systems Management



Guest Editors Taho Yang Mu-Chen Chen Felix T S ChanChiwoon Cho and Vikas Kumar

Copyright copy 2015 Hindawi Publishing Corporation All rights reserved

This is a special issue published in ldquoMathematical Problems in Engineeringrdquo All articles are open access articles distributed under theCreative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

Editorial Board

Mohamed Abd El Aziz EgyptFarid Abed-Meraim FranceSilvia Abrahao SpainPaolo Addesso ItalyClaudia Adduce ItalyRamesh Agarwal USAJuan C Aguero AustraliaRicardo Aguilar-Lopez MexicoTarek Ahmed-Ali FranceHamid Akbarzadeh CanadaMuhammad N Akram NorwayMohammad-Reza Alam USASalvatore Alfonzetti ItalyFrancisco Alhama SpainJuan A Almendral SpainSaiied Aminossadati AustraliaLionel Amodeo FranceIgor Andrianov GermanySebastian Anita RomaniaRenata Archetti ItalyFelice Arena ItalySabri Arik TurkeyFumihiro Ashida JapanHassan Askari CanadaMohsen Asle Zaeem USAFrancesco Aymerich ItalySeungik Baek USAKhaled Bahlali FranceLaurent Bako FranceStefan Balint RomaniaAlfonso Banos SpainRoberto Baratti ItalyMartino Bardi ItalyAzeddine Beghdadi FranceAbdel-Hakim Bendada CanadaIvano Benedetti ItalyElena Benvenuti ItalyJamal Berakdar GermanyEnrique Berjano SpainJean-Charles Beugnot FranceSimone Bianco ItalyDavid Bigaud FranceJonathan N Blakely USAPaul Bogdan USADaniela Boso Italy

Abdel-Ouahab Boudraa FranceFrancesco Braghin ItalyMichael J Brennan UKMaurizio Brocchini ItalyJulien Bruchon FranceJavier Bulduu SpainTito Busani USAPierfrancesco Cacciola UKSalvatore Caddemi ItalyJose E Capilla SpainAna Carpio SpainMiguel E Cerrolaza SpainMohammed Chadli FranceGregory Chagnon FranceChing-Ter Chang TaiwanMichael J Chappell UKKacem Chehdi FranceXinkai Chen JapanChunlin Chen ChinaFrancisco Chicano SpainHung-Yuan Chung TaiwanJoaquim Ciurana SpainJohn D Clayton USACarlo Cosentino ItalyPaolo Crippa ItalyErik Cuevas MexicoPeter Dabnichki AustraliaLuca DrsquoAcierno ItalyWeizhong Dai USAPurushothaman Damodaran USAFarhang Daneshmand CanadaFabio De Angelis ItalyStefano de Miranda ItalyFilippo de Monte ItalyXavier Delorme FranceLuca Deseri USAYannis Dimakopoulos GreeceZhengtao Ding UKRalph B Dinwiddie USAMohamed Djemai FranceAlexandre B Dolgui FranceGeorge S Dulikravich USABogdan Dumitrescu FinlandHorst Ecker AustriaKaren Egiazarian Finland

Ahmed El Hajjaji FranceFouad Erchiqui CanadaAnders Eriksson SwedenGiovanni Falsone ItalyHua Fan ChinaYann Favennec FranceRoberto Fedele ItalyGiuseppe Fedele ItalyJacques Ferland CanadaJose R Fernandez SpainSimme Douwe Flapper NetherlandsThierry Floquet FranceEric Florentin FranceFrancesco Franco ItalyTomonari Furukawa USAMohamed Gadala CanadaMatteo Gaeta ItalyZoran Gajic USACiprian G Gal USAUgo Galvanetto ItalyAkemi Galvez SpainRita Gamberini ItalyMaria Gandarias SpainArman Ganji CanadaXin-Lin Gao USAZhong-Ke Gao ChinaGiovanni Garcea ItalyFernando Garcıa SpainLaura Gardini ItalyAlessandro Gasparetto ItalyVincenzo Gattulli ItalyJurgen Geiser GermanyOleg V Gendelman IsraelMergen H Ghayesh AustraliaAnna M Gil-Lafuente SpainHector Gomez SpainRama S R Gorla USAOded Gottlieb IsraelAntoine Grall FranceJason Gu CanadaQuang Phuc Ha AustraliaOfer Hadar IsraelMasoud Hajarian IranFrederic Hamelin FranceZhen-Lai Han China

Thomas Hanne SwitzerlandTakashi Hasuike JapanXiao-Qiao He ChinaMarıa I Herreros SpainVincent Hilaire FranceEckhard Hitzer JapanJaromir Horacek Czech RepublicMuneo Hori JapanAndrs Horvth ItalyGordon Huang CanadaSajid Hussain CanadaAsier Ibeas SpainGiacomo Innocenti ItalyEmilio Insfran SpainNazrul Islam USAPayman Jalali FinlandReza Jazar AustraliaKhalide Jbilou FranceLinni Jian ChinaBin Jiang ChinaZhongping Jiang USANingde Jin ChinaGrand R Joldes AustraliaJoaquim Joao Judice PortugalTadeusz Kaczorek PolandTamas Kalmar-Nagy HungaryTomasz Kapitaniak PolandHaranath Kar IndiaKonstantinos Karamanos BelgiumC Masood Khalique South AfricaDo Wan Kim KoreaNam-Il Kim KoreaOleg Kirillov GermanyManfred Krafczyk GermanyFrederic Kratz FranceJurgen Kurths GermanyKyandoghere Kyamakya AustriaDavide La Torre ItalyRisto Lahdelma FinlandHak-Keung Lam UKAntonino Laudani ItalyAimersquo Lay-Ekuakille ItalyMarek Lefik PolandYaguo Lei ChinaThibault Lemaire FranceStefano Lenci ItalyRoman Lewandowski PolandQing Q Liang Australia

Panos Liatsis UKWanquan Liu AustraliaYan-Jun Liu ChinaPeide Liu ChinaPeter Liu TaiwanJean J Loiseau FrancePaolo Lonetti ItalyLuis M Lopez-Ochoa SpainVassilios C Loukopoulos GreeceValentin Lychagin NorwayF M Mahomed South AfricaYassir T Makkawi UKNoureddine Manamanni FranceDidier Maquin FrancePaolo Maria Mariano ItalyBenoit Marx FranceGeerard A Maugin FranceDriss Mehdi FranceRoderick Melnik CanadaPasquale Memmolo ItalyXiangyu Meng CanadaJose Merodio SpainLuciano Mescia ItalyLaurent Mevel FranceY V Mikhlin UkraineAki Mikkola FinlandHiroyuki Mino JapanPablo Mira SpainVito Mocella ItalyRoberto Montanini ItalyGisele Mophou FranceRafael Morales SpainAziz Moukrim FranceEmiliano Mucchi ItalyDomenico Mundo ItalyJose J Muoz SpainGiuseppe Muscolino ItalyMarco Mussetta ItalyHakim Naceur FranceHassane Naji FranceDong Ngoduy UKTatsushi Nishi JapanBen T Nohara JapanMohammed Nouari FranceMustapha Nourelfath CanadaSotiris K Ntouyas GreeceRoger Ohayon FranceMitsuhiro Okayasu Japan

Eva Onaindia SpainJavier Ortega-Garcia SpainAlejandro Ortega-Moux SpainNaohisa Otsuka JapanErika Ottaviano ItalyAlkiviadis Paipetis GreeceAlessandro Palmeri UKAnna Pandolfi ItalyElena Panteley FranceManuel Pastor SpainPubudu N Pathirana AustraliaFrancesco Pellicano ItalyMingshu Peng ChinaHaipeng Peng ChinaZhike Peng ChinaMarzio Pennisi ItalyMatjaz Perc SloveniaFrancesco Pesavento ItalyM do Rosario Pinho PortugalAntonina Pirrotta ItalyVicent Pla SpainJavier Plaza SpainJean-Christophe Ponsart FranceMauro Pontani ItalyStanislav Potapenko CanadaSergio Preidikman USAChristopher Pretty New ZealandCarsten Proppe GermanyLuca Pugi ItalyYuming Qin ChinaDane Quinn USAJose Ragot FranceK Ramamani Rajagopal USAGianluca Ranzi AustraliaSivaguru Ravindran USAAlessandro Reali ItalyGiuseppe Rega ItalyOscar Reinoso SpainNidhal Rezg FranceRicardo Riaza SpainGerasimos Rigatos GreeceJose Rodellar SpainRosana Rodriguez-Lopez SpainIgnacio Rojas SpainCarla Roque PortugalAline Roumy FranceDebasish Roy IndiaR Ruiz Garcıa Spain

Antonio Ruiz-Cortes SpainIvan D Rukhlenko AustraliaMazen Saad FranceKishin Sadarangani SpainMehrdad Saif CanadaMiguel A Salido SpainRoque J Saltaren SpainFrancisco J Salvador SpainAlessandro Salvini ItalyMaura Sandri ItalyMiguel A F Sanjuan SpainJuan F San-Juan SpainRoberta Santoro ItalyIlmar Ferreira Santos DenmarkJose A Sanz-Herrera SpainNickolas S Sapidis GreeceE J Sapountzakis GreeceThemistoklis P Sapsis USAAndrey V Savkin AustraliaValery Sbitnev RussiaThomas Schuster GermanyMohammed Seaid UKLotfi Senhadji FranceJoan Serra-Sagrista SpainLeonid Shaikhet UkraineHassan M Shanechi USASanjay K Sharma IndiaBo Shen GermanyBabak Shotorban USAZhan Shu UKDan Simon USALuciano Simoni ItalyChristos H Skiadas GreeceMichael Small Australia

Francesco Soldovieri ItalyRaffaele Solimene ItalyRuben Specogna ItalySri Sridharan USAIvanka Stamova USAYakov Strelniker IsraelSergey A Suslov AustraliaThomas Svensson SwedenAndrzej Swierniak PolandYang Tang GermanySergio Teggi ItalyRoger Temam USAAlexander Timokha NorwayRafael Toledo-Moreo SpainGisella Tomasini ItalyFrancesco Tornabene ItalyAntonio Tornambe ItalyFernando Torres SpainFabio Tramontana ItalySebastien Tremblay CanadaIrina N Trendafilova UKGeorge Tsiatas GreeceAntonios Tsourdos UKVladimir Turetsky IsraelMustafa Tutar SpainEfstratios Tzirtzilakis GreeceFilippo Ubertini ItalyFrancesco Ubertini ItalyHassan Ugail UKGiuseppe Vairo ItalyKuppalapalle Vajravelu USARobertt A Valente PortugalRaoul van Loon UKPandian Vasant Malaysia

M E Vazquez-Mendez SpainJosep Vehi SpainKalyana C Veluvolu KoreaFons J Verbeek NetherlandsFranck J Vernerey USAGeorgios Veronis USAAnna Vila SpainRafael J Villanueva SpainU E Vincent UKMirko Viroli ItalyMichael Vynnycky SwedenJunwu Wang ChinaShuming Wang SingaporeYan-WuWang ChinaYongqi Wang GermanyJeroen A S Witteveen NetherlandsYuqiang Wu ChinaDash Desheng Wu CanadaGuangming Xie ChinaXuejun Xie ChinaGen Qi Xu ChinaHang Xu ChinaXinggang Yan UKLuis J Yebra SpainPeng-Yeng Yin TaiwanIbrahim Zeid USAHuaguang Zhang ChinaQingling Zhang ChinaJian Guo Zhou UKQuanxin Zhu ChinaMustapha Zidi FranceAlessandro Zona Italy

Contents

Mathematical Problems in Emerging Manufacturing SystemsManagement Taho Yang Mu-Chen ChenFelix T S Chan Chiwoon Cho and Vikas KumarVolume 2015 Article ID 680121 2 pages

Clustering Ensemble for Identifying Defective Wafer Bin Map in Semiconductor ManufacturingChia-Yu HsuVolume 2015 Article ID 707358 11 pages

AMultiple Attribute Group Decision Making Approach for Solving Problems with the Assessment ofPreference Relations Taho Yang Yiyo Kuo David Parker and Kuan Hung ChenVolume 2015 Article ID 849897 10 pages

Integrated Supply Chain Cooperative Inventory Model with Payment Period Being Dependent onPurchasing Price under Defective Rate Condition Ming-Feng Yang Jun-Yuan Kuo Wei-Hao Chenand Yi LinVolume 2015 Article ID 513435 20 pages

Joint Optimization Approach of Maintenance and Production Planning for a Multiple-ProductManufacturing System Lahcen Mifdal Zied Hajej and Sofiene DellagiVolume 2015 Article ID 769723 17 pages

Impacts of Transportation Cost on Distribution-Free Newsboy Problems Ming-Hung ShuChun-Wu Yeh and Yen-Chen FuVolume 2014 Article ID 307935 10 pages

Undesirable Outputsrsquo Presence in Centralized Resource Allocation Model Ghasem TohidiHamed Taherzadeh and Sara HajihaVolume 2014 Article ID 675895 6 pages

The Integration of Group Technology and Simulation Optimization to Solve the Flow Shop with HighlyVariable Cycle Time Process A Surgery Scheduling Case Study T K Wang F T S Chan and T YangVolume 2014 Article ID 796035 10 pages

EditorialMathematical Problems in Emerging ManufacturingSystems Management

Taho Yang1 Mu-Chen Chen2 Felix T S Chan3 Chiwoon Cho4 and Vikas Kumar5

1 Institute of Manufacturing Information and Systems National Cheng Kung University Tainan 70101 Taiwan2Department of Transportation and Logistics Management National Chiao Tung University Taipei 100 Taiwan3Department of Industrial and Systems Engineering The Hong Kong Polytechnic University Hong Kong4Department of Industrial Engineering University of Ulsan Ulsan 680-749 Republic of Korea5Bristol Business School University of the West of England Bristol BS16 1QY UK

Correspondence should be addressed to Taho Yang tyangmailnckuedutw

Received 8 April 2015 Accepted 8 April 2015

Copyright copy 2015 Taho Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This special issue aims to address the mathematical problemsassociated with the management of innovative emergingmanufacturing systems The scope of innovative manufac-turing systems management in this special issue addressesthe emerging issues from production and operations man-agement manufacturing strategy leanagile manufacturingsupply chain and logistics management healthcare systemsmanagement and so forth The contributions gathered inthis special issue offer a snapshot of different interestingresearches problems and solutions In the following webriefly highlight these topics and synthesize the content ofeach paper

The paper ldquoImpacts of Transportation Cost onDistribution-Free Newsboy Problemsrdquo by M-H Shu etal addresses a distribution-free newsboy problem (DFNP)for a vendor to decide a productrsquos stock quantity in asingle-period inventory system to sustain its least maximum-expected profits The transportation cost is formulated as afunction of shipping quantity and is modeled as a nonlinearregression form An optimal solution of the order quantity iscomputed on the basis of Newtonrsquos approach to ameliorate itscomplexity of computation The empirical results are quitecompetitive with the results from the existing literature

The paper ldquoThe Integration of Group Technology andSimulation Optimization to Solve the Flow Shop with HighlyVariable Cycle Time Process A Surgery Scheduling CaseStudyrdquo by T K Wang et al introduces a case of healthcare

system application It proposes an algorithm that allowsthe estimation of the mean effective process time and thecoefficient of variation It also develops a group technologybased takt time A simulation model is combined with thecase study and the capacity buffers are optimized against theremaining variability for each group The empirical resultsfrom a practical application are quite promising

The paper ldquoUndesirable Outputsrsquo Presence in CentralizedResource Allocation Modelrdquo by G Tohidi et al extendsthe existing Data Envelopment Analysis (DEA) literatureand proposes a new Centralized Resource Allocation (CRA)model to assess the overall efficiency of system consisting ofDecisionMakingUnits (DMUs) by using directional distancefunction when DMUs produce desirable and undesirableoutputs

The paper ldquoA Multiple Attribute Group Decision MakingApproach for Solving Problems with the Assessment ofPreference Relationsrdquo by T Yang et al proposes to usea fuzzy preference relations matrix which satisfies additiveconsistency in solving a multiple attribute group decisionmaking (MAGDM) problem It takes a heterogeneous groupof experts into consideration A numerical example is used totest the proposed approach and the results illustrate that themethod is simple effective and practical

The paper ldquoIntegrated Supply Chain Cooperative Inven-tory Model with Payment Period Being Dependent on Pur-chasing Price under Defective Rate Conditionrdquo byM-F Yang

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 680121 2 pageshttpdxdoiorg1011552015680121

2 Mathematical Problems in Engineering

et al aims at finding the maximum of the joint expectedtotal profit and at coming up with a suitable inventorypolicy It solves the trade-off between increased postponedpayment deadline and the decreased profit for a buyer andvice versa for a vendor Its numerical illustrations provideuseful managerial insights

The paper ldquoClustering Ensemble for IdentifyingDefectiveWafer Bin Map in Semiconductor Manufacturingrdquo by C-YHsu proposes a clustering ensemble approach to facilitatewafer bin map defect detection problem from semiconductormanufacturing It adopts a series of algorithms to solvethe proposed problem such as mountain function 119896-meansparticle swarm optimization and neural network modelThenumerical results are promising

The paper ldquoJoint Optimization Approach of Maintenanceand Production Planning for a Multiple-Product Manufac-turing Systemrdquo by L Mifdal et al deals with the problemof maintenance and production planning for randomly fail-ing multiple-product manufacturing system It establishessequentially an economical production plan and an optimalmaintenance strategy taking into account the influence of theproduction rate on the systemrsquos degradation Analytical mod-els are developed in order to minimize sequentially the totalproductioninventory cost and then the total maintenancecost Finally a numerical example is presented to illustrate theusefulness of the proposed approach

The paper ldquoThe Dynamics of Bertrand Model with Tech-nological Innovationrdquo by FWang et al studied the dynamicsof a Bertrand duopoly game with technology innovationwhich contains bounded rational and naive players Thestability of the equilibrium point the bifurcation and chaoticbehavior of the dynamic system have been analyzed It con-cludes that technology innovation can enlarge the stabilityregion of the speed and control the chaos of the dynamicsystem effectively

Acknowledgments

The guest editors would like to deeply thank all the authorsthe reviewers and the Editorial Board involved in thepreparation of this issue

Taho YangMu-Chen ChenFelix T S ChanChiwoon ChoVikas Kumar

Research ArticleClustering Ensemble for Identifying Defective WaferBin Map in Semiconductor Manufacturing

Chia-Yu Hsu

Department of Information Management and Innovation Center for Big Data amp Digital Convergence Yuan Ze UniversityChungli Taoyuan 32003 Taiwan

Correspondence should be addressed to Chia-Yu Hsu cyhsusaturnyzuedutw

Received 30 October 2014 Revised 27 January 2015 Accepted 28 January 2015

Academic Editor Chiwoon Cho

Copyright copy 2015 Chia-Yu HsuThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wafer bin map (WBM) represents specific defect pattern that provides information for diagnosing root causes of low yield insemiconductor manufacturing In practice most semiconductor engineers use subjective and time-consuming eyeball analysis toassess WBM patterns Given shrinking feature sizes and increasing wafer sizes various types of WBMs occur thus relying onhuman vision to judge defect patterns is complex inconsistent and unreliable In this study a clustering ensemble approach isproposed to bridge the gap facilitating WBM pattern extraction and assisting engineer to recognize systematic defect patternsefficiently The clustering ensemble approach not only generates diverse clusters in data space but also integrates them in labelspace First the mountain function is used to transform data by using pattern density Subsequently k-means and particle swarmoptimization (PSO) clustering algorithms are used to generate diversity partitions and various label results Finally the adaptiveresponse theory (ART) neural network is used to attain consensus partitions and integration An experiment was conducted toevaluate the effectiveness of proposed WBMs clustering ensemble approach Several criterions in terms of sum of squared errorprecision recall and F-measure were used for evaluating clustering results The numerical results showed that the proposedapproach outperforms the other individual clustering algorithm

1 Introduction

To maintain their profitability and growth despite con-tinual technology migration semiconductor manufacturingcompanies provide wafer manufacturing services generatingvalue for their customers through yield enhancement costreduction on-time delivery and cycle time reduction [1 2]The consumer market requires that semiconductor productsexhibiting increasing complexity be rapidly developed anddelivered to market Technology continues to advance andrequired functionalities are increasing thus engineers havea drastically decreased amount of time to ensure yieldenhancement and diagnose defects [3]

The lengthy process of semiconductor manufacturinginvolves hundreds of steps in which big data includingthe wafer lot history recipe inline metrology measurementequipment sensor value defect inspection and electrical testdata are automatically generated and recorded Semicon-ductor companies experience challenges integrating big data

from various sources into a platform or data warehouse andlack intelligent analytics solutions to extract useful manufac-turing intelligence and support decision making regardingproduction planning process control equipment monitor-ing and yield enhancement Scant intelligent solutions havebeen developed based on data mining soft computing andevolutionary algorithms to enhance the operational effective-ness of semiconductor manufacturing [4ndash7]

Circuit probe (CP) testing is used to evaluate each dieon the wafer after the wafer fabrication processes Waferbin maps (WBMs) represent the results of a CP test andprovide crucial information regarding process abnormalitiesfacilitating the diagnosis of low-yield problems in semicon-ductor manufacturing In WBM failure patterns the spatialdependences across wafers express systematic and randomeffects Various failure patterns are required these patterntypes facilitate rapidly identifying the associate root causes oflow yield [8] Based on the defect size shape and locationon the wafer the WBM can be expressed as specific patterns



such as rings circles edges and curves Defective dies causedby random particles are difficult to completely remove andtypically exhibit nonspecific patterns Most WBM patternsconsisted of a systematic pattern and a random defect [8ndash10]

In practice thousands ofWBMs are generated for inspec-tion and engineers must spend substantial time on patternjudgment rather than determining the assignable causes oflow yield Grouping similar WBMs into the same clustercan enable engineers to effectively diagnose defects Thecomplicated processes and diverse products fabricated insemiconductor manufacturing can yield variousWBM typesmaking it difficult to detect systematic patterns by using onlyeyeball analysis

Clustering analysis is used to partition data into severalgroups in which the observations are homogeneous withina group and heterogeneous between groups Clusteringanalysis has been widely applied in applications such asgrouping [11] and pattern extraction [12] However mostconventional clustering algorithms influence the result basedon the data type algorithm parameter settings and priorinformation For example the 119896-means algorithm is used toanalyze substantial amount of data that exhibit time com-plexity [13] However the results of the 119896-means algorithmdepend on the initially selected centroid and predefinednumber of clusters To address the disadvantages of the 119896-means algorithm evolutionary methods have been developedto conduct data clustering such as the genetic algorithm(GA) and particle swarm optimization (PSO) [14] PSO isparticularly advantageous because it requires less parameteradjustment compared with the GA [15]

Combining results by applying distinct algorithms tothe same data set or algorithm by using various parametersettings yields high-quality clusters Based on the criteria ofthe clustering objectives no individual clustering algorithmis suitable for whole problem and data type Compared withindividual clustering algorithms clustering ensembles thatcombine multiple clustering results yield superior clusteringeffectiveness regarding robustness and stability incorpo-rating conflicting results across partitions [16] Instead ofsearching for an optimal partition clustering ensemblescapture a consensus partition by integrating diverse partitionsfrom various clustering algorithms Clustering ensembleshave been developed to improve the accuracy robustnessand stability of clustering such ensembles typically involvetwo steps The first step involves generating a basic set ofpartitions that can be similar to or distinct from those ofvarious parameters and cluster algorithms [17] The secondstep involves combining the basic set of partitions by usinga consensus function [18] However with the shrinkingintegrated circuit feature size and complicatedmanufacturingprocess the WBM patterns become more complex becauseof various defect density die size and wafer rotation It isdifficult to extract defect pattern by single specific cluster-ing approach and needs to incorporate different clusteringaspects for various complicated WBM patterns

To bridge the need in real setting this study proposes aWBMclustering ensemble approach to facilitateWBMdefectpattern extraction First the target bin value is categorizedinto binary value and the wafer maps are transformed from

two-dimensional to one-dimensional data Second 119896-meansand PSO clustering algorithms are used to generate variousdiversity partitions Subsequently the clustering results areregarded as label representations to facilitate aggregatingthe diversity partition by using an adaptive response theory(ART) neural network To evaluate the validity of the pro-posedmethod an experimental analysis was conducted usingsix typical patterns found in the fabrication of semiconduc-tor wafers Using various parameter settings the proposedcluster ensembles that combine diverse partitions instead ofusing the original features outperform individual clusteringmethods such as 119896-means and PSO

The remainder of this study is organized as followsSection 2 introduces a fundamentalWBM Section 3 presentsthe proposed approach to the WBM clustering problemSection 4 provides experimental comparisons applying theproposed approach to analyze the WBM clustering problemSection 5 offers a conclusion and the findings and futureresearch directions are discussed

2 Related Work

A WBM is a two-dimensional failure pattern Based onvarious defects types random systematic and mixed fail-ure patterns are primary types of WBMs generated duringsemiconductor fabrication [19 20] Random failure patternsare typically caused by random particles or noises in themanufacturing environment In practice completely elimi-nating these random defects is difficult Systematic failurepatterns show the spatial correlation across wafers such asrings crescentmoon edge and circles Figure 1 shows typicalWBM patterns which are transformed into binary values forvisualization and analysis The dies that pass the functionaltest are denoted as 0 and the defective dies are denoted as1 Based on the systematic patterns domain engineers canrapidly determine the assignable causes of defects [8] Mixedfailure patterns comprise the random and systematic defectson a wafer The mixed pattern can be identified if the degreeof the random defect is slight

Defect diagnosis of facilitating yield enhancement iscritical in the rapid development of semiconductor manu-facturing technology An effective method of ensuring thatthe causes of process variation are assignable is analyz-ing the spatial defect patterns on wafers WBMs providecrucial guidance enabling engineers to rapidly determinethe potential root causes of defects by identifying patternsMost studies have used neural network and model-basedapproaches to extract common WBM patterns Hsu andChien [8] integrated spatial statistical analysis and an ARTneural network to conduct WBM clustering and associatedthe patterns with manufacturing defects to facilitate defectdiagnosis In addition to ART neural network Liu andChien [10] applied moment invariant for shape clusteringof WBMs Model-based clustering algorithms are used toconstruct a model for each cluster and compare the like-lihood values between clusters to identify defect patternsWang et al [21] used model-based clustering applying aGaussian expectation maximization algorithm to estimatedefect patterns Hwang and Kuo [22] modeled global defects

Mathematical Problems in Engineering 3

(a) (b) (c)

(d) (e) (f)

Figure 1 Typical WBM patterns

and local defects in clusters exhibiting ellipsoidal patternsand local defects in clusters exhibiting linear or curvilinearpatterns Yuan and Kuo [23] used Bayesian inference toidentify the patterns of spatial defects in WBMs Drivenby continuous migration of semiconductor manufacturingtechnology the more complicated types of WBM patternshave been occurred due to the increase of wafer size andshrinkage of critical dimensions on specific aspect of complexWBM pattern and little research has evaluated using theclustering ensemble approach to analyze WBMs and extractfailure patterns

3 Proposed Approach

The terminologies and notations used in this study are asfollows

119873119892 number of gross dies119873119908 number of wafers119873119901 number of particles119873119888 number of clusters119873119887 number of bad dies119894 wafer index 119894 = 1 2 119873119908119895 dimension index 119895 = 1 2 119873119892119896 cluster index 119896 = 1 2 119873119888119897 particle index 119897 = 1 2 119873119901119902 clustering result index 119902 = 1 2 119872119903 bad die index 119903 = 1 2 119873119887119904 clustering subobjective in PSO clustering 119904 =

1 2 3119880 uniform random number in the interval [0 1]120596V inertia weight of velocity update120596119904 weight of clustering subobjective119888119901 personal best position acceleration constants

119888119892 global best position acceleration constants120573 a normalization factor119898 a constant for approximate density shape inmoun-tain function119910119903 the 119903th bad die on a wafer119899119896 the number of WBMs in the 119896th cluster119899119897119896 the number of WBMs in the 119896th cluster of 119897thparticle119862119896 subset of WBMs in the 119896th cluster119909max maximum value in the WBM data

m119896 vector of the 119896th cluster centroidm119896 = [1198981198961 1198981198962

119898119896119873119892]

m119897119896 vector centroid of the 119896th cluster of 119897th particlep119897 vector centroids of the 119897th particle p119897 = [1198981198971 1198981198972

119898119897119896]120579119897119895 position of the 119897th particle at the 119895th dimension119881119897119895 velocity of the 119897th particle at the 119895th dimension120595119897119895 personal best position (119901best) of the 119897th particle at119895th dimension120595119892119895 global best position (119892best) at the 119895th dimensionx119894 vector of the 119894th WBM x119894 = [1199091198941 1199091198942 119909119894119873119892

]

Θ119897 vector position of the 119897th particle Θ119897 = [1205791198971 1205791198972

120579119897119873119892]

V119897 vector velocity of the 119897th particle V119897 = [1198811198971 1198811198972

119881119897119873119892]

120595119897 vector personal best of the 119897th particle 120595

119897= [1205951198971

1205951198972 120595119897119873119892]

120595119892 vector global best position 120595

119892= [1205951198921 1205951198922

120595119892119873119892]


Consensuspartition

Final clusteringresults

WBMs

1 clustering

q clustering

2 clustering

First stage data space Second stage label space

Labels1205871Labels 1205872

Labels120587 q

Figure 2 A framework for WBMs clustering ensemble

31 Problem Definition of WBM Clustering Ensemble Clus-tering ensembles can be regarded as two-stage partitions inwhich various clustering algorithms are used to assess thedata space at the first stage and consensus function is used toassess the label space at the second stage Figure 2 shows thetwo-stage clustering perspective Consensus function is usedto develop a clustering combination based on the diversity ofthe cluster labels derived at the first stage

Let X = x1 x2 x119873119908 denote a set of 119873119908 WBMsand Π = 1205871 1205872 120587119872 denote a set of partitions basedon 119872 clustering results The various partitions of 120587119902(119909119894)

represent a label assigned to 119909119894 by the 119902th algorithm Eachlabel vector 120587119902 is used to construct a representation Πin which the partitions of X comprise a set of labels foreach wafer x119894 119894 = 1 119873119908 Therefore the difficulty ofconstructing a clustering ensemble is locating a new partitionΠ that provides a consensus partition satisfying the labelinformation derived from each individual clustering result ofthe original WBM For each label 120587119902 a binary membershipindicator matrix119867

(119902) is constructed containing a column foreach cluster All values of a row in the119867(119902) are denoted as 1 ifthe row corresponds to an object Furthermore the space ofa consensus partition changes from the original 119873119892 featuresinto 119873119908 features For example Table 1 shows eight WBMsgrouped using three clustering algorithms (1205871 1205872 1205873) thethree clustering results are transformed into clustering labelsthat are transformed into binary representations (Table 2)Regarding consensus partitions the binarymembership indi-cator matrix 119867

(119902) is used to determine a final clusteringresult using a consensus model based on the eight features(V1 V2 V8)

32 Data Transformation The binary representation of goodand bad dies is shown in Figure 3(a) Although this binaryrepresentation is useful for visualisation displaying the spa-tial relation of each bad die across a wafer is difficult

To quantify the spatial relations and increase the densityof a specific feature the mountain function is used to trans-form the binary value into a continuous valueThe mountainmethod is used to determine the approximate cluster centerby estimating the probability density function of a feature[24] Instead of using a grid node a modified mountain

Table 1 Original label vectors

1205871

1205872

1205873

x1

1 1 1x2

1 1 1x3

1 1 1x4

2 2 1x5

2 2 2x6

3 1 2x7

3 1 2x8

3 1 2

Table 2 Binary representation of clustering ensembles

Clustering results V1

V2

V3

V4

V5

V6

V7

V8

119867(1)

ℎ11

1 1 1 0 0 0 0 0ℎ12

0 0 0 1 1 0 0 0ℎ13

0 0 0 0 0 1 1 1

119867(2) ℎ

211 1 1 0 0 1 1 1

ℎ22

0 0 0 1 1 0 0 0

119867(3) ℎ

311 1 1 1 0 0 0 0

ℎ32

0 0 0 0 1 1 1 1

function can employ data points by using a correlation self-comparison [25] The modified mountain function for a baddie 119903 on a wafer119872(119910119903) is defined as follows

119872(119910119903) =

119873119887

sum

119903=1

119890minus119898120573119889(119910119903 119910119904) 119903 = 1 2 3 119873119887 (1)

where

120573 = (119889 (119910119903 minus 119910wc)

119873119887

)

minus1

(2)

and 119889(119910119903 119910119904) is the distance between dices 119903 and 119904 Parameter120573 is the normalization factor for the distance between baddie 119903 and the wafer centroid 119910wc Parameter 119898 is a constantParameter 119898120573 determines the approximate density shape ofthewafer Figure 3(b) shows an example ofWBMtransforma-tion Two types of data are used to generate a basic set of par-titions Moreover each WBM must sequentially transform


(1) Randomly select 119896 data as the centroid of cluster(2) Repeat

For each data vector assign each data into the group with respect to the closest centroid byminimum Euclidean distancerecalculate the new centroid based on all data within the group

end for(3) Steps 1 and 2 are iterated until there is no data change

Procedure 1 119896-means algorithm

(a) Binary value

51015202530

(b) Continuous value

Figure 3 Representation of wafer bin map by binary value and continuous value

from a two-dimensional map into a one-dimensional datavector [8] Such vectors are used to conduct further clusteringanalysis

33 Diverse Partitions Generation by 119896-Means and PSO Clus-tering Both 119896-means andPSO clustering algorithms are usedto generate basic partitions To consider the spatial relationsacross awafer both the binary and continuous values are usedto determine distinct clustering results by using 119896-means andPSO clustering Subsequently various numbers of clusters areused for comparison

119870-means is an unsupervised method of clustering analy-sis [13] used to group data into several predefined numbersof clusters by employing a similarity measure such as theEuclidean distance The objective function of the 119896-meansalgorithm is tominimize the within-cluster difference that isthe sum of the square error (SSE) which is determined using(3) The 119896-means algorithm consists of the following steps asshown in Procedure 1

SSE =

119873119888

sum

119896=1

sum

x119894isin119862119896(x119894 minusm119896)

2 (3)

Data clustering is regarded as an optimisation problemPSO is an evolutionary algorithm [14] which is used to searchfor optimal solutions based on the interactions amongstparticles it requires adjusting fewer parameters comparedwith using other evolutionary algorithms van derMerwe andEngelbrecht [26] proposed a hybrid algorithm for clusteringdata in which the initial swarm is determined using the119896-means result and PSO is used to refine the cluster results

A single particle p119897 represents the 119896 cluster centroidvectors p119897 = [1198981198971 1198981198972 119898119897119896] A swarm defines a numberof candidate clusters To consider the maximal homogeneitywithin a cluster and heterogeneity between clusters a fitnessfunction is used to maximize the intercluster separation andminimize the intracluster distance and quantisation error

119891 (p119894Z119897) = 1205961 times 119869119890 + 1205962 times 119889max (p119897Z119897) + 1205963

times (119883max minus 119889min (p119897)) (4)

where Z119897 is a matrix representing the assignment of theWBMs to the clusters of the 119897th particle The followingquantization error equation is used to evaluate the level ofclustering performance

119869119890 =sum119873119888

119896=1lfloorsumforallx119894isin119862119896 119889 (x119894 119898119896) 119899119896rfloor

119870 (5)

In addition

119889max (p119894Z119897) = max119896=12119873119888

[[

[

sum

forallx119894isin119862119897119896

119889 (x119894m119897119896)119899119897119896

]]

]

(6)

is the maximum average Euclidean distance of particle to theassigned clusters and

119889min (p119897) = minforall119906V119906 =V

[119889 (m119897119906m119897V)] (7)

is the minimum Euclidean distance between any pair ofclusters Procedure 2 shows the steps involved in the PSOclustering algorithm


(1) Initialize each particle with 119896 cluster centroids(2) For iteration 119905 = 1 to 119905 = max do

For each particle 119897 doFor each data pattern x

119894

calculate the Euclidean distance to all cluster centroids and assign pattern x119894to cluster 119888

119896

which has the minimum distanceend forcalculate the fitness function 119891(p

119894Z119897)

end forfind the personal best and global best positions of each particleupdate the cluster centroids by the update velocity equation (i) and update coordinate equation (ii)V119894(119905 + 1) = 120596VV119894(119905) + 119888

119901119906(120595119897(119905) minusΘ

119897(119905)) + 119888

119892119906(120595119892(119905) minusΘ

119897(119905)) (i)

Θ119897(119905 + 1) = Θ

119897(119905) + V

119897(119905 + 1) (ii)

end for(3) Step 2 is iterated until these is no data change

Procedure 2 PSO clustering algorithm

34 Consensus Partition by Adaptive Response Theory ARThas been used in numerous areas such as pattern recognitionand spatial analysis [27] Regarding the unstable learningconditions caused by new data ART can be used to addressstability and plasticity because it addresses the balancebetween stability and plasticity match and reset and searchand direct access [8] Because the input labels are binarythe ART1 neural network [27] algorithm is used to attain aconsensus partition of WBMs

The consensus partition approach is as follows

Step 1 Apply 119896-means and PSO clustering algorithms anduse various parameters (eg various numbers of clusters andtypes of input data) to generate diverse clusters

Step 2 Transform the original clustering label into binaryrepresentationmatrix119867 as an input forART1 neural network

Step 3 Apply ART1 neural network to aggregate the diversepartitions

4 Numerical Experiments

In this section this study conducts a numerical study todemonstrate the effectiveness of the proposed clusteringensemble approach Six typical WBM patterns from semi-conductor fabrication were used such as moon edge andsector In the experiments the percentage of defective diesin six patterns is designed based on real casesWithout losinggenerality of WBM patterns the data have been systemati-cally transformed for proprietary information protection ofthe case company Total 650 chips were exposed on a waferBased on various degrees of noise each pattern type was usedto generate 10 WBMs for estimating the validity of proposedclustering ensemble approach The noise in WBM could becaused from random particles across a wafer and test bias inCP test which result in generating bad die randomly on awafer and generating good die within a group of bad dies Itmeans that some bad dices are shown as good dice and the

1012

1518

2315221370

1184 1098945

02004006008001000120014001600

0

5

10

15

20

25

03 04 05 06 07

SSE

Clus

ter n

umbe

r

ART1 vigilance threshold

Clustering numberSSE

Figure 4 Comparison of various ART1 vigilance threshold

density of bad die could be sparse For example the value ofdegree of noise is 002 which represents total 2 good die andbad dies are inverse

The proposed WBM clustering ensemble approach wascompared with 119896-means PSO clustering method and thealgorithm proposed by Hsu and Chien [8] Six numbers ofclusters were used for single 119896-means methods and singlePSO clustering algorithms Table 3 showed the parametersettings for PSO clustering The number of clusters extractedbyART1 neural network is sensitive to the vigilance thresholdvalue The high vigilance threshold is used to produce moreclusters and the similarity within a cluster is high In contrastthe low vigilance threshold results in fewer numbers ofclusters However the similarity within a cluster could below To compare the parameter setting of ART1 vigilancethreshold various values were used as shown in Figure 4Each clustering performance was evaluated in terms of theSSE and number of clusters The SSE is used to compare thecohesion amongst various clustering results and a small SSEindicates that theWBMwithin a cluster is highly similarThenumber of clusters represents the effectiveness of the WBMgrouping According to the objective of clustering is to group


Table 3 Parameter settings for PSO clustering

Parameter Value Parameter Value119898 20 120596 1119883

max 1 1198861

04119888119901

2 1198862

03119888119892

2 1198863

03Iteration 500

Table 4 Results of clustering methods by SSE

Methods Noise degree002 004 006 008 010

Hsu and Chien [8] 1184 1192 1203 1248 1322

Individualclustering

KB 2889 3092 3003 4083 3570KC 3331 2490 2603 3169 2603PB 5893 3601 6566 5839 6308PC 4627 4873 3330 3787 6112

Clusteringensemble

KB and PB 1827 1280 1324 1801 2142KC and PC 2272 2363 2400 1509 1718KB and PC 1368 1459 2400 1509 2597KC and PB 2100 2048 1421 1928 2043KB and PB andKC and PC 1586 1550 1541 1571 1860

the WBM into few clusters in which the similarities amongthe WBMs within a cluster are high as possible Thereforethe setting of ART1 vigilance threshold value is used as 050in the numerical experiments

WBM clustering is to identify the similar type of WBMinto the same cluster To consider only six types ofWBMs thatwere used in the experiments the actual number of clustersshould be six Based on the various degree of noise in WBMgeneration as shown in Table 4 several individual clusteringmethods including ART1 [8] 119896-means clustering and PSOclustering were used for evaluating clustering performanceTable 4 shows that the ART1 neural network yielded a lowerSSE compared with the other methods However the ART1neural network separates the WBM into 15 clusters as shownin Figure 5 The ART1 neural network yields unnecessarypartitions for the similar type of WBM pattern In order togenerate diverse clustering partitions for clustering ensemblemethod four combinations with various data scale andclustering algorithms including 119896-means by binary value(KB) 119896-means by continuous value (KC) PSO by binaryvalue (PB) and PSO by continuous value (PC) are usedRegardless of the individual clustering results based on sixnumbers of clusters using 119870-means clustering and PSOclustering individually yielded larger SSE values than usingART1 only

Table 4 also shows the clustering ensembles that usevarious types of input data For example the clusteringensemble method KBampPB integrates the six results includingthe 119896-means algorithm by three kinds of clusters (ie 119896 =

5 6 7) and PSO clustering by three kinds of clusters (ie119896 = 5 6 7) respectively to form the WBM clustering via

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9

Group 10

Group 11

Group 12

Group 13

Group 14

Group 15

Figure 5 Clustering result by ART1 (15 clusters)

label space In general the clustering ensembles demonstratesmaller SSE values than do individual clustering algorithmssuch as the 119896-means or PSO clustering algorithms

In addition to compare the similarity within the clusteran index called specificity was used to evaluate the efficiencyof the evolved cluster over representing the true clusters [28]The specificity is defined as follows

specificity =119905119888

119879119890

(8)

where 119905119888 is the number of true WBM patterns covered by thenumber of evolvedWBM patterns and 119879119890 is the total numberof evolved WBM patterns As shown in the ART1 neuralnetwork clustering results the total number of evolvedWBMclusters is 15 and number of true WBM clusters is 6 Thenthe specificity is 04 Table 5 shows the results of specificity


(a) (b) (c)

(d) (e) (f)

Figure 6 Six types of WBM patterns

Table 5 Results of clustering methods by specificity


Hsu and Chien [8] 04 04 04 04 04


KB 10 10 10 10 10KC 10 10 10 10 10PB 10 10 10 10 10PC 10 10 10 10 10

Clusteringensemble


among clusteringmethodsTheART1 neural network has thelowest specificity due to the large number of clusters Thespecificity of individual clustering is 1 because the number ofevolved WBM patterns is fixed as 6 Furthermore comparedwith individual clustering algorithms combining variousclustering ensembles yields not only smaller SSE values butalso smaller numbers of clusters Thus the homogeneitywithin a cluster can be improved using proposed approachThe threshold of ART1 neural network yields maximal clus-ter numbers Therefore the proposed clustering ensembleapproach considering diversity partitions has better resultsregarding the SSE and number of clusters than individualclustering methods

To evaluate the results among various clustering ensem-bles and to assess cluster validity WBM class labels areemployed based on six pattern types as shown in Figure 6

Thus the indices including precision and recall are two classi-fication-oriented measures [29] defined as follows

precision =TP

TP + FP

recall = TPTP + FN

(9)

where TP (true positive) is the number of WBMs correctlyclassified into WBM patterns FP (false positive) is the num-ber of WBMs incorrectly classified and FN (false negative)is the number of WBMs that need to be classified but not tobe determined incorrectly The precision measure is used toassess how many WBMs classified as Pattern (a) are actuallyPattern (a) The recall measure is used to assess how manysamples of Pattern (a) are correctly classified

However a trade-off exists between precision and recalltherefore when one of these measures increases the otherdecreasesThe119865-measure is a harmonicmeanof the precisionand recall which is defined as follows

119865 =2 times precision times recallprecision + recall

=2TP

FP + FN + 2TP (10)

Specifically the 119865-measure represents the interactionbetween the actual and classification results (ie TP) If theclassification result is close to the actual value the 119865-measureis high

Tables 6 7 and 8 show a summary of various metricsamong six types ofWBM in precision recall and 119865-measurerespectively As shown in Figure 6 Patterns (b) and (c) aresimilar in the wafer edge demonstrating smaller averageprecision and recall values compared with the other patternsThe clustering ensembles which generate partitions by using119896-means make it difficult to identify in both Patterns (b)and (c) Using a mountain function transformation enables


Table 6 Clustering result on the index of precision

Hsu and Chien [8] Clustering ensembleKB and PB KC and PC KB and PC KC and PB KB and PB and KC and PC

Precision

A 070 084 092 092 092 098B 050 066 096 092 062 096C 060 064 100 100 060 100D 070 098 092 092 098 100E 060 094 082 082 098 098F 080 098 076 076 098 098

Avg 065 084 090 089 085 098

Table 7 Clustering result on the index of recall


Recall

A 100 100 100 093 100 100B 100 097 07 078 083 100C 100 094 067 084 067 097D 100 081 100 100 100 100E 100 079 100 100 100 100F 100 100 100 100 100 100

Avg 100 092 090 093 092 100

Table 8 Clustering result on the index of 119865-measure


119865-measure

A 082 091 096 092 096 099B 067 079 081 084 071 098C 075 076 08 091 063 098D 082 089 096 096 099 100E 075 086 090 090 099 099F 089 099 086 086 099 099

Avg 078 087 088 090 088 099

considering the defect density of the spatial relations betweenthe good and bad dies across awafer Based on the119865-measurethe clustering ensembles obtained using all generated parti-tions exhibit larger precision and recall values and superiorlevels of performance regarding each pattern compared withthe other methods Thus the partitions generated by using119896-means and PSO clustering in various data types must beconsidered

The practical viability of the proposed approach wasexamined The results show that the ART1 neural networkperforming into data space directly leads to worse clusteringperformance in terms of precision However the true types ofWBM can be identified through transforming original dataspace into label space and performing consensus partitionby ART1 neural network The proposed cluster ensembleapproach can get better performance with fewer numbersof clusters than other conventional clustering approachesincluding 119896-means PSO clustering and ART1 neural net-work

5 Conclusion

WBMs provide important information for engineers torapidly find the potential root cause by identifying patternscorrectly As the driven force for semiconductor manufac-turing technology WBM identification to the correct patternbecomes more difficult because the same type of patterns isinfluenced by various factors such as die size pattern densityand noise degree Relying on only engineersrsquo experiencesof visual inspections and personal judgments in the mappatterns is not only subjective and inconsistent but also verytime-consuming and inefficient Therefore grouping similarWBM quickly helps engineer to use more time to diagnosethe root cause of low yield

Considering the requirements of clustering WBMs inpractice a cluster ensemble approach was proposed tofacilitate extracting the common defect pattern of WBMsenhancing failure diagnosis and yield enhancement Theadvantage of the proposed method is to yield high-qualityclusters by applying distinct algorithms to the same data


set and by using various parameter settings The robustnessof clustering ensemble is higher than individual clusteringmethod because the clustering fromvarious aspects includingalgorithms and parameter setting is integrated into a consen-sus result

The proposed clustering ensemble has two stages At thefirst stage diversity partitions are generated using two typesof input data various cluster numbers and distinct clusteringalgorithms At the second stage a consensus partition isattained using these diverse partitions The numerical anal-ysis demonstrated that the clustering ensemble is superiorto using individual 119896-means or PSO clustering algorithmsThe results demonstrate that the proposed approach caneffectively group the WBMs into several clusters based ontheir similarity in label space Thus engineers can have moretime to focus the assignable cause of low yield instead ofextracting defect patterns

Clustering is an exploratory approach In this study weassume that the number of clusters is known Evaluating theclustering ensemble approach prior information is requiredregarding the cluster numbers Further research can be con-ducted regarding self-tuning the cluster number in clusteringensembles

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is supported by National Science CouncilTaiwan (NSC 102-2221-E-155-093 MOST 103-2221-E-155-029-MY2) The author would like to thank Mr Tsu-An Chaofor his kind assistance The author also wishes to thankthe editors and two anonymous referees for their insightfulcomments and suggestions

References

[1] R C Leachman S Ding and C-F Chien ldquoEconomic efficiencyanalysis of wafer fabricationrdquo IEEE Transactions on AutomationScience and Engineering vol 4 no 4 pp 501ndash512 2007

[2] C-F Chien and C-H Chen ldquoA novel timetabling algorithmfor a furnace process for semiconductor fabrication with con-strained waiting and frequency-based setupsrdquo OR Spectrumvol 29 no 3 pp 391ndash419 2007

[3] C-F Chien W-C Wang and J-C Cheng ldquoData mining foryield enhancement in semiconductor manufacturing and anempirical studyrdquo Expert Systems with Applications vol 33 no1 pp 192ndash198 2007

[4] C-F Chien Y-J Chen and J-T Peng ldquoManufacturing intelli-gence for semiconductor demand forecast based on technologydiffusion and product life cyclerdquo International Journal of Pro-duction Economics vol 128 no 2 pp 496ndash509 2010

[5] C-J Kuo C-F Chien and J-D Chen ldquoManufacturing intel-ligence to exploit the value of production and tool data toreduce cycle timerdquo IEEE Transactions on Automation Scienceand Engineering vol 8 no 1 pp 103ndash111 2011

[6] C-F Chien C-YHsu andC-WHsiao ldquoManufacturing intelli-gence to forecast and reduce semiconductor cycle timerdquo Journalof Intelligent Manufacturing vol 23 no 6 pp 2281ndash2294 2012

[7] C-F Chien C-Y Hsu and P-N Chen ldquoSemiconductor faultdetection and classification for yield enhancement and man-ufacturing intelligencerdquo Flexible Services and ManufacturingJournal vol 25 no 3 pp 367ndash388 2013

[8] S-C Hsu and C-F Chien ldquoHybrid data mining approach forpattern extraction fromwafer binmap to improve yield in semi-conductor manufacturingrdquo International Journal of ProductionEconomics vol 107 no 1 pp 88ndash103 2007

[9] C-F Chien S-C Hsu and Y-J Chen ldquoA system for onlinedetection and classification of wafer bin map defect patterns formanufacturing intelligencerdquo International Journal of ProductionResearch vol 51 no 8 pp 2324ndash2338 2013

[10] C-W Liu and C-F Chien ldquoAn intelligent system for wafer binmap defect diagnosis an empirical study for semiconductormanufacturingrdquo Engineering Applications of Artificial Intelli-gence vol 26 no 5-6 pp 1479ndash1486 2013

[11] C-F Chien and C-Y Hsu ldquoA novel method for determiningmachine subgroups and backups with an empirical study forsemiconductor manufacturingrdquo Journal of Intelligent Manufac-turing vol 17 no 4 pp 429ndash439 2006

[12] K-S Lin and C-F Chien ldquoCluster analysis of genome-wideexpression data for feature extractionrdquo Expert Systems withApplications vol 36 no 2 pp 3327ndash3335 2009

[13] J A Hartigan and M A Wong ldquoA K-means clustering algo-rithmrdquo Applied Statistics vol 28 no 1 pp 100ndash108 1979

[14] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[15] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[16] A Strehl and J Ghosh ldquoCluster ensemblesmdasha knowledge reuseframework for combining multiple partitionsrdquo The Journal ofMachine Learning Research vol 3 no 3 pp 583ndash617 2002

[17] A L V Coelho E Fernandes and K Faceli ldquoMulti-objectivedesign of hierarchical consensus functions for clusteringensembles via genetic programmingrdquoDecision Support Systemsvol 51 no 4 pp 794ndash809 2011

[18] A Topchy A K Jain and W Punch ldquoClustering ensemblesmodels of consensus and weak partitionsrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 27 no 12 pp1866ndash1881 2005

[19] C H Stapper ldquoLSI yield modeling and process monitoringrdquoIBM Journal of Research and Development vol 20 no 3 pp228ndash234 1976

[20] W Taam and M Hamada ldquoDetecting spatial effects fromfactorial experiments an application from integrated-circuitmanufacturingrdquo Technometrics vol 35 no 2 pp 149ndash160 1993

[21] C-H Wang W Kuo and H Bensmail ldquoDetection and clas-sification of defect patterns on semiconductor wafersrdquo IIETransactions vol 38 no 12 pp 1059ndash1068 2006

[22] J Y Hwang andW Kuo ldquoModel-based clustering for integratedcircuit yield enhancementrdquo European Journal of OperationalResearch vol 178 no 1 pp 143ndash153 2007

[23] T Yuan andWKuo ldquoSpatial defect pattern recognition on semi-conductor wafers using model-based clustering and Bayesianinferencerdquo European Journal of Operational Research vol 190no 1 pp 228ndash240 2008


[24] R R Yager and D P Filev ldquoApproximate clustering via themountain methodrdquo IEEE Transactions on Systems Man andCybernetics vol 24 no 8 pp 1279ndash1284 1994

[25] M-S Yang and K-L Wu ldquoA modified mountain clusteringalgorithmrdquo Pattern Analysis and Applications vol 8 no 1-2 pp125ndash138 2005

[26] D W van der Merwe and A P Engelbrecht ldquoData cluster-ing using particle swarm optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) pp 215ndash220December 2003

[27] G A Carpenter and S Grossberg ldquoTheARTof adaptive patternrecognition by a self-organization neural networkrdquo Computervol 21 no 3 pp 77ndash88 1988

[28] C Wei and Y Dong ldquoA mining-based category evolutionapproach to managing online document categoriesrdquo in Pro-ceedings of the 34th Annual Hawaii International Conference onSystem Sciences January 2001

[29] L Rokach and O Maimon ldquoData mining for improvingthe quality of manufacturing a feature set decompositionapproachrdquo Journal of Intelligent Manufacturing vol 17 no 3 pp285ndash299 2006

Research ArticleA Multiple Attribute Group Decision Making Approach forSolving Problems with the Assessment of Preference Relations

Taho Yang1 Yiyo Kuo2 David Parker3 and Kuan Hung Chen1

1 Institute of Manufacturing Information and Systems National Cheng Kung University Tainan 70101 Taiwan2Department of Industrial Engineering and Management Ming Chi University of Technology New Taipei City 24301 Taiwan3The University of Queensland Business School Brisbane QLD 4072 Australia

Correspondence should be addressed to Yiyo Kuo yiyomailmcutedutw

Received 19 June 2014 Revised 21 October 2014 Accepted 23 October 2014

Academic Editor Mu-Chen Chen


A number of theoretical approaches to preference relations are used for multiple attribute decision making (MADM) problemsand fuzzy preference relations is one of them When more than one person is interested in the same MADM problem it thenbecomes a multiple attribute group decision making (MAGDM) problem For both MADM and MAGDM problems consistencyamong the preference relations is very important to the result of the final decision The research reported in this paper is based ona procedure that uses a fuzzy preference relations matrix which satisfies additive consistency This matrix is used to solve multipleattribute group decision making problems In group decision problems the assessment provided by different experts may divergeconsiderably Therefore the proposed procedure also takes a heterogeneous group of experts into consideration Moreover themethods used to construct the decision matrix and determine the attribution of weight are both introduced Finally a numericalexample is used to test the proposed approach and the results illustrate that the method is simple effective and practical

1 Introduction

There are many situations in daily life and in the workplacewhich pose a decision problem Some of them involve pickingthe optimum solution from amongmultiple available alterna-tives Therefore in many domain problems multiple attributedecision making methods such as simple additive weighting(SAW) the technique for order preference by similarity toideal solution (TOPSIS) analytical hierarchy process (AHP)data envelopment analysis (DEA) or grey relational analysis(GRA) [1ndash5] are usually adopted for example layout design[6ndash8] supply chain design [9] pushpull junction pointselection [10] pacemaker location determination [11] workin process level determination [12] and so on

If more than one person is involved in the decision thedecision problem becomes a group decision problem Manyorganizations have moved from a single decision maker orexpert to a group of experts (eg Delphi) to accomplish thistask successfully [13 14] Note that an ldquoexpertrdquo represents an

authorized person or an expert who should be involved inthis decision making process However no single alternativeworks best for all performance attributes and the assessmentof each alternative given by different decision makers maydiverge considerably As a consequence multiple attributegroup decision making (MAGDM) is more difficult thancases where a single decision maker decides using a multipleattribute decision making method

MAGDMis one of themost common activities inmodernsociety which involves selecting the optimal one from afinite set of alternatives with respect to a collection ofthe predefined criteria by a group of experts with a highcollective knowledge level on these particular criteria [15]When a group of experts wants to choose a solution fromamong several alternatives preference relations is one typeof assessment that experts could provide Preference relationsare comparisons between two alternatives for a particularattribute A higher preference relation means that there is ahigher degree of preference for one alternative over another



However different expertsmay use different assessment typesto express the preference relation It is possible that in groupdecision making different experts express their preference indifferent formats [16ndash21]

In addition after experts have provided their assessmentof the preference relation the appropriateness of the compar-ison from each expert must be tested Consistency is one ofthe important properties for verifying the appropriateness ofchoices [22] If the comparison from an expert is not logicallyconsistent for a specific attribute it means that at leastone preference relation provided by the expert is defectiveTherefore the lack of consistency in decisionmaking can leadto inconsistent conclusions

Quite apart from the type of assessment there can beconsiderable variation between experts as to their evaluationof the degree of the preference relation In general it would bepossible to aggregate the preferences of experts by taking theweight assigned by every expert into consideration Howeverheterogeneity among experts should also be considered [23]For example if the expert who assigns the greatest weightto a preference relation also makes choices that are notappropriate and quite different from the evaluations of theother experts who assign lower weights then the groupdecision procedure can be distorted and imperfect

Moreover the determination of attribute weight is also animportant issue [24] In some decision cases some attributesare considered to be more important in the expertsrsquo profes-sional judgment However for these important attributes thepreference relation provided by experts may be quite similarfor all alternatives Even for the attribute with the highestweight the degree of influence on the final decision wouldbe very small in this case In this way this kind of attributecan become unimportant to the final decision [25]

Therefore during the multiple attribute group decisionprocess 5 aspects should be noted

(i) considering different assessment types simultane-ously

(ii) insuring the preference relations provided by expertsare consistent

(iii) taking heterogeneous experts into consideration(iv) deciding the weight of each attribute(v) ranking all alternatives

Group decision making has been addressed in the lit-erature In recent years Olcer and Odabasi [23] proposeda fuzzy multiple attribute decision making method to dealwith the problem of ranking and selecting alternativesExperts provide their opinion in the form of a trapezoidalfuzzy number These trapezoidal fuzzy numbers are thenaggregated and defuzzified into a MADM Finally TOPSISis used to rank and select alternatives In the method expertscan provide their opinion only by trapezoidal fuzzy number

Boran et al [26] proposed a TOPSIS method combinedwith intuitionistic fuzzy set to select appropriate supplierin group decision making environment Intuitionistic fuzzyweighted averaging (IFWA) operator is utilized to aggre-gate individual opinions of decision makers for rating the

importance of criteria and alternatives Cabrerizo et al [27]presented a consensus model for group decision makingproblems with unbalanced fuzzy linguistic information Thisconsensus model is based on both a fuzzy linguistic method-ology to deal with unbalanced linguistic term sets and twoconsensus criteriamdashconsensus degrees and proximity mea-sures Chuu [28] builds a group decisionmakingmodel usingfuzzy multiple attributes analysis to evaluate the suitability ofmanufacturing technology The proposed approach involveddeveloping a fusion method of fuzzy information which wasassessed using both linguistic and numerical scales

Lu et al [29] developed a software tool for support-ing multicriteria group decision making When using thesoftware after inputting all criteria and their correspondingweights and the weighting for all the experts all the expertscan assess every alternative against each attribute Then theranking of all alternatives can be generated In the softwareonly one assessment type is allowed and there is no functionthat can be used to ensure that the preference relationsprovided by experts are consistent Zhang and Chu [30]proposed a group decision making approach incorporatingtwo optimization models to aggregate these multiformat andmultigranularity linguistic judgments Fuzzy set theory isutilized to address the uncertainty in the decision makingprocess

Cabrerizo et al [14] proposed a consensus model to dealwith group decision making problems in which experts useincomplete unbalanced fuzzy linguistic preference relationsto provide their preference However the model requiresthat preference relations should be assessed in the sameway and no allowance is made for heterogeneous expertsCebi and Kahraman [31] proposed a methodology for groupdecision support The methodology consists of eight stepswhich are (1) definition of potential decision criteria possiblealternatives and experts (2) determining the weighting ofexperts (3) identifying the importance of criteria (4) assign-ing alternatives (5) aggregating expertsrsquo preferences (6)

identifying functional requirements (7) calculating informa-tion contents and (8) calculating weighted total informationcontents and selecting the best alternative The methodologydoes not include a check on the consistency of preferencerelations provided by the experts

The novelty of the present study is that it proposes amultiple attribute group decision making methodology inwhich all of the five issues mentioned above are addressedA review of the literature related to this research suggeststhat no previous research has addressed all of the issuessimultaneously For managers who are not experts in fuzzytheory group decision making MADM and so on thisresearch can provide a complete guideline for solving theirmultiple attribute group decision making problem

The remainder of this paper is organized as followsIn Section 2 all the issues set out above are discussed andappropriate methodologies for dealing with them are pro-posed Then an overall approach is proposed in Section 3The proposedmodel is tested and examined with a numericalexample in Section 4 Finally Section 5 contains the discus-sion and conclusions


2 Multiple Attribute GroupDecision Making Methodology

21 Assessment and Transformation of Preference RelationsThere are two types of preference relations that are widelyused One is fuzzy preference relations in which 119903119894119895 denotesthe preference degree or intensity of the alternative 119894 over 119895[32ndash35] If 119903119894119895 = 05 it means that alternatives 119894 and 119895 areindifferent if 119903119894119895 = 1 it means that alternative 119894 is absolutelypreferred to 119895 and if 119903119894119895 gt 05 it means that alternative 119894 ispreferred to 119895 119903119894119895 is reciprocally additive that is 119903119894119895 + 119903119895119894 = 1

and 119903119894119894 = 05 [35 36]The other widely used type of preference relations is mul-

tiplicative preference relations in which 119886119894119895 indicates a ratioof preference intensity for alternative 119894 to that of alternative 119895that is it is interpreted asmeaning that alternative 119894 is 119886119894119895 timesas good as alternative 119895 [17] Saaty [3] suggested measuring119886119894119895 on an integer scale ranging from 1 to 9 If 119886119894119895 = 1 itmeans that alternatives 119894 and 119895 are indifferent if 119886119894119895 = 9 itmeans that alternative 119894 is absolutely preferred to 119895 and if8 ge 119903119894119895 ge 2 it means that alternative 119894 is preferred to 119895 Inaddition 119886119894119895 times 119886119895119894 = 1 and 119886119894119895 = 119886119894119896 times 119886119896119895

For these two preference types Chiclana et al [17] pro-posed an equation to transform the multiplicative preferencerelation into the fuzzy preference relation as shown by

119903119894119895 = 05 (1 + log9119886119894119895) (1)

However for both preference types it is possible thatsome experts would not wish to provide their preferencerelation in the form of a precise value In the fuzzy preferencerelations experts can use the following classifications

(i) a precise value for example ldquo07rdquo(ii) a range for example (03 07) the value is likely to

fall between 03 and 07(iii) a fuzzy number with triangular membership func-

tion for example (04 06 08) the value is between04 and 08 and is most probably 06

(iv) a fuzzy number with trapezoidal membership func-tion for example (03 05 06 08) the value isbetween 03 and 08 most probably between 05 and06

In this paper the four classifications set out above areunified by transferring them into trapezoidal membershipfunctions Thus 07 becomes (07 07 07 07) (03 07)becomes (03 03 07 07) and (04 06 08) then becomes(04 06 06 08) If experts provide their assessment inthe format of multiplicative preference relations it will betransformed into a trapezoidal membership function firstand then using (1) it will be further transformed into theformat of fuzzy preference relations For example (3 4 56) can be transferred into (075 082 087 091) by using(1) Therefore this paper will mention only fuzzy preferencerelations in what follows

22 The Generation of Consistent Preference Relations Theproperty of consistency has been widely used to establish

a verification procedure for preference relations and it isvery important for designing good decision making models[22] In the analytical hierarchy process for example inorder to avoid potential comparative inconsistency betweenpairs of categories a consistency ratio (CR) an index forconsistency has been calculated to assure the appropriatenessof the comparisons [3] If the CR is small enough there isno evidence of inconsistency However if the CR is too highthen the experts should adjust their assessments again andagain until the CR decreases to a reasonable value For fuzzypreference relations Herrera-Viedma et al [22] designeda method for constructing consistent preference relationswhich satisfy additive consistency Using this method allexperts need only to provide preference relations betweenalternatives 119894 and 119894 + 1 119903119894(119894+1) and the remaining preferencerelations can be calculated using (2) if 119894 gt 119895 and (3) if 119894 lt 119895

119903119894119895 =119894 minus 119895 + 1

2minus 119903119895(119895+1) minus 119903(119895+1)(119895+2) minus sdot sdot sdot minus 119903(119894minus1)119894 forall119894 gt 119895

(2)

119903119894119895 = 1 minus 119903119895119894 forall119894 lt 119895 (3)

To illustrate the generation of preferential relations weprovide an empirical example of four alternatives as followsFirst the expert provides the three preference relations as11990312 = 03 11990323 = 06 and 11990334 = 08

According to (2)

11990321 = 1 minus 03 = 07

11990331 = 15 minus 03 minus 06 = 06

11990341 = 2 minus 03 minus 06 minus 08 = 03

11990332 = 1 minus 06 = 04

11990342 = 15 minus 06 minus 08 = 01

11990343 = 1 minus 08 = 02

(4)

According to (3)

11990313 = 1 minus 06 = 04

11990314 = 1 minus 03 = 07

11990324 = 1 minus 01 = 09

(5)

Therefore the preference relations matrix PR is

PR =[[[

[

05 03 04 07

07 05 06 09

06 04 05 08

03 01 02 05

]]]

]

(6)

In general experts are asked to evaluate all pairs ofalternatives and then construct a preference matrix with fullinformation However it is difficult to obtain a consistentpreference matrix in practice especially when measuringpreferences on a set with a large number of alternatives [22]


23 Assessment Aggregation for a Heterogeneous Group ofExperts For each comparison between a pair of alternativesthe preference relations provided by different experts wouldvary Hsu and Chen [37] proposed an approach to aggregatefuzzy opinions for a heterogeneous group of experts ThenChen [38]modified the approach and Olcer andOdabasi [23]present it as the following six-step procedure

(1) Calculate the Degree of Agreement between Each Pairof Experts For a comparison between two alternatives letthere be 119864 experts in the decision group (1198861 1198862 1198863 1198864) and(1198871 1198872 1198873 1198874) are the preference relations provided by experts119886 and 119887 1 le 119886 le 119864 1 le 119887 le 119864 and 119886 = 119887 The similaritybetween these two trapezoidal fuzzy numbers 119878119886119887 can bemeasured by

119878119886119887 = 1 minus

10038161003816100381610038161198861 minus 11988711003816100381610038161003816 +

10038161003816100381610038161198862 minus 11988721003816100381610038161003816 +

10038161003816100381610038161198863 minus 11988731003816100381610038161003816 +

10038161003816100381610038161198864 minus 11988741003816100381610038161003816

4 (7)

(2) Construct the Agreement Matrix After all the agreementdegrees between experts are measured the agreement matrix(AM) can be constructed as follows

AM =

[[[[

[

1 11987812 sdot sdot sdot 119878111986411987821 1 sdot sdot sdot 1198782119864

119878119886119887

1198781198641 1198781198642 sdot sdot sdot 1

]]]]

]

(8)

in which 119878119886119887 = 119878119887119886 and if 119886 = 119887 then 119878119886119887 = 1

(3) Calculate the AverageDegree of Agreement for Each ExpertThe average degree of agreement for expert 119886 (AA119886) can becalculated by

AA119886 =1

119864 minus 1

119864

sum

119887=1119886 =119887

119878119886119887 forall119886 (9)

(4) Calculate the RelativeDegree of Agreement for Each ExpertAfter calculating the average degree of agreement for allexperts the relative degree of agreement for expert 119886 (RA119886)can be calculated by

RA119886 =AA119886

sum119864

119886=1AA119886

forall119886 (10)

(5) Calculate the Coefficient for the Degree of Consensusfor Each Expert Let ew119886 be the weight of expert 119886 andsum119864

119886=1ew119886 = 1 The coefficient of the degree of consensus for

expert 119886 (CC119886) can be calculated by

CC119886 = 120573 sdot ew119886 + (1 minus 120573) sdot RA119886 forall119886 (11)

in which 120573 is a relaxation factor of the proposed method and0 le 120573 le 1 It represents the importance of ew119886 over RA119886

When 120573 = 0 it means that the group of experts is consideredto be homogeneous

(6) Calculate the Aggregation Result Finally the aggregationresult of the comparison between two alternatives 119894 and 119895 is119903119894119895 where

119903119894119895 = CC1 otimes 119903119894119895 (1) oplus CC2 otimes 119903119894119895 (2) oplus sdot sdot sdot oplus CC119886

otimes 119903119894119895 (119886) oplus sdot sdot sdot oplus CC119864 otimes 119903119894119895 (119864)

(12)

In (12) 119903119894119895(119886) is the preference relation between alterna-tives 119894 and 119895 provided by expert 119886 and 119903119894119895 = (119903

1

119894119895 1199032

119894119895 1199033

119894119895 1199034

119894119895)

Moreover otimes and oplus are the fuzzy multiplication operator andthe fuzzy addition operator respectively

Let there be 119873 alternatives Since each expert onlyprovides preference relations between alternatives 119894 and 119894 +

1 the aggregation process for a heterogeneous group ofexperts must be executed 119873 minus 1 times in order to generate119873 minus 1 aggregated trapezoidal fuzzy numbers These 119873 minus

1 trapezoidal fuzzy numbers can then be converted into aprecise value by the use of

119903119894119895 =1199031

119894119895+ 2 (119903

2

119894119895+ 1199033

119894119895) + 1199034

119894119895

6 (13)

After the aggregation procedure using (2) and (3) anaggregated preference relations matrix for attribute 119896 isconstructed as follows

PR119896 =[[[[

[


1

1199031198731 1199031198732 sdot sdot sdot 1

]]]]

]

(14)

24 AttributeWeightDetermination In a preference relationsmatrix of attribute 119896 119903119894119895 indicates the degree of preferenceof alternative 119894 over 119895 when attribute 119896 was consideredTherefore sum119873

119895=1119895 =119894119903119894119895 indicates total degree of preference of

alternative 119894 over the other 119873 minus 1 alternatives In the sameway sum119873

119895=1119895 =119894119903119895119894 indicates the total degree of preference of the

other119873minus1 alternatives over alternative 119894 Fodor and Roubens[39] proposed (15) to define 120575119894119896 the net degree of preferenceof alternative 119894 over the other 119873 minus 1 alternatives by attribute119896 and the bigger 120575119894119896 is the better alternative 119894 by attribute 119896is

120575119894119896 =

119873

sum

119895=1119895 =119894

119903119894119895 minus

119873

sum

119895=1119895 =119894

119903119895119894 forall119894 119896 (15)

Thus the problem is reduced to a multiple attributedecision making problem

DM =

[[[[

[


1205751198731 1205751198732 sdot sdot sdot 120575119873119872

]]]]

]

(16)


For the decision matrix constructed in Section 24 Wangand Fan [25] proposed two approaches absolute deviationmaximization (ADM) and standard deviation maximization(SDM) to determine the weight of all attributes For a certainattribute if the difference of the net degree of preferenceamong all alternatives shows a wide variation this means thisattribute is quite important ADM and SDM used absolutedeviation (AD) and standard deviation (SD) to measure thedegree of variation An attribute with a bigger value of ADand SD will be a more important attribute

When ADM was adopted the weight of attribute 119896 aw119896was calculated by using (17) while if SDM was adopted (18)was used for calculating the weight of attribute 119896

aw119896 =(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119896 minus 120575119895119896

10038161003816100381610038161003816)1(119901minus1)

sum119872

119897=1(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119897 minus 120575119895119897

10038161003816100381610038161003816)1(119901minus1)

forall119896 119901 gt 1 (17)

aw119896 =(sum119873

119894=11205752

119894119896)12(119901minus1)

sum119872

119897=1(sum119873

119894=11205752119894119897)12(119901minus1)

forall119896 119901 gt 1 (18)

where 119901 is the parameter of these two functions for calcu-lating weights Setting the variable to different values willlead to different weights and when 119901 = infin all weightswill be equal Therefore in order to reflect the differencesamong the attribute weights Wang and Fang [25] suggestedpreferring a small value for parameter 119901 Further details ofthe demonstration of the use of ADM and SDM can be foundin the paper by Wang and Fan [25]

25 Alternative Ranking Once the weights of all attributesare determined by (17) or (18) the multiple attribute decisionmaking problem constructed by (16) can be solved by theapplication of a multiple attribute decision making methodsuch as SAW TOPSIS ELECTRE or GRA [1 2 5] Accordingto Kuo et al [40] different MADM methods would lead todifferent results but similar ranking of alternatives In thisresearch SAW was selected for the MADM problem Sincethe weight calculated by (17) and (18) has been normalizedand sum

119872

119896=1aw119896 = 1 the score of alternatives 119894 119862119894 can be

calculated directly by

119862119894 =

119872

sum

119896=1

aw119896120575119894119896 119894 = 1 2 119873 (19)

The bigger the119862119894 is the better the alternative 119894 is After thescores of all alternatives have been calculated the alternativescan be ranked by 119862119894

3 The Proposed Approach

Following from the consideration of issues whichwere set outin the Introduction and further developed in Section 2 thisresearch proposes a 5-step procedure for multiple attributegroup decision making problems as shown in Figure 1

In Step 1 experts provide their preference relations forall attributes using their preferred format of expression In

transformation

heterogeneous group of experts

relations

(1) Preference relations assessment and

(2) Assessment aggregation for

(3) The generation of consistent preference

(4) Attribute weight determination

(5) Alternatives ranking

Figure 1 The proposed MAGDM procedure

order to ensure the additive consistency of these preferencerelations only the preference relations between alternatives 119894and 119894+1 are assessedThen these preference relations providedby the experts are transformed into trapezoidal membershipfunctions If the preference relations are multiplicative pref-erence relations (1) is used to transform them into fuzzypreference relations

In Step 2 in order to take the heterogeneity of the expertsinto consideration the trapezoidal membership function offuzzy preference relations for all experts is aggregated by a six-step procedure given by Olcer andOdabasi [23]Then (2) and(3) are used to calculate the remaining preference relationswhich had not been provided by the experts and these arethen used to construct preference relationmatrixes which areadditively consistent in Step 3

In Step 4 these preference relation matrixes are trans-formed into a traditional multiple attribute decision matrixand used to determine the weight of all attributes using (17)and (18) Finally all the scores of alternatives can be calculatedusing (19) and the alternatives can be ranked in Step 5

4 Numerical Example

The proposed MAGDM methodology allows two types ofpreference relations fuzzy reference relations andmultiplica-tive preference relations which are explained in Section 21The former ones are transformed to numerical numberthrough fuzzy membership functions and the latter onesdirectly use numerical numbers They are then aggregatedthrough the proposed aggregation and ranking procedure asdiscussed in Sections 22 to 25 Due to both the transforma-tion and aggregation procedures the resulting numbers arereal numbers


In this section we provide a numerical example toillustrate the implementation of the proposed methodologyConsider four alternatives three experts and two attributeMAGDM problems as follows

Step 1 (preference relations assessment and transformation)The preference relations assessments of Attribute 1 providedby these three experts were given as follows in which 119877119886119896 isthe assessment of attribute 119896 provided by expert 119886

11987711 =[[[

[

minus Low minus minus

minus minus Low minus

minus minus minus Mediumminus minus minus minus

]]]

]

11987721 =[[[

[

minus More low minus minus

minus minus Medium minus


]]]

]

11987731 =

[[[[[[

[

minus1

3minus minus

minus minus1

4minus

minus minus minus 1

minus minus minus minus

]]]]]]

]

(20)

In this example Experts 1 and 2 preferred to provideassessment by fuzzy preference relations and Expert 3 pre-ferred to provide assessment by multiplicative preferencerelations However Expert 1 used the membership functionas shown in Figure 2 Expert 2 used themembership functionas shown in Figure 3 and Expert 3 used precise values forproviding hisher preference relations All assessments arethen transformed into the type of trapezoidal membershipfunction as shown below

11987711 =[[[

[

minus 0125 0225 0325 0425 minus minus

minus minus 0125 0225 0325 0425 minus

minus minus minus 0350 0450 0550 0650


]]]

]

11987721 =[[[

[





]]]

]

11987731 =

[[[[[[

[

minus1

31

31

31

3minus minus

minus minus1

41

41

41

4minus



]]]]]]

]

(21)

The preference relationsrsquo assessments of Attribute 2 whichhave been transformed into the type of trapezoidal member-ship function were given as follows

11987712 =[[[

[





]]]

]

11987722 =[[[

[





]]]

]


11987732 =

[[[[[[

[

minus1

41

41

41

4minus minus


minus minus minus1

31

31

31

3


]]]]]]

]

(22)

Using (1) themultiplicative preference relations in11987731 and11987732 can be transformed into fuzzy preference relations and

then become 119877101584031

and 1198771015840

32as follows 11987731 and 11987732 were then

replaced by 119877101584031and 1198771015840

32for the rest of the analysis

1198771015840

31=

[[[

[





]]]

]

1198771015840

32=

[[[

[





]]]

]

(23)

Step 2 (assessment aggregation for heterogeneous group ofexperts) In this example the weights of Experts 1 2 and 3are 03 03 and 04 respectively Following the method setout in Section 23 the six steps can be used to aggregate theassessments provided by the heterogeneous group of expertsLet the relaxation factor 120573 = 05 The results are thensummarized in Table 1

Therefore the aggregated preference relations matrixesPR1 and PR2 are as shown in the following

PR1 =[[[

[

minus 0290 minus minus

minus minus 0311 minus

minus minus minus 0500


]]]

]

PR2 =[[[

[





]]]

]

(24)

Step 3 (the generation of consistent preference relations) InStep 3 the results in PR1 and PR2 are incomplete Equations(2) and (3) are then used to calculate the remaining preferencerelations and to construct additively consistent preference

relation matrixes The complete preference relation matrixesPR10158401and PR1015840

2are

PR10158401=

[[[

[

0500 0290 0100 0100

0710 0500 0311 0311

0900 0689 0500 0500

0900 0689 0500 0500

]]]

]

PR10158402=

[[[

[

0500 0218 0265 0055

0782 0500 0547 0337

0735 0453 0500 0290

0945 0663 0710 0500

]]]

]

(25)

According to the proposition and proof from Herrera-Viedma et al [22] a fuzzy preference relation PR = (119903119894119895) isconsistent if and only if 119903119894119895 + 119903119895119896 + 119903119896119894 = 32 forall119894 le 119895 le 119896 It canbe found that above PR1015840

1and PR1015840

2are consistent

Step 4 (attribute weight determination) Using (15) to calcu-late all 120575119894119896 the decision matrix DM can be constructed asfollows

DM =[[[

[

minus2019 minus1923

minus0336 0331

1178 minus0045

1178 1637

]]]

]

(26)

According to the constructed decision matrix whenADM and SDM were adopted the weight of Attributes 1 and2 can be calculated by (17) and (18) respectively A valueof 119901 = 2 has been adopted arbitrarily for the sake of thisdemonstration If ADM is adopted the weights of Attributes 1and 2 are 0501 and 0499 respectively If SDM is adopted theweights of Attributes 1 and 2 are 0509 and 0491respectively


Table 1 Aggregation of heterogeneous group of experts for Attribute 1

11990312

11990323

11990334

Expert 1 (0125 0225 0325 0425) (0125 0225 0325 0425) (0350 0450 0550 0650)Expert 2 (0200 0300 0400 0500) (0350 0450 0550 0650) (0350 0450 0550 0650)Expert 3 (0250 0250 0250 0250) (0185 0185 0185 0185) (0500 0500 0500 0500)Degree of agreement (119878

119886119887)

11987812

0925 0775 100011987813

0900 0880 090011987823

0875 0685 0900Average degree of agreement of expert 119886 (AA

119886)

AA1 0913 0828 0950AA2 0900 0730 0950AA3 0888 0783 0900

Relative degree of agreement of expert 119886 (RA119886)

RA1 0338 0354 0339RA2 0333 0312 0339RA3 0329 0334 0321

Consensus degree coefficient of expert 119886 (CC119886) for

120573 = 05

CC1 0319 0327 0320CC2 0317 0306 0320CC3 0364 0367 0361

Aggregated results 11990312= (019 026 032 038) 119903

23= (022 028 034 041) 119903

34= (040 047 053 060)

Converted results 11990312= 0290 119903

23= 0311 119903

34= 0500

Fuzzy preference relation

Very highHighMediumLow

Very low

02 04 06 1008

02

04

06

08

10

Mem

bers

hip

valu

e

Figure 2 Membership functions adopted by Expert 1


Very low

More highMore

low

02

04

06

08

10

Mem

bers

hip

valu

e

Fuzzy preference relation 02 04 06 1008


Table 2The scoring results byweight determinationmethodsADMand SDM

Alternative 119894 120575119894119896

119862119894(ADM) 119862

119894(SDM) Ranking results

1 minus2019 minus1923 minus1971 minus1972 42 minus0336 0331 minus0003 minus0009 33 1178 minus0045 0567 0577 24 1178 1637 1407 1403 1aw119896by ADM 0501 0499

aw119896by SDM 0509 0491

Step 5 (ranking alternatives) After generating the weights ofAttributes 1 and 2 using SAW the score of all alternatives119862119894 can be calculated by (9) The scoring results are as shownin Table 2 In Table 2 119862119894 (ADM) and 119862119894 (SDM) indicate thescores of all alternatives using attribute weight determiningapproaches ADM and SDM respectively The bigger valuesof 119862119894 indicate that the alternative 119894 is better In the case ofthe values of 119862119894 (ADM) for example because 1198624 (ADM)gt 1198623 (ADM) gt 1198622 (ADM) gt 1198621 (ADM) the groupdecision selected Alternative 4 as the first priority Moreoveraccording to the values of 119862119894 (SDM) the results also showAlternative 4 as the first priority

Although the theoretical development involves com-plicated technical details the implementation is relativelystraightforward in light of the numerical implementation


Therefore the proposedmethodology is applicable for a prac-tical application Its contribution can be justified accordingly

5 Conclusion

This paper proposes a procedure for solvingmultiple attributegroup decision making problems In the proposed proce-dure the transformation of assessment type the propertyof consistency the heterogeneity of a group of experts thedetermination of weight and scoring of alternatives are allconsidered It would be a useful tool for decision makers indifferent industries A review of the literature related to thisresearch suggests that no previous research has addressedall of the issues simultaneously The proposed procedure hasseveral important properties as follows

(i) Experts can provide their preference relations invarious formats which can then be transformed intoa standard type

(ii) Because all preference relation types are transformedinto fuzzy preferences and experts only providepreference relations between alternatives 119894 and 119894 + 1 itis possible to construct preference relations matrixesthat satisfy the property of additive consistency

(iii) Experts who are highly divergent from the groupmean will have their weights reduced

(iv) The weights of each attribute depend on the degree ofvariation the higher the variation of the attribute thehigher its weight

(v) Decisionmakers can select suitableMADMmethodssuch as SAW GRA or TOPSIS for the final rankingstep

In the proposed procedure all the steps are adopted inresponse to observations made in the related literature andare understood by managers who are not experts in fuzzytheory group decision making MADM or similar issues Anumerical example was described to illustrate the proposedprocedure It was demonstrated that the proposed procedureis simple and effective and can be easily applied to othersimilar practical problems

The proposed procedure has some weaknesses in severalof its properties The weight of each expert depends on thedivergence of his (or her) assessment from the opinionsof other experts Sometimes the real expert provides themost accurate assessment but is highly divergent from themean of group This characteristic would reduce the qualityof the group decision Moreover the proposed procedureassumes that an attribute is quite important if the differenceof the net degree of preference among all alternatives showsa wide variation However if an attribute is very importantand has a relatively high weight any small divergence inthe assessment of the attribute can influence the rankingproduced by the group decision These weaknesses canprovide the opportunity for future work


The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the National ScienceCouncil of Taiwan under Grants NSC-101-2221-E-131-043 andNSC-101-2221-E-006-137-MY3

References

[1] K Yoon and C L Hwang Multiple Attribute Decision MakingAn Introduction Sage Thousand Oaks Calif USA 1995

[2] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications vol 186 of Lecture Notes in Economicsand Mathematical Systems Springer New York NY USA 1981

[3] T L Saaty The Analytical Hierarchical Process John Wiley ampSons New York NY USA 1980

[4] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[5] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982

[6] T Yang and C Kuo ldquoA hierarchical AHPDEA methodologyfor the facilities layout design problemrdquo European Journal ofOperational Research vol 147 no 1 pp 128ndash136 2003

[7] T Yang and C C Hung ldquoMultiple-attribute decision mak-ing methods for plant layout design problemrdquo Robotics andComputer-Integrated Manufacturing vol 23 no 1 pp 126ndash1372007

[8] T Yang Y-C Chang and Y-H Yang ldquoFuzzy multiple attributedecision-makingmethod for a large 300-mm fab layout designrdquoInternational Journal of Production Research vol 50 no 1 pp119ndash132 2012

[9] T Yang Y-F Wen and F-F Wang ldquoEvaluation of robustnessof supply chain information-sharing strategies using a hybridTaguchi and multiple criteria decision-making methodrdquo Inter-national Journal of Production Economics vol 134 no 2 pp458ndash466 2011

[10] J-C Lu T Yang and C-T Suc ldquoAnalysing optimum pushpulljunction point location using multiple criteria decision-makingformultistage stochastic production systemrdquo International Jour-nal of Production Research vol 50 no 19 pp 5523ndash5537 2012

[11] T Yang and J-C Lu ldquoThe use of a multiple attribute decision-making method and value streammapping in solving the pace-maker location problemrdquo International Journal of ProductionResearch vol 49 no 10 pp 2793ndash2817 2011

[12] J C Lu T Yang and C Y Wang ldquoA lean pull systemdesign analysed by value stream mapping and multiple criteriadecision-making method under demand uncertaintyrdquo Interna-tional Journal of Computer IntegratedManufacturing vol 24 no3 pp 211ndash228 2011

[13] JMa J Lu andG Zhang ldquoDecider a fuzzymulti-criteria groupdecision support systemrdquo Knowledge-Based Systems vol 23 no1 pp 23ndash31 2010

[14] F J Cabrerizo I J Perez and E Herrera-Viedma ldquoManagingthe consensus in group decisionmaking in an unbalanced fuzzylinguistic context with incomplete informationrdquo Knowledge-Based Systems vol 23 no 2 pp 169ndash181 2010


[15] J Guo ldquoHybrid multicriteria group decision making methodfor information system project selection based on intuitionisticfuzzy theoryrdquoMathematical Problems in Engineering vol 2013Article ID 859537 12 pages 2013

[16] F Chiclana F Herrera and E Herrera-Viedma ldquoIntegratingthree representation models in fuzzy multipurpose decisionmaking based on fuzzy preference relationsrdquo Fuzzy Sets andSystems vol 97 no 1 pp 33ndash48 1998

[17] F Chiclana F Herrera and E Herrera-Viedma ldquoIntegratingmultiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relationsrdquo Fuzzy Setsand Systems vol 122 no 2 pp 277ndash291 2001

[18] E Herrera-Viedma F Herrera and F Chiclana ldquoA consensusmodel for multiperson decision making with different pref-erence structuresrdquo IEEE Transactions on Systems Man andCybernetics Part ASystems and Humans vol 32 no 3 pp 394ndash402 2002

[19] Z-P Fan S-H Xiao and G-F Hu ldquoAn optimization methodfor integrating two kinds of preference information in groupdecision-makingrdquo Computers and Industrial Engineering vol46 no 2 pp 329ndash335 2004

[20] Z-P Fan J Ma Y-P Jiang Y-H Sun and L Ma ldquoA goalprogramming approach to group decision making based onmultiplicative preference relations and fuzzy preference rela-tionsrdquo European Journal of Operational Research vol 174 no1 pp 311ndash321 2006

[21] J Zeng M An and N J Smith ldquoApplication of a fuzzy baseddecision making methodology to construction project riskassessmentrdquo International Journal of Project Management vol25 no 6 pp 589ndash600 2007

[22] E Herrera-Viedma F Herrera F Chiclana and M LuqueldquoSome issues on consistency of fuzzy preference relationsrdquoEuropean Journal of Operational Research vol 154 no 1 pp 98ndash109 2004

[23] A I Olcer and A Y Odabasi ldquoA new fuzzy multiple attributivegroup decision making methodology and its application topropulsionmanoeuvring system selection problemrdquo EuropeanJournal of Operational Research vol 166 no 1 pp 93ndash114 2005

[24] S Bozoki ldquoSolution of the least squares method problem ofpairwise comparison matricesrdquo Central European Journal ofOperations Research (CEJOR) vol 16 no 4 pp 345ndash358 2008

[25] Y-M Wang and Z-P Fan ldquoFuzzy preference relations aggre-gation and weight determinationrdquo Computers amp IndustrialEngineering vol 53 no 1 pp 163ndash172 2007

[26] F E Boran S Genc M Kurt and D Akay ldquoA multi-criteriaintuitionistic fuzzy groupdecisionmaking for supplier selectionwith TOPSISmethodrdquoExpert Systemswith Applications vol 36no 8 pp 11363ndash11368 2009

[27] F J Cabrerizo S Alonso and E Herrera-Viedma ldquoA consensusmodel for group decision making problems with unbalancedfuzzy linguistic informationrdquo International Journal of Informa-tion Technology and Decision Making vol 8 no 1 pp 109ndash1312009

[28] S J Chuu ldquoGroup decision-makingmodel using fuzzymultipleattributes analysis for the evaluation of advanced manufactur-ing technologyrdquo Fuzzy Sets and Systems vol 160 no 5 pp 586ndash602 2009

[29] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[30] Z Zhang and X Chu ldquoFuzzy group decision-making for multi-format and multi-granularity linguistic judgments in qualityfunction deploymentrdquo Expert Systems with Applications vol 36no 5 pp 9150ndash9158 2009

[31] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[32] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986

[33] J Kacprzyk and M Robubnes Non-Conventional PreferenceRelations in Decision Making Springer Berlin Germany 1988

[34] L Kitainik Fuzzy Decision Procedures with Binary RelationsTowards a UnifiedTheory vol 13 Kluwer Academic PublishersDordrecht The Netherlands 1993

[35] T Tanino ldquoFuzzy preference orderings in group decisionmakingrdquo Fuzzy Sets and Systems vol 12 no 2 pp 117ndash131 1984

[36] S A Orlovsky ldquoDecision-making with a fuzzy preferencerelationrdquo Fuzzy Sets and Systems vol 1 no 3 pp 155ndash167 1978

[37] HMHsu andC T Chen ldquoAggregation of fuzzy opinions undergroup decision-makingrdquo Fuzzy Sets and Systems vol 79 no 3pp 279ndash285 1996

[38] S M Chen ldquoAggregating fuzzy opinions in the group decision-making environmentrdquo Cybernetics and Systems vol 29 no 4pp 363ndash376 1998

[39] J Fodor and M Roubens Fuzzy Preference Modelling andMulticriteria Decision Support Kluwer Academic PublishersDordrecht The Netherlands 1994

[40] Y Kuo T Yang and G-W Huang ldquoThe use of grey relationalanalysis in solving multiple attribute decision-making prob-lemsrdquo Computers and Industrial Engineering vol 55 no 1 pp80ndash93 2008

Research ArticleIntegrated Supply Chain Cooperative Inventory Model withPayment Period Being Dependent on Purchasing Price underDefective Rate Condition

Ming-Feng Yang1 Jun-Yuan Kuo2 Wei-Hao Chen3 and Yi Lin4

1Department of Transportation Science National Taiwan Ocean University Keelung City 202 Taiwan2Department of International Business Kainan University Taoyuan 338 Taiwan3Department of Shipping and Transportation Management National Taiwan Ocean University Keelung City 202 Taiwan4Graduate Institute of Industrial and Business Management National Taipei University of Technology No 1Sec 3 Zhongxiao E Road Taipei City 106 Taiwan

Correspondence should be addressed to Ming-Feng Yang yang60429mailntouedutw

Received 18 August 2014 Revised 7 November 2014 Accepted 18 November 2014


Copyright copy 2015 Ming-Feng Yang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In most commercial transactions the buyer and vendor may usually agree to postpone payment deadline During such delayedperiod the buyer is entitled to keep the products without having to pay the sale price However the vendor usually hopes toreceive full payment as soon as possible especially when the transaction involves valuable items yet the buyer would offer a higherpurchasing price in exchange of a longer postponementTherefore we assumed such permissible delayed period is dependent on thepurchasing price As for the manufacturing side defective products are inevitable from time to time and not all of those defectiveproducts can be repaired Hence we would like to add defective production and repair rate to our proposed model and discusshow these factors may affect profits In addition holding cost ordering cost and transportation cost will also be considered as wedevelop the integrated inventory model with price-dependent payment period under the possible condition of defective productsWe would like to find the maximum of the joint expected total profit for our model and come up with a suitable inventory policyaccordingly In the end we have also provided a numerical example to clearly illustrate possible solutions

1 Introduction

Inventory occurs in every stage of the supply chain thereforemanaging inventory in an effective and efficient way becomesa significant task for managers in the course of supply chainmanagement (SCM) Fogarty [1] pointed out that the purposeof inventory is to retrieve demand and supply in an uncertainenvironment Frankel [2] considered supply chain to beclosely related to controlling and preserving stocks A goodinventory policy should contain a right venue to order tomanufacture and to distribute accurate supply quantities atthe right moment which will then store inventory at the rightplace to minimize total cost Another reason for the needto collaborate with other members in the supply chain isto remain competitive Better collaboration with customersand suppliers will not only provide better service but also

reduce costs [3] Beheshti [4] considered inventory policyas the key to affect conditions during the supply chainand applying inappropriate inventory policy would resultin great loss Therefore it is crucial for SCM practice togenerate suitable inventory policy Since the EOQ modelproposed byHarris [5] and researchers aswell as practitionershave shown interest in optimal inventory policy Harris [5]focused on inventory decisions of individual firms yet fromthe SCM perspective collaborating closely with membersof the supply chain is certainly necessary Goyal [6] is thefirst researcher to point out the importance of performancewhen integrating a supplier and a customerrsquos inventorypolicies The single-supplier single-customer model showedthe total relevant cost reduction compared with traditionalindependent inventory strategy Jammernegg and Reiner [7]pointed out that effective inventorymanagement can enhance



the value of the full supply chain Olson and Xie [8] proposedpurchasers and sellers should have a common inventorysystem when they cooperate with each other Since supplychain is formed with multiple firms focusing on a vendorand a buyerrsquos inventory problem is not sufficient In otherwords multiechelon inventory problem is one of the leadingissues in SCM Huang et al [9] developed an inventorymodel as three-level dynamic noncooperative game by usingthe Nash equilibrium Giannoccaro and Pontrandolfo [10]developed an inventory forecast for three-echelon supplychain to minimize the joint total cost Cardenas-Barron etal [11] made complements to some shortcomings in themodel proposed by Sana [12] and then introduced alternativealgorithm to obtain shorter CPU time and fewer total cost [3]Sana [12] coordinated production and inventory decisionsacross the supplier the manufacture and the customerto maximize the total expected profits Chung et al [13]combined deteriorating items with two levels of trade creditunder three-layer condition in the supply chain system Anew economic production quantity (EPQ) inventory is thenproposed to minimize the total cost Yang and Tseng [14]assumed that defective products occurred in the supplier andthe manufacturer stage and then backorder is allowed todevelop a three-echelon inventory model Permissible delayin payments and controllable lead time are also considered inthe model

Yield rate is an important factor in manufacturing indus-try Production can be imperfect which may have resultedfrom insufficient process control wrongly planned main-tenance inadequate work instructions or damages duringhandling (Rad et al [15]) High defective rate will increasenot only production costs but also inspecting costs andrepair costs which may likely cause shortage during theprocess In early researches defective production was rarelyconsidered in economic ordering quantity (EOQ) modelhowever defective production is a common condition inreal practice Schwaller [16] added fixed defective rate andinspecting costs to the traditional EOQ model Paknejadet al [17] developed an imperfect inventory model underrandom demands and fixed lead time Liu and Yang [18]developed an imperfect inventory model which includedgood products repairable products and scrap to maximizethe joint total profits Salameh and Jaber [19] indicatedthat all products should be divided into good productsand defective products they found that EOQ will increaseas defective products increase Eroglu and Ozdemir [20]extended Salameh and Jaberrsquos [19] model who indicatedhow defective rate affects economic production quantity(EPQ) with defective products and permissible shortageAll defective products can be inspected and sold separatelyfrom good products Pal et al [21] developed a three-layerintegrated production-inventory model considering out-of-control quality may occur in the supplier and manufacturerstage The defective products are reworked at a cost afterthe regular production time Using Stakelbergrsquos approach wecan see that the integrated expected average profit was beingcompared with the total expected average profits Sarkar etal [22] extended such work and developed three inventorymodels considering that the proportion of products could

follow different probability distribution uniform triangularand beta The models allowed planned backorders and thedefective products to be reworked [23]The comparison tablewas made to show that the minimum cost is obtained in thecase of triangular distribution Soni and Patel [24] assumedthat an arrival order lot may contain defective items and thenumber of defective items is a random variable which followsbeta distribution in a numerical example The demand issensitive to retail price and the production rate will react todemand

Recently permissible delay in payments has become acommon commercial strategy between the vendor and thebuyer It will bring additional interests or opportunity coststo each other as permissible delayed period varies hencedelayed period is a critical issue that researchers shouldconsider when developing inventory models In traditionalEOQ assumptions the buyer has to pay upon productdelivery however in actual business transactions the vendorusually gives a fixed delayed period to reduce the stress ofcapital During such period the buyer can make use of theproducts without having to pay to the vendor both partiescan earn extra interests from sales Goyal [25] developed anEOQ model with delays in payments Two situations werediscussed in the research (1) time interval between successiveorders was longer than or equal to permissible delay insettling accounts (2) time interval between successive orderswas shorter than permissible delay in settling accountsAggarwal and Jaggi [26] quoted Goyalrsquos [25] assumptionsto develop a deteriorating inventory model under fixeddeteriorating rate Jamal et al [27] extended Aggarwal andJaggirsquos [26] model and added shortage condition Teng [28]also amended Goyalrsquos [25] EOQ model and acquired twoconclusions (1) The EOQ decreases and the order cycleperiod shortens It is different from Goyalrsquos [25] conclusion(2) If the supplier wants to decrease the stocks the supplierhas to set higher interest rate to the retailer unpaid paymentsafter the payment periods are overdue but this will cause theEOQ to be higher than traditional EOQ model Huang et al[29] developed a vendor-buyer inventory model with orderprocessing cost reduction and permissible delay in paymentsThey considered applying information technologies to reduceorder processing cost as long as the vendor and the buyer arewilling to pay additional investment costs They also showedthat Ha and Kimrsquos [30] model is actually a special case Louand Wang [31] extended Huangrsquos [32] integrated inventorymodel which discussed the relationship between the vendorand the buyer in trade credit financing They relaxed theassumption that the buyerrsquos interest earned is always lessthan or equal to the interests charged They also establisheda discrimination term to determine whether the buyerrsquosreplenishment cycle time is less than the permissible delayperiod Li et al [33] extended the model of Meca et al [34]by adding permissible payment delays into the correspondinginventory game They also showed that the core of theinventory game is nonempty and the grand coalition is stablein amyopic perspective therefore largest consistent set (LCS)is applied to improve the grand coalition While most ofEOQmodels are considered with infinite replenishment rateSarkar et al [35] developed EOQ model for various types of


time-dependent demand when delay in payment and pricediscount are permitted by suppliers in order to obtain theoptimal cycle time with finite replenishment rate

The main purpose of this paper is to maximize theexpected joint total profits Based on Yang and Tsengrsquos[14] model we also considered the fact that some defec-tive products can be repaired Furthermore we proposedfunctions between purchasing costs and permissible delayedpayment period to balance the opportunity costs and interestsincome when we promote cooperation We first defined theparameters and assumptions in Section 2 and thenwe startedto develop the integrated inventory model in Section 3 InSection 4 we tried to solve the model to get the optimalsolution A series of numerical examples would be discussedto observe the variations of decision variables by changingparameters in Section 5 In the end we summarized thevariation and present conclusions

2 Notations and Assumptions

We first develop a three-echelon inventory model withrepairable rate and include permissible delay in paymentsdependent on sale price The expected joint total annualprofits of the model can be divided into three parts theannual profit of the supplier the manufacturer and theretailer We then observe how purchasing cost may affectpermissible delayed period EOQ the number of delivery perproduction run and the expected joint total annual profitsunder different manufacturerrsquos production rate and defectiverate

21 Notations To establish the mathematical model thefollowing notations and assumptions are used The notationsare shown as follows

The Parameters and the Decision Variable

119876119894 Economic delivery quantity of the 119894th model 119894 =1 2 3 4 a decision variable119899119894 The number of lots delivered in a production cyclefrom themanufacturer to the retailer of 119894th model 119894 =1 2 3 4 a positive integer and a decision variable

(i) Supplier Side

119862119904 Supplierrsquos purchasing cost per unit119860 119904 Supplierrsquos ordering cost per orderℎ119904 Supplierrsquos annual holding cost per unit119868sp Supplierrsquos opportunity cost per dollar per year119868se Supplierrsquos interest earned per dollar per year

(ii) Manufacturer Side

119875 Manufacturerrsquos production rate119883 Manufacturerrsquos permissible delayed period119862119898 Manufacturerrsquos purchasing cost per unit119860119898 Manufacturerrsquos ordering cost per order

119885 The probability of defective products from manu-facturer119877 The probability of defective products can berepaired119882 Manufacturerrsquos inspecting cost per unit119862rm Manufacturerrsquos repair cost per unit119866 Manufacturerrsquos scrap cost per unit119905119904 The time for repairing all defective products atmanufacturer119865119898 Manufacturerrsquos transportation cost per shipmentℎ119898 Manufacturerrsquos annual holding cost per unit119871119898 The length of lead time of manufacturer119868mp Manufacturerrsquos opportunity cost per dollar peryear119868me Manufacturerrsquos interest earned per dollar peryear

(iii) Retailer Side

119863 Average annual demand per unit time119884 Retailerrsquos permissible delayed period119875119903 Retailerrsquos selling price per unit119862119903 Retailerrsquos purchasing cost per unit119860119903 Retailerrsquos ordering cost per order119865119903 Retailerrsquos transportation cost per shipmentℎ119903 Retailerrsquos annual holding cost per unit119871119903 The length of lead time of retailer119868rp Retailerrsquos opportunity cost per dollar per year119868re Retailerrsquos interest earned per dollar per yearTP119904 Supplierrsquos total annual profitTP119898 Manufacturerrsquos total annual profitTP119903 Retailerrsquos total annual profitEJTP119894 The expected joint total annual profit 119894 =1 2 3 4

Note ldquo119894rdquo represents four different cases due to the relationshipof lead time and permissible payment period ofmanufacturerand the relationship of lead time and permissible paymentperiod of retailer We will have more detailed discussions inSection 3

22 Assumptions

(1) This supply chain system consists of a single suppliera single manufacturer and a single retailer for a singleproduct

(2) Economic delivery quantitymultiplied by the numberof deliveries per production run is economic orderquantity (EOQ)

(3) Shortages are not allowed


(4) The sale price must not be less than the purchasingcost at any echelon 119875119903 ge 119862119903 ge 119862119898 ge 119862119904

(5) Defective products only happened in the manu-facturer and can be inspected and separated intorepairable products and scrap immediately

(6) Scrap cannot be recycled so the manufacturer has topay to throw away

(7) The seller provides a permissible delayed period (119883and 119884) During the period the purchaser keepsselling the products and earning the interest by sellingrevenueThe purchaser pays to the seller at the end ofthe time period If the purchaser still has stocks it willbring capital cost

(8) The lead time of manufacturer is equal to the cycletime (119871119898 = 119899119876119863) The lead time of supplier is equalto the cycle time (119871119903 = 119876119863)

(9) The purchasing cost is in inverse to the permissibledelayed period Itmeans that the cheaper the purchas-ing cost the longer the permissible delayed period

(10) The time horizon is infinite

3 Model Formulation

In this section we have discussed the model of suppliermanufacture and retailer and we combined them all into anintegrated inventory model We extended Yang and Tsengrsquos[14] research to compute opportunity costs and interestsincome Finally we used the function between purchasingcosts and the permissible delayed payment period to discussand observe the variation of the expected joint total annualprofits

31 The Supplierrsquos Total Annual Profit In each productionrun the supplierrsquos revenue includes sales revenue and interestincome the supplierrsquos includes ordering cost holding costand opportunity cost Under the condition of permissibledelay in payments if the payment time of the manufacturer(119883) is longer than the lead time of the manufacturer (119871119898)it will bring additional interests income based on its interestrate (119868me) to the manufacturer On the other hand it causesthe supplier to pay additional opportunity cost based on itsinterest rate (119868sp) If the payment time of the manufacturer(119883) is shorter than the lead time of the manufacturer (119871119898)it will bring not only additional interests income but alsothe opportunity costs based on its interest rate (119868me and 119868sp)separately to the manufacturer because of the rest of stockshowever it causes the supplier to pay additional opportunitycosts but gains additional interests income based on itsinterest rate (119868sp and 119868se) separately

Before we start to establish the inventory model we haveto discuss how defective rate (119885) and repair rate (119877) can affectyield rate In each production run the manufacturer outputsdefective products because of the imperfect production lineIn other words yield rate is (1 minus119885) There is fixed proportionto repair these defective products which means that theproportion of repaired products is (119885119877) Since the repaired

Repaired products

Defective products

Normal products

Figure 1 Three kinds of products in the production run

products are counted in the yield products we have to reviseyield rate by adding the proportion of repaired productsFigure 1 showed the relationship of defective rate repair rateand yield rate So revised yield rate is (1minus119885(1minus119877)) In order tosatisfy the demand in each production run the manufacturerwill request the supplier to deliver (119899119876)[1 minus 119885(1 minus 119877)]

Figure 2 showed the supplier manufacturer and retailerrsquosinventory level As mentioned before the retailer needs (119899119876)to satisfy the demand while the manufacturer produces(119899119876)[1 minus 119885(1 minus 119877)] due to defective rate and repair rate andthe supplier would need to prepare (119899119876)[1 minus 119885(1 minus 119877)] toprevent storage

Case 1 (119871119898 lt 119883) If 119871119898 lt 119883 the manufacturer will earninterests income but themanufacturerrsquos interests incomewillbe transferred into opportunity costs for the supplier (seeFigure 3) Consider the following

(i) Sales revenue =119863(119862119898 minus 119862119904)(1 minus 119885(1 minus 119877))(ii) Ordering cost = 119860 119904119863119899119894119876119894

(iii) Holding cost = ℎ1199041198631198991198941198761198942119875[1 minus 119885(1 minus 119877)]2

(iv) Transfer opportunity cost = 119862119904119868sp(2119863119883 minus 119899119894119876119894)2[1 minus119885(1 minus 119877)]

Thus TP1199041 is given by

TP1199041 = sales revenue minus ordering cost minus holding cost

minus transfer opportunity cost

=119863 (119862119898 minus 119862119904)

1 minus 119885 (1 minus 119877)minus119860 119904119863

119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]

(1)

Case 2 (119871119898 ge 119883) If 119871119898 ge 119883 the manufacturer will not onlyearn interests income but also pay the opportunity costs dueto the rest of stocksThemanufacturerrsquos interests income andopportunity costs will be transferred into opportunity costsand interests income for the supplier (see Figure 4) Considerthe following

(i) Transfer opportunity cost = 119862119904119868sp(2119863119883 minus 119899119894119876119894)2[1 minus119885(1 minus 119877)]


nQ

1 minus Z(1 minus R)

nQD

nQD

nQD

nQ

nQ

P[1 minus Z(1 minus R)]

Z(1 minus R)nQ

1 minus Z(1 minus R)ts

nZQ


nQ


P

Q

t

t

t

Q

Q

Q

Q

P

QD

QD (n minus 1)Q

D

nRZQ


Figure 2 The inventory pattern for the three firms

(ii) Transfer interest income = 119862119898119868se(119899119894119876119894 minus119863119883)22119899[1 minus

119885(1 minus 119877)]119876119894



minus transfer opportunity cost + interest income

=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]+119862119898119868se (119899119894119876119894 minus 119863119883)

2

2119899 [1 minus 119885 (1 minus 119877)]119876119894

(2)

32 The Manufacturerrsquos Total Annual Profit In each pro-duction run the manufacturerrsquos revenue includes sales rev-enue and interests income the manufacturerrsquos cost includesordering costs holding costs transportation costs inspectingcosts repair costs scrap costs and opportunity costs Wehave discussed the relationship between the lead time of themanufacturer (119871119898) and the payment time of the manufac-turer (119883) This relationship can be also used to discuss theretailerrsquos lead time (119871119903) and the payment time (119884) thereforethe manufacturerrsquos total annual profit has four different casesIn themiddle of Figure 2 is themanufacturerrsquos inventory levelwhich has been the effect of defective rate and repair rate

Case 1 (119871119898 lt 119883 119871119903 lt 119884) If 119871119898 lt 119883 and 119871119903 lt 119884both the manufacturer and the retailer will earn interestsincome but the retailerrsquos interests income will be transferred


nQ


Lm =nQ

D

X

Q

t

Interest income

Figure 3 119871119898lt 119883

Lm =nQ

D

nQ


X

Q

Interest income

Opportunity cost

t

Figure 4 119871119898ge 119883

into opportunity costs for the manufacturer Consider thefollowing

(i) Sales revenue =119863[119862119903 minus 119862119898(1 minus 119885(1 minus 119877))]

(ii) Ordering cost = 119860119898119863119899119894119876119894

(iii) Holding cost = ℎ119898119863119876119894[(119899119894 minus1)2119863+ 1minus2[1minus119885(1minus119877)]1198991198942119875[1minus119885(1minus119877)]

2+1119875]minus119905119904119885119877119899119894(1minus119885(1minus119877))

(iv) Transportation cost = 119865119898119863119899119894119876119894

(v) Inspecting cost =119882119863(1 minus 119885(1 minus 119877))

(vi) Repair cost =119882119863(1 minus 119885(1 minus 119877))

(vii) Scrap cost = 119866119885(1 minus 119877)119863(1 minus 119885(1 minus 119877))

(viii) Interest income =119862119903119868me(2119863119883minus119899119894119876119894)2[1minus119885(1minus119877)]

(ix) Transfer opportunity cost = 119862119898119868mp(119863119884 minus 1198761198942)


TP1198981

= sales revenue minus ordering cost minus holding cost

minus transportation cost minus inspecting cost minus repair cost

minus scrap cost + interest income


= 119863[119862119903 minus119862119898

1 minus 119885 (1 minus 119877)] minus

119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1

2119863+1 minus 2 [1 minus 119885 (1 minus 119877)] 119899119894

2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]minus 119862119898119868mp (119863119884 minus

119876119894

2)

(3)

Case 2 (119871119898 lt 119883 119871119903 lt 119884) If 119871119898 lt 119883 and 119871119903 ge 119884 themanufacturer will earn interests incomewhile the retailer willnot due to the rest of stocks but the retailerrsquos interests incomeand opportunity costs will be transferred into opportunitycosts and interests income for the manufacturer

Interest income =119862119903119868me (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)] (4)

Consider the following

(i) Transfer opportunity cost = 119862119898119868mp(119863119884)22119876119894

(ii) Transfer interest income = 119862119903119868me(119876119894 minus 119863119884)22119876119894


TP1198982




minus transfer opportunity cost + transfer interest income

= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)


minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]minus119862119898119868mp (119863119884)

2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(5)

Case 3 (119871119898 ge 119883 119871119903 lt 119884) If 119871119898 ge 119883 and 119871119903 lt 119884the manufacturer will not earn interests income but also payopportunity costs and the retailer will earn interests incomebut such incomewill be transferred into opportunity costs forthe manufacturer Consider the following

(i) Opportunity cost = 119862119898119868mp(119899119894119876119894 minus 119863119883)22[1 minus 119885(1 minus

119877)]119899119894119876119894(ii) Interest income = 119862119903119868me(119863119883)

22[1 minus 119885(1 minus 119877)]119899119894119876119894

(iii) Transfer opportunity cost = 119862119898119868mp(119863119884 minus 1198761198942)


TP1198983



minus scrap cost minus opportunity cost + interest income


= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119894119876119894 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus 119862119898119868mp (119863119884 minus119876119894

2)

(6)

Case 4 (119871119898 ge 119883 119871119903 ge 119884) If 119871119898 ge 119883 and 119871119903 ge 119884both the manufacturer and the retailer will not earn interestsincome but need to pay opportunity costs and the retailerrsquosinterests income and opportunity costs will be transferredinto opportunity costs for the manufacturer Consider thefollowing


119877)]119899119894119876119894

(ii) Interest income = 119862119903119868me(119863119883)22[1 minus 119885(1 minus 119877)]119899119894119876119894

(iii) Transfer opportunity cost = 119862119898119868mp(119863119884)22119876119894

(iv) Transfer interest income = 119862119903119868me(119876119894 minus 119863119884)22119876119894


TP1198984





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119876 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus119862119898119868mp (119863119884)

2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(7)

33 The Retailerrsquos Total Annual Profit In each produc-tion run the retailerrsquos revenue includes sales revenue andinterests income the retailerrsquos costs include ordering costsholding costs transportation costs and opportunity costsThe relationship between the retailerrsquos lead time (119871119903) andpayment time (119884) has been discussed before The retailermay gain additional interests incomeor pay opportunity costsaccording to two different cases shown as follows

Case 1 (119871119903 lt 119884) If 119871119903 lt 119884 the retailer will earn interestincome Consider the following

(i) Sales revenue =119863(119875119903 minus 119862119903)

(ii) Ordering cost = 119860119903119863119899119894119876119894

(iii) Holding cost = ℎ1199031198761198942


(v) Interest income = 119875119903119868re(119863119884 minus 1198761198942)



TP1199031


minus transportation cost + interest income

= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

+ 119875119903119868re (119863119884 minus119876119894

2)

(8)Case 2 (119871119903 ge 119884) If 119871119903 ge 119884 the retailer will not only earninterests income but also pay opportunity costs due to the restof stocks Consider the following

(i) Opportunity cost = 119862119903119868rp(119876119894 minus 119863119884)22119876119894

(ii) Interest income = 119875119903119868re(119863119884)22119876119894


TP1199032


minus transportation cost minus opportunity cost

+ interest income

= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

minus119862119903119868rp (119876119894 minus 119863119884)

2

2119876119894

+119875119903119868re (119863119884)

2

2119876119894

(9)

34 The Expected Joint Total Annual Profit According todifferent conditions the expected joint total annual profitfunction EJTP(119876119894 119899119894) can be expressed as

EJTP119894 (119876119894 119899119894)

=

EJTP1 (1198761 1198991) = TP1199041 + TP1198981 + TP1199031if 119871119898 lt 119883 119871119903 lt 119884

EJTP2 (1198762 1198992) = TP1199041 + TP1198982 + TP1199032if 119871119898 lt 119883 119871119903 ge 119884

EJTP3 (1198763 1198993) = TP1199042 + TP1198983 + TP1199031if 119871119898 ge 119883 119871119903 lt 119884

EJTP4 (1198764 1198994) = TP1199042 + TP1198984 + TP1199032if 119871119898 ge 119883 119871119903 ge 119884

(10)

whereEJTP1 (1198761 1198991)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198761 [1198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198991

1 minus 119885 (1 minus 119877) minus

ℎ1199031198761

2minus

ℎ11990411986311989911198761

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198761

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989911198761)

2 [1 minus 119885 (1 minus 119877)]

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198761

2)

EJTP2 (1198762 1198992)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198762 [1198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198992


ℎ1199031198762

2minus

ℎ11990411986311989921198762

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989921198762)

2 [1 minus 119885 (1 minus 119877)]

+(119862119903119868me minus 119862119903119868rp) (1198762 minus 119863119884)

2

21198762

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198762

EJTP3 (1198763 1198993)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198763 [1198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198993


ℎ1199031198763

2minus

ℎ11990411986311989931198763

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198763

2)

EJTP4 (1198764 1198994)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]


minus ℎ1198981198631198764 [1198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198994


ℎ1199031198764

2minus

ℎ11990411986311989941198764

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198764

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119898119868se minus 119862119898119868mp) (11989941198764 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119903119868me minus 119862119903119868rp) (1198764 minus 119863119884)

2

21198764

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198764

(11)

4 Solution Procedure

41 Determination of the Optimal Delivery Quantity 119876119894 forAny Given 119899119894 We would like to find the maximum value ofthe expected total profit EJTP(119876119894 119899119894) For any 119899119894 we will takethe first and second partial derivations of EJTP(119876119894 119899119894) withrespect to 119876119894 We have

120597EJTP1 (1198761 1198991)1205971198761

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198762

1

minus ℎ1198981198631198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198991

2119875 [1 minus 119885 (1 minus 119877)]2minus(119862119903119868me minus 119862119904119868sp) 1198991

2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp)

2

(12)

120597EJTP2 (1198762 1198992)1205971198762

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

2

minus ℎ1198981198631198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198992


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987622

+(119862119903119868me minus 119862119903119868rp) [119876

2

2minus (119863119884)

2]

211987622

(13)

120597EJTP3 (1198763 1198993)1205971198763

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198762

3

minus ℎ1198981198631198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198993

2119875 [1 minus 119885 (1 minus 119877)]2

minus(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

+(119862119898119868se minus 119862119898119868mp) [(11989931198763)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

minus(119875119903119868re minus 119862119898119868mp)

2

(14)

120597EJTP4 (1198764 1198994)1205971198764

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198762

4

minus ℎ1198981198631198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198994

2119875 [1 minus 119885 (1 minus 119877)]2minus(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987624

+(119862119898119868se minus 119862119898119868mp) [(11989941198764)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

+(119862119903119868me minus 119862119903119868rp) [119876

2

4minus (119863119884)

2]

211987624

(15)


102

1025

103

1035

104

1045

105

2295 2305 2315 2325 2335 2345 2355

Q1

Manufacturerrsquos purchasing cost Cm

(a) The value of1198761 by changing 119862119898 under different 119875

0

100

200

300

400

500

600

700

2295 2305 2315 2325 2335 2345 2355Manufacturerrsquos purchasing cost Cm

Q2

(b) The value of1198762 by changing 119862119898 under different 119875

777879808182838485

235 236 237 238 239 240

Q3

P = 1100

P = 1200

P = 1300


(c) The value of1198763 by changing 119862119898 under different 119875

0

200

400

600

800

1000

1200

235 236 237 238 239 240

Q4

P = 1100

P = 1200

P = 1300


(d) The value of1198764 by changing 119862119898 under different 119875

Figure 5 The value of delivery quantity by changing 119862119898in 119876119894 for 119894 = 1 2 3 4

1205972EJTP1 (1198761 1198991)

12059711987621

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198763

1

lt 0

(16)

1205972EJTP2 (1198762 1198992)

12059711987622

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198763

2

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987632

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987632

lt 0

(17)

1205972EJTP3 (1198763 1198993)

12059711987623

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

3

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

lt 0

(18)

1205972EJTP4 (1198764 1198994)

12059711987624

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198763

4

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987634

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987634

lt 0

(19)

Because (16) (17) (18) and (19)lt 0 therefore EJTP(119876119894 119899119894)is concave function in 119876119894 for fixed 119899119894 We can finda unique value of 119876119894 that maximize EJTP(119876119894 119899119894) Let


60000

60500

61000

61500

62000

62500

63000

63500

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1


(a) The value of EJTP1 by changing 119862119898 under different 119875

30000

35000

40000

45000

50000

55000

60000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2


(b) The value of EJTP2 by changing 119862119898 under different 119875

43000432004340043600438004400044200444004460044800

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

3


(c) The value of EJTP3 by changing 119862119898 under different 119875

60008000

10000120001400016000180002000022000

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

4


(d) The value of EJTP4 by changing 119862119898 under different 119875

Figure 6 The value of profit by changing 119862119898in EJTP

119894 for 119894 = 1 2 3 4

120597EJTP119894(119876119894 119899119894)120597119876119894 = 0 in (16) (17) (18) and (19) so we canget that 119876119894 are as follows

The original equations are too long so in order to shortenthem we let [1 minus119885(1minus119877)] = 119880 (119862119903119868me minus119862119904119868sp) = 119872 (119875119903119868re minus119862119898119868mp) = 119882 (119862119903119868meminus119862119903119868rp) = 119861 (119862119898119868seminus119862119898119868mp) = 119864Thenwe substitute them into the original equations

119876lowast

1= ((2119863119875119880

2(119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991))

times (1198992 119875119880 [119880 (ℎ119898 (1198991 minus 1) + ℎ119903 +119882) +1198721198991]

+119863 [1198991 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(20)

119876lowast

2= ((119875119880

2[2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

minus 1198992 (119861 +119882) (119863119884)2])

times (1198992 119875119880 [119880 (ℎ119898 (1198992 minus 1) + ℎ119903 minus 119861) +1198721198992]

+119863 [1198992 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(21)

119876lowast

3= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

minus (119872 + 119864) (119863119883)2])

times (1198993 119875119880 [119880 (ℎ119898 (1198993 minus 1) + ℎ119903 +119882) minus 119864]

+119863 [1198993 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(22)

119876lowast

4= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

minus (119872 + 119864) (119863119883)2minus 1198801198994 (119861 +119882) (119863119884)

2])

times (1198994 119875119880 [119880 (ℎ119898 (1198994 minus 1) + ℎ119903 minus 119861) minus 119864]

+119863 [1198994 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(23)

Algorithm To summarize the above arguments we estab-lished the algorithm to obtain the optimal values ofEJTP(119899119894 119876119894)

Equation (10) shows the situations of each case obviouslyeach case is mutual exclusive In other words before we start


102

1025

103

1035

104

1045

105

239 240 241 242 243 244 245 246

Q1

Retailerrsquos purchasing cost Cr


560565570575580585590595600605610

245 246 247 248 249 250 251

Q2



200210220230240250260270280290300

239 240 241 242 243 244 245 246

Q3

P = 1100

P = 1200

P = 1300



500550600650700750800850900

245 246 247 248 249 250 251

Q4

P = 1100

P = 1200

P = 1300




to find the optimal solutions we have to recognize whichequations should be used first

Step 1 Examine the relationship of 119871119898 119883 and 119871119903 119884 to usecorresponding equations

Step 2 Let 119899119894 = 1 and substitute into (20) (21) (22) or (23)to find 1198761 1198762 1198763 or 1198764

Step 3 Find EJTP119894 by substituting 119899119894 119876119894 and different pro-duction rate (119875)

Step 4 Let 119899 = 119899119894 + 1 and repeat Step 2 to Step 3 untilEJTP119894(119899119894) gt EJTP119894(119899119894+1)

5 Numerical Example

In Section 5 we will observe the variation of119876119894 119899119894 and EJTP119894by changing119862119898 and119862119903 separately under different productionrate or defective rate We consider an inventory system withthe following data

Consider119863 = 1000 unityear 119862119904 = 200 per unit 119860 119904 = 80per order ℎ119904 = 20 per unit 119868sp = 0025 per year 119868se = 00254per year 119862119898 = 235 per unit 119860119898 = 100 per order ℎ119898 = 23per unit 119882 = 5 per unit 119862rm = 10 per unit 119866 = 10 per

unit 119865119898 = 100 per time 119885 = 01 119877 = 09 119905119904 = 00055 year119868mp = 00256 per year 119868me = 002 per year 119862119903 = 245 per unit119860119903 = 120 per order ℎ119903 = 25 per unit 119865119903 = 150 per time119875119903 = 280 per unit 119868rp = 002 per year and 119868re = 0021 peryear

51 The Variation under Different 119875 In Section 51 we sup-posed that the maximum of the production rate is 1300The manufacturer can change the production rate under anycondition furthermore the extra payment by changing therate is ignored Let us observe the value of delivery quantityand profit with 119875 = 1100 119875 = 1200 and 119875 = 1300 bychanging the manufacturerrsquos purchasing costs and we set thefunction of 119871119898 and 119883 is 119883 = 3000119862119898 or changing theretailerrsquos purchasing costs and we set the function of 119871119903 and119884 is 119884 = 3000119862119903

511 The Permissible Period 119883 and EJTP We have changed119862119898 by 05 per unit In order to find out which condition ismore beneficial to the proposed inventory model we formedthe details shown in Table 1 and the solution results areillustrated in Figures 5 and 6

We have discussed that if the payment time is longerthan the lead time it will bring additional interests income


Table 1 The value of profit in different condition by changing 119862119898

119875 = 1100 119875 = 1200 119875 = 1300

119862119898

2300sim2350 2300sim2350 2300sim23501198991

2 2 21198761

10219sim10229 10339sim10349 10444sim10454EJTP1

6278289sim6124925 6293018sim6139667 lowast6305673sim6152333119862119898

2300sim2350 2300sim2350 2300sim23501198992

1 1 11198762

18902sim57786 19404sim59321 19862sim60721EJTP2

5846523sim3350315 5877907sim3446259 5905128sim3529477119862119898

2355sim240 2355sim240 2355sim2401198993

14 13 131198763

7873sim7772 8404sim8295 8415sim8306EJTP3

4463785sim4357297 4466066sim4359922 4468691sim4362513119862119898

2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355sim2365) 2 (119862

119898= 2355) 1

1 (119862119898= 2355sim2365) 1 (119862

119898= 2360sim2365)

1198764

64172sim67519 (119862119898= 2355sim2365) 65370 (119862

119898= 2355) 90178sim104684

90662sim99636 (119862119898= 2370sim2400) 89788sim102277 (119862

119898= 2355sim2365)

EJTP4

1800704sim835320 1900021sim1000745 1990857sim1144218lowastOptimal solution of EJTP119894

59500

60000

60500

61000

61500

62000

62500

63000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



33000

34000

35000

36000

37000

38000

39000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



34000

36000

38000

40000

42000

44000

46000

240 2405 241 2415 242 2425 243 2435 244 2445 245

P = 1100

P = 1200

P = 1300

EJTP

3



200002200024000260002800030000320003400036000

2455 246 2465 247 2475 248 2485 249 2495 250

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4


1018102

1022102410261028

103103210341036

229 230 231 232 233 234 235 236

Q1



100150200250300350400450500550600

229 230 231 232 233 234 235 236

Q2



707274767880828486

235 236 237 238 239 240 241

Q3

Z = 01

Z = 02

Z = 03



0

200

400

600

800

1000

1200

235 236 237 238 239 240 241

Q4

Z = 01

Z = 02

Z = 03




to the buyer However if the payment time is shorter thanthe lead time it will bring additional interests income andopportunity costs to the buyer due to the rest of stocks Aftercomputing and comparing the results in Table 1 we havefound that the optimal profits will occur in EJTP1(1198761 1198991)under the manufacturerrsquos production rate being 1300 unitsper year Also the worst profit will occur in EJTP4(1198764 1198994)under themanufacturerrsquos production rate being 1100 units peryear

512 The Permissible Time 119883 and EJTP In Section 512 wechanged the retailerrsquos purchasing cost to observe the value ofprofit the solution results are illustrated in Figures 7 and 8and the detailed result is shown in Table 2

From Table 2 we have found that the optimal profits willoccur in EJTP1(1198761 1198991) under the manufacturerrsquos productionrate being 1300 units per year which is the same as inSection 511 Also theworst profitwill occur in EJTP4(1198764 1198994)under themanufacturerrsquos production rate being 1100 units peryear

52 The Variation under Different 119885 In Section 52 wesupposed that the maximum of defective rate is 03 Themanufacturer can change the production rate under anycondition also the extra payment by changing the rate isignored

521 The Permissible Period 119883 and EJTP We have changedmanufacturerrsquos purchasing cost 119862119898 by 05 per unit In orderto compare which condition is more beneficial we formeddetailed results in Table 3 The solution results are illustratedin Figures 9 and 10

From Table 3 we have found that the optimal profitswill occur in EJTP1(1198761 1198991) under the manufacturerrsquos defec-tive rate being 01 Also the worst profits will occur inEJTP4(1198764 1198994) under the manufacturerrsquos defective rate being03

522 The Permissible Period 119884 and EJTP We have changedretailerrsquos purchasing costs 119862119903 by 05 per unit In order toknow which condition is more beneficial we formed detailedresults in Table 4 The solution results are illustrated inFigures 11 and 12

From Table 4 we have found the optimal profits willoccur in EJTP1(1198761 1198991) under the manufacturerrsquos defec-tive rate being 01 Also the worst profits will occur inEJTP4(1198764 1198994) under the manufacturerrsquos defective rate being03

53 Observation (See Figures 5ndash12 and Tables 1ndash4) InSection 51 we observed the variation of quantity per deliverynumbers of delivery and EJTP by changing manufacturerrsquos



119875 = 1100 119875 = 1200 119875 = 1300

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761

lowastlowast10229 lowastlowastlowast10349 lowastlowastlowastlowast10454EJTP1 5993540sim6124925 6008294sim6139666 lowast6020970sim6152333

119862119903 2455sim2500 2455sim2500 2455sim25001198992 1 1 11198762 57670sim56645 59202sim58148 60598sim59517

EJTP2 3370094sim3546836 3465934sim3640884 3548895sim3722454


EJTP3 3425530sim4221153 3464154sim4251420 3497160sim4277287

119862119903 2455sim2500 2455sim2500 2455sim2500

1198994

2 (119862119903= 2455sim2465)

1 (119862119903= 2470sim2500)

2 (119862119903= 2455sim2460)

1 (119862119903= 2355sim2500)

2 (119862119903= 2455)

1 (119862119903= 2360sim2500)

1198764

61574sim59822 (119862119903= 2455sim2465)

77530sim66375 (119862119903= 2470sim2500)

62723sim61837(119862119903= 2455sim2460)

81338sim68135 (119862119903= 2465sim250)

63748 (119862119903= 2455)

85010sim69738 (119862119903= 246sim250)

EJTP4 2021540sim2977979 2116825sim3088182 2198873sim3183760

lowastOptimal solution of EJTP119894lowastlowastlowastlowastlowastlowastlowastlowastWe cannot observe the variation because of low increasing rate in fact1198761 will decrease slightly when 119862119903 increases

52000

54000

56000

58000

60000

62000

64000

66000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



0

10000

20000

30000

40000

50000

60000

70000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



3000032000340003600038000400004200044000460004800050000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

3



02000400060008000

100001200014000160001800020000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4


1021022102410261028

1031032103410361038

104

239 240 241 242 243 244 245 246

Q1



576578580582584586588590592594

245 246 247 248 249 250 251

Q2



6065707580859095

100

239 240 241 242 243 244 245 246

Q3

Z = 01

Z = 02

Z = 03



500

550

600

650

700

750

800

850

245 246 247 248 249 250 251

Q4

Z = 01

Z = 02

Z = 03





119885 = 01 119885 = 02 119885 = 03


EJTP1

lowast6293018sim6139667 5978353sim5825030 5657147sim5503852119862119898 2300sim2350 2300sim2350 2300sim23501198992 1 1 11198762 19404sim59321 19348sim59150 19290sim58973

EJTP2 5877907sim3446259 5561648sim3126491 5238827sim2800005


EJTP3 4466066sim4359922 4147330sim4041153 3822615sim3716446

119862119898 2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

1198764

65370 (119862119898= 2355)

89788sim102277 (119862119898= 236sim240)

65203 (119862119898= 2355)

89751sim102171 (119862119898= 236sim240)

65029 (119862119898= 2355)

89708sim102058 (119862119898= 236sim240)

EJTP4 1900021sim1000745 1567517sim667383 1227857sim362920

lowastOptimal solution of EJTP119894


4600048000500005200054000560005800060000620006400066000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



2500027000290003100033000350003700039000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



3000032000340003600038000400004200044000460004800050000

240 2405 241 2415 242 2425 243 2435 244 2445 245

Z = 01

Z = 02

Z = 03

EJTP

3



10000

15000

20000

25000

30000

35000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4

purchasing costs 119862119898 or retailerrsquos purchasing costs 119862119903 underdifferent production rate Obviously higher production ratewill yield higher profits All EJTP of each case decreases when119862119898 increases In Section 511 the optimal profits occur inEJTP1(1198761 1198991) under 119875 = 1300 in Section 512 the optimalprofits also occur in EJTP1(1198761 1198991) under 119875 = 1300

In Section 52 the observations are shown under differentdefective rate consideration Surely higher defective rateleads manufacturer to pay more costs to rework defectiveitems and deal with scrap As 119862119898 increases all EJTP of eachcase decreases nevertheless increasing C119903 brings decreasingEJTP contrarily In Section 521 the optimal profits occur inEJTP1(1198761 1198991) under 119885 = 01 in Section 522 the optimalprofits also occur in EJTP1(1198761 1198991) under 119885 = 01

Because of the relationship between the price and pay-ment period the decision-makers can get different paymentperiod by varying the price When the supply chain issuccessfully integrated this variation can lead to unnecessarycosts reduction or enhance the performance

6 Conclusions and Future Works

Themain purpose of every firm is to maximize profits Thereare two ways to enhance profits one is to raise the productsrsquoselling price and the other is to lower the relevant costs insupply chain To raise the productsrsquo selling price firms have toenhance productsrsquo quality and show uniqueness to convince

customers Alternatively firms can provide proper strategiesto reduce relevant costs such as purchasing costs productioncosts holding costs and transportation costs

Permissible delay in payments is a common commercialstrategy in real business transactions since the purpose ofbusiness strategies is to enhance the flexibility of capital Inother words firms can obtain additional interests incomefrom sales revenue during the payment period yet upstreamfirms simply grant loans to downstream firms without anyinterestsThus it is of great importance to decide the length ofpayment period in an SCM setting There are many ways tobalance the costs or revenue of each firm From the rewardperspective providing discounts is a direct way to attractdownstream firms in accepting shorter payment period Onthe other hand which is from the punishment perspectivedownstream firms must pay extra costs if they wish to enjoya longer payment period Whether it is from the rewardsor the punishments perspective the purpose is always aboutshortening the payment period In this paper we have useddifferent ways to determine the payment period We setthe relationship of purchasing costs and payment period asinverse proportion that is payment period is floating andhigher purchasing costs will bring shorter payment periodFrom the results in Section 5 decision-makers should negoti-ate with their upstream or downstream firms to enhance sup-ply chain performance From the supplier andmanufacturerrsquos



119885 = 01 119885 = 02 119885 = 03

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761

lowast10349 lowastlowast10283 lowastlowastlowast10216EJTP1


EJTP2 3465846sim3640884 3146216sim3322506 2819872sim2997440

119862119903 2400sim245 2400sim245 2400sim245

1198993

17 (119862119903= 240sim241)

16 (119862119903= 2415sim2425)

15 (119862119903= 243sim2435)

14 (119862119903= 244sim2445)

13 (119862119903= 245)

19 (119862119903= 240)

18 (119862119903= 2405sim2415)

17 (119862119903= 242sim2425)

16 (119862119903= 243sim2435)

15 (119862119903= 244sim2445)

14 (119862119903= 245)

21 (119862119903= 240)

20 (119862119903= 2405sim241)

19 (119862119903= 2415sim242)

18 (119862119903= 2425)

17 (119862119903= 243sim2435)

16 (119862119903= 244sim2445)

15 (119862119903= 245)

1198763

8295sim7997 (119862119903= 240sim241)

8277sim811 (119862119903= 2415sim2425)

8221sim8032 (119862119903= 243sim2435)

8325sim8112 (119862119903= 244sim2445)

8417 (119862119903= 245)

7453 (119862119903= 240)

7684sim7399 (119862119903= 2405sim2415)

7629sim7471 (119862119903= 242sim2425)

7709sim7533 (119862119903= 243sim2435)

7582 (119862119903= 244sim2445)

7834 (119862119903= 245)

6762 (119862119903= 240)

6940sim6814 (119862119903= 2405sim241)

6998sim6860 (119862119903= 2415sim242)

7048 (119862119903= 2425)

7252sim7088 (119862119903= 243sim2435)

7296sim7113 (119862119903= 244sim2445)

7323 (119862119903= 245)

EJTP3 3823707sim4477773 3503787sim4159040 3178546sim3834324

119862119903 2455sim250 2455sim250 2455sim250

1198994

2 (119862119903= 2455)

1 (119862119903= 246sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

1198764

62723 (119862119903= 2455)

61837sim68135 (119862119903= 246sim250)

62565sim61676 (119862119903= 2455)

2465sim250 (119862119903= 246sim250)

62400sim61508 (119862119903= 2455sim246)

81257sim67924 (119862119903= 246sim250)

EJTP4 2116836sim3088182 1785043sim2763725 1446121sim2432425


viewpoint EJTP moves up when the purchasing costs ofmanufacturer go down However there is a contrary result onthemanufacturer and supplierrsquos side Higher purchasing costsof the supplier will lead to lower profits Decision-makersshould know where their firms are positioned in the supplychain and may thus make appropriate decisions

Defective rate is also an important factor in the man-ufacturing process The higher the probability of defectiveproduct occurrence the higher the cost and more time willbe spent by the manufacturer these may include reorderingthe materials and reproducing repairing and declaring thescrap Additionally defective rate is one of the direct factorsto affect the amount of storage If retailers do not have enoughstocks to satisfy customersrsquo needs customers may lose theirpatience and therefore choose other retailers Surely it isimportant to accurately grasp the situation of productionlines

From what has been discussed above we developed athree-echelon inventory model to determine optimal jointtotal profits Firstly we have developed four inventorymodelsin Section 3 according to different permissible delay payment

period and lead time Secondly we computed the decisionvariables economical delivery quantity and the number ofdeliveries per production run from the manufacturer to theretailer Finally we observed and found the optimal profits byvarying the manufacturerrsquos purchasing costs or the supplierrsquospurchasing costs

Compared with Yang and Tsengrsquos [14] article althoughthey considered the defective products to occur in the threeechelons we only assumed the defective products occur inthe manufacturing process In this paper we also focusedon the relationship between materialsfinished productrsquos saleprice and the permissible delay period We assumed thatthe relationship is inverse proportion and developed thefunction while Yang and Tsengrsquos [14] simply focused onvariable lead time and assumed that the permissible delayperiod is constant

In the future we can addmore conditions or assumptionssuch as ignoring the backorder and variable lead time whichwere considered by Yang and Tsengrsquos [14] The assumptionscan be added again to develop more practical inventorymodels Besides multiple sellers or multiple purchasers are


not unusual situations in commerce Moreover the param-eters in this paper are fixed while some of them (such asdemand or defective rate) may be unfixed in practice byusing fuzzy theory The fuzzy variables can lead to betterresults The issue regarding deteriorating products is worthyof deliberation in the inventory model since all productswould face deterioration (ie rust or decay) sooner or laterWe look forward to illustrating real-world numerical exam



References

[1] R Fogarty ldquoTen ways to integrate curriculumrdquo EducationalLeadership vol 49 no 2 pp 61ndash65 1991

[2] R Frankel ldquoThe role and relevance of refocused inventorysupply chainmanagement solutionsrdquo Business Horizons vol 49no 4 pp 275ndash286 2006

[3] M Ben-Daya R AsrsquoAd and M Seliaman ldquoAn integratedproduction inventory model with raw material replenishmentconsiderations in a three layer supply chainrdquo InternationalJournal of Production Economics vol 143 no 1 pp 53ndash61 2013

[4] H M Beheshti ldquoA decision support system for improvingperformance of inventory management in a supply chainnetworkrdquo International Journal of Productivity and PerformanceManagement vol 59 no 5 pp 452ndash467 2010

[5] F W Harris ldquoHow many parts to make at oncerdquo OperationsResearch vol 38 no 6 pp 947ndash950 1913

[6] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[7] W Jammernegg and G Reiner ldquoPerformance improvement ofsupply chain processes by coordinated inventory and capacitymanagementrdquo International Journal of Production Economicsvol 108 no 1-2 pp 183ndash190 2007

[8] D L Olson and M Xie ldquoA comparison of coordinated supplychain inventory management systemsrdquo International Journal ofServices and Operations Management vol 6 no 1 pp 73ndash882010

[9] Y Huang G Q Huang and S T Newman ldquoCoordinatingpricing and inventory decisions in a multi-level supply chaina game-theoretic approachrdquo Transportation Research Part ELogistics and Transportation Review vol 47 no 2 pp 115ndash1292011

[10] I Giannoccaro and P Pontrandolfo ldquoInventory managementin supply chains a reinforcement learning approachrdquo Interna-tional Journal of Production Economics vol 78 no 2 pp 153ndash161 2002

[11] L E Cardenas-Barron J-T Teng G Trevino-Garza H-MWee andK-R Lou ldquoAn improved algorithmand solution on anintegrated production-inventory model in a three-layer supplychainrdquo International Journal of Production Economics vol 136no 2 pp 384ndash388 2012

[12] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[13] K-J Chung L Eduardo Cardenas-Barron and P-S Ting ldquoAninventory model with non-instantaneous receipt and exponen-tially deteriorating items for an integrated three layer supplychain system under two levels of trade creditrdquo InternationalJournal of Production Economics vol 155 pp 310ndash317 2014

[14] M F Yang and W C Tseng ldquoThree-echelon inventory modelwith permissible delay in payments under controllable leadtime and backorder considerationrdquo Mathematical Problems inEngineering vol 2014 Article ID 809149 16 pages 2014

[15] M A Rad F Khoshalhan and C H Glock ldquoOptimizinginventory and sales decisions in a two-stage supply chain withimperfect production and backordersrdquo Computers amp IndustrialEngineering vol 74 pp 219ndash227 2014

[16] R L Schwaller ldquoEOQ under inspection costsrdquo Production andInventory Management Journal vol 29 no 3 pp 22ndash24 1988

[17] M J Paknejad F Nasri and J F Affisco ldquoDefective units ina continuous review (s Q) systemrdquo International Journal ofProduction Research vol 33 no 10 pp 2767ndash2777 1995

[18] J J Liu and P Yang ldquoOptimal lot-sizing in an imperfect pro-duction system with homogeneous reworkable jobsrdquo EuropeanJournal of Operational Research vol 91 no 3 pp 517ndash527 1996

[19] M K Salameh andM Y Jaber ldquoEconomic production quantitymodel for items with imperfect qualityrdquo International Journal ofProduction Economics vol 64 no 1 pp 59ndash64 2000

[20] A Eroglu and G Ozdemir ldquoAn economic order quantity modelwith defective items and shortagesrdquo International Journal ofProduction Economics vol 106 no 2 pp 544ndash549 2007

[21] B Pal S S Sana and K Chaudhuri ldquoThree-layer supplychainmdasha production-inventory model for reworkable itemsrdquoApplied Mathematics and Computation vol 219 no 2 pp 530ndash543 2012

[22] B Sarkar L E Cardenas-Barron M Sarkar and M L SinggihldquoAn economic production quantity model with random defec-tive rate rework process and backorders for a single stageproduction systemrdquo Journal of Manufacturing Systems vol 33no 3 pp 423ndash435 2014

[23] L E Cardenas-Barron ldquoEconomic production quantity withrework process at a single-stage manufacturing system withplanned backordersrdquoComputers and Industrial Engineering vol57 no 3 pp 1105ndash1113 2009

[24] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[25] S K Goyal ldquoEconomic order quantity under conditions ofpermissible delay in paymentsrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 335ndash338 1985

[26] S P Aggarwal and C K Jaggi ldquoOrdering policies of deteriorat-ing items under permissible delay in paymentsrdquo Journal of theOperational Research Society vol 46 pp 658ndash662 1995

[27] A M M Jamal B R Sarker and S Wang ldquoAn ordering policyfor deteriorating items with allowable shortage and permissibledelay in paymentrdquo Journal of the Operational Research Societyvol 48 no 8 pp 826ndash833 1997

[28] J-T Teng ldquoOn the economic order quantity under conditionsof permissible delay in paymentsrdquo Journal of the OperationalResearch Society vol 53 no 8 pp 915ndash918 2002

[29] C K Huang D M Tsai J C Wu and K J Chung ldquoAn inte-grated vendor-buyer inventory model with order-processingcost reduction and permissible delay in paymentsrdquo EuropeanJournal of Operational Research vol 202 no 2 pp 473ndash4782010


[30] D Ha and S-L Kim ldquoImplementation of JIT purchasingan integrated approachrdquo Production Planning amp Control TheManagement of Operations vol 8 no 2 pp 152ndash157 1997

[31] K-R Lou and W-C Wang ldquoA comprehensive extension ofan integrated inventory model with ordering cost reductionand permissible delay in paymentsrdquo Applied MathematicalModelling vol 37 no 7 pp 4709ndash4716 2013

[32] C-K Huang ldquoAn integrated inventory model under conditionsof order processing cost reduction and permissible delay inpaymentsrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol34 no 5 pp 1352ndash1359 2010

[33] J Li H Feng and Y Zeng ldquoInventory games with permissibledelay in paymentsrdquo European Journal of Operational Researchvol 234 no 3 pp 694ndash700 2014

[34] A Meca J Timmer I Garcia-Jurado and P Borm ldquoInventorygamesrdquo European Journal of Operational Research vol 156 no1 pp 127ndash139 2004

[35] B Sarkar S S Sana and K Chaudhuri ldquoAn inventory modelwith finite replenishment rate trade credit policy and price-discount offerrdquo Journal of Industrial Engineering vol 2013Article ID 672504 18 pages 2013

Research ArticleJoint Optimization Approach of Maintenance and ProductionPlanning for a Multiple-Product Manufacturing System

Lahcen Mifdal12 Zied Hajej1 and Sofiene Dellagi1

1LGIPM Universite de Lorraine Ile de Saulcy 57045 Metz Cedex 01 France2Ecole Polytechnique drsquoAgadir Universiapolis Bab Al Madina Tilila 80000 Agadir Morocco

Correspondence should be addressed to Lahcen Mifdal lahcenmifdaluniv-lorrainefr

Received 31 October 2014 Accepted 2 December 2014

Academic Editor Felix T S Chan

Copyright copy 2015 Lahcen Mifdal et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the problem of maintenance and production planning for randomly failing multiple-product manufacturingsystem The latter consists of one machine which produces several types of products in order to satisfy random demandscorresponding to every type of product At any given time the machine can only produce one type of product and thenswitches to another one The purpose of this study is to establish sequentially an economical production plan and an optimalmaintenance strategy taking into account the influence of the production rate on the systemrsquos degradation Analytical modelsare developed in order to find the production plan and the preventive maintenance strategy which minimizes sequentially the totalproductioninventory cost and then the total maintenance cost Finally a numerical example is presented to illustrate the usefulnessof the proposed approach

1 Introduction

Manufacturing companies must manage several functionalcapacities successfully such as production maintenancequality and marketing One of the keys to success consists intreating all these services simultaneously On the other handthe customer satisfaction is one of the first objectives of acompany In fact the nonsatisfaction of the customer on timeis often due to a random demand or a sudden failure of pro-duction system Therefore it is necessary to develop main-tenance policies relating to production reducing the totalproduction and maintenance cost One of the first actions ofdecision-making hierarchy of a company is the developmentof an economical production plan and an optimal mainte-nance strategy

It is necessary to find the best production plan and thebest maintenance strategy required by the company to satisfycustomers This is a complex task because there are variousuncertainties due to external and internal factors Externalfactorsmay be associated with the inability to precisely definethe behaviour of the application during periods of produc-tion Internal factorsmay be associatedwith the availability of

hardware resources of the company In this context Filho [1]treated a stochastic scheduling problem in terms of produc-tion under the constraints of the inventory

Establishing an optimal production planning and main-tenance strategy has always been the greatest challenge forindustrial companies Moreover during the last few decadesthe integration of production andmaintenance policies prob-lem has received much research attention In this contextNodem et al [2] developed a method to find the optimalproduction replacementrepair and preventive maintenancepolicies for a degraded manufacturing system Gharbi et al[3] assumed that failure frequencies can be reduced throughpreventive maintenance and developed joint production andpreventivemaintenance policies depending on produced partinventory levels An analytical model and a numerical proce-dure which allow determining a joint optimal inventory con-trol and an age based on preventive maintenance policy fora randomly failing production system was presented by Rezget al [4]

This work examined a problem of the optimal productionplanning formulation of a manufacturing system consistingof one machine producing several products in order to



meet several random demands This type of problem wasstudied by Kenne et al [5] They presented an analysis ofproduction control and corrective maintenance problem in amultiple-machine multiple-product manufacturing systemThey obtained a near optimal control policy of the systemthrough numerical techniques by controlling both produc-tion and repair rates Feng et al [6] developed amultiproductmanufacturing systems problem with sequence dependentsetup times andfinite buffers under seven scheduling policiesSloan and Shanthikumar [7] presented a Markov decisionprocess model that simultaneously determines maintenanceand production schedules for a multiple-product single-machine production system accounting for the fact thatequipment condition can affect the yield of different producttypes differently Filho [8] developed a stochastic dynamicoptimization model to solve a multiproduct multiperiodproduction planning problem with constraints on decisionvariables and finite planning horizon

Looking at the literature on integrated maintenancepolicies we noticed that the influence of the production rateon the degradation system over a finite planning horizon wasrarely addressed in depth Recently Zied et al [9ndash11] took intoaccount the influence of production plan on the equipmentdegradation in the case of a system composed of singlemachine producing one type of product under randomlyfailing and satisfying a random demand over a finite horizonIn the same context Kenne and Nkeungoue [12] proposed amodel where the failure rate of a machine depends on its agehence the corrective and preventivemaintenance policies aremachine-age dependent

Motivated by the work in the Zied et al [9ndash11] we treatthe production and maintenance problem in another contextthat we consider a more complex and real industrial systemcomposed of one machine that produces several productsduring a finite horizon divided into subperiods This studydisplays that it has a novelty and originality relative to thistype of problem which considers the influence of severalproducts on the degradation degree of the consideredmachine and consequently on the average number of failureas well as on the maintenance strategy

This paper is organized as follows Section 2 states theproblem Section 3 presents the notations The productionand maintenance models are developed respectively in Sec-tions 4 and 5 A numerical example and sensitivity study arepresented respectively in Sections 6 and 7 Finally theconclusion is included in Section 8

2 Statement of the Industrial Problem

This study treated an industrial case The problem concernsa textile company located in North Africa specialized inclothing manufacturing The companyrsquos production systemconsists of a conversion of three types of fiber into yarn thenfabric and textiles These are then fabricated into clothes orother artefacts The production machine is called the loomand it uses a jet of air or water to insert the weft The loomensures pattern diversity and faultless fabrics by a flexibleand gentle material handling process Fabrics can be in one

2

1

Product 1

Product 2

Stock

Stock

Stock

Machine

Randomdemand 1

Random

Random

demand 2

demand n

Product n

n

Figure 1 Problem description

plain color with or without a simple pattern or they can havedecorative designs

Based on the industrial example described this study wasconducted to deal with the problem of an optimal productionand maintenance planning for a manufacturing system Thesystem is composed of a single machine which produces sev-eral products in order to meet corresponding several randomdemands The problem is presented in (Figure 1)

The considered equipment is subject to random failuresThe degradation of the equipment increases with time andvaries according to the production rate The machine is sub-mitted to a preventive maintenance policy in order to reducethe occurrence of failures In the literature the influence ofthe production rate on thematerial degradation is rarely stud-ied In this study this influence was taken into considerationin order to establish the optimal maintenance strategy

The model developed in this study is based on the worksof Zied et al [9ndash11] These studies seek to determine aneconomical production plan followed by an optimal mainte-nance policy but for the case of only one product

Firstly for a randomly given demand an optimal pro-duction plan was established to minimize the average totalstorage and production costs while satisfying a service levelSecondly using the obtained optimal production plan andconsidering its influence on themanufacturing system failurerate an optimal maintenance schedule is established tominimize the total maintenance cost

3 Notations

In this paper we shall as far as possible use the notationsummarized as follows

Cp(119894) unit production cost of product 119894Cs(119894) holding cost of one unit of product 119894 during Δ119905St(119894) setup cost of product 119894Mc corrective maintenance action cost


Mp preventive maintenance action cost119867 total number of periods119899 total number of products119901 total number of subperiods during each periodΔ119905 production period duration119880119894 nom nominal production quantity of product 119894

during Δ119905120579119894 probabilistic index (related to customer satisfac-tion) of product 119894119889119894(119896) demand of product 119894 during period 119896119878119894(119896times119901)minus(119901minus119895) inventory level of product 119894 at the end ofsubperiod 119895 of period 119896119885(119880) the total expected cost of production andinventory over the finite horizonVar(119889119894(119896)) the demand variance of product 119894 at period119896120593(120579119894) cumulative Gaussian distribution function120593minus1(120579119894) inverse distribution function

Γ(119873) the total cost of maintenance120582(119896times119901)minus(119901minus119895)(sdot) failure rate function at subperiod 119895 ofthe period 119896120582119899(sdot) nominal failure rate120601(sdot) the average number of failures119879 intervention period for preventive maintenanceactions

Decision Variables

119880119894119895119896 production quantity of product 119894 during subpe-riod 119895 of period 119896120575(119896times119901)minus(119901minus119895) duration of subperiod 119895 at period 119896119910119894119895119896 a binary variable which is equal to 1 if product119894 is produced in subperiod 119895 of the period 119896 and 0otherwise119873 number of preventive maintenance actions duringthe finite horizon

4 Production Policy

In this section we developed an analytical model whichminimizes the total cost of production and storageThe deci-sion variables are the production quantities 119880119894119895119896 the binaryvariable 119910119894119895119896 and the duration of subperiods 120575(119896times119901)minus(119901minus119895)Our objective consists in determining an economical pro-duction plan 119880

lowast(119880lowast

= 119880lowast

119894119895119896 119910lowast

119894119895119896and 120575

lowast

(119896times119901)minus(119901minus119895)forall119894 =

1 119899 119895 = 1 119901 119896 = 1 119867) for a finite timehorizon 119867 times Δ119905 The production plan must satisfy randomdemands under the requirement of a given level of servicewhile minimizing the cost of production and storage Theproduction of each product 119894 will take place at the beginningof subperiods and delivery to the customer will be at the endof periods

Period 1

Δt Δt

j = 1 j = 2 j = 3

1205751 1205752 1205753

Period k

120575(klowastp)minus(pminusj)

Subperiod j

Figure 2 Production plan

The state of the stock is determined at the end of eachsubperiod Figure 2 shows an example of a production plan

41 Stochastic Model of the Problem To develop this sectionthe following assumptions are specifically made

(i) holding and production costs of each product areknown and constant

(ii) only a single product can be produced in eachsubperiod

(iii) as described in (Figure 2) we have divided the period119896 into 119901 equal subperiods with 119901 = 119899 (the totalnumber of products)

(iv) the standard deviation of demand 120590(119889119894) and theaverage demand 119889119894 for each product and each period119896 are known and constant

The model has the following basic structure

To Minimize [(production cost) + (Holding cost)] (1)

under the constraints below

(i) the inventory balance equation(ii) the service level(iii) the admissibility of production plan(iv) the maximum production capacity

Formally

(i) The Cost Functions Consider

Production cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + Cp (119894) times 119880119894119895119896)

Holding cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

Cs (119894) times120575(119896times119901)minus(119901minus119895)

Δ119905times 119878119894(119896times119901)minus(119901minus119895)

(2)

(ii) The Inventory Balance Equation The available stock at theend of each subperiod 119895 of period 119896 for each product 119894 is


formulated in the form of flow balance constraints (inflow =outflow)

119878119894(119896times119901)minus(119901minus119895) = 119878119894(119896times119901)minus(119901minus119895)minus1 + (119910119894119895119896 times 119880119894119895119896)

minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(3)

where 1198781198940 is the initial stock level of product 119894This equation shows that the stock of product 119894 at the end

of each subperiod 119895 of each period 119896 ((119896 times 119901) minus (119901 minus 119895)) isdetermined by the state of the stock of product 119894 at the end ofthe subperiod (119896 times 119901) minus (119901 minus 119895) minus 1

(iii) The Admissibility of Production Plan and Service LevelConstraints The service level of product 119894 is determined bythe probability constraint on the stock level at the end of eachperiod 119896

Prob (119878119894(119896times119901) ge 0 ) ge 120579119894 forall 119896 = 1 119867 119894 = 1 119899

(4)

We can transform the probabilistic constraint of stock level toa deterministic constraint

Formally the function becomes

119896

sum

119904=1

119863 (119894 119904) + Stock min (119894 119896)

le

119896

sum

119904=1

119901

sum

119895=1

(119910(119894119895119904) times 119880119894119895119904) + stock init (119894 119904 = 0)

forall 119894 = 1 119899

(5)

where119863(119894 119904) is the estimated demand of product 119894 during theperiod 119904 Stock min(119894 119896) is the minimum stock level of prod-uct 119894 required at the end of period 119896 and stock init(119894 119904 = 0)

is the initial stock level of product 119894

(iv) The Maximum Production Capacity The productionquantity of the machine for each product 119894 119894 = 1 119899 islimited and is presented as follows

0 le 119880119894119895119896 le120575(119896times119901)minus(119901minus119895)

Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(6)

The term ⟨⟨120575(119896times119901)minus(119901minus119895)Δ119905⟩⟩ allows taking into account theinfluence of duration of subperiods 120575(119896times119901)minus(119901minus119895) on the max-imum quantity of production If 120575(119896times119901)minus(119901minus119895) tends to 0 themaximum quantity of production tends also to 0 and if120575(119896times119901)minus(119901minus119895) tends to Δ119905 the maximum quantity of productiontends to 119880119894 nom (with 119880119894 nom Nominal production quantity ofproduct 119894 during Δ119905)

However the term ⟨⟨(120575119905(119896times119901)minus(119901minus119895)Δ119905) times 119880119894 nom⟩⟩ repre-sents the maximum production quantity of product 119894 duringthe subperiod 119895 of period 119896

42 Problem Formulation We recall that in this study weassume that the horizon is divided into 119867 equal periodsand each period is divided into 119901 subperiods with differentdurations Figure 2 shows the distribution of the productionplan for the finite horizon119867timesΔ119905 Each product 119894 is producedin a single subperiod 119895 in each period 119896 The demand of eachproduct 119894 is satisfied at the end of each period 119896

The mathematical formulation of the proposed problemis based on the extension of themodel described by Zied et al[11] for the one product case study

Their problem is defined as follows

Min[Cs times 119864 [119878 (119867)2]

+

119867minus1

sum

119896=0

(Cs times 119864 [119878 (119896)2] + Cp times 119864 [119906 (119896)

2])]

(7)

where Cp is unit production cost and Cs is holding cost of aproduct unit during the period 119896

Formally our stochastic production problem is defined asfollows

Min (Ζ (119880))

119880 = 119880119894119895119896 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(8)with119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))

+ (Cs (119894) times120575(119896times119901)minus(119901minus119895)

Δ119905

times 119864 [(119878119894(119896times119901)minus(119901minus119895))2

])]

(9)where 119864[sdot] is the mathematical expectation

Under the following constraints


minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(10)

Prob (119878119894(119896times119901) ge 0) ge 120579119894 forall 119894 = 1 119899 119896 = 1 119867

(11)


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(12)

119901

sum

119895=1

120575(119896times119901)minus(119901minus119895) = Δ119905 forall 119896 = 1 119867 (13)


The first constraint stands for the inventory balance equationfor each product 119894 119894 = 1 119899 during each subperiod 119895119895 = 1 119901 of period 119896 119896 = 1 119867 Equation (11) refersto the satisfaction level of demand of product 119894 in each period119896 Constraint (12) defines the upper production quantity ofthe machine for each product 119894 The aim of (13) is to divideeach period 119896 into 119901 different subperiods

The constraints below should also be taken into account

119899

sum

119894=1

119910119894119895119896 = 1 forall 119895 = 1 119901 for 119896 = 1 119867

119901

sum

119895=1

119910119894119895119896 = 1 forall 119894 = 1 119899 for 119896 = 1 119867

(14)

119910119894119895119896 isin0 1 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(15)

Equation (14) indicates that only one type of product will beproduced in subperiod 119895 of period 119896 Constraint (15) statesthat 119910119894119895119896 is a binary variable We note that 119910119894119895119896 is equal to 1if product 119894 is produced in subperiod 119895 of the period 119896 and 0otherwise

For each subperiod 119895 of period 119896 the equation of the stockstatus is determined by the first constraint This equationremains random because of the uncertainty of fluctuatingdemand Therefore the variables of production and storageare stochastic Their statistics depend on a probabilistic dis-tribution function of demand It is therefore necessary to useconstraint (11) for decision variables These constraints canhelp us to analyse the various production scenarios toimprove the performance of the production system

43 The Deterministic Production Model We admit that afunction 119891(119894119895119896) forall119894 = 1 119899 119895 = 1 119901 119896 = 1 119867represents the cost of storage and productionwhich is relativeto the proposed plan and 119864[sdot] represents the value of themathematical expectation The quantity stocked of product119894 at the end of the subperiod 119895 of period 119896 is stood for by119878119894(119896times119901)minus(119901minus119895) The production quantity required to satisfy thedemand of product 119894 at the end of period 119896 is 119880119894119895119896 where119895 represents the subperiod during which the product 119894 isproduced

Thus the problem formulation can be presented asfollows

119880lowast= Min [119864 [119891(119894119895119896) (119880119894119895119896 119878119894(119896times119901)minus(119901minus119895))]] (16)

The purpose then is to determine the decision variables(119880119894119895119896 119910119894119895119896 and 120575(119896times119901)minus(119901minus119895)) required to satisfy economicallythe various demands under the constraints seen in theprevious subsection

The resolution of the stochastic problem under theseassumptions is generally difficult Thus its transformationinto a deterministic problem facilitates its resolution

(i) Inventory Balance Equation The stochastic inventorybalance equation is


minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(17)

with 1198781198940 being the initial stock level of product 119894We suppose that the means and variance of demand are

known and constant for each product 119894 in each period 119896Therefore

119864 [119889119894 (119896)] = 119889119894 (119896) Var [119889119894 (119896)] = 1205902(119889119894 (119896))

forall 119894 = 1 119899 119896 = 1 119867

(18)

The inventory equation 119878119894(119896times119901)minus(119901minus119895) is statistically describedby its means

119864 [119878119894(119896times119901)minus(119901minus119895)] = 119878119894(119896times119901)minus(119901minus119895)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(19)

We note that

119864 [119880119894119895119896] = 119894119895119896 = 119880119894119895119896 (20)

because 119880119894119895119896 is constant for each interval 120575(119896times119901)minus(119901minus119895)And

Var (119880119894119895119896) = 0

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(21)

Then the balance equation (10) can be converted into anequivalent inventory balance equation


minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(22)

with 1198781198940 being the average initial stock level of product

(ii) Service Level Constraint The second step is to convert theservice level constraint into a deterministic equivalent con-straint by specifying certain minimum cumulative produc-tion quantities that depend on the service level requirements

Lemma 1 Consider the following119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) ge Var (119889119894 (119896)) times 120593minus1

(120579119894) + 119889119894 (119896) minus 119878119894(119896minus1)times119901

forall 119894 = 1 119899 119896 = 1 119867

(23)


Proof We know that


(24)

119878119894(119896times119901) = 119878119894(119896minus1)times119901 +

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

forall 119894 = 1 119899 119896 = 1 119867

997904rArr Prob(119878119894(119896minus1)times119901

+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896) ge 0) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) ge 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

997904rArr Prob(119878119894(119896minus1)times119901 +

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

ge 119889119894 (119896) minus 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

997904rArr Prob(119878119894(119896minus1)times119901 + sum

119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))

ge119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896))) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(25)

Noting that

119883 =119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896)) (26)

119883 is a Gaussian random variable for demand 119889119894(119896)Hence

Prob(119878119894(119896minus1)times119901 + sum

119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))ge 119883) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(27)

We recall that 120579119894 represents the probabilistic index (related tocustomer satisfaction) of product 119894 and Var(119889119894(119896)) representsthe demand variance of product 119894 at period 119896

The distribution function is invertible because it is anincreasing and differentiable function

Hence

119878119894(119896minus1)times119901 +

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896) ge Var (119889119894 (119896)) times 120593minus1

(120579119894)

forall 119894 = 1 119899 119896 = 1 119867

(28)

Therefore

119901

sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(29)

(iii) The Expression of the Total Production and Storage CostIn this step we proceed to a simplification of the expectedcost of production and storage

The expression of the total cost of production is presentedas follows

Lemma 2 Consider the following

119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))

+ Cs (119894) times120575(119896times119901)minus(119901minus119895)

Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))

+ (119878119894(119896times119901)minus(119901minus119895))2

]

(30)

Proof See Appendix A


(iv) In Summary The deterministic optimization problembecomes as follows

(a) The Objective Function Consider

119880lowast= Min

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]]

(31)

(b) The Constraints Consider


minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(120579119894)

+ 119889119894 (119896) minus 119878119894(119896minus1)times119901

forall 119894 = 1 119899 119896 = 1 119867


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1

120575(119896times119901)minus(119901minus119895) = Δ119905 forall 119896 = 1 119867

(32)

5 Maintenance Strategy

51 Description of the Maintenance Strategy The mainte-nance strategy adopted in this study is known as preventivemaintenance with minimal repair The actions of preventivemaintenance are practiced in the period 119902 times 119879 (119902 = 1 2 )The replacement rule for this policy is to replace the systemwith another new system (as good as new) at each period 119902 times

q = 1 q = 2

j = 1j = 2 j = p

Deg

rada

tion

rate

k = 1 k = 2 k = 3

T t2T

1205822 1205822p1205821

120582p+1

120575p+1

Figure 3 Degradation rate

119879 At each failure between preventive maintenance actionsonly one minimal repair is implemented If we note Mcthe cost of corrective maintenance actions and Mp the costof preventive maintenance actions and degradation of themachine is linear the total cost of maintenance is expressedas follows

Γ (119873) = Mc times 120601(119873119880) +Mp times 119873 (33)

To develop the analytical model it was assumed that

(i) durations of maintenance actions are negligible

(ii) Mp and Mc costs incurred by the preventive and cor-rective maintenance actions are known and constantwith Mc ≫ Mp

(iii) preventivemaintenance actions are always performedat the end of the subperiods of production

The aim of this maintenance strategy is to find the optimalnumber of preventivemaintenance actions119873lowast (119873 = 1 2 )

minimizing the total cost of maintenance over a givenhorizon119867timesΔ119905 The existence of an optimal number of parti-tions119873lowast and therefore the optimal preventive maintenanceperiod 119879

lowast is proven in the literature It has been proven that119879lowast exists if the failure rate is increasing [13]Before determining the analytical model minimizing the

total cost of maintenance we need first to develop theexpression of the failure rate 120582(119896times119901)minus(119901minus119895)(119905) and then theaverage number of failures expression 120601(119880119873) during the finitehorizon119867 times Δ119905

52 Expression of Failure Rate Recall that the key of thisstudy is the influence of the variation of the production rateson the failure rate

Figure 3 represents the general description of the evolu-tion of the failure rate which depends on both the productionrate and the failure rate of the previous period

As presented in Figure 3 the failure rate is reset after each119902 times 119879 with 119902 = 1 119873 + 1


(q minus 1) times T

Period k minus 1 Period k Period k + m Period k + m + 1

120575(ktimesp)minus(pminus1)

T

120575ktimesp q times T

1

2

3

Δt

120575((k+m)timesp)

Figure 4 The evolution of the failure rate during the interval [(119902 minus 1) times 119879 119902 times 119879]

Thus the expression of the failure rate depending on timeand production rate can be written as follows

120582(119896times119901)minus(119901minus119895) (119905)

= [(120582(119896times119901)minus(119901minus119895)minus1 (120575(119896times119901)minus(119901minus119895)minus1))

times (1 minus In[(119896 times 119901) minus (119901 minus 119895 + 1)

119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896

120575(119896times119901)minus(119901minus119895)

times1

119880119894 nomΔ119905times 120582119899 (119905)]

forall119905 isin [0 120575(119896times119901)minus(119901minus119895)] forall 119896 = 1 119867 119895 = 1 119901

(34)

The term ⟨⟨119880119894119895119896120575(119896times119901)minus(119901minus119895)⟩⟩ represents the production rateof product 119894 during subperiod 119895 of period 119896

The term ⟨⟨119880119894 nomΔ119905⟩⟩ represents the nominal produc-tion rate of product 119894 during Δ119905

Therefore




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905

119880119894 nom times 120575(119896times119901)minus(119901minus119895)

times 120582119899 (119905)]


(35)

The aim of the expression (1minusIn[((119896times119901)minus(119901minus119895))(119902times119879)]) isto reset the failure rate after each 119902 times 119879 with 119902 = 1 119873 + 1

Note that

119902 = In[(119896 times 119901) minus (119901 minus 119895 + 2)

119879] + 1 (36)

where In[119909] is the integer part of number 119909



= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905

119880119894max times 120575(119876times119901)minus(119901minus119897)

times 120582119899 (120575(119876times119901)minus(119901minus119897))

+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905


times 120582119899 (120575(119896times119901)minus(119901minus119897)))


119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]

119905 isin [0 120575(119896times119901)minus(119901minus119895)] forall 119896 = 1 119867 119895 = 1 119901

(37)

Proof See Appendix B

53 Expression of the Average Number of Failures In order toreduce the complexity of the generation of the optimal num-ber of preventive maintenance we assume that interventionsare made at the end of subperiods

Hence the function of the period of intervention ispresented as follows

119879 = Round [119867 times 119901

119873] (38)

where Round[119909] is a round number of 119909To determine the average number of failures expression

120601(119880119873) during the finite horizon 119867 times Δ119905 we will focus onthe calculation of the average number of failures during the


interval [(119902minus1)times119879 119902times119879] which we designate 120601119879(119880119873)

Hencewe have to calculate the three surfaces 1 2 and 3

mentioned in Figure 4

Therefore the average number of failures expressionduring the interval [(119902 minus 1) times 119879 119902 times 119879] is presented as fol-lows

120601119879

(119880119873)= [

[

119901

sum

119895=((119902minus1)times119879+1)minus(In[((119902minus1)times119879)Δ119905]times119901)int

120575(In[((119902minus1)times119879)Δ119905]+1)times119901minus(119901minus119895)

0

120582(In[((119902minus1)times119879)Δ119905]+1)times119901minus(119901minus119895) (119905) 119889119905

+

In[(119902times119879)Δ119905]

sum

119896=Insup[((119902minus1)times119879+1)Δ119905]+1

119901

sum

119895=1

int


0

120582(119896times119901)minus(119901minus119895) (119905) 119889119905

+

119902times119879minusIn[(119902times119879)Δ119905]times119901

sum

119895=1

int

120575(In[(119902times119879)Δ119905]+1)times119901minus(119901minus119895)

0

120582(In[(119902times119879)Δ119905]+1)times119901minus(119901minus119895) (119905) 119889119905]

]

(39)

where Insup[119909] is the superior integer part of number 119909Thus the average number of failures expression 120601(119880119873)

during the finite horizon119867 times Δ119905 is defined by120601(119880119873)

=

119873+1

sum

119902=1

120601119879

(119880119873) (40)

Therefore we have the following lemma


120601(119880119873)

=

119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(41)

Note that119873 = 1 2

54 Expression of the Total Cost of Maintenance We recallthat the initial expression of the total cost of maintenancepresented in (33) is


Using the average number of failures 120601(119880119873) established inLemma 4 we can deduce that the analytical expression of thetotal maintenance cost is expressed as follows

Γ (119873) = [

[

Mc times119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

+Mp times 119873]

]

(43)


The goal is to find the optimal number of preventive main-tenance actions 119873

lowast that minimizes the total cost of main-tenance Γ(119873) Using this decision variable we can deducethe optimal period of intervention 119879

lowast knowing that 119879lowast =

Round[(119867 times 119901)119873lowast]

55 Existence of an Optimal Solution The following equationdetermines analytically the optimal solution

120597Γ (119873)

120597119873= 0 (44)

Since it is difficult to solve analytically the expression ofmaintenance cost we use numerical procedure

We start by proving the existence of a local minimumWe have the followingLimits at the terminals of Γ(119873) are

lim119873rarr1

Γ (119880119873) = lim119873rarr1

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarrconstant+ 119872119901 times 119873⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarrconstant

)

= 119872119888 times 120601 (119880 1) + 119872119901

lim119873rarr+infin

Γ (119880119873) = lim119873rarr+infin

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr0

+ 119872119901 times 119873⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr+infin

)

= +infin

(45)

Note that 120601(119880119873) is the average number of failures Mc andMp represent respectively the corrective and the preventivemaintenance costs

Moreover

Γ (119880119873 + 1) minus Γ (119880119873) ge 0

997904rArr [119872119888 times 120601 (119880 (119873 + 1)) + 119872119901 times (119873 + 1)]

minus [119872119888 times 120601 (119880119873) + 119872119901 times 119873] ge 0

997904rArr 119872119888 times (120601 (119880 (119873 + 1)) minus 120601 (119880119873)) + 119872119901 ge 0

997904rArr 120601 (119880 (119873 + 1)) minus 120601 (119880119873) le119872119901

119872119888

(46)

In addition

Γ (119880119873) minus Γ (119880119873 minus 1) le 0

997904rArr [119872119888 times 120601 (119880119873) + 119872119901 times (119873)]

minus [119872119888 times 120601 (119880 (119873 minus 1)) + 119872119901 times (119873 minus 1)] le 0

997904rArr 119872119888 times (120601 (119880119873) minus 120601 (119880 (119873 minus 1))) minus 119872119901 le 0

997904rArr 120601 (119880119873) minus 120601 (119880 (119873 minus 1)) ge119872119901

119872119888

(47)

In summary there is an optimal number of partition 119873lowast

which is unique and satisfies the previous relations (46) and(47) The following lemma ensures the existence of a localminimum


exist119873lowast119904119894 120585119873 le

119872119901

119872119888

le 120585119873minus1 (48)

with

120585119873 = 120601 (119880119873) minus 120601 (119880 (119873 + 1)) (49)

Therefore there exists an optimal number of partition 119873lowast

which satisfies the following expressions

119873lowastexist119904119894

120601 (119880 (119873 + 1)) minus 120601 (119880119873) ge 0

120601 (119880119873) minus 120601 (119880 (119873 minus 1)) le 0

lim119873rarr1

Γ (119880119873) = 119862119900119899119904119905119886119899119905

lim119873rarr+infin

Γ (119880119873) = +infin

(50)

The resolution of this maintenance policy using a numer-ical procedure is performed by incrementing the numberof maintenance intervals until an 119873

lowast satisfying the twofirst relations in Lemma 5 and minimizing the total cost ofmaintenance Γ(119873) described by (43)

6 Numerical Example

From the industrial example presented in Section 2 we haveconsidered a system producing 3 types of fiber in orderto meet three random demands according to every type ofproduct Using the analytical models developed in previoussections we start by establishing the optimal production planand then we determine the optimal maintenance strategyexpressed as optimal number of preventive maintenanceminimizing the total cost of maintenance over a finiteplanning horizon119867 = 8 trimesters (two years) We note thatthe optimal maintenance strategy is obtained while consid-ering of the influence of the production plan on the systemdegradation We supposed that the standard deviation ofdemand of product 119894 is the same for all periods The datarequired to run this model are given in sequence

61 Numerical Example

(i) The Data Relating to Production The mean demands (inbobbins) as shown in Table 1

1198891 = 200 120590 (1198891) = 15

1198892 = 110 120590 (1198892) = 09

1198893 = 320 120590 (1198893) = 12

(51)

The other data are presented as shown in Table 2

(ii) The Data Relating to System Reliability System reliabilityand costs related to maintenance actions are defined by thefollowing data

(1) the law of failure characterizing the nominal condi-tions is Weibull It is defined by


Table 1

DemandsTrim 1 Trim 2 Trim 3 Trim 4 Trim 5 Trim 6 Trim 7 Trim 8

Product 1 201 199 198 199 201 202 200 199Product 2 111 119 108 111 112 110 110 119Product 3 321 322 323 319 321 317 320 319

Table 2

Initial stock level1198781198940(up)

Nominal production quantities119880119894 nom (up)

Unit production costsCp(119894) (um)

Unit holding costsCs(119894) (umut)

Satisfaction rates120579119894()

Product 1 110 750 13 3 87Product 2 85 530 17 5 95Product 3 145 1150 9 2 90

Table 3 The optimal production plan

Trimester 1 Trimester 2 Trimester 3 Trimester 41205751

1205752

1205753

1205754

1205755

1205756

1205757

1205758

1205759

12057510

12057511

12057512

085 071 144 119 120 061 081 118 101 043 074 183Product 1 0 169 0 388 0 0 0 321 0 0 151 0Product 2 150 0 0 0 185 0 134 0 0 0 0 312Product 3 0 0 507 0 0 230 0 0 387 158 0 0


12057514

12057515

12057516

12057517

12057518

12057519

12057520

12057521

12057522

12057523

12057524


(a) scale parameter (120573) 12 months(b) shape parameter (120572) 2(c) position parameter (120574) 0

(2) the initial failure rate 1205820 = 0

These parameters provide information on the evolution of thefailure rate in time

This failure rate is increasing and linear over time Thusthe function of the nominal failure rate is expressed by

120582119899 (119905) =120572

120573times (

119905

120573)

120572minus1

=2

12times (

119905

12) (52)

The preventive and corrective maintenance costs are respec-tively Mp = 800mu and Mc = 1 500mu

62 Determination of the Economic Production Plan Theeconomic production plan obtained is presented in Table 3

63 Determination of the Optimal Maintenance Plan Asdescribed in Figure 5 the optimal maintenance strategy isobtained based on the optimal production plan given in theprevious section

Figure 6 shows the curve of the total cost of maintenanceaccording to119873 (number of preventive maintenance actions)

We conclude that the optimal number of preventive mainte-nance actions that minimizes the total cost of maintenanceduring the finite horizon (two years) is119873lowast = 2 times Hencethe optimal period to intervene for the preventive mainte-nance is 119879

lowast= 12 months and the minimal total cost of

maintenance Γlowast(119873) = 3316mu

7 The Economical Profit of the Study

We recall that the specificity of this study is that it consideredthe impact of the production rate variation on the systemdegradation and consequently on the optimal maintenancestrategy adopted in the case of multiple product In order toshow the significance of our study we will consider in thissection the case of not considering the influence of theproduction rate variation on the systemrsquos degradationThat isto say we assume that the manufacturing system is exploitedat its maximal production rate every time Analytically wewill consider the nominal failure rate which depends only ontime The results of this study are presented in Table 4

The optimal number of preventive maintenance obtainedin the case when we did not consider the variation of produc-tion rate is119873lowast = 3 times and it corresponds to a total cost ofmaintenance during the finite horizon (two years) Γlowast(119873) =

3 704mu We recall that in our case study when we consider


Optimization ofproduction policy

Optimization ofmaintenance strategy

Nlowast

d = di k ( )

Ulowast= Uijk ( )

k =

i =

k =i =j =

1 H1 n

1 p1 H1 n

Figure 5 Sequential production and maintenance optimization

0 2 4 6 8 10

4000

5000

6000

7000

8000

The number of preventive maintenance actions (N)

The t

otal

cost

of m

aint

enan

ceΓ

(N)

Figure 6 The total cost of maintenance depending to119873

Table 4 The sensitivity study based on the variation of productionrate

Γlowast(119873) (um) 119873

lowast (times)Case 1 considering variation ofproduction rate 3 316 2

Case 2 not considering the variation ofproduction rate 3 704 3

the variation of production rate we have obtained 119873lowast

=

2 and Γlowast(119873) = 3 316mu We can easily note that we have

reduced the optimal number of preventive maintenance withperforming an economical gain estimated at 10

Several studies have addressed issues related to produc-tion and maintenance problem But the consideration of themateriel degradation according to the production rate in thecase of multiple-product has been rarely studied

This study was conducted to deal with the problem of anoptimal production and maintenance planning for a manu-facturing systemThe significance of the present study is thatwe took into account the influence of the production planon the system degradation in order to establish an optimalmaintenance strategy The considered system is composed ofa single machine which produces several products in order tomeet corresponding several random demands

8 Conclusion

In this paper we have discussed the problem of integratedmaintenance to production for a manufacturing system con-sisting of a single machine which produces several types ofproducts to satisfy several random demands As the machine

is subject to random failures preventive maintenance actionsare considered in order to improve its reliability At failure aminimal repair is carried out to restore the system into theoperating state without changing its failure rate

At first we have formulated a stochastic productionproblem To solve this problem we have used a productionpolicy to achieve a level of economic output This policy ischaracterized by the transformation of the problem to a deter-ministic equivalent problem in order to obtain the economicproduction plan In the second step taking into account theeconomic production plan obtained we have studied andoptimized the maintenance policy This policy is defined bypreventive actions carried out at constant time intervals Theobjective of this policy is to determine the optimal number ofpreventivemaintenance and the optimal intervention periodsover a finite horizon This policy is characterized by a failurerate for a linear degradation of the equipment consideringthe influence of production rate variation on the systemdegradation and on the optimal maintenance plan in the caseof multiple products represents

The promising results obtained in this thesis can lead tointeresting perspectives A perspective that we are looking forat the short term is to consider maintenance durations Werecall that throughout our study we neglected the durationsof actions of preventive and correctivemaintenance It is clearthat the consideration of these durations impacts the optimalmaintenance plan and the established production plan Inthe medium term it is interesting to concretely consider theimpact of logistics service on the study It is clear that thein-maintenance logistics are absent in most researches Thecombination of maintenance logistics and production repre-sents a motivating perspective in this field of study

Another interesting perspective specifying the manufac-tured product can be explored

Appendices

A Expression of the Total Production andStorage Cost

We have119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A1)

Also


minus Ent [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A2)

Therefore

119864 [(119878119894(119896times119901)minus(119901minus119895) minus 119878119894(119896times119901)minus(119901minus119895))2

]

= 119864[([ 119878119894(119896times119901)minus(119901minus119895)minus1 + (119910119894119895119896 times 119880119894119895119896)

minus Ent [119895

119901] times 119889119894 (119896)]

minus [ 119878119894(119896times119901)minus(119901minus119895)minus1 + (119910119894119895119896 times 119880119894119895119896)

minusEnt [119895

119901] times 119889119894 (119896) ])

2

]

997904rArr 119864[(119878119894(119896times119901)minus(119901minus119895) minus 119878119894(119896times119901)minus(119901minus119895))2

]

= 119864[([119878119894(119896times119901)minus(119901minus119895)minus1 minus 119878119894(119896times119901)minus(119901minus119895)minus1]

minus [Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896))])

2

]


]

= 119864[(119878119894(119896times119901)minus(119901minus119895)minus1 minus 119878119894(119896times119901)minus(119901minus119895)minus1)2

minus 2 times [(119878119894(119896times119901)minus(119901minus119895)minus1 minus 119878119894(119896times119901)minus(119901minus119895)minus1)

times (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]


]

= [119864 [(119878119894(119896times119901)minus(119901minus119895)minus1 minus 119878119894(119896times119901)minus(119901minus119895)minus1)2

]

minus 2 times 119864 [(119878119894(119896times119901)minus(119901minus119895)minus1 minus 119878119894(119896times119901)minus(119901minus119895)minus1)

times(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ 119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A3)

119878119894(119896times119901)minus(119901minus119895)minus1 and 119889119894(119896) are independent random variablesso we can deduce


]


]

minus 2 times 119864 [(119878119894(119896times119901)minus(119901minus119895)minus1 minus 119878119894(119896times119901)minus(119901minus119895)minus1)]

times 119864 [(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A4)

On the other hand we note that

119864 [(119878119894(119896times119901)minus(119901minus119895)minus1 minus 119878119894(119896times119901)minus(119901minus119895)minus1)]

= 119864 [(119878119894(119896times119901)minus(119901minus119895)minus1)] minus 119864 [(119878119894(119896times119901)minus(119901minus119895)minus1)] = 0

(A5)

Therefore


]


]

+(Ent [119895

119901])

2

times 119864 [(119889119894 (119896) minus 119889119894 (119896))2

]]

(A6)

We know that

119864 [(119909119896 minus 119909119896)2] = Var (119909119896)

(Int [119895

119901])

2

= Int [119895

119901] because 0 le

119895

119901le 1

(A7)

Therefore


]

= Var (119878119894(119896times119901)minus(119901minus119895)minus1) + Ent [119895

119901] times Var (119889119894 (119896))

(A8)


Finally

Var (119878119894(119896times119901)minus(119901minus119895)) = Var (119878119894(119896times119901)minus(119901minus119895)minus1)

+ Ent [119895

119901] times Var (119889119894 (119896))

(A9)

Consequently

(i) for 119896 = 1

(a) 119895 = 1

Var (1198781198941) = Var (1198781198940) + (Ent [ 1

119901]) times Var (119889119894 (1))

(A10)

(b) 119895 = 2

Var (1198781198942) = Var (1198781198940) +2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A11)

(c) 119895 = 119901

Var (119878119894119901) = Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A12)

(ii) for 119896 = 2

(a) 119895 = 1

Var (119878119894119901+1) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+ Ent [ 1

119901] times Var (119889119894 (2))]

(A13)

(b) 119895 = 2

Var (119878119894119901+2) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (1)) + Ent [ 1

119901]

times Var (119889119894 (2)) + Ent [ 2

119901] times Var (119889119894 (2))]

(A14)

(c) 119895 = 119901

Var (119878119894(2times119901)) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+

119875

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119901))]

(A15)

(iii) for any value of 119896

(a) 119895 = 1

Var (119878119894(119896times119901)minus(119901minus1)) = [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

1

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A16)

(b) 119895 = 2


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A17)

(c) for any value of 119895

997904rArr Var (119878119894(119896times119901)minus(119901minus119895))

= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A18)

On the other hand


]

= 119864 [ (119878119894(119896times119901)minus(119901minus119895))2

minus 2 times 119878119894(119896times119901)minus(119901minus119895)

times 119878119894(119896times119901)minus(119901minus119895) + (119878119894(119896times119901)minus(119901minus119895))2

]


]

= [119864 [(119878119894(119896times119901)minus(119901minus119895))2

]

minus 119864 [2 times 119878119894(119896times119901)minus(119901minus119895) times 119878119894(119896times119901)minus(119901minus119895)]

+119864 [(119878119894(119896times119901)minus(119901minus119895))2

]]

(A19)

We know that

119864 [(119878119894(119896times119901)minus(119901minus119895))2

] = (119878119894(119896times119901)minus(119901minus119895))2

(A20)


Hence


]


] minus 2 times 119878119894(119896times119901)minus(119901minus119895)

times 119864 [119878119894(119896times119901)minus(119901minus119895)] + (119878119894(119896times119901)minus(119901minus119895))2

]



]

= [ 119864 [(119878119894(119896times119901)minus(119901minus119895))2

] minus 2 times (119878119894(119896times119901)minus(119901minus119895))2

times119864 [(119878119894(119896times119901)minus(119901minus119895))2

] + (119878119894(119896times119901)minus(119901minus119895))2

]

(A21)

Consequently


]

= 119864 [(119878119894(119896times119901)minus(119901minus119895))2

] minus (119878119894(119896times119901)minus(119901minus119895))2

(A22)

Noting that


]

= Var (119878119894(119896times119901)minus(119901minus119895))




(A23)

we deduce from (A18) and (A23) that



= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

997904rArr 119864[(119878119894(119896times119901)minus(119901minus119895))2

]

= [ Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896)) + (119878119894(119896times119901)minus(119901minus119895))

2

]

(A24)

Consequently


]

= [ 1205902(1198781198940) +

119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times 1205902(119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times 1205902(119889119894 (119896)) + (119878119894(119896times119901)minus(119901minus119895))

2

]

(A25)

Substituting (A25) in the expected cost expression (9)

119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(A26)

B Expression of Failure Rate

Equation (A9) is expressed as follows for the differentsubperiods

(i) for 119896 = 1

(a) 119895 = 1

1205821 (119905) = (1205820) times (1 minus In [0

119902 times 119879]) +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (119905)

(B1)

(b) 119895 = 2

1205822 (119905) = 1205821 (1205751) times (1 minus In [1

119902 times 119879])

+

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)


1205822 (119905) = (1205820 +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (120575(1)))

times (1 minus In [1

119902 times 119879]) +

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)

(B2)

(c) 119895 = 119901

120582119901 (119905) = (120582119901minus1 (120575119901minus1)) times (1 minus In [119901 minus 1

119902 times 119879])

+

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)

120582119901 (119905) = [(1205820 +

119901minus1

sum

119897=1

119899

sum

119894=1

1198801198941198971 times Δ119905

119880119894 nom times 120575119897

times 120582119899 (120575(119897)))

times(1 minus In [119901 minus 1

119902 times 119879]) +

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)]

(B3)

(ii) for any value of 119896

(a) 119895 = 1

120582((119896minus1)times119901)+1 (119905)

= [(120582(119896minus1)times119901 (120575(119896minus1)times119901)) times (1 minus In[((119896 minus 1) times 119901)

119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905

119880119894 nom times 120575((119896minus1)times119901)+1

times 120582119899 (119905)]

120582((119896minus1)times119901)+1 (119905)

= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



times (1 minus In[((119896 minus 1) times 119901)

119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

(B4)

(b) for any value of 119895


= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(B5)



References

[1] O S S Filho ldquoStochastic production planning problem underunobserved inventory systemrdquo in Proceedings of the AmericanControl Conference (ACC rsquo07) pp 3342ndash3347 New York NYUSA July 2007

[2] F I D Nodem J P Kenne and A Gharbi ldquoSimultaneous con-trol of production repairreplacement and preventive mainte-nance of deteriorating manufacturing systemsrdquo InternationalJournal of Production Economics vol 134 no 1 pp 271ndash2822011

[3] A Gharbi J-P Kenne and M Beit ldquoOptimal safety stocks andpreventive maintenance periods in unreliable manufacturingsystemsrdquo International Journal of Production Economics vol 107no 2 pp 422ndash434 2007

[4] N Rezg S Dellagi and A Chelbi ldquoOptimal strategy of inven-tory control and preventive maintenancerdquo International Journalof Production Research vol 46 no 19 pp 5349ndash5365 2008

[5] J P Kenne E K Boukas andA Gharbi ldquoControl of productionand corrective maintenance rates in a multiple-machine multi-ple-product manufacturing systemrdquo Mathematical and Com-puter Modelling vol 38 no 3-4 pp 351ndash365 2003

[6] W Feng L Zheng and J Li ldquoThe robustness of schedulingpolicies in multi-product manufacturing systems with sequ-ence-dependent setup times and finite buffersrdquo Computersand Industrial Engineering vol 63 no 4 pp 1145ndash1153 2012


[7] TW Sloan and J G Shanthikumar ldquoCombined production andmaintenance scheduling for a multiple-product single-machine production systemrdquo Production and OperationsManagement vol 9 no 4 pp 379ndash399 2000

[8] O S S Filho ldquoA constrained stochastic production planningproblem with imperfect information of inventoryrdquo in Proceed-ings of the 16th IFACWorld Congress vol 2005 Elsevier SciencePrague Czech Republic

[9] Z Hajej S Dellagi and N Rezg ldquoAn optimal produc-tionmaintenance planning under stochastic random demandservice level and failure raterdquo in Proceedings of the IEEE Interna-tional Conference onAutomation Science andEngineering (CASErsquo09) pp 292ndash297 Bangalore India August 2009

[10] ZHajejContribution au developpement de politiques demainte-nance integree avec prise en compte du droit de retractation et duremanufacturing [These de doctorat] Universite Paul VerlaineMetz France 2010

[11] Z Hajej S Dellagi and N Rezg ldquoOptimal integrated mainte-nanceproduction policy for randomly failing systems withvariable failure raterdquo International Journal of ProductionResearch vol 49 no 19 pp 5695ndash5712 2011

[12] J P Kenne and L J Nkeungoue ldquoSimultaneous control ofproduction preventive and corrective maintenance rates of afailure-prone manufacturing systemrdquo Applied Numerical Math-ematics vol 58 no 2 pp 180ndash194 2008

[13] T Nakagawa and S Mizutani ldquoA summary of maintenancepolicies for a finite intervalrdquo Reliability Engineering and SystemSafety vol 94 no 1 pp 89ndash96 2009

Research ArticleImpacts of Transportation Cost onDistribution-Free Newsboy Problems

Ming-Hung Shu1 Chun-Wu Yeh2 and Yen-Chen Fu3

1 Department of Industrial Engineering amp Management National Kaohsiung University of Applied Sciences415 Chien Kung Road Kaohsiung 80778 Taiwan

2Department of Information Management Kun Shan University 195 Kunda Road Yongkang District Tainan 71003 Taiwan3Department of Industrial and Information Management National Cheng Kung University 1 University Road Tainan 70101 Taiwan

Correspondence should be addressed to Yen-Chen Fu r3897101mailnckuedutw

Received 27 June 2014 Revised 3 September 2014 Accepted 13 September 2014 Published 30 September 2014

Academic Editor Vikas Kumar

Copyright copy 2014 Ming-Hung Shu et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A distribution-free newsboy problem (DFNP) has been launched for a vendor to decide a productrsquos stock quantity in a single-period inventory system to sustain its least maximum-expected profits when combating fierce and diverse market circumstancesNowadays impacts of transportation cost ondetermination of optimal inventory quantity have become attentive where its influenceon the DFNP has not been fully investigated By borrowing an economic theory from transportation disciplines in this paperthe DFNP is tackled in consideration of the transportation cost formulated as a function of shipping quantity and modeled as anonlinear regression form from UPSrsquos on-site shipping-rate data An optimal solution of the order quantity is computed on thebasis of Newtonrsquos approach to ameliorating its complexity of computation As a result of comparative studies lower bounds of themaximal expected profit of our proposed methodologies surpass those of existing work Finally we extend the analysis to severalpractical inventory cases including fixed ordering cost random yield and a multiproduct condition

1 Introduction

Anewsboy (newsvendor) problemhas been initiated to deter-mine the stock quantity of a product in a single-period inven-tory system when the product whose demand is stochastichas a single chance of procurement prior to the beginning ofselling period Aiming to maximize expected profit decisivequantity trades off between the risk of underordering whichfails to gain more profit and the loss of overordering whichcompels release below the unit purchasing cost

Traditional models for the newsboy problem assumethat a single vendor encounters the demand of a productcomplying with a particular probability distribution func-tion with known parameters such as a normal Schmeiser-Deutsch beta gamma or Weibull distribution [1] Withthis assumption several recent studies have to a certainextent succeeded in resolution of certain practical problemsFor example Chen and Ho [2] and Ding [3] analyzedthe optimal inventory policy for newsboy problems withfuzzy demand and quantity discounts Arshavskiy et al [4]

performed experimental studies by implementing the classi-cal newsvendor problem in practice Ozler et al [5] studieda multiproduct newsboy problem under value-at-risk con-straint with loss-averse preferences Wang [6] introduced aproblem of multinewsvendors who compete with inventoriessetting from a risk-neutral supplier When confronting myr-iad conditions in markets however in many occasions thisdesignated distributional demand failed to best safeguard thevendorrsquos profit

To cope with the failure models for the distribution-free newsboy problem (DFNP) have been broadly introducedover the past twodecadesGallego andMoon [7] first outlineda compacted analysis procedure for arranging optimal orderquantities to certain inventory models such as the singleproduct fixed ordering random yield and a multiproductcase Alfares and Elmorra [8] further employed the procedurefor the inventory model which considers shortage penaltycost Moon and Choi [9] derived an ordering rule for thebalking-inventory control model where probability of perunit sold declines as inventory level falls below balking level



More recently Cai et al [10] provided measurements fordeployment of multigenerational product development withthe project cost accrued fromdifferent phases of a product lifecycle such as development service and associated risks Leeand Hsu [11] and Guler [12] developed an optimal orderingrule when an effect of advertising expenditure was reckonedon the inventory model Kamburowski [13] presented newtheoretical foundations for analyzing the best-case andworst-case scenarios Due to prevalence of purchasing onlineMostard et al [14] studied a resalable-return model forthe distant selling retailers receiving internet orders fromcustomers who have right to return their unfit merchandisein a stipulated period

Over the past few years energy prices have risen signif-icantly and become more volatile transportation of goodshas become the highest operational expense as noted byBarry [15] Many evidences indicate that in the US inboundfreight costs for domestically sourced products and importedproducts typically range from 2 to 4 and from 6 to 12 ofgross sales respectively and outbound transportation coststypically average 6 to 8 of net sales In addition Swensethand Godfrey [16] reported that depending on the estimatesutilized upwards of 50 of the total annual logistic cost of aproduct could be attributed to transportation and that thesecosts were going up UPS recently announced a 49 increasein its net average shipping rate Ostensibly the expenditure ofthe inboundoutbound material transportation has becomea critical component of a total annual logistic cost functionfor determining purchase quantities Effects of transportationhave gained substantial recognition in vendor-buyer coordi-nation problems

Swenseth and Godfrey [16] unified two freight ratefunctions into a total annual cost function to understand theirbrunt on purchasing decisions For integration of inventoryand inboundoutbound transportation decisions Cetinkayaand Lee [17] enabled an optimal inventory policy and Toptalet al [18] carried out ideal cargo capacity and minimal costsToptal and Cetinkaya [19] further studied a coordinationproblem between a vendor and a buyer under explicittransportation considerationMore recently Zhang et al [20]generalized a standard newsboy model to the freight costproportional to the number of the containers used Toptal [21]studied exponentiallyuniformly distributed demands andtrucking costs Mutlu and Cetinkaya [22] developed an opti-mal solution when inventory replenishment and shipmentscheduling under common dispatch costs are considered

Although impacts of the transportation cost on determi-nation of the optimal inventory quantity have become atten-tive its influence on theDFNPhas not been fully investigatedTo bridge the gap this paper develops analytical and efficientprocedures to acquire optimal policies for theDFNP inwhichthe transportation cost function is explicitly joined into thevendorrsquos expected profit structure We borrowed the ideafrom the transportation management models [23] that thetransportation cost ismodeled as a function of delivery quan-tities as a result of the computational studies our proposedoptimal-ordering rules increase lower bound of maximizedexpected profit as much as 4 on average as opposed tothe optimal policies recommended by Gallego andMoon [7]

Moreover in order to determine and implement the optimalpolicies in practice we perform comprehensive sensitivityanalyses for the vital parameters such as the demand meanand variance unit cost of product and transportation cost

Lastly this paper is organized as follows Section 2describes our model formulation for the DFNP in presenceof transportation cost whose optimal order quantity119876lowast alongwith lower bound of maximized expected profit 119864(119876lowast) isresolved in Section 3 In Section 4 we study sensitivityanalyses and comparative studies A fixed-ordering costcase is analyzed in Section 5 while a random-yield case isconsidered in Section 6 In Section 7 we further contemplatea multiproduct case with budget constraint Conclusions andImplications make up Section 8

2 Model Formulation for the DFNPwith Transportation Cost

For investigating impacts of theDFNP in consideration of thetransportation cost we briefly depict its model assumptionsand notations used in this paper Demand rate from a specificbuyer is denoted by119863 whose distribution119866 is unknownwithmean 120583 and variance 1205902 Note that the unknown distribution119866 is equal to or better off the worst possible distribution120599 With a productrsquos unit cost 119888 a vendor orders size of 119876which arrive before delivering to the buyer Intuitively in onereplenishment cycle min119876119863 units are sold with unit price119901 and the unsold items (119876 minus 119863)+ are salable with unit salvagevalue 119904 where 119904 lt 119901 where (119876 minus 119863)+ defined as the positivepart of 119876 minus 119863 are equivalent to max119876 minus 119863 0 This implies119876 = min119876119863 + (119876 minus 119863)+

Furthermore we assume transportation cost is a functionof the order quantity119876 denoted by tc(119876) We further assumethe transportation cost is in a general form of the tapering(or proportional) function for example tc(119876) = 119886 + 119887 ln119876for 119886 119887 ge 0 where 119886 and 119887 represent fixed and variabletransportation cost Intuitively high volume corresponds tolower per unit rate of transportation reflecting that theinequality [tc(119876)119876]1015840 le 0 holds true That is [tc(119876)119876]1015840 =(119887 minus 119886 minus 119887 ln119876)1198762 le 0 or equivalently 119876 ge exp(1 minus 119886119887)where the regulatedminimal quantity level of delivery is119876119904 =exp(1 minus 119886119887) and 119876 ge 119876119904

The assumption is based on the following observationsfrom the existing works and UPSrsquos on-site data set Firstoff economic trade-off for the optimal transportation costlies between provided service level and shipped quantity[17] Secondly in the shipment more weight signifies largerdelivery quantity and higher shipment cost [19] Thirdly thetransportation management models proposed by Swensethand Godfrey [16] and Toptal et al [18] indicated that optimalshipping quantity renders minimum of the transportationcost Finally we display the on-site shipping data set collectedfrom the UPS worldwide expedited service at zone 7 shownin Figure 1

Now we are ready to combat the DFNP in presence ofthe transportation cost Our purpose is to decide an optimalstock quantity in a single-period inventory system for avendor to sustain its least maximum-expected profits when


16

14

12

10

08

06

Ship

men

t cos

t (lowast$100)

5 10 15 20Shipment weight (kg)

Actual rate data036 + 042 ln Q

R2= 0926

Figure 1 The fitted regression model for the data set of UPSworldwide expedited service at zone 7

encountering fierce and diverse market circumstances Firstwe construct the vendorrsquos expected profit 119864(119876)

119864 (119876) = 119901119864 (min 119876119863) + 119904119864(119876 minus 119863)+ minus 119888119876

minus 119886 + 119887 ln [119864 (min 119876119863)]

minus 119886 + 119887 ln [119864(119876 minus 119863)+]

= 119901119864 (min 119876119863) + 119904119864(119876 minus 119863)+ minus 119888119876 minus 2119886

minus 119887 ln 119864 (min 119876119863) 119864(119876 minus 119863)+

(1)

Then according to the relationships of min119876119863 = 119863 minus(119863 minus 119876)

+ and (119876 minus 119863)+ = (119876 minus 119863) + (119863 minus 119876)+ we furtherrewrite (1)

119864 (119876) = (119901 minus 119904) 120583 minus (119901 minus 119904) 119864(119863 minus 119876)+

minus (119888 minus 119904)119876 minus 2119886

minus 119887 ln [120583 minus 119864(119863 minus 119876)+] [119876 minus 120583 + 119864(119863 minus 119876)+] (2)

For developing an optimal order quantity for the vendorto sustain its lower bound of maximized expected profit119864(119876) we consider 119866 the distribution of 119863 to be under theworst possible distribution 120599Therefore based onGallego and

Moonrsquos Lemma 1 in [7] we have the lower bound of expectedprofit 119864(119876) for the vendor

119864 (119876) ge (119901 minus 119904) 120583 minus (119901 minus 119904)

times[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2

minus (119888 minus 119904)119876 minus 2119886 + 2119887 ln 2

minus 119887 ln minus1205832 minus 1 + 1198762 + 2120583[1205902 + (119876 minus 120583)2]12

(3)

Lemma 1 (see [7]) Under the worst possible distribution 120599 theupper bound of expected value for the positive part of 119876 minus 119863 is

119864(119863 minus 119876)+le[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2

(4)

Let the right-hand side term of (3) be a continuous functionwith respect to119876 then first and second derivatives of 119864(119876) areelaborately derived as follows

119889119864 (119876)

119889119876=119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876 minus 120583)

2[1205902 + (119876 minus 120583)2]12

minus 1198872119876 + 2120583 (119876 minus 120583) [120590

2+ (119876 minus 120583)

2]minus12

minus1 minus 1205832 + 1198762 + 4120583[1205902 + (119876 minus 120583)2]12

(5)

1198892119864 (119876)

1198891198762= minus

(119901 minus 119904) 1205902

2[1205902 + (119876 minus 120583)2]32

minus 119887

minus 2 + 21205832minus 21198762

+ 4120583[1205902+ (119876 minus 120583)

2]12

+2120583 (minus1 minus 120583

2+ 4120583119876 minus 3119876

2)

[1205902 + (119876 minus 120583)2]12

minus81205832(119876 minus 120583)

2

1205902 + (119876 minus 120583)2

+(119876 minus 120583)

2(21205833+ 2120583 minus 2120583119876

2)

[1205902 + (119876 minus 120583)2]32

sdot minus1 minus 1205832+ 1198762+ 4120583[120590

2+ (119876 minus 120583)

2]12

minus2

(6)

Obviously 1198892119864(119876)1198891198762 in (6) is not necessarily being negativeIt implies that the generally explicit and analytical close formfor the optimal order quantity max119876lowast 119876119904 with the least ofmaximized expected profits is not available Therefore there isa need to develop an efficient search procedure to obtain theoptimal order quantity 119876lowast and its corresponding lower boundof maximized expected profit 119864(119876lowast)


Table 1 The optimal order quantity using Newtonrsquos optimization approach

Iteration 119894 119876119894

1198911015840(119876119894) 119891

10158401015840(119876119894) 119891

1015840(119876119894)11989110158401015840(119876119894) 119876

119894+1

0 9 minus0695 minus2471 0281 87191 8719 0030 minus1729 minus0017 87362 8736 minus0014 minus1804 0008 87283 8728 0005 minus1772 minus0003 87314 8731 minus0000 minus1785 0000 8731

3 An Efficient SolutionProcedure for 119876lowast and 119864(119876lowast)

Step 1 Start from 119894 = 0 let initial order quantity 1198760 = 120583and set the allowable tolerance 120576 for example the acceptableldquoprecisionrdquo or ldquoaccuracyrdquo selected by the decision maker forthe optimal decision policy

Step 2 Perform Newtonrsquos approach (see Hillier and Lieber-man [24 pp 555ndash557]) to seeking the optimal order quantityof 119876

Let119876119894+1 = 119876119894 minus (1198911015840(119876119894)119891

10158401015840(119876119894)) According to (5) we set

1198911015840(119876119894) =

119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876119894 minus 120583)

2[1205902 + (119876119894 minus 120583)2]12

minus 1198872119876119894 + 2120583 (119876119894 minus 120583) [120590

2+ (119876119894 minus 120583)

2]minus12

1198762119894minus 1205832 minus 1 + 2120583[1205902 + (119876119894 minus 120583)

2]12

(7)

From (6) we set

11989110158401015840(119876119894) = minus

(119901 minus 119904) 1205902

2[1205902 + (119876119894 minus 120583)2]32

minus 119887 minus 21198762

119894minus 2 + 2120583

2+ 4120583[120590

2+ (119876119894 minus 120583)

2]12

+2120583 (4120583119876119894 minus 3119876

2

119894minus 1205832minus 1)

[1205902 + (119876119894 minus 120583)2]12

+(119876119894 minus 120583)

2(21205833+ 2120583 minus 2120583119876

3

119894)

[1205902 + (119876119894 minus 120583)2]32

minus81205832(119876119894 minus 120583)

2

1205902 + (119876119894 minus 120583)2

(8)

Stop the search when |119876119894+1 minus 119876119894| le 120576 so the optimal orderquantity 119876lowast can be found at the value 119876119894+1

Step 3 For verifying adequacy of Newtonrsquos method substitute119876lowast into (6) if 1198892119864(119876lowast)119889119876lowast2 lt 0meaning Newtonrsquos method

is satisfactory then the final solution is 119876lowast whose 119864(119876lowast)is the vendorrsquos lower bound of maximized expected profit

otherwise go to Step 4 to perform the bisection optimizationmethod

Step 4 Select 119897 a quantifiable order quantity Start 119894 = 0 and let[119876119904

0 119876lowast

0] be the initial searching interval where119876119904

0= exp(1minus

119886119887) is the regulatedminimal quantity level of delivery for thetransportation cost tc(119876) = 119886 + 119887 ln119876 and 119876lowast

0= 119876lowast

Step 5 If |119876lowast119894minus 119876119904

119894| lt 119897 then stop the optimal order quantity

is119876lowastlowast119894= (119876119904

119894+119876lowast

119894)2 along with the lower bound of maximal

expected profit 119864(119876lowastlowast119894) otherwise let 119876119887

119894= (119876119904

119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast

119894) then 119876lowast

119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894

otherwise 119876lowast119894+1= 119876lowast

119894and 119876119904

119894+1= 119876119887

119894 Go back Step 5 with

119894 = 119894 + 1To demonstrate the efficient solution procedure for

the DFNP incorporating the explicit transportation cost anumerical example is illustrated

31 Finding 119876lowast and 119864(119876lowast) A chosen product has demandmean 120583 = 9 kg and standard deviation 120590 = 05 Its unitcost is 119888 = $35kg unit selling price 119901 = $5kg andunit salvage value 119904 = $25kg Including fuel and handlingcharges on-site data of the transportation cost collected fromUPS worldwide expedited service at zone 7 from Europe toTaiwan are 058 069 077 085 093 100 106 112 118124 131 137 143 149 155 161 164 165 166 and 166 forshipment weight of 1 2 20 kg respectively For clarity ofdescription the costs considered here are all roundeddown toa 45-hundred US dollar-scale By fitting the data through thenonlinear regression model we have an empirical tamperingfunction tc(119876) = 036 + 042 ln119876 shown in Figure 1 with1198772=0926We conclude that the fitted function provides high

fidelity to represent the actual dataThen we follow the proposed search procedure

Step 1 From 119899 = 0 and 119894 = 0 set 1198760 = 120583 = 9 and 120576 = 10minus3

Step 2 When 119899 = 1 we have 1198761 = 1198760 minus (1198911015840(1198760)119891

10158401015840(1198760)) =

9211 In this case |1198761 minus 1198760| gt 0001 so continue Newtonrsquossearch until reaching |119876119894+1 minus 119876119894| le 0001 Then the optimalorder quantity 119876lowast = 119876119894+1 The searching details are listed inTable 1

Step 3 The optimal order quantity 119876lowast = 8731 (the condition1198892119864(119876lowast)119889119876lowast2= minus1783 lt 0 holds true) Substituting

119876lowast = 8731 and known parameters into (5) we obtain lower

bound of maximized expected profit 119864(119876lowast) which is $11899


Table 2 The computational results with fixed values of 119901 = 5 and 119904 = 25

Policy Parameters setting Our proposed policy Gallego and Moon [7] Profit gain120583 120590 119888 tc(119876) 119864(119876

lowast) 119864(119876

lowast) ()

1 7 04 3 036 + 042ln119876 12659(6824) 12454(7300) 1622 11 04 3 036 + 042ln119876 20470(10863) 20262(11300) 1023 7 06 3 036 + 042ln119876 12236(698) 12086(7450) 1224 11 06 3 036 + 042ln119876 20040(10700) 19893(11450) 0735 7 04 4 036 + 042ln119876 5939(6503) 5533(7082) 6846 11 04 4 036 + 042ln119876 9736(10516) 9339(11082) 4087 7 06 4 036 + 042ln119876 5476(6509) 5102(7122) 6828 11 06 4 036 + 042ln119876 9271(10509) 8907(11122) 3939 7 04 3 031 + 056ln119876 12764(6705) 12411(7300) 27610 11 04 3 031 + 056ln119876 20511(10754) 20155(11300) 17311 7 06 3 031 + 056ln119876 12250(6858) 11988(7450) 21312 11 06 3 031 + 056ln119876 19987(10881) 19731(11450) 12813 7 04 4 031 + 056ln119876 6160(6444) 5561(7082) 97414 11 04 4 031 + 056ln119876 9889(10435) 9303(11082) 59315 7 06 4 031 + 056ln119876 5605(6428) 5075(7122) 94616 11 06 4 031 + 056ln119876 9331(10414) 8814(11122) 554

Average 405

12

10

8

6

4

2

0

5 10 15 20

Order quantity Q

Expe

cted

pro

fitE

(Q)

Figure 2 Illustration of the expected profit with respect to orderquantity 119876

Figure 2 concavely exhibits119864(119876lowast)with respect to awide rangeof 119876lowast

32 Models Comparison For models comparison we imple-ment theDFNPbased onGallego andMoon [7] whosemodeldoes not reckon the transportation cost and perform thesimilar searching procedure described in Section 3 Theirmodel obtains the optimal order quantity119876lowast = 8731 with thelower bound of maximized expected profit 119864(119876lowast) = $11752In this case our proposed model in consideration of the

transportation cost has manifested (11899minus11752)11899 =12 of gains in 119864(119876lowast)

4 Sensitivity Analyses andComparative Studies

Furthermore we apply a 24 factorial design to investigatesensitivity of parameters They are set as follows Let theunit selling price be 119901 = $5kg and the unit salvage valuebe 119904 = $25kg two levels are selected for each of the fourparameters that is mean 120583 isin [7 11] standard deviation 120590 isin[01 1] unit product cost 119888 isin [3 4] and the transportationcost tc(119876) isin [036 + 042 ln119876 031 + 056 ln119876] whoseselected levels are based on fitting another data set gatheredfrom UPSrsquos transportation cost (worldwide express saver atzone 7 from Europe to Taiwan) US$ 066 078 088 098108 115 123 131 139 147 155 162 170 179 187 194201 209 217 and 225 respectively for shipment weight of1 2 3 20 kg

Table 2 lists 119864(119876lowast) along with 119876lowast for our proposedmodel in the 6th column and Gallego and Moonrsquos model[7] in the 7th column First this sensitivity analysis demon-strates significant correlations among the parameters whosesimultaneous consideration is imperative for the proposedoptimal policy Moreover in contrast to Gallego and Moonrsquosmodel the percentages of the profit gain obtained from ourproposed model are listed in the 8th column Apparently ourproposed model outperforms Gallego and Moonrsquos model inevery policy especially in the ordering policies 13 and 15the profit advance can be more than 94 on average ourproposed policy provides the return gain as much as 4 asopposed to that of the Gallego and Moonrsquos model

In views of the impact of transportation cost on theDFNPas well as the gains elicited from our proposed policies we


then extend contemplation of the transportation cost intoseveral practical inventory cases such as fixed ordering costrandom yield and a multiproduct case

5 The Fixed Ordering Cost Case withTransportation Cost

Let a vendor have an initial inventory 119868 (119868 ge 0) prior toplacing an order 119876 gt 0 where ordering cost 119860 is fixed forany size of order Let 119903 denote the reorder point known as aninventory level when the order is submitted Let 119878 = 119868 + 119876be end inventory level an inventory level after receiving theorder

Similarly min119878 119863 units are sold 119878 minus 119863 units aresalvaged For an (119903 119878) inventory replenishment policy inconsideration of the transportation cost expected profit 119864(119878)is constructed as

119864 (119878) = 119901119864 (min 119878 119863) + 119904119864(119878 minus 119863)+

minus 119888 (119878 minus 119868) minus 1198601[119878gt119868] minus 119886 + 119887 ln [119864 (min 119878 119863)]

minus 119886 + 119887 ln [119864(119878 minus 119863)+]


minus (119888 minus 119904) 119878 + 119888119868 minus 119860119868[119878gt119868] minus 2119886


where 119868[119878gt119868] = 1 if 119878gt1198680 otherwise

According to Lemma 1 the expression can be simplifiedas min119878ge119868119860119868[119878gt119868] + 119869(119878) where

119869 (119878) = minus (119901 minus 119904) 120583 + (119901 minus 119904)[1205902+ (119878 minus 120583)

2]12

minus (119878 minus 120583)

2

+ (119888 minus 119904) 119878 minus 119888119868 + 2119886 minus 2119887 ln 2

+ 119887 ln minus1205832 minus 1 + 1198782 + 2120583[1205902 + (119878 minus 120583)2]12

(10)

The relationship of 119878 = 119868 + 119876 implies that acquiring theoptimal end inventory level of 119878 for the fixed ordering costmodel is equivalent to having optimal order quantity of119876 forthe single-product model Clearly because 119868 lt 119878 119869(119868) gt 119860 +119869(119878) For determining the optimal reorder point of 119903 119869(119903) =119860 + 119869(119878) is set Then we have

119901 minus 119904

2[1205902+ (119903 minus 120583)

2]12

minus 119903 + (119888 minus 119904) 119903


minus 119860 minus119901 minus 119904

2[1205902+ (119878 minus 120583)

2]12

minus 119878 minus (119888 minus 119904) 119878


= 0

(11)

Furthermore we develop a solution procedure to deter-mine the optimal reorder point

Step 1 By performing the solution procedure for the optimalorder quantity in Section 3 we first obtain119876lowastThen let119876lowast bethe end inventory level 119878 where 119868 is set to be 0 for brevity

Step 2 Start 119894 = 0 set the initial reorder point 1199030 to be 119878 anddetermine the allowable tolerance 120576 for accuracy of the finalresult

Step 3 Perform Newtonrsquos search (see Grossman [25 pp228])to compute the optimal reorder level of 119903 That is 119903119894+1 = 119903119894 minus(119891(119903119894)119891

1015840(119903119894)) where

119891 (119903119894) =119901 minus 119904

2[1205902+ (119903119894 minus 120583)

2]12

minus 119903119894 + (119888 minus 119904) 119903119894

+ 119887 ln minus1205832 minus 1 + 1199032119894+ 2120583[120590

2+ (119903119894 minus 120583)

2]12


2[1205902+ (119878 minus 120583)

2]12



1198911015840(119903119894) =

2119888 minus 119901 minus 119904

2+

(119901 minus 119904) (119903119894 minus 120583)

2[1205902 + (119903119894 minus 120583)2]12

+ 1198872119903119894 + 2120583 (119903119894 minus 120583) [120590

2+ (119903119894 minus 120583)

2]minus12

minus1205832 minus 1 + 1199032119894+ 2120583[1205902 + (119903119894 minus 120583)

2]12

(12)

Stop the search when |119903119894+1 minus 119903119894| le 120576 Then the optimal orderquantity is 119903119894+1

Step 4The optimal policy is to order up to 119878 units if the initialinventory is less than 119903 and not to order otherwise

51 An Example Continuing the numerical example inSection 3 we assume that the ordering cost is given by 119860 =$03 Using the above solution procedure we find that theoptimal reorder level of 119903 is 8210 and the end inventory level119878 = 8731

6 The Random Yield Case withTransportation Cost

Suppose randomvariable119866(119876) expresses the number of goodunits produced from ordered quantity 119876 where each goodunit being ordered or produced has an equal probability of 120588Thus 119866(119876) is a binomial random variable with mean119876120588 andvariance119876120588119902 where 119902 = 1minus120588 Let119898 be the pricemarkup rateand 119889 the discount rate so unit selling price 119901 = (1 + 119898)119888120588


and salvage value 119904 = (1 minus 119889)119888120588 Thus the expected profit in(1) can be rewritten as

119864 (119876) = 119901119864 (min 119866 (119876) 119863) + 119904119864(119866 (119876) minus 119863)+ minus 119888119876 minus 2119886

minus 119887 ln 119864 (min 119866 (119876) 119863) 119864(119866 (119876) minus 119863)+

=119888

120588(119898 + 119889) 120583 minus (119898 + 119889) 119864[119863 minus 119866 (119876)]

+

minus (120588 + 119889 minus 1)119876 minus 2119886

minus 119887 ln [120583 minus 119864[119863 minus 119866 (119876)]+]

times [119876 minus 120583 + 119864[119863 minus 119866 (119876)]+]

(13)

Applying Lemma 1 to this case we have

119864[119863 minus 119866 (119876)]+le[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

(14)

Substituting the above relationship into (13) we have lowerbound of the expected profit in this case Consider

119864 (119876) ge119888

120588

(119898 + 119889) 120583 minus (119898 + 119889)

times[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

minus (120588 + 119889 minus 1)119876

minus 2119886 + 2119887 ln 2

minus 119887 ln 2 (120588119876 + 120583) (1 minus 120588)119876 minus 1205902 minus 120588119902119876

+ 2 (120583 + 120588119876 minus 119876)

times [1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

(15)

The right-hand side of (15) is a continuous function interms of 119876 Then first and second derivatives of 119864(119876) can bederived as119889119864 (119876)

119889119876

= minus119888 (119898 + 119889)

2[1

2119883minus12(119902 minus 2120583 + 2120588119876) minus 1] minus

119888

120588(120588 + 119889 minus 1)

minus 119887 (2 (1 minus 120588) (120583 + 2120588119876) minus 120588119902 minus 2 (1 minus 120588)11988312

+120588 (120583 + 120588119876 minus 119876) (119902 minus 2120583 + 2120588119876)119883minus12)

times (2 (120588119876 + 120583) (1 minus 120588)119876 minus 1205902

minus120588119902119876 + 2 (120583 + 120588119876 minus 119876)11988312)minus1

(16)

where119883 = 1205902 + 120588119902119876 + (120588119876 minus 120583)2

1198892119864 (119876)

1198891198762= minus119888 (119898 + 119889)

2[minus120588

4(119902 minus 2120583 + 2120588119876)

2119883minus32

+ 120588119883minus12]

minus 1198871198841015840119885 minus 119884119885

1015840

1198852

(17)where119884 = 2 (1 minus 120588) (120583 + 2120588119876) minus 120588119902 minus 2 (1 minus 120588)119883

12

+ 120588 (120583 + 120588119876 minus 119876) (119902 minus 2120583 + 2120588119876)119883minus12

119885 = 2 (120588119876 + 120583) (1 minus 120588)119876 minus 1205902minus 120588119902119876

+ 2 (120583 + 120588119876 minus 119876)11988312

1198841015840= 4120588 (1 minus 120588) minus 2120588

times [(1 minus 120588) (119902 minus 2120583 + 2120588119876) minus 120588 (120583 + 120588119876 minus 119876)]119883minus12

minus1205882

2(120583 + 120588119876 minus 119876) (119902 minus 2120583 + 2120588119876)

2119883minus32

1198851015840= 2 (1 minus 120588) (2120588119876 + 120583) minus 120588119902

minus 2 (1 minus 120588)11988312+120588

4(120583 + 120588119876 minus 119876)

times (119902 minus 2120583 + 2119876)119883minus12

(18)

Obviously 1198892119864(119876)1198891198762 is not necessarily being negativeSimilarly we develop a solution procedure to find the

optimal order quantity in this random yield case

Step 1 Start 119894 = 0 and 1198760 = 120583 Set the allowable tolerance 120576

Step 2 Perform Newtonrsquos search (see Hillier and Lieberman[24] pp555ndash557) to compute the optimal order quantity 119876That is 119876119894+1 = 119876119894 minus (119891

1015840(119876119894)119891

10158401015840(119876119894)) where 119891

1015840(119876119894) and

11989110158401015840(119876119894) stand for (16) and (17) respectively Stop the search

when |119876119894+1 minus 119876119894| le 120576 The optimal order quantity is 119876119894+1

Step 3 For verifying adequacy of Newtonrsquos method substitute119876119894+1 into (19) if 119889

2119864(119876119894+1)119889119876

2

119894+1lt 0 representing Newtonrsquos

method is satisfactory then the final solution is 119876lowast = 119876119894+1whose 119864(119876lowast) is the vendorrsquos lower bound of the maximizedexpected profit otherwise go to Step 4 to perform thebisection optimization method


0 119876lowast


0= exp(1minus


0= 119876lowast



is 119876lowastlowast119894= (119876119904

119894+ 119876lowast

119894)2 along with 119864(119876lowastlowast

119894) the lower bound of

maximal expected profit otherwise let 119876119887119894= (119876119904

119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887


119894 = 119894 + 1


61 An Example We continue Section 3 We assume thatfor each unit of 119876 the probability of being good is 120588 = 09We find the optimal order quantity119876lowast=10403 and the lowerbound of the maximum expected profit 119864(119876lowast) is 14573 Thecondition 1198892119864(119876119894+1)119889119876

2

119894+1= minus0916 lt 0 is satisfactory

In contrast the order quantity placed on the product withperfect quality can be computed as much as 8731 which issmaller than119876lowast= 10403 Apparently in therandom yield casethe order quantity is increased to provide safeguard against apossible shortage

7 The Multiproduct Case withTransportation Cost

We now study a multiproduct newsboy problem in thepresence of a budget constraint also known as the stochasticproduct-mixed problem [26] Suppose that each product 119895for 119895 = 1 119873 has order quantity 119876119895 received fromeither purchasing or manufacturing where a limited budgetis allocated due to the limited production capacity in thesystemThat is the total purchasing ormanufacturing cost forall the 119873 competing products cannot exceed allotted budget119861 Denote that each itemrsquos unit cost of the 119895th product is 119888119895 itsunit selling price is 119901119895 and its unit salvage value is 119904119895 For the119895th productrsquos demand its mean and variance are denoted by120583119895 and 120590

2

119895 respectively

In the sequel under the distribution-free demand jointedwith the explicit transportation cost the vendor is in needof deciding the optimal order quantities for 119873 competingproducts whose total purchasing or manufacturing cost doesnot exceed the allocated budge 119861 where heshe guarantees topossess the least of all possible maximum expected profits

For solving this problem we first extend the singleproduct case in (3) to have lower bound of expected profit119864(1198761 119876119873) for the vendor provided that the individualorder quantity of11987611198762 and119876119873 is affected by the budgetconstraint 119861 For the vendor to secure the least amount of themaximum expected profit over various situations of marketwe maximize (19) with a budget constraint expressed in (20)to determine the optimal order quantities 119876lowast

1 119876lowast2 and

119876lowast

119873

max1198761 119876119873

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2


minus 119887 lnminus 1205832119895minus 1 + 119876

2

119895

+2120583[1205902

119895+ (119876119895 minus 120583119895)

2

]12

(19)

Subject to119873

sum

119895=1

119888119895119876119895 le 119861 (20)

We further transfer the problem into an unconstrainedoptimization equation

119871 (1198761 119876119873 120582)

=

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583119895[1205902

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582(

119873

sum

119895=1

119888119895119876119895 minus 119861)

(21)

where 120582 is the Lagrange multiplier Hence we have

120597119871 (1198761 119876119873 120582)

120597119876119895

=119901119895 + 119904119895 minus 2119888119895

2minus(119901119895 minus 119904119895) (119876119895 minus 120583119895)

2[1205902119895+ (119876119895 minus 120583119895)

2

]12

minus 119887

2119876119895 + 2120583119895 (119876119895 minus 120583119895) [1205902

119895+ (119876119895 minus 120583119895)

2

]minus12

minus1 minus 1205832 + 1198762119895+ 4120583119895[120590

2

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582119888119895

(22)

To find the optimal order quantities119876lowast1119876lowast2 and119876lowast

119873with

maximum 119871 we set 120597119871120597119876119895 = 0 In this case a line searchprocedure is developed

Step 1 For multiple products119873 let 119895 = 1 119873

Step 2 Let 120582 = 0 and perform the solution procedureproposed in Section 3 to find 119876lowast

119895 If (20) is satisfied go to

Step 6 otherwise go to Step 3

Step 3 Substituting each of119876lowast1119876lowast2 and119876lowast

119873into (22) their

corresponding 120582 can be obtained

Step 4 Start from the smallest nonnegative 120582 let its corre-sponding optimal order quantity be 0 (others are intact) andcheck the condition of (20)

Step 5 If the condition is satisfactory then we have thefinal solution 119876lowast

1 119876lowast2 and 119876lowast

119873 otherwise select the next

smallest nonnegative120582 to perform the sameprocedure in Step4 until (20) is satisfied

Step 6 Find the least amount of themaximum expected profit119864(1198761lowast 119876119873lowast)


71 An Example The total budget is $80 for the four itemsThe relevant data are as follows 119888 = (35 25 28 05) 119901 = (54 32 06) 119904 = (25 12 15 02) 120583 = 119888(9 8 12 23) and 120590 =119888(05 1 07 1) Performing the above procedure we have thefollowing

Step 1 Let 119895 = 1 2 3 4

Step 2 Let 120582 = 0 We solve the four order quantities by usingthe solution procedure introduced in Section 3 The optimalorder quantities 119876lowast

1= 8731 119876lowast

2= 7762 119876lowast

3= 11072 and 119876lowast

4

= 21243 Check sum4119895=1119888119895119876lowast

119895= $92 gt $80 where (20) is not

satisfied so we go to Step 3

Step 3 Performing a simple line search we increase theoptimal value of the Lagrangian multiplier until 120582 = 0147In this case its corresponding 119876lowast

3is set to 0

Step 4 Since sum4119895=1119888119895119876lowast

119895= $61 lt $80 (20) is satisfied

Step 5 The optimal order quantities are 8731 7762 0 and21243 and the lower bound of the maximum expected profitis $21667

8 Conclusions and Implications

Models for the distribution-free newsboy problem have beenwidely introduced over the past two decades to provide theoptimal order quantity for securing the vendor with theleast amount of the maximum expected profit when facinga variety of situations in modern business environment

Over the past few years energy prices have risen sig-nificantly so that the transportation of goods has becomea vital component for the vendorrsquos logistic-cost function todetermine its required purchase quantities However impactsof the transportation cost on previous models for the DFNPwere inattentive by either overlooking or deeming it as partof implicit components of ordering cost In this paper threemain contributions along with their managerial implicationhave been done

First we develop the DFNP incorporating the explicittransportation cost into the expected profit function Inparticular the transportation cost is modeled based onthe economic theory from transportation disciplines andfitted a nonlinear regression via actual rate data collectedfrom the shipper In practice this way has implied that (1)economic trade-off for the optimal transportation cost liesbetween provided service level and shipped quantity (2) inthe shipment more weight signifies larger delivery quantityand higher shipment cost and (3) optimal shipping quantityrenders minimum of the transportation cost

Secondly since the expected profit function is neitherconcave nor convex the optimization problem underlyingthis generalization is challenging therefore we developedanalytical and efficient procedures to acquire the optimalpolicy As a result of the computational studies our proposedoptimal ordering rules in comparisonwith the optimal policyrecommended by Gallego and Moon [7] increased the lowerbound of the maximal expected profit by as much as 4 on

average This result has demonstrated that the expenditure ofthe inboundoutbound material transportation has becomea critical component of a total annual logistic cost functionfor determining purchase quantities Effects of transportationhave gained substantial recognition in the DFNP

Thirdly according to the results of sensitivity analy-ses the parameters such as demand mean and varianceproductrsquos unit cost and transportation cost are the keydecision variables whose joint reckoning is imperative forthe optimal policy proposed Moreover we proceed toanalyses of several practical inventory cases including fixedordering cost random yield and multiproduct case Thesestudies further demonstrate the impacts of transportationcost as well as the realized-least profit gains drawn fromour recommended policies on the DFNP that explicitlyincorporates the transportation cost into consideration Inaddition these numerical findings have implied that jointdecision coordinated operation or integrated managementis crucial in lowering the vendor-and-buyer operating cost aswell as balancing a supply-chain operation and structure

Finally based on the shipping data sets collected fromUnited Parcel Service (UPS) the transportation cost ismodeled using a natural logarithm for a nonlinear regressionfunction in this paper For future studies other functionalforms may be reckoned to model different transportationcosts such as a step function or a logistic function to validatea wide variety of applications Besides using our proposedmodel as a basis model in a couple of more advancedstudies with certain circumstances such as the multiproductnewsboy under a value-at-risk and the multiple newsvendorswith loss-averse preferences is intriguing

Highlights

(i) We extend previous work on the distribution-freenewsboy problem where the vendorrsquos expected profitis in presence of transportation cost

(ii) The transportation cost is formulated as a functionof shipping quantity and modeled as a nonlinearregression form based on UPSrsquos on-site shipping-ratedata

(iii) The comparative studies have demonstrated signifi-cant positive impacts by using our proposed method-ology whose profit gains in comparison with priorresearch can be as much as 9 and 4 on average

(iv) The sensitivity analyses jointly reckon the imperativeparameters for the optimal policy

(v) We expand our methodology to several practicalinventory cases including fixed ordering cost randomyield and a multiproduct condition




References

[1] M Khouja ldquoThe single-period (news-vendor) problem litera-ture review and suggestions for future researchrdquoOmega vol 27no 5 pp 537ndash553 1999

[2] S-P Chen and Y-H Ho ldquoOptimal inventory policy for thefuzzy newsboy problem with quantity discountsrdquo InformationSciences vol 228 pp 75ndash89 2013

[3] S B Ding ldquoUncertain random newsboy problemrdquo Journal ofIntelligent and Fuzzy Systems vol 26 no 1 pp 483ndash490 2014

[4] V Arshavskiy V Okulov and A Smirnova ldquoNewsvendorproblem experiments riskiness of the decisions and learningby experiencerdquo International Journal of Business and SocialResearch vol 4 no 5 pp 137ndash150 2014

[5] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009

[6] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[7] G Gallego and I Moon ldquoDistribution free newsboy problemreview and extensionsrdquo Journal of the Operational ResearchSociety vol 44 no 8 pp 825ndash834 1993

[8] H K Alfares and H H Elmorra ldquoThe distribution-freenewsboy problem extensions to the shortage penalty caserdquoInternational Journal of Production Economics vol 93-94 pp465ndash477 2005

[9] I Moon and S Choi ldquoThe distribution free newsboy problemwith balkingrdquo Journal of the Operational Research Society vol46 no 4 pp 537ndash542 1995

[10] X Cai S K Tyagi and K Yang ldquoActivity-based costing modelfor MGPDrdquo in Improving Complex Systems Today pp 409ndash416Springer London UK 2011

[11] C-M Lee and S-L Hsu ldquoThe effect of advertising on thedistribution-free newsboy problemrdquo International Journal ofProduction Economics vol 129 no 1 pp 217ndash224 2011

[12] M G Guler ldquoA note on lsquothe effect of optimal advertising onthe distribution-free newsboy problemrsquordquo International Journal ofProduction Economics vol 148 pp 90ndash92 2014

[13] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014

[14] J Mostard R de Koster and R Teunter ldquoThe distribution-freenewsboy problem with resalable returnsrdquo International Journalof Production Economics vol 97 no 3 pp 329ndash342 2005

[15] J Barry Rising Transportation Costs-and What to do aboutThem Article and White Papers F Curtis Barry amp Company2013

[16] S R Swenseth and M R Godfrey ldquoIncorporating transporta-tion costs into inventory replenishment decisionsrdquo Interna-tional Journal of Production Economics vol 77 no 2 pp 113ndash1302002

[17] S Cetinkaya and C-Y Lee ldquoOptimal outbound dispatch poli-cies modeling inventory and cargo capacityrdquo Naval ResearchLogistics vol 49 no 6 pp 531ndash556 2002

[18] A Toptal S Cetinkaya and C-Y Lee ldquoThe buyer-vendorcoordination problem modeling inbound and outbound cargocapacity and costsrdquo IIE Transactions vol 35 no 11 pp 987ndash1002 2003

[19] A Toptal and S Cetinkaya ldquoContractual agreements for coordi-nation and vendor-managed delivery under explicit transporta-tion considerationsrdquo Naval Research Logistics vol 53 no 5 pp397ndash417 2006

[20] J-L Zhang C-Y Lee and J Chen ldquoInventory control problemwith freight cost and stochastic demandrdquo Operations ResearchLetters vol 37 no 6 pp 443ndash446 2009

[21] A Toptal ldquoReplenishment decisions under an all-units discountschedule and stepwise freight costsrdquo European Journal of Oper-ational Research vol 198 no 2 pp 504ndash510 2009

[22] F Mutlu and S Cetinkaya ldquoAn integrated model for stockreplenishment and shipment scheduling under common carrierdispatch costsrdquo Transportation Research E Logistics and Trans-portation Review vol 46 no 6 pp 844ndash854 2010

[23] S-D Lee and Y-C Fu ldquoJoint production and shipment lot siz-ing for a delivery price-based production facilityrdquo InternationalJournal of Production Research vol 51 no 20 pp 6152ndash61622013

[24] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 2010

[25] S L Grossman Calculus Harcourt Brace New York NY USA5th edition 1993

[26] L Johnson andDMontgomeryOperations Research in Produc-tion Planning Scheduling and Inventory Control John Wiley ampSons New York NY USA 1974

Research ArticleUndesirable Outputsrsquo Presence in CentralizedResource Allocation Model

Ghasem Tohidi Hamed Taherzadeh and Sara Hajiha

Department of Mathematics Islamic Azad University Central Branch Tehran Iran

Correspondence should be addressed to Hamed Taherzadeh htaherzadehhotmailcom

Received 15 July 2014 Revised 25 August 2014 Accepted 28 August 2014 Published 15 September 2014


Copyright copy 2014 Ghasem Tohidi et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Data envelopment analysis (DEA) is a common nonparametric technique to measure the relative efficiency scores of the individualhomogenous decision making units (DMUs) One aspect of the DEA literature has recently been introduced as a centralizedresource allocation (CRA) which aims at optimizing the combined resource consumption by all DMUs in an organization ratherthan considering the consumption individually through DMUs Conventional DEA models and CRA model have been basicallyformulated on desirable inputs and outputsThe objective of this paper is to present newCRAmodels to assess the overall efficiencyof a system consisting of DMUs by using directional distance function when DMUs produce desirable and undesirable outputsThis paper initially reviewed a couple of DEA approaches for measuring the efficiency scores of DMUs when some outputs areundesirableThen based upon these theoretical foundations we develop the CRAmodel when undesirable outputs are consideredin the evaluation Finally we apply a short numerical illustration to show how our proposed model can be applied

1 Introduction

Data envelopment analysis (DEA) was introduced in 1978DEA includes many models for assessing the efficiencyscore in the variety of conditions Many researchers usethis technique to evaluate the efficiency and inefficiencyscores of decision making units (DMUs) Two of the mostcommon DEA models are CCR (Charnes Cooper andRhodes) and BCC (Banker Charnes and Cooper) whichwere introduced by Charnes et al [1] and Banker et al [2]respectively In addition there are other important modelssuch as additive (ADD) model which was introduced byCharnes et al [3] and SMB model (slack-based measure)which was introduced by Tone [4] Classical DEA models(such as CCR BCC ADD and SMB) rely on the assumptionthat inputs have to beminimized and outputs have to bemax-imized In authentic situations however it is possible thatthe production process consumes undesirable inputs andorgenerates undesirable outputs [5 6] Consequently classicalDEA models need to be modified in order to deal with thesituation because undesirable outputs should notmaximize atall

There frequently exist undesirable inputs andor outputsin the real applications Many studies have been done on theundesirable data Broadly we can divide these studies intotwo parts The first part involves some methods which usetransformation data For instance Koopman [6] suggesteddata transformation Although the reflection function wasused in this method it caused the positive data to turninto negative data and it was not straightforward to defineefficiency score for negative data at that time Some of therelated methods had been suggested by Iqbal Ali and Seiford[7] Pastor [8] Scheel [9] and Seiford and Zhu [10] HoweverGolany and Roll [11] and Lovell and Pastor [12] attemptedto introduce another form of transformation which wasmultiplicative inverse Being a nonlinear transformation itsbehaviors were even more complicated to deal with (Scheel[13])Therefore the approaches based on data transformationmay unexpectedly produce unfavorable results such as thosediscussed by Liu and Sharp [14] The second part consistsof many methods which can avoid data transformation Asan initial attempt Liu and Sharp [14] suggested consideringundesirable outputs as desirable inputs but undesirable inputsas desirable outputs This method is currently used as an



attractive one in studying operational efficiency because of itssimplicity and elegance

In many authentic situations there are cases in whichall DMUs are under the control of a centralized decisionmaker (DM) that oversees them and tends to increase theefficiency of all of the system instead of increasing theefficiency of each unit separatelyThese situations occurwhenall of the units belong to the same organization (publicandor private) which provides the units with the necessaryresources to obtain their outputs such as bank branchesrestaurant chains hospitals university departments andschools Thus DMrsquos goal is to optimize the resource utiliza-tion of all DMUs across the total entity Lozano and Villa[15] first introduced the meaning of centralized resourceallocation They presented the envelopment and multiplierform of BCC model with regard to centralized meaningMar-Molinero et al [16] demonstrated that the centralizedresource allocation model proposed by Lozano and Villa [15]can be substantially simplified There are some other similarresearches done by Korhonen and Syrjanen [17] Du et al[18] and Asmild et al [19] Multiple-objective model hasbeen used in order to optimize the efficiency of system byKorhonen and Syrjanen [17] and Du et al [18] proposedanother approach for optimization in centralized scenarioAsmild et al [19] reformulated the centralized model pro-posed by Lozano and Villa [15] considering adjustments ofinefficient units Hosseinzadeh Lotfi et al [20] and Yu et al[21] are other researchers engaged in centralized resourceallocation

In this paper we discuss a DEA model in centralizedresource allocation when some of the inputs or outputs areundesirable This paper is organized as follows In Section 2research motivation of this study is given Section 3 brieflypresents some methods for measuring the efficiency scoreswhen some of the outputs are undesirable Section 4 discussesthe centralized resource allocation model and its advantagesWe develop the centralized resource allocation model in theundesirable outputsrsquo presence in Section 5 An illustration isgiven in Section 6 and Section 7 provides the conclusion ofthe paper

2 Research Motivation

Traditional DEA models are consecrated to the performanceevaluation of DMUs in different situations Although unde-sirable outputs treatments have been studied by interestedresearchers centralized resource allocation has never dealtwith undesirable outputs Moreover in many real situationsthe production of undesirable outputs is unavoidable hencedecision makers need scientific methods to deal with theundesirable outputsrsquo production and decrease them whenall of DMUs are under their control Here we will answerthe following question scientifically how can centralizedresource allocation model be modified in order to evaluatethe performance of a system involving several DMUs whichproduce both desirable and undesirable outputs

3 Undesirable Output Models

Most researchers recently analyze closely the structure ofthe undesirable data Undesirable outputs such as air purifi-cation sewage treatment and wastewater can be jointlyproduced with desirable outputs When the undesirable out-puts are taken into account the efficiency scorersquos evaluationof DMUs is different Therefore traditional DEA modelsshould be modified Briefly we review a couple of methodsto measure the efficiency scores when some of the dataare undesirable and we address some papers for evaluatingundesirable data

Seiford and Zhu [10] showed that the traditional DEAmodel is used to improve the performance through increas-ing the desirable outputs and decreasing undesirable outputsby the classification invariance property useTheir model canalso be applied to a situationwhen inputs need to be increasedto improve the performance This model is as follows

max 120601

st 120582119883 le 119909119863

119900

120582119884119863ge 120601119910119863

119900

120582119884119880

ge 120601119910119900119880

119890120582 = 1

120582 ge 0

(1)

in which 119910119900119880= minus119884

119880+ V gt 0 Hadi Vencheh et al [22]

proposed a model for treating undesirable factors in theframework of DEA as follows

max 120601

st 120582119883119863le (1 minus 120601) 119909

119863

119900

120582119883119880

le (1 minus 120601) 119909119900119880

120582119884119863ge (1 + 120601) 119910

119863

119900

120582119884119880

ge (1 + 120601) 119910119900119880

119890120582 = 1

120582 ge 0

(2)

in which 119910119900119880= minus119884

119880+ V gt 0 and 119883

119880

= minus119883119880+ 119908 gt 0

(Seiford and Zhu [10]) Model (2) evaluates the efficiencylevel of each DMU by considering desirable and undesirablefactors In fact model (2) expands desirable outputs andcontracts undesirable outputs A similar discussion holds forthe inputs Jahanshahloo et al [23] presented an alternativemethod to deal with desirable and undesirable factors (inputsand outputs) in nonradial DEA models They demonstrated


that their proposed model is feasible bounded and unitinvariant The model is given as follows

min 1 minus [

[

119908119900 +1

119898 + 119904(sum

119894isin119868119863

119905minus119863

119894+ sum

119903isin119874119863

119905+119863

119903)]

]

st119899

sum

119895=1

120582119895119909119863

119894119895+ 119905minus119863

119894= 119909119863

119894119900minus 119908119900 119894 isin 119868119863

119899

sum

119895=1

120582119895119909119880

119894119895+ 119905minus119880

119894= 119909119880

119894119900+ 119908119900 119894 isin 119868119880

119899

sum

119895=1

120582119895119910119863

119903119895minus 119905+119863

119903= 119910119863

119903119900+ 119908119900 119903 isin 119874119863

119899

sum

119895=1

120582119895119910119880

119903119895minus 119905+119880

119903= 119910119880

119903119900minus 119908119900 119903 isin 119874119880

119899

sum

119895=1

120582119895 = 1

(3)

in which all variables are restricted to be nonnegative Inmodel (3) 119868119863 119868119880 119874119863 and 119874119880 stand for desirable inputsundesirable inputs desirable outputs and undesirable out-puts respectively Recently Wu and Guo [24] suggested amodel for measuring the efficiency score which is formulatedbased on that inputs and undesirable outputs are decreasedproportionally This model is as follows

min 120579

st119899

sum

119895=1

120582119895119909119894119895 le 120579119909119894119900 forall119894 isin 119868

119899

sum

119895=1

120582119895119910119863

119903119895ge 119910119863

119903119900forall119903 isin 119874

119863

119899

sum

119895=1

120582119895119910119880

119903119895le 120579119910119880

119903119900forall119903 isin 119874

119880

120582119895 ge 0 forall119895 isin 119873

(4)

Inmodel (4) 119868119874119863 and119874119880 refer to inputs desirable outputsand undesirable outputs sets respectively The studies ofScheel [9] and Amirteimoori et al [25] are another twostudies Indeed Scheel [9] proposed new efficiency measureswhich are oriented to desirable and undesirable outputsrespectively They are based on the assumption that anychange of output levels involves both desirable and unde-sirable outputs Amirteimoori et al [25] presented a DEAmodel which can be used to improve the relative performancevia increasing undesirable inputs and decreasing undesirableoutputs

4 Centralized Resource Allocation Model

Measuring the performance plays an important role for a DMproviding its weaknesses for the subsequent improvementWorking on the usual DEA framework assume that thereare 119899 DMUs (DMU119895 119895 = 1 119899) which consume 119898 inputs(119909119894 119894 = 1 119898) to produce 119904 outputs (119910119903 119903 = 1 119904) Thefirst phase of CRA input-oriented model (CRA-I) developedby Lozano and Villa [15] measures the efficiency of systemthrough solving the following linear program

min 120579

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le 120579

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 ge

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(5)

In Phase II of CRA model an additional reduction of anyinputs or expansion of any outputs is followed Phase II isformulated to remove any possible input excesses and anyoutput shortfalls as follows

max119898

sum

119894=1

119904minus

119894+

119904

sum

119903=1

119905+

119903

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 + 119904minus

119894= 120579lowast

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 minus 119905+

119903=

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

119904minus

119894ge 0 119905

+

119903ge 0 119894 = 1 119898 119903 = 1 119904

120582119895119896 ge 0 119896 119895 = 1 119899

(6)

Model (5) was formulated based on two important purposesFirst instead of reducing the inputs of each DMU the aimis to reduce the total amount of input consumption of theDMUs Second after solving the problem in Phase II theprojection of all DMUs will be onto the efficient frontierof production possibility set Indeed the efficiency scoreof system is more important than efficiency score of eachunit in the centralized scenario For that reason decisionmanager (DM) tries to reallocate resources to have a moreefficient system Toward this end some of the inputs can betransferred fromoneDMU to otherDMUs It is not necessaryto keep the total value of inputs or outputs in original levelbecause the overall consumption may be decreased and theoverall production may be increased


The improvement activity of DMU119900 which is obtained bythe maximum slack solution and is located on the efficiencyfrontier of production possibility set is defined as follows

119909119894119900 =

119899

sum

119895=1

120582119900lowast

119895119909119894119895 = 120579

lowast119909119894119900 minus 119904

minuslowast

119894119894 = 1 119898

119910119903119900 =

119899

sum

119895=1

120582119900lowast

119895119910119903119895 = 119910119903119900 + 119905

+lowast

119903119903 = 1 119904

(7)

The difference between the total consumption of improvedactivity and the original DMUs in each input and output canbe found by the following relationship

119878119894 =

119899

sum

119895=1

119909119894119895 minus

119899

sum

119895=1

119909119894119895 ge 0 119894 = 1 119898

119879119903 =

119899

sum

119895=1

119910119903119895 minus

119899

sum

119895=1

119910119903119895 ge 0 119903 = 1 119904

(8)

The dual formulation of the envelopment form of the CRAinput oriented model to find the common input and outputweights which maximize the relative efficiency score of avirtual DMU with the average inputs and outputs can bewritten as follows

max119899

sum

119895=1

119904

sum

119903=1

119906119903119910119903119895 +

119899

sum

119896=1

120577119896

st119899

sum

119895=1

119898

sum

119894=1

V119894119909119894119895 = 1

119904

sum

119903=1

119906119903119910119903119895 minus

119898

sum

119894=1

V119894119909119894119895 + 120577119896 le 0 119895 119896 = 1 119899

119906119903 ge 0 119903 = 1 119904

V119894 ge 0 119894 = 1 119898

(9)

The above model has 1198992 + 1 constraints and 119898 + 119904 +

119899 variables Solving model (9) derives the common set ofweights (CSW) It is worth mentioning that we can use thiscommon set of weights to evaluate the absolute efficiency ofeach efficientDMU inorder to rank themThe ranking adoptsthe CSW generated from model (9) which makes sensebecause a DM objectively chooses the common weights forthe purpose of maximizing the group efficiency For instancethe government is interested inmeasuring the performance ofDEA efficient banks The government would determine onecommon set of weights based upon the group performance ofthe DEA efficient banks

5 Proposed Method

Proposing the model in this study we used the distancedirectional function to measure the overall efficiency scoreof each system Throughout this method we deal with119899 DMU119904 (119895 = 1 119899) having 119898 inputs (119894 = 1 119898)

and 119904 outputs The outputs are divided into two sets oneas desirable outputs and one as undesirable outputs Let theinputs and desirable and undesirable outputs be as follows

119883 = 119909119894119895 isin 119877119898times119899

+ 119884

119863= 119910119863

119903119895 isin 119877119904119863times119899

+

119884119880= 119910119880

119905119895 isin 119877119904119880times119899

+

(10)

where 119883 119884119863 and 119884119880 are input desirable output and unde-sirable output matrices respectively In our proposed modelwe apply the distance directional function to reformulate thecentralized resource allocationmodel when some outputs areundesirable In addition we consider undesirable outputs asinputs in evaluation The model is as follows

max 120593

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le

119899

sum

119895=1

119909119894119895 minus 120593119877119909119894 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119863

119903119895ge

119899

sum

119895=1

119910119863

119903119895+ 120593119877119910

119863

119903119903 = 1 119904

119863

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119880

119905119895le

119899

sum

119895=1

119910119880

119903119895minus 120593119877119910

119880

119905119905 = 1 119904

119880

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(11)

where119877119909119894119877119910119863

119903 and119877119910119880

119905are parameters also 119904119863 and 119904119880 stand

for the number of desirable outputs and undesirable outputsrespectively The objective of model (11) is to decrease inputsand undesirable outputs level and increase desirable outputslevel with regard to the (119877119909119894 119877119910

119863

119903 119877119910119880

119905) direction Here we

use the ideal point to assign to the (119877119909119894 119877119910119863

119903 119877119910119880

119905) vector as

follows

119877119909119894 =

119899

sum

119895=1

119909119894119895 minus 119899 (min 119909119894119895119895=1119899) 119894 = 1 119898

119877119910119863

119903=

119899

sum

119895=1

119910119863

119903119895minus 119899 (max 119910119863

119903119895119895=1119899

) 119903 = 1 119904119863

119877119910119880

119905=

119899

sum

119895=1

119910119880

119905119895minus 119899 (min 119910119880

119905119895119895=1119899

) 119905 = 1 119904119880

(12)

The optimal objective value of model (11) measures sys-tem inefficiency score It is worth mentioning that anotheralternative for the directional vector (119877119909119894 119877119910

119863

119903 119877119910119880

119905) can be

chosen as follows

(119877119909119894 119877119910119863

119903 119877119910119880

119905) = (

119899

sum

119895=1

119909119894119895

119899

sum

119895=1

119910119863

119903119895

119899

sum

119895=1

119910119880

119905119895) (13)

The purposes of model (11) are to reduce the total consump-tion of inputs reduce the total production of undesirable


Table 1 Data set with undesirable outputs

Inputs Desirable outputs Undesirable outputsI1 I2 O1 O2 UO1 UO2

DMU 1 5 8 9 15 4 3DMU 2 7 5 12 19 9 7DMU 3 5 4 18 21 4 3DMU 4 6 8 14 11 10 6DMU 5 7 7 11 14 8 8DMU 6 8 3 10 17 4 9DMU 7 5 5 16 10 6 5DMU 8 4 9 19 9 5 2Sum 47 49 109 116 50 43

Projection pointsDMU 1 5 8 9 15 4 3DMU 2 7 5 12 19 9 7DMU 3 5 4 18 21 4 3DMU 4 6 8 14 11 10 6DMU 5 7 7 11 14 8 8DMU 6 8 3 10 17 4 9DMU 7 5 5 16 10 6 5DMU 8 4 9 19 9 5 2Sum 392 36 1448 1584 328 232

Table 2 Current and optimized levels of the entire system

Inputs Desirable outputs Undesirable outputsI1 I2 O1 I1 I2 O1

Current level 47 49 109 116 50 43Optimal level 392 36 1448 1584 328 232Rate of reduction or increase 165 265 247 267 344 46

outputs and increase the overall production of desirableoutputs in the direction of (119877119909119894 119877119910

119863

119903 119877119910119880

119905) simultaneously It

should be pointed out that undesirable outputs are consideredas inputs in the proposed model

6 Numerical Example

To illustrate the proposed model (11) consider that a systemconsists of 8 DMUs and that each DMU consumes twoinputs to produce four outputs (twodesirable outputs and twoundesirable outputs) Table 1 shows the data

The efficiency score of the entire system can be readilyobtained by using model (11) which is 48 Moreover theprojection points are shown in Table 1 As can be seenfrom Table 2 we can compare the observed system with theprojected system For instance model (11) suggests 165and 265 saving (reduction) in the first and second inputsrespectively In addition by using model (11) to project allof DMUs onto the efficient frontier DM could have 247and 267 increases in producing the desirable output 1 andoutput 2 respectively

Increasing the production of desirable output 1 from 109(current level) to 1448 (optimum level) and increasing theproduction of desirable output 2 from 116 (current level) to

1584 (optimum level) are meaningful Model (11) also has asignificant reduction plan in both undesirable outputs thatis decreasing the production level of undesirable output 1from 50 to 328 (344 reduction) and decreasing the levelof production of undesirable output 2 from 43 to 232 (46reduction)

7 Conclusion

The issue of dealing with undesirable data in CRA is animportant topicThe existing CRAmodels have been focusedon desirable inputs and outputs In this paper we developedan approach proposed by Lozano and Villa [15] for dealingwith undesirable outputs by using distance directional func-tion The CRA model presented here can be used for theanalysis of any real situations where a significant number ofdesirable and undesirable outputs are included

Moreover the proposed model is able to suggest amanagerial point of view to DM to make decision and comeup with a plan for the system In a similar way the proposedmodel can be reformulated to deal with undesirable inputsrsquotreatment in centralized resource allocation scenario On thebasis of the promising findings presented in this paper workon the remaining issues is continuing and will be presented


in future papers Clearly in our future research we intendto concentrate on CRA model with imprecise interval andfuzzy data


The authors have no conflict of interests to disclose

References


[2] R D Banker A Charnes and W W Cooper ldquoSome methodsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[4] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[5] K Allen ldquoDEA in the ecological context an overviewrdquo in DataEnvelopment Analysis in the Service Sector G Wesermann Edpp 203ndash235 Gabler Wiesbaden Germany 1999

[6] T C Koopman ldquoAnalysis of production as an efficient com-bination of activitiesrdquo in Activity Analysis of Production andAllocation Cowles Commission T C Koopmans Ed pp 33ndash97Wiley New York NY USA 1951

[7] A Iqbal Ali and L M Seiford ldquoTranslation invariance in dataenvelopment analysisrdquoOperations Research Letters vol 9 no 6pp 403ndash405 1990

[8] J T Pastor ldquoTranslation invariance in data envelopment analy-sis a generalizationrdquo Annals of Operations Research vol 66 pp93ndash102 1996

[9] H Scheel ldquoUndesirable outputs in efficiency valuationsrdquo Euro-pean Journal of Operational Research vol 132 no 2 pp 400ndash410 2001

[10] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[11] B Golany and Y Roll ldquoAn application procedure for DEArdquoOmega vol 17 no 3 pp 237ndash250 1989

[12] C A K Lovell and J T Pastor ldquoUnits invariant and translationinvariant DEAmodelsrdquo Operations Research Letters vol 18 no3 pp 147ndash151 1995

[13] H Scheel ldquoEfficiency measurement system DEA for windowsrdquoSoftware Operations Research and Wirtschafts-informatikUniveritat Dortmund 1998

[14] W Liu and J Sharp ldquoDEA models via goal programmingrdquoin Data Envelopment Analysis in the Service Sector G West-ermann Ed pp 79ndash101 Deutscher Universitatsverlag Wies-baden Germany 1999

[15] S Lozano and G Villa ldquoCentralized resource allocation usingdata envelopment analysisrdquo Journal of Productivity Analysis vol22 no 1-2 pp 143ndash161 2004

[16] C Mar-Molinero D Prior M-M Segovia and F Portillo ldquoOncentralized resource utilization and its reallocation by usingDEArdquo Annals of Operations Research 2012

[17] P Korhonen and M Syrjanen ldquoResource allocation based onefficiency analysisrdquoManagement Science vol 50 no 8 pp 1134ndash1144 2004

[18] J Du L Liang Y Chen and G B Bi ldquoDEA-based productionplanningrdquo Omega vol 38 no 1-2 pp 105ndash112 2010

[19] M Asmild J C Paradi and J T Pastor ldquoCentralized resourceallocation BCC modelsrdquo Omega vol 37 no 1 pp 40ndash49 2009

[20] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo JGerami and M R Mozaffari ldquoCentralized resource allocationfor enhanced Russell modelsrdquo Journal of Computational andApplied Mathematics vol 235 no 1 pp 1ndash10 2010

[21] M-M Yu C-C Chern and B Hsiao ldquoHuman resource right-sizing using centralized data envelopment analysis evidencefrom Taiwanrsquos airportsrdquo Omega vol 41 no 1 pp 119ndash130 2013

[22] A Hadi Vencheh R Kazemi Matin and M Tavassoli KajanildquoUndesirable factors in efficiency measurementrdquoAppliedMath-ematics and Computation vol 163 no 2 pp 547ndash552 2005

[23] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoUndesirable inputs and outputs in DEAmodelsrdquo Applied Mathematics and Computation vol 169 no 2pp 917ndash925 2005

[24] J Wu and D Guo ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling vol 58 no 5-6 pp 1102ndash1109 2013

[25] A Amirteimoori S Kordrostami andM Sarparast ldquoModelingundesirable factors in data envelopment analysisrdquo AppliedMathematics and Computation vol 180 no 2 pp 444ndash4522006

Research ArticleThe Integration of Group Technology and SimulationOptimization to Solve the Flow Shop with Highly Variable CycleTime Process A Surgery Scheduling Case Study

T K Wang1 F T S Chan2 and T Yang1

1 Institute of Manufacturing Information and Systems National Cheng Kung University Tainan 70101 Taiwan2Department of Industrial and Systems Engineering The Hong Kong Polytechnic University Hung Hom Hong Kong

Correspondence should be addressed to T Yang tyangmailnckuedutw



Copyright copy 2014 T K Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Surgery scheduling must balance capacity utilization and demand so that the arrival rate does not exceed the effective productionrate However authorized overtime increases because of random patient arrivals and cycle timesThis paper proposes an algorithmthat allows the estimation of the mean effective process time and the coefficient of variation The algorithm quantifies patient flowvariability When the parameters are identified takt time approach gives a solution that minimizes the variability in productionrates and workload as mentioned in the literature However this approach has limitations for the problem of a flow shop with anunbalanced highly variable cycle time process The main contribution of the paper is to develop a method called takt time whichis based on group technology A simulation model is combined with the case study and the capacity buffers are optimized againstthe remaining variability for each group The proposed methodology results in a decrease in the waiting time for each operatingroom from 46 minutes to 5 minutes and a decrease in overtime from 139 minutes to 75 minutes which represents an improvementof 89 and 46 respectively

1 Introduction

Currently the US healthcare system spends more money totreat a given patientwhenever the system fails to provide goodquality and efficient care As a result healthcare spending inthe US will reach 25 trillion dollars by 2015 which is nearly20 of the gross domestic product (GDP) A similar trendis observed by the Organization for Economic Cooperationand Development (OECD) which included Taiwan Thecost of increased healthcare spending will become moreimportant in the coming years One way to decrease the costof healthcare is to increase efficiency

The demand for surgery is increasing at an average rateof 3 per year To increase access operating rooms (ORs)must invest in related training for specialized nursing andmedical staff ORs will be a hospitalrsquos largest expense atapproximately $10ndash30min and will account for more than40 of hospital revenue [1] Two types of surgical services

are provided by ORs reaction to unpredictable events inthe emergency department (ED) and elective cases wherepatients have an appointment for a surgical procedure on aparticular day This paper considers elective cases because animportant part of the variance can be controlled by reducingflow variability [2] The efficiency of ORs not only has animpact on the bed capacity andmedical staff requirement butalso impacts the ED [3] Therefore increasing OR efficiencyis the motivation for this study

Utilization is usually the key performance indicatorfor OR scheduling Maximum productivity requires highutilization However in combination with high variabilityhigh utilization results in a long cycle time according toLittlersquos Law [4] as shown in Figure 1 High utilization andlow cycle times can be achieved by reducing the flowvariability as shown in Figure 2 Therefore the identificationand reduction of the main sources of variability are keys tooptimizing the compromise between throughput and cycle



20 40 60 80 100Utilization ()

Cycle

tim

eIn

crea

sing

Figure 1 Cycle time versus utilization

20 40 60 80 100

Low variability

Utilization ()

High variability

Cycle

tim

eIn

crea

sing

te

to

Figure 2 The corrupting influence of variability

time Unfortunately a few measures for flow variability areused in ORs Such a measure would be highly valuable inreducing variability and would allow more efficient study

The flow variability determines the average cycle timeThere are different sources of variability such as resourcebreakdown setup time and operator availability Anapproach proposed by Hopp and Spearman used the VUTequation to describe the relationship between the waitingtime as the cycle time in queue (CT119902) variability (119881)utilization (119880) and process time (119879) for a single processcenter [5] The VUT is written in its most general form as(1) This study determines the parameters and the solutionsof this equation

CT119902 = 119881119880119879 (1)

This paper is structured as follows The analytical VUTequation is applied to a workstation with real surgicalscheduling dataThe algorithm quantifies the patient flow forthe entire OR system and makes the cycle time longer thanpredicted due to several parameters An example then showsthe potential of the VUT algorithm for use in cycle timereduction programs The solution depends on finding the

parameters that cause the cycle time variability A simulationmodel is used to demonstrate the feasibility of the solutionFinally the main conclusions and some remarks on futurework are given

2 Literature Review

Timeframe-based classification schemes generally includelong intermediate and short term processes as follows(1) capacity planning (2) process reengineeringredesign(3) the surgical services portfolio (4) estimation of theprocedural duration (5) schedule construction and (6)schedule execution monitoring and control [6] This studyfocuses on short-term aspects because the shop floor controlmakes adjustments when the process flow is disrupted bythe variability of patientsrsquo late arrivals surgery durations andresource unavailability in the real world

The sequencing decision which can be thought of as alist of elements with a particular order and its impact on ORefficiency are addressed in the literature [7 8] Most of thestudies use a variety of algorithms to improve the utilizationunder the assumption that the cycle time is determinis-tic Studies developed a stochastic optimization model andheuristics to computeOR schedules that reduce theOR teamrsquoswaiting idling and overtime costs [9 10] Goldman et al[11] used a simulation model to evaluate three schedulingpolicies (ie FIFO longest-case first and shortest-case first)and concluded that the longest-case first approach is superiorto the other two

Scheduling always struggles to balance capacity utiliza-tion and demand in order to let the arrival rate 119903119886 not exceedthe effective production rate 119903119890 [12ndash14] Then the utilizationat each station is given by the ratio of the throughput to thestation capacity (119906 = 119903119886119903119890) Under the assumption that thereis no variability which includes the assumption that casesare always available at their designated start time the surgerydurations are deterministic and resources never break downHowever it is not possible to predict which patients or staffwill arrive late precisely how long a case will take to performor what unexpected problems may delay care [15] This iswhy none of a variety of research models has had widespreadimpact on the actual practice of surgery scheduling over thepast 55 years [6]Therefore this study will consider these flowvariability issues

Studies show that themanagement of variability is criticalto the efficiency of an OR system McManus et al [16] notedthat natural variability can be used to optimize the allocationof resources but no empirical model was included in thestudy Managing the variability of patient flow has an effecton nurse staffing quality of care and the number of inpatientbeds for ED admission and solves the overcrowding problem[17 18] However there is a lack of quantitative analysisto demonstrate which flow variability parameter causes theimpact In summary this study quantitatively analyzes flowvariability determines which parameters have an impact andprovides relevant solutions for empirical illustration

Womack et al [19] stated that high utilization withrelatively low cycle time requires a minimum variability


Although this originates from the Toyota Production System(TPS) its potential applications and in-depth philosophyare not well defined [20] Different industries apply theseprinciples and develop customized approaches to optimizeshop floor processes The methodology of the study refersto Ohno [21] Monden [22] and Liker [23] for details ofdevelopment The five-step process is as follows

The first step defines the current needs for improvementKey performance indicators are selected Performance mea-sures for the OR system fall into two main categories patientwaiting time and staff overtime Patient waiting is associatedwith two activities patients waiting for the preparation of aroom and waiting for surgery There is no waiting time forthe recovery process because recovery begins immediatelyafter surgery Late closure results in overtime costs for nursesand other staff members A reduction in overtime has apositive effect on the quality of care decreases surgeonsrsquo dailyhours produces annualized cost savings makes inpatientbeds available for ED admission and positively affects EDovercrowding [17]

The second step incorporates an in-depth analysis ofthe production line Before starting detailed time studiesstandardmovements are observed andmapped Value streammapping (VSM) is used to design and analyze anORrsquos processlayer [24] VSM has a wide perspective and does not examineindividual processesThe average cycle time is determined byvariability but VSM does not provide quantifiable evidenceand fails to determine how methods can be made moreviable Hopp and Spearman proposed the use of the VUTequation Equation (2) represents the variability as the sumof the squared coefficients of the variation in the interarrivaltimes 1198622

119886 the squared coefficients of the variation in the

effective process time 1198622119890 the utilization 119880 and the squared

coefficients of the variation in departure 1198622119889 The squared

coefficient of variation is defined as the quotient of thevariance and the mean squared Therefore 1198622

119886= 1205902

1198861199052

119886and

1198622

119890= 1205902

1198901199052

119890 where 119905119886 and 119905119890 are the mean interarrival time

and themean process time respectivelyThe effective processtime paradigms 119905119890 and 119862

2

119890 include the effects of operational

time losses due to machine downtime setup rework andother irregularities Compared with the theoretical processtime 119905119900 119905119890 gt 119905119900 and 119862

2

119890gt 1198622

119900 1198622119890is considered low

when it is less than 05 moderate when it is between 05and 175 and high if more than 175 Equation (3) showsthat for low utilization the flow variability of the departingflow equals the variability of the arriving flow and forhigh utilization the flow variability of the departing flowequals the effective process time variability The equationsgive quantifiable evidence of variability

CT119902 = (1198622

119886+ 1198622

119890

2)(

119906

1 minus 119906) 119905119890 (2)

1198622

119889= 11990621198622

119890+ (1 minus 119906

2) 1198622

119886 (3)

The third step consolidates the current performance dataand determines the baseline for efficiency improvementBecause the period of operating time for this study is from

800 am to 500 pm the total overtime after 500 pm asthe baseline per day is 3336 minutes

The fourth step defines implementation methods thatsatisfy the abovementioned subtargets and use the detailedtime studies and data analysis from earlier steps In summary(2) and (3) clearly show the contribution of variability Theleveling approach minimizes the variability in productionrates and work load [25] However a leveling approach thatonly considers a single production level is not applicableto the problem of low volume and high mix production[26] Only a few papers outline leveling approaches for flowshop environments [27] The flow shop with an unbalancedhighly variable cycle time process can be solved by takttime grouping [28] However this method assumes that theprocess time for each batch is the same and is not applicableto this studyThis study uses a newmethod of takt time basedon group technology to implement the flow environment

When all of the improvement items are chosen thefifth step ensures their sustainable implementation Discrete-event simulation is used to model the behavior of a complexsystem By simulating the process the system behavior isobserved and the potential improvements after changes canbe evaluated [29] However grouping and leveling are stillrequired to achieve the optimal solution for a given problem

3 Case Description by the Current-StateVSM and VUT Equation

31 The Current-State VSM The case studied in this paper isfrom aTaiwanesemedical center that has 21350 surgical casesper year The surgical department consists of 24 operatingrooms 15 of which are for specialty procedures In identifyingthe overall flow shop procedure using the current-state VSMwhich includes the processing time for each process boxesare used to understand the type of activities that occur in theORs VSM allows a visualization of the processes for an entireservice rather than just one particular process This result isplotted in Figure 3 The current value stream mapping showsthe cycle time which includes value-added time and non-value-added time The non-value-added time is the waitingtime which is 46 minutes

32 The VUT Equation Analysis To describe the perfor-mance of a single workstation the following parameters areassumed

119905119900 the mean natural process time119903119886 the arrival rate120590119900 the standard deviation for the natural processtime119888119900 the coefficient of variability for the natural processtime119873119904 the average number of cases between setups119905119904 the mean setup time120590119904 the standard deviation for the setup time119905119890 the mean effective process time


Patient

Patient for surgical prep inoperating room

Surgery start suture andfinish

Home

10 min24 min

20 min11 min11 min

98 min0 min

10 min0 min

3 min3 min0 min

10 min

Scheduling system Billing and coding

Waiting time = 46 min

Cycle time = 200 min

Clean room

Cycle time = 3 Cycle time = 10Cycle time = 20

Cycle time = 98 Cycle time = 10 Cycle time = 3 Cycle time = 10

Move in preparative room

ORemergence

time

Patient out of room

Inpatient room

Start anesthesia care(Mj)

(Nj)

WW W W W W

Figure 3 The current-state VSM

1205902

119890 the variance of the effective process time

1198882

119890 the squared coefficient of the variation in the

effective process time1198882

119886 the squared coefficient of the variation in demand

arrivals

The daily surgical scheduling has 80 elective cases onaverage according to the effective capacity from 800 am to500 pm Namely the arrival rate 119903119886 is 89 caseshour Eachpatient will go through the two series of stage (119882119894) whichincluded the process of preparation (1198821) and operation (1198822)For the worst-case example at the starting time patientsmove into the OR system from wards when the operatingroom (1198822) is ready Because the ward and the surgicaldepartment are far from each other the interarrival timeis assumed to be exponential (1198882

119886= 1) The characterizing

flow in the ORsrsquo system passes through the two stages (119882119894)shown in Figure 4 The first stage (1198821) checks the patientrsquosdocumentation nursing history and laboratory data Thenatural process time mean 119905119900 is 20 minutes and the naturalstandard deviation 120590119900 is 2 minutes These result in a naturalCV of 119888119900 = 120590119900119905119900 = 01 The capacity of the preparationroom (119872119895) in the first stage is 12 which is less than the valueof 24 for the second stage (119873119895) and this is so for all casesUsing a dispatching rule of first-come-first-served (FCFS) inthe first stage (1198821) the first stage (1198821) can breakdown undercertain conditions (eg the patient does not arrive at the starttime when the preparation room (119872119895) is ready or when thenumber of patients is greater than 12) These situations arecalled nonpreemptive outages Specifically 1198821 has a meantime to failure (MTTF)119898119891 of 60minutes and amean time to

repair (MTTR)119898119903 of 35 minutes MTTF is the elapsed timebetween failures of a system during operation and MTTR isthe average time required to repair a failed operation Theaverage capacity of 1198821 for nonpreemptive outages can becalculated using (4) where the availability119860 = 60(60 + 35)=

063 The effective mean process time 119905119890 calculated using(5) is 3175 minutes The utilization of the first stage (1198821) iscalculated using (6) to be 027 and 119888

2

119890is calculated using (7)

as 083

119860 =119898119891

119898119891 + 119898119903

(4)

119905119890 =119905119900

119860 (5)

119906 =119903119886

119903119890

=119903119886119905119890

119898 (6)

1198622

119890= 1198622

119900+ 2119860 (1 minus 119860)

119898119903

119905119900

(7)

After the previous patient has left the operating room andfollowing the setup time the current patient then starts atthe second stage (1198822) Both the process time and setup timeare stochastic and will be commensurate with the complexityof the disease The natural mean process time 119905119900 is 12017minutes and the natural standard deviation 120590119900 is 8025minutes The setup time is regarded as a preemptive outagewhen they occur due to changes in the following surgeryTrends in the setup time are associated with the type ofsurgery and the mean of the setup time 119905119904 is 2526 minutesand the standard deviation of the setup time 120590119904 1543minutes


Specialty 1dispatchqueue



Specialty 1

FCFS

Specialty 2

Specialty 15

T dayward Preparative room

Operating room

Recoverroom

First stage (W1)

Second stage (W2)

(Mj)

(Nj)

M1

M2

M12

N1

N2

N3

N4

N5

N24

Figure 4 The charactering flow in the ORsrsquo system

The effective mean process time 119905119890 from (8) is 14543 minutesThe capacity is 99 caseshour The utilization of1198822 by (6) is089 Using (9) we can compute 1198882

119890= 749 From the VUT

equation we conclude that this is a stable system in the flowshop with an unbalanced high variation cycle time processConsider

119905119890 = 119905119900 +119905119904

119873119904

(8)

1205902

119890= 1205902

119900+1205902

119904

119873119904

+119873119904 minus 1

1198732119904

1199052

119904

1198882

119890=1205902

119890

1199052119890

(9)

33 The Baseline for Efficiency Improvement The third stepconsolidates the current performance data and determinesthe baseline for efficiency improvement Then the VUTequation for computing queue time CT119902 of 1198821 is 1081minutes and 1198882

119889is 099 however CT119902 of 1198822 is 76474minutes

After analysis of the VUT (2) we found that the relativedifferences among the mean of the effective process time 119905119890and utilization compared to the variability are small Thevalue of 424 comes from two parts the first is 1198882

119890= 749

which is highly variable based on the process time in thesecond stage (1198822) the second is 1198882

119886= 099 which is equal

to 1198882119889from the first stage (1198821) The departure variability of

1198822 depends on the arrival variability of1198821 The 1198882119890= 083 in

the1198821 due to the nonpreemptive outages which are causedby the interarrival rate from the inpatient ward to the ORsrsquosystem Equations (2) and (3) provide useful models for a

deeper understanding of the worst case of natural and flowvariability when access to resources is limiting In practicebalancing the average utilization and the systemic stressesresults in a smoother patient flow Consider

CT119902 =1198622

119886+ 1198622

119890

2

119906

1 minus 119906119905119890

=(099 + 749)

2(

089

1 minus 089) 14543

= (424) (809) (14543)

(10)

These are some assumptions in this case study

(i) The data in analysis of surgical-specific proceduretime is the year of 2002

(ii) Each preparation room (119872119895) and operating room(119873119895) can process only one case at a time

(iii) For this study there should be totally 24 rooms strictlyassigned to the different surgical cases Each case canbe carried out in any of the 24 rooms but each roommust be assigned one group at most

(iv) The period of opening of operating room is from 800am to 500 pm and the overtime is counted after500 pm

(v) Emergency surgeries are not considered Eitherpatients must have appointments on certain OR daysfor a medical reason or any period during whichsurgeons cannot perform is ignored In other wordsno surgeries are cancelled or added


(vi) There is no constraint to surgeons or other staff avail-ability In other words surgeons are available at anyperiod of the day (ie when a case is moved from themorning to the afternoon)

(vii) Each physician can only accept one patient at a timeOnce the surgery is started the operation is notallowed to be interrupted or cancelled Surgical break-downs are not considered

4 Proposed Methodology

The fourth step defines implementation methods that satisfythe abovementioned subtargets and uses the detailed timestudies and data analysis from earlier steps Leveling basedon group technology consists of two fundamental stepsIn the first step families are formed for leveling based onsimilarities Clustering techniques are used to group familiesaccording to their similarities Using these families a levelingpattern is created in the second step Every family and everyinterval is arranged for a monthly period

41 Group Technology Approach It has been shown thatvariability affects the efficiency of the system Groupingsurgeries minimizes the duration variability of surgery [30]Of these approaches cluster analysis is the most flexible andtherefore the most reasonable method to employ here K-means is a well-known and widely used clustering method[31] This method is fast but cannot easily determine thenumber of groups If the group is arranged randomly therewill be no obvious difference between each group Anderberg[32] recommended a two-stage cluster analysis methodologyWardrsquos minimum variance method is used at first followedby the K-means method This is a hierarchical process thatforms the initial clusters Wardrsquos method can minimize thevariance through merging the most similar pair of clustersamong119873 elements Perform those steps until all clusters aremerged The Ward objective is to find out the two clusterswhose merger gives the minimum error sum of squares Itdetermines a number of clusters and then starts the next stepK-means clustering uses the coefficient of variation which isdefined as the ratio of the standard deviation to the meanas measured by (11) The software SPSS was used for clusteranalysis Consider

Coefficient of variation = 120590

120583 (11)

42 Takt Time Approach Leveling allocates the volume andvariety of surgeries among the ORsrsquo resources to fulfill thepatient demand over a defined period of time The first stepin leveling is to calculate the takt time which is measuredby (12) The takt time is a function of time that determineshow fast a process must run to meet customer demand [28]The second step is a pacemaker process selection and levelingof production by both volume and product mix [33] Thepacemaker process must be the only scheduling point inthe production system and dictates the production rhythmfor the rest of the system where the pace is based on a

supermarket pull system further upstream from this point aswell as First In First Out (FIFO) systems further downstream[34ndash37] According to the theory of constraints (TOC) oneof the most important points to consider is the bottleneckThus the pacemaker process selection must be located inthe second stage (1198822) However the number of resources foreach groupingmust still be determined to achieve the optimalsolution for a given problem Consider

Takt time =Available monthly work timeTotal monthly volume required

(12)

43 Simulation Modeling and Optimization The fifth stepensures sustainable implementation The simulation toolchecks the feasibility of integrating the methods into thecurrent system Simulation is useful in evaluating whetherthe implementation of the method is justified [38] RockwellArena a commercial discrete-event simulator has been usedfor many studies [39] To evaluate potential improvementsdue to the implementation of takt time based on grouptechnology Rockwell Arena 1351 was used to build thegeneral simulation model for the OR system Depending onthe nature and the goal of the simulation study it is classifiedas either a terminating or a steady-state simulationThis studyis a terminating simulation which signifies that the systemhas starting and stopping conditions [40]

This study optimizes the capacity buffers against theremaining variability of each surgical group to minimize ORovertime (ie work after 500 pm) Optimization finds thebest solution to the problem that can be expressed in theform of an objective function and a set of constraints [41]Therefore the difference between the model that representsthe system and the procedure that is used to solve theoptimization problems is defined within this model Theoptimization procedure uses the outputs from the simulationmodel as an input and the results of the optimization arefed into the next simulation This process iterates untilthe stopping criterion is met The interaction between thesimulation model and the optimization is shown in Figure 5[42]

5 Empirical Results

51 Takt Time Based on a Group Technology Approach Clus-tering Method This study focuses on 263 surgical-specificprocedures using a Pareto analysis of a total of 1198 typesof surgical-specific procedure times in the year 2002 Wardrsquosminimum variance method gives the number of clustersas 5 The following step is segmented into 5 groups basedon Wardrsquos minimum variance method and then K-meansclustering to give the time expression shown in Table 1

52 Takt Time Mechanism Leveling is used to calculate thetakt time for each surgery group The surgical departmentorganizes the working time according to a monthly timeschedule The monthly time available is 10800 minutes asthere are 9 hours a day and 5 days in a week in this case Themonthly volume was measured and the takt time for eachgroup is shown in Table 2


Table 1 The five groups

Categories 1 2 3 4 5Expression minus0001 + ERLA (287 2) minus0001 + LOGN (119 226) 5 + WEIB (91 0856) 5 + WEIB (162 12) 5 + GAMM (943 151)

Table 2 The monthly volume and takt time of each group

Group Monthly time available (minutes) Monthly volume of surgeries (units) Takt time (minutes)

1 10800 813 10800

813≒ 13

2 10800 159 10800

159≒ 68

3 10800 134 10800

134≒ 81

4 10800 346 10800

346≒ 31

5 10800 185 10800

185≒ 58

Input

Output

Optimizationprocedure

Simulationmodel

Figure 5 Relationship between simulationmodel and optimization

53 Simulation Model Rockwell Arena 1351 was used tobuild the simulation model that represents the OR systemsThe computer-based module logic design establishes anexperimental platform that allows a decisionmaker to quicklyunderstand the conditions of the system

When the simulation model is constructed we wantedto tighten precision cover on the population mean (119906) thesmaller the confidence interval the larger the number ofrequired simulation replications The length of one replica-tion is set as one month The coefficient of variation (CV)which is defined as the ratio of the sample standard deviationto the sample mean is used as an indicator of the magnitudeof the variance The value of the CV stabilizes when thenumber of replications reaches 35 as shown in Figure 6 [43]We generated the input values from probability distributionsin Arena The simulation model used the time expressionwith the run length of 1 month and 35 replications Eachreplication starts with a both empty and idle system Theindividual replication result is independent and identicallydistributed (IID) we could form a confidence interval forthe true expected performance measure 120583 In this study themean daily cycle time (120583) and the 95 confidence intervalare adopted as the system performance measure We have aninitial set of replications 35 we compute a sample averagecycle time 21428 minutes and then a confidence intervalwhose half width is 192 minutes It is noted that the halfwidth of this interval (192) is pretty small compared to thevalue of the center (21428) The mathematical basis for theabove discussion is that in the 95 of the cases of making 35simulation replications as we did the interval formed like thiswill contain the true expected value of total population

Table 3 The error between the real system and simulation

Compare (average) System Simulation Error ()Waiting time 4614 4310 7

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

0020018001601400120010008000600040002

0

CV

Number of replications

WIP

Figure 6 The CV chart

In this study simulation models for verification andvalidation are both used Verification ensures that the modelbehaves as intended and validation ensures that the modelbehaves like the real system As shown in Table 3 the errorbetween the simulation and the real system in terms of thedaily waiting time in each OR is 7

54 The Optimal Solution Identification of the optimal sce-nario uses one week in July which in practice is usually 5days On each day each group 119894 is available and has anexpression time OptQuest is utilized in conjunction withArena to provide the optimal solutionThe required notationsfor the formulation are defined as follows

Parameters

119894 = an index for the groups of surgeries 119894 isin 119868119868 = 1 2 3 4 5119895 = an index for the number of operating rooms119895 isin 119869 119869 = 1 2 3 24


Patient

Start anesthesia carein preparative room

Patient ready for surgical prep in operating room Surgery start

suture and finish

Home

Leveling

Billing and coding

Waiting time= 5 min


Patient out of roomand clean room

Emergency time in recovery

Inpatient room

W W W W

Dispatchqueue (Mj) (Nj)

Cycle time = 10Cycle time = 10 Cycle time = 10

Cycle time = 20 Cycle time = 98

10 min 10 min20 min0 min 0 min

10 min0 min

98 min0 min5 min

Figure 7 The future-state VSM

Intermediate variables

119874119895 = the overtime associated with the ORs

Decision variables

119860 119894119895 = a binary assignment whether the surgerygroup 119894 is assigned to operating room 119895 (119860 119894119895 =1) or not (119860 119894119895 = 0)119862119894 = an index for the number of operating roomsthat are allocated to the surgery group 119894

The optimization model solves

Minimize24

sum

119895=1

119874119895 (13)

subject to the following constraints

5

sum

119894=1

119860 119894119895 = 1 forall119895 (14)

119862119894 ge 1 forall119894 (15)

5

sum

119894=1

119862119894 = 24 (16)

119860 119894119895 isin 0 1 forall119894119895 (17)

The objective function minimizes the total amount ofovertime Constraint (14) specifies that each operating roommust be assigned to one group at most Constraint (15)ensures that each group is allocated at least in one operatingroom Constraint (16) sets the limitation of operating roomsfor all groups Constraint (17) as a binary assignment iswhether the surgery group 119894 is assigned to operating room119895

55 The Result The results are plotted in Figure 7 Thecapacity buffers optimized against the remaining variabilityof each group are 1198621 = 2 1198622 = 2 1198623 = 8 1198624 = 9 and1198625 = 3 In the optimized solution the computational resultsshow that the waiting time and overtime for each operationroom decrease from 46 minutes to 5 minutes and from 139minutes to 75 minutes respectively which is a respectiveimprovement of 89 and 46 as shown in Table 4

56 Conclusions and Further Research Maximizing the effi-ciency of the OR system is important because it impacts theprofitability of the facility and the medical staff OR schedul-ing must balance capacity utilization and demand so that thearrival rate 119903119886 does not exceed the effective production rate119903119890 However authorized overtime is increasing due to therandomness of patient arrivals and cycle times This paperdiffers from the existing literature and makes a number ofcontributions It focuses on shop floor control and uses aVUT algorithm that quantifies and explains flow variabilityWhen the parameters are identified the impact on the


Table 4 Optimal results

Overtime per operating room (minute) Waiting time (minute) Cycle time (minute)Average Standard deviation Average Standard deviation Average Standard deviation

Original system 139 26 46 16 200 22Optimal solution 75 2 5 1 153 2Improvement () 46 89 24

surgery schedule using leveling based on group technologyis illustrated A more robust model of surgical processesis achieved by explicitly minimizing the flow variability Asimulation model is combined with the case study to opti-mize the capacity buffers against the remaining variability ofeach group The computational result shows that overtime isreduced from 139 minutes to 75 minutes per operating room

The most significant managerial implications can besummarized as follows

(i) To achieve a higher return on investment highutilization and reasonable cycle times which dependon the level of variability are necessary The identifi-cation and reduction of themain sources of variabilityare keys to optimizing the performance instead ofutilization

(ii) This study solves OR scheduling using various heuris-tic methods and provides the anticipated start timesfor each case and each operating room Howevermost real cases violate the assumptions (eg allcases are not ready at the start time cycle times arestochastic and resources do not break down etc)The schedule cannot be accurately predicted once theassumptions are violated

(iii) Sequencing patients using takt time based on grouptechnology reduces the flow variability and waitingtime by 89

(iv) The empirical illustration shows that natural variabil-ity is prevented by optimizing the capacity buffers andreducing overtime by 46

In practice there are additional constraints that affect theresults and these require further study

(i) Although the duration of surgery is analyzed for 263types of surgical categories and for 340 surgeons eachhospital is different For example some hospitals havea higher proportion of complex surgeries and shouldmake comparisons among institutions

(ii) The tests ofmodel accuracy were performed using theyear of 2002 they do account for diurnal variationHowever the year variation should be reflected

(iii) Additional constraints may arise due to the availabil-ity of surgeons or other staff For example surgeonsmay not be available when the case is moved fromthe morning to the afternoon because they haveoutpatient clinics or other obligations

(iv) This study applies to facilities at which the surgeonand patient choose the day and the case is not allowedto be allocated to another day even if performancemay be increased by rescheduling

(v) Additional constraints may arise due to the availabil-ity of the recovery room



Acknowledgments

Thework described in this paper was substantially supportedby a grant from The Hong Kong Polytechnic UniversityResearch Committee under the Joint Supervision Schemewith the Chinese Mainland and Taiwan andMacao Universi-ties 201011 (Project no G-U968)This workwas also partiallysupported by the National Science Council of Taiwan underGrant NSC-101-2221-E-006-137-MY3

References

[1] L R Farnworth D E Lemay T Wooldridge et al ldquoA com-parison of operative times in arthroscopic ACL reconstructionbetween orthopaedic faculty and residents the financial impactof orthopaedic surgical training in the operating roomrdquo TheIowa Orthopaedic Journal vol 21 pp 31ndash35 2001

[2] J Belien E Demeulemeester and B Cardoen ldquoA decisionsupport system for cyclic master surgery scheduling withmultiple objectivesrdquo Journal of Scheduling vol 12 no 2 pp 147ndash161 2009

[3] E Litvak M C Long A B Copper and M L McManusldquoEmergency department diversion causes and solutionsrdquo Aca-demic Emergency Medicine vol 8 no 11 pp 1108ndash1110 2001

[4] J D C Little ldquoLittlersquos Law as viewed on its 50th anniversaryrdquoOperations Research vol 59 no 3 pp 536ndash549 2011

[5] W J Hopp and M L Spearman Factory Physics McGraw-HillEducation Boston Mass USA 3rd edition 2011

[6] J H May W E Spangler D P Strum and L G VargasldquoThe surgical scheduling problem current research and futureopportunitiesrdquoProduction andOperationsManagement vol 20no 3 pp 392ndash405 2011

[7] B Denton J Viapiano and A Vogl ldquoOptimization of surgerysequencing and scheduling decisions under uncertaintyrdquoHealthCare Management Science vol 10 no 1 pp 13ndash24 2007

[8] B Cardoen E Demeulemeester and J Belien ldquoOptimizing amultiple objective surgical case sequencing problemrdquo Interna-tional Journal of Production Economics vol 119 no 2 pp 354ndash366 2009


[9] B T Denton A S Rahman H Nelson and A C BaileyldquoSimulation of a multiple operating room surgical suiterdquo inProceedings of the Winter Simulation Conference pp 414ndash424Monterey Calif USA December 2006

[10] M Lamiri X Xie and A Dolgui ldquoA stochastic model foroperating room planning with elective and emergency demandfor surgeryrdquo European Journal of Operational Research vol 185no 3 pp 1026ndash1037 2008

[11] J Goldman H A Knappenberger and E W Moore Jr ldquoAnevaluation of operating room scheduling policiesrdquo HospitalManagement vol 107 no 4 pp 40ndash51 1969

[12] E Marcon S Kharraja and G Simonnet ldquoThe operatingtheatre planning by the follow-up of the risk of no realizationrdquoInternational Journal of Production Economics vol 85 no 1 pp83ndash90 2003

[13] D Gupta and B Denton ldquoAppointment scheduling in healthcare challenges and opportunitiesrdquo IIETransactions vol 40 no9 pp 800ndash819 2008

[14] Y-J Chiang and Y-C Ouyang ldquoProfit optimization in SLA-aware cloud services with a finite capacity queuing modelrdquoMathematical Problems in Engineering vol 2014 Article ID534510 11 pages 2014

[15] M D Basson T W Butler and H Verma ldquoPredicting patientnonappearance for surgery as a scheduling strategy to optimizeoperating room utilization in a Veteransrsquo Administration Hos-pitalrdquo Anesthesiology vol 104 no 4 pp 826ndash834 2006

[16] M L McManus M C Long A Cooper et al ldquoVariabilityin surgical caseload and access to intensive care servicesrdquoAnesthesiology vol 98 no 6 pp 1491ndash1496 2003

[17] E Litvak ldquoOptimizing patient flow by managing its variabilityrdquoin Front Office to Front Line Essential Issues for Health CareLeaders pp 91ndash111 Joint Commission Resources OakbrookTerrace Ill USA 2005

[18] E Litvak P I Buerhaus F Davidoff M C Long M LMcManus and D M Berwick ldquoManaging unnecessary vari-ability in patient demand to reduce nursing stress and improvepatient safetyrdquo Joint Commission Journal on Quality and PatientSafety vol 31 no 6 pp 330ndash338 2005

[19] J P Womack D T Jones and D Roos The Machine thatChanged The World Free Press New York NY USA 1990

[20] M Holweg ldquoThe genealogy of lean productionrdquo Journal ofOperations Management vol 25 no 2 pp 420ndash437 2007

[21] T Ohno Toyota Production System Beyond Large-Scale Produc-tion Productivity Press New York NY USA 1988

[22] Y Monden Toyota Production System An Integrated Approachto Just-in-Time CRS Press Florida Fla USA 4th edition 1998

[23] J K LikerThe Toyota Way 14 Management Principles from theWorldrsquos Greatest Manufacturer McGraw- Hill Education NewYork NY USA 2004

[24] J-C Lu T Yang and C-Y Wang ldquoA lean pull systemdesign analysed by value stream mapping and multiple criteriadecision-making method under demand uncertaintyrdquo Interna-tional Journal of Computer IntegratedManufacturing vol 24 no3 pp 211ndash228 2011

[25] J Miltenburg ldquoLevel schedules for mixed-model assembly linesin just-in-time production systemsrdquo Management Science vol35 no 2 pp 192ndash207 1989

[26] N Boysen M Fliedner and A Scholl ldquoThe product ratevariation problem and its relevance in real world mixed-modelassembly linesrdquo European Journal of Operational Research vol197 no 2 pp 818ndash824 2009

[27] P R McMullen ldquoThe permutation flow shop problem with justin time production considerationsrdquo Production Planning andControl vol 13 no 3 pp 307ndash316 2002

[28] M A Millstein and J S Martinich ldquoTakt Time Groupingimplementing kanban-flow manufacturing in an unbalancedhigh variation cycle-time process with moving constraintsrdquoInternational Journal of Production Research 2014

[29] P T Vanberkel and J T Blake ldquoA comprehensive simulation forwait time reduction and capacity planning applied in generalsurgeryrdquo Health Care Management Science vol 10 no 4 pp373ndash385 2007

[30] E Hans G Wullink M van Houdenhoven and G KazemierldquoRobust surgery loadingrdquo European Journal of OperationalResearch vol 185 no 3 pp 1038ndash1050 2008

[31] Y Yin I Kaku J Tang and J M Zhu Data Mining ConceptsMethods and Applications in Management and EngineeringDesign Springer London UK 2011

[32] M R Anderberg Cluster Analysis for Applications AcademicPress New York NY USA 1973


[34] M Rother and J Shook Learning to See Value StreamMappingto Add Value and Eliminate Muda Lean Enterprise InstituteCambridge Mass USA 2003

[35] T Yang C-H Hsieh and B-Y Cheng ldquoLean-pull strategy in are-entrant manufacturing environment a pilot study for TFT-LCD array manufacturingrdquo International Journal of ProductionResearch vol 49 no 6 pp 1511ndash1529 2011

[36] J-C Lu T Yang and C-T Su ldquoAnalysing optimum pushpulljunction point location using multiple criteria decision-makingformultistage stochastic production systemrdquo International Jour-nal of Production Research vol 50 no 19 pp 5523ndash5537 2012

[37] T Yang Y F Wen and F F Wang ldquoEvaluation of robustnessof supply chain information-sharing strategies using a hybridTaguchi and multiple criteria decision-making methodrdquo Inter-national Journal of Production Economics vol 134 no 2 pp458ndash466 2011

[38] R B Detty and J C Yingling ldquoQuantifying benefits of con-version to lean manufacturing with discrete event simulationa case studyrdquo International Journal of Production Research vol38 no 2 pp 429ndash445 2000

[39] J Banks J S Carson B L Nelson and D M Nicol Discrete-Event System Simulation Prentice Hall New Jersey NJ USA2000

[40] W D Kelton R P Sadowski and N B Swets Simulationwith Arena McGraw-Hill Education Boston Mass USA 5thedition 2010

[41] E Erdem X Qu and J Shi ldquoRescheduling of elective patientsupon the arrival of emergency patientsrdquo Decision SupportSystems vol 54 no 1 pp 551ndash563 2012

[42] F Glover J P Kelly and M Laguna ldquoNew advances andapplications of combining simulation and optimizationrdquo inProceedings of the 28th Conference on Winter Simulation pp144ndash152 Coronado Calif USA December 1996

[43] T Yang H-P Fu and K-Y Yang ldquoAn evolutionary-simulationapproach for the optimization of multi-constant work-in-process strategymdasha case studyrdquo International Journal of Produc-tion Economics vol 107 no 1 pp 104ndash114 2007







Editorial Board










Contents




























Acknowledgments




Chia-Yu Hsu







1 Introduction














2 Related Work




(a) (b) (c)

(d) (e) (f)



3 Proposed Approach






119898119896119873119892]



]


120579119897119873119892]


119881119897119873119892]


119897= [1205951198971

1205951198972 120595119897119873119892]


119892= [1205951198921 1205951198922

120595119892119873119892]


Consensuspartition


WBMs

1 clustering

q clustering

2 clustering



Labels120587 q










1205871

1205872

1205873

x1

1 1 1x2

1 1 1x3

1 1 1x4

2 2 1x5

2 2 2x6

3 1 2x7

3 1 2x8

3 1 2



V2

V3

V4

V5

V6

V7

V8

119867(1)

ℎ11

1 1 1 0 0 0 0 0ℎ12

0 0 0 1 1 0 0 0ℎ13

0 0 0 0 0 1 1 1

119867(2) ℎ

211 1 1 0 0 1 1 1

ℎ22

0 0 0 1 1 0 0 0

119867(3) ℎ

311 1 1 1 0 0 0 0

ℎ32

0 0 0 0 1 1 1 1


119872(119910119903) =

119873119887

sum

119903=1

119890minus119898120573119889(119910119903 119910119904) 119903 = 1 2 3 119873119887 (1)

where

120573 = (119889 (119910119903 minus 119910wc)

119873119887

)

minus1

(2)







(a) Binary value

51015202530






SSE =

119873119888

sum

119896=1

sum

x119894isin119862119896(x119894 minusm119896)

2 (3)






119869119890 =sum119873119888


119870 (5)

In addition

119889max (p119894Z119897) = max119896=12119873119888

[[

[

sum

forallx119894isin119862119897119896

119889 (x119894m119897119896)119899119897119896

]]

]

(6)



[119889 (m119897119906m119897V)] (7)





119894


119896


119894Z119897)


119901119906(120595119897(119905) minusΘ

119897(119905)) + 119888

119892119906(120595119892(119905) minusΘ

119897(119905)) (i)

Θ119897(119905 + 1) = Θ

119897(119905) + V

119897(119905 + 1) (ii)










1012

1518

2315221370

1184 1098945

02004006008001000120014001600

0

5

10

15

20

25

03 04 05 06 07

SSE

Clus

ter n

umbe

r









max 1 1198861

04119888119901

2 1198862

03119888119892

2 1198863

03Iteration 500



Hsu and Chien [8] 1184 1192 1203 1248 1322


KB 2889 3092 3003 4083 3570KC 3331 2490 2603 3169 2603PB 5893 3601 6566 5839 6308PC 4627 4873 3330 3787 6112

Clusteringensemble






Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9

Group 10

Group 11

Group 12

Group 13

Group 14

Group 15





119879119890

(8)



(a) (b) (c)

(d) (e) (f)




Hsu and Chien [8] 04 04 04 04 04


KB 10 10 10 10 10KC 10 10 10 10 10PB 10 10 10 10 10PC 10 10 10 10 10

Clusteringensemble





precision =TP

TP + FP

recall = TPTP + FN

(9)




=2TP

FP + FN + 2TP (10)






Precision

A 070 084 092 092 092 098B 050 066 096 092 062 096C 060 064 100 100 060 100D 070 098 092 092 098 100E 060 094 082 082 098 098F 080 098 076 076 098 098

Avg 065 084 090 089 085 098



Recall

A 100 100 100 093 100 100B 100 097 07 078 083 100C 100 094 067 084 067 097D 100 081 100 100 100 100E 100 079 100 100 100 100F 100 100 100 100 100 100

Avg 100 092 090 093 092 100



119865-measure

A 082 091 096 092 096 099B 067 079 081 084 071 098C 075 076 08 091 063 098D 082 089 096 096 099 100E 075 086 090 090 099 099F 089 099 086 086 099 099

Avg 078 087 088 090 088 099



5 Conclusion









Acknowledgment


References







































1 Introduction





























119903119894119895 = 05 (1 + log9119886119894119895) (1)









119903119894119895 =119894 minus 119895 + 1


(2)

119903119894119895 = 1 minus 119903119895119894 forall119894 lt 119895 (3)


According to (2)

11990321 = 1 minus 03 = 07

11990331 = 15 minus 03 minus 06 = 06


11990332 = 1 minus 06 = 04

11990342 = 15 minus 06 minus 08 = 01

11990343 = 1 minus 08 = 02

(4)

According to (3)

11990313 = 1 minus 06 = 04

11990314 = 1 minus 03 = 07

11990324 = 1 minus 01 = 09

(5)


PR =[[[

[

05 03 04 07

07 05 06 09

06 04 05 08

03 01 02 05

]]]

]

(6)





119878119886119887 = 1 minus

10038161003816100381610038161198861 minus 11988711003816100381610038161003816 +

10038161003816100381610038161198862 minus 11988721003816100381610038161003816 +

10038161003816100381610038161198863 minus 11988731003816100381610038161003816 +

10038161003816100381610038161198864 minus 11988741003816100381610038161003816

4 (7)


AM =

[[[[

[


119878119886119887

1198781198641 1198781198642 sdot sdot sdot 1

]]]]

]

(8)



AA119886 =1

119864 minus 1

119864

sum

119887=1119886 =119887

119878119886119887 forall119886 (9)


RA119886 =AA119886

sum119864

119886=1AA119886

forall119886 (10)










(12)


1

119894119895 1199032

119894119895 1199033

119894119895 1199034

119894119895)





119903119894119895 =1199031

119894119895+ 2 (119903

2

119894119895+ 1199033

119894119895) + 1199034

119894119895

6 (13)


PR119896 =[[[[

[


1

1199031198731 1199031198732 sdot sdot sdot 1

]]]]

]

(14)






120575119894119896 =

119873

sum

119895=1119895 =119894

119903119894119895 minus

119873

sum

119895=1119895 =119894

119903119895119894 forall119894 119896 (15)


DM =

[[[[

[


1205751198731 1205751198732 sdot sdot sdot 120575119873119872

]]]]

]

(16)




aw119896 =(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119896 minus 120575119895119896

10038161003816100381610038161003816)1(119901minus1)

sum119872

119897=1(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119897 minus 120575119895119897

10038161003816100381610038161003816)1(119901minus1)

forall119896 119901 gt 1 (17)

aw119896 =(sum119873

119894=11205752

119894119896)12(119901minus1)

sum119872

119897=1(sum119873

119894=11205752119894119897)12(119901minus1)

forall119896 119901 gt 1 (18)



119872



119862119894 =

119872

sum

119896=1

aw119896120575119894119896 119894 = 1 2 119873 (19)





transformation


relations










4 Numerical Example





11987711 =[[[

[




]]]

]

11987721 =[[[

[




]]]

]

11987731 =

[[[[[[

[

minus1

3minus minus

minus minus1

4minus

minus minus minus 1


]]]]]]

]

(20)


11987711 =[[[

[





]]]

]

11987721 =[[[

[





]]]

]

11987731 =

[[[[[[

[

minus1

31

31

31

3minus minus

minus minus1

41

41

41

4minus



]]]]]]

]

(21)


11987712 =[[[

[





]]]

]

11987722 =[[[

[





]]]

]


11987732 =

[[[[[[

[

minus1

41

41

41

4minus minus


minus minus minus1

31

31

31

3


]]]]]]

]

(22)


then become 119877101584031

and 1198771015840


replaced by 119877101584031and 1198771015840


1198771015840

31=

[[[

[





]]]

]

1198771015840

32=

[[[

[





]]]

]

(23)



PR1 =[[[

[





]]]

]

PR2 =[[[

[





]]]

]

(24)



2are

PR10158401=

[[[

[

0500 0290 0100 0100

0710 0500 0311 0311

0900 0689 0500 0500

0900 0689 0500 0500

]]]

]

PR10158402=

[[[

[

0500 0218 0265 0055

0782 0500 0547 0337

0735 0453 0500 0290

0945 0663 0710 0500

]]]

]

(25)


1and PR1015840

2are consistent


DM =[[[

[

minus2019 minus1923

minus0336 0331

1178 minus0045

1178 1637

]]]

]

(26)




11990312

11990323

11990334


119886119887)

11987812

0925 0775 100011987813

0900 0880 090011987823


119886)

AA1 0913 0828 0950AA2 0900 0730 0950AA3 0888 0783 0900


RA1 0338 0354 0339RA2 0333 0312 0339RA3 0329 0334 0321


120573 = 05

CC1 0319 0327 0320CC2 0317 0306 0320CC3 0364 0367 0361


23= (022 028 034 041) 119903

34= (040 047 053 060)


23= 0311 119903

34= 0500



Very low

02 04 06 1008

02

04

06

08

10

Mem

bers

hip

valu

e



Very low

More highMore

low

02

04

06

08

10

Mem

bers

hip

valu

e




Alternative 119894 120575119894119896

119862119894(ADM) 119862



aw119896by SDM 0509 0491





5 Conclusion











Acknowledgments


References


















































1 Introduction

















(i) Supplier Side





(iii) Retailer Side



22 Assumptions












3 Model Formulation




Repaired products

Defective products

Normal products











=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]

(1)




nQ


nQD

nQD

nQD

nQ

nQ


Z(1 minus R)nQ


nZQ


nQ


P

Q

t

t

t

Q

Q

Q

Q

P

QD

QD (n minus 1)Q

D

nRZQ




119885(1 minus 119877)]119876119894




=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)


2

2119899 [1 minus 119885 (1 minus 119877)]119876119894

(2)




nQ


Lm =nQ

D

X

Q

t

Interest income

Figure 3 119871119898lt 119883

Lm =nQ

D

nQ


X

Q

Interest income

Opportunity cost

t

Figure 4 119871119898ge 119883



(ii) Ordering cost = 119860119898119863119899119894119876119894










TP1198981





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


119876119894

2)

(3)



2 [1 minus 119885 (1 minus 119877)] (4)





TP1198982





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)


minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(5)




22[1 minus 119885(1 minus 119877)]119899119894119876119894



TP1198983





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119894119876119894 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus 119862119898119868mp (119863119884 minus119876119894

2)

(6)



119877)]119899119894119876119894





TP1198984





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119876 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus119862119898119868mp (119863119884)

2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(7)




(ii) Ordering cost = 119860119903119863119899119894119876119894






TP1199031



= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

+ 119875119903119868re (119863119884 minus119876119894

2)





TP1199032



+ interest income

= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

minus119862119903119868rp (119876119894 minus 119863119884)

2

2119876119894

+119875119903119868re (119863119884)

2

2119876119894

(9)


EJTP119894 (119876119894 119899119894)

=





(10)

whereEJTP1 (1198761 1198991)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198761 [1198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198991


ℎ1199031198761

2minus

ℎ11990411986311989911198761

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198761

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989911198761)

2 [1 minus 119885 (1 minus 119877)]

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198761

2)

EJTP2 (1198762 1198992)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198762 [1198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198992


ℎ1199031198762

2minus

ℎ11990411986311989921198762

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989921198762)

2 [1 minus 119885 (1 minus 119877)]

+(119862119903119868me minus 119862119903119868rp) (1198762 minus 119863119884)

2

21198762

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198762

EJTP3 (1198763 1198993)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198763 [1198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198993


ℎ1199031198763

2minus

ℎ11990411986311989931198763

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198763

2)

EJTP4 (1198764 1198994)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]


minus ℎ1198981198631198764 [1198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198994


ℎ1199031198764

2minus

ℎ11990411986311989941198764

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198764

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119898119868se minus 119862119898119868mp) (11989941198764 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119903119868me minus 119862119903119868rp) (1198764 minus 119863119884)

2

21198764

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198764

(11)



120597EJTP1 (1198761 1198991)1205971198761

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198762

1

minus ℎ1198981198631198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198991


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp)

2

(12)

120597EJTP2 (1198762 1198992)1205971198762

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

2

minus ℎ1198981198631198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198992


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987622

+(119862119903119868me minus 119862119903119868rp) [119876

2

2minus (119863119884)

2]

211987622

(13)

120597EJTP3 (1198763 1198993)1205971198763

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198762

3

minus ℎ1198981198631198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198993

2119875 [1 minus 119885 (1 minus 119877)]2

minus(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

+(119862119898119868se minus 119862119898119868mp) [(11989931198763)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

minus(119875119903119868re minus 119862119898119868mp)

2

(14)

120597EJTP4 (1198764 1198994)1205971198764

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198762

4

minus ℎ1198981198631198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198994


2

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987624

+(119862119898119868se minus 119862119898119868mp) [(11989941198764)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

+(119862119903119868me minus 119862119903119868rp) [119876

2

4minus (119863119884)

2]

211987624

(15)


102

1025

103

1035

104

1045

105

2295 2305 2315 2325 2335 2345 2355

Q1



0

100

200

300

400

500

600

700


Q2


777879808182838485

235 236 237 238 239 240

Q3

P = 1100

P = 1200

P = 1300



0

200

400

600

800

1000

1200

235 236 237 238 239 240

Q4

P = 1100

P = 1200

P = 1300




1205972EJTP1 (1198761 1198991)

12059711987621

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198763

1

lt 0

(16)

1205972EJTP2 (1198762 1198992)

12059711987622

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198763

2

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987632

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987632

lt 0

(17)

1205972EJTP3 (1198763 1198993)

12059711987623

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

3

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

lt 0

(18)

1205972EJTP4 (1198764 1198994)

12059711987624

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198763

4

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987634

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987634

lt 0

(19)



60000

60500

61000

61500

62000

62500

63000

63500

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



30000

35000

40000

45000

50000

55000

60000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



43000432004340043600438004400044200444004460044800

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

3



60008000

10000120001400016000180002000022000

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4



119876lowast

1= ((2119863119875119880

2(119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991))

times (1198992 119875119880 [119880 (ℎ119898 (1198991 minus 1) + ℎ119903 +119882) +1198721198991]

+119863 [1198991 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(20)

119876lowast

2= ((119875119880

2[2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

minus 1198992 (119861 +119882) (119863119884)2])


+119863 [1198992 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(21)

119876lowast

3= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

minus (119872 + 119864) (119863119883)2])


+119863 [1198993 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(22)

119876lowast

4= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

minus (119872 + 119864) (119863119883)2minus 1198801198994 (119861 +119882) (119863119884)

2])


+119863 [1198994 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(23)




102

1025

103

1035

104

1045

105

239 240 241 242 243 244 245 246

Q1



560565570575580585590595600605610

245 246 247 248 249 250 251

Q2



200210220230240250260270280290300

239 240 241 242 243 244 245 246

Q3

P = 1100

P = 1200

P = 1300



500550600650700750800850900

245 246 247 248 249 250 251

Q4

P = 1100

P = 1200

P = 1300









5 Numerical Example









119875 = 1100 119875 = 1200 119875 = 1300

119862119898

2300sim2350 2300sim2350 2300sim23501198991

2 2 21198761

10219sim10229 10339sim10349 10444sim10454EJTP1


2300sim2350 2300sim2350 2300sim23501198992

1 1 11198762

18902sim57786 19404sim59321 19862sim60721EJTP2

5846523sim3350315 5877907sim3446259 5905128sim3529477119862119898

2355sim240 2355sim240 2355sim2401198993

14 13 131198763

7873sim7772 8404sim8295 8415sim8306EJTP3

4463785sim4357297 4466066sim4359922 4468691sim4362513119862119898

2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355sim2365) 2 (119862

119898= 2355) 1

1 (119862119898= 2355sim2365) 1 (119862

119898= 2360sim2365)

1198764

64172sim67519 (119862119898= 2355sim2365) 65370 (119862

119898= 2355) 90178sim104684

90662sim99636 (119862119898= 2370sim2400) 89788sim102277 (119862

119898= 2355sim2365)

EJTP4


59500

60000

60500

61000

61500

62000

62500

63000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



33000

34000

35000

36000

37000

38000

39000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



34000

36000

38000

40000

42000

44000

46000

240 2405 241 2415 242 2425 243 2435 244 2445 245

P = 1100

P = 1200

P = 1300

EJTP

3



200002200024000260002800030000320003400036000

2455 246 2465 247 2475 248 2485 249 2495 250

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4


1018102

1022102410261028

103103210341036

229 230 231 232 233 234 235 236

Q1



100150200250300350400450500550600

229 230 231 232 233 234 235 236

Q2



707274767880828486

235 236 237 238 239 240 241

Q3

Z = 01

Z = 02

Z = 03



0

200

400

600

800

1000

1200

235 236 237 238 239 240 241

Q4

Z = 01

Z = 02

Z = 03















119875 = 1100 119875 = 1200 119875 = 1300

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3370094sim3546836 3465934sim3640884 3548895sim3722454


EJTP3 3425530sim4221153 3464154sim4251420 3497160sim4277287

119862119903 2455sim2500 2455sim2500 2455sim2500

1198994

2 (119862119903= 2455sim2465)

1 (119862119903= 2470sim2500)

2 (119862119903= 2455sim2460)

1 (119862119903= 2355sim2500)

2 (119862119903= 2455)

1 (119862119903= 2360sim2500)

1198764

61574sim59822 (119862119903= 2455sim2465)

77530sim66375 (119862119903= 2470sim2500)

62723sim61837(119862119903= 2455sim2460)

81338sim68135 (119862119903= 2465sim250)

63748 (119862119903= 2455)

85010sim69738 (119862119903= 246sim250)

EJTP4 2021540sim2977979 2116825sim3088182 2198873sim3183760


52000

54000

56000

58000

60000

62000

64000

66000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



0

10000

20000

30000

40000

50000

60000

70000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



3000032000340003600038000400004200044000460004800050000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

3



02000400060008000

100001200014000160001800020000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4


1021022102410261028

1031032103410361038

104

239 240 241 242 243 244 245 246

Q1



576578580582584586588590592594

245 246 247 248 249 250 251

Q2



6065707580859095

100

239 240 241 242 243 244 245 246

Q3

Z = 01

Z = 02

Z = 03



500

550

600

650

700

750

800

850

245 246 247 248 249 250 251

Q4

Z = 01

Z = 02

Z = 03





119885 = 01 119885 = 02 119885 = 03


EJTP1


EJTP2 5877907sim3446259 5561648sim3126491 5238827sim2800005


EJTP3 4466066sim4359922 4147330sim4041153 3822615sim3716446

119862119898 2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

1198764

65370 (119862119898= 2355)

89788sim102277 (119862119898= 236sim240)

65203 (119862119898= 2355)

89751sim102171 (119862119898= 236sim240)

65029 (119862119898= 2355)

89708sim102058 (119862119898= 236sim240)

EJTP4 1900021sim1000745 1567517sim667383 1227857sim362920



4600048000500005200054000560005800060000620006400066000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



2500027000290003100033000350003700039000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



3000032000340003600038000400004200044000460004800050000

240 2405 241 2415 242 2425 243 2435 244 2445 245

Z = 01

Z = 02

Z = 03

EJTP

3



10000

15000

20000

25000

30000

35000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4










119885 = 01 119885 = 02 119885 = 03

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3465846sim3640884 3146216sim3322506 2819872sim2997440

119862119903 2400sim245 2400sim245 2400sim245

1198993

17 (119862119903= 240sim241)

16 (119862119903= 2415sim2425)

15 (119862119903= 243sim2435)

14 (119862119903= 244sim2445)

13 (119862119903= 245)

19 (119862119903= 240)

18 (119862119903= 2405sim2415)

17 (119862119903= 242sim2425)

16 (119862119903= 243sim2435)

15 (119862119903= 244sim2445)

14 (119862119903= 245)

21 (119862119903= 240)

20 (119862119903= 2405sim241)

19 (119862119903= 2415sim242)

18 (119862119903= 2425)

17 (119862119903= 243sim2435)

16 (119862119903= 244sim2445)

15 (119862119903= 245)

1198763

8295sim7997 (119862119903= 240sim241)

8277sim811 (119862119903= 2415sim2425)

8221sim8032 (119862119903= 243sim2435)

8325sim8112 (119862119903= 244sim2445)

8417 (119862119903= 245)

7453 (119862119903= 240)

7684sim7399 (119862119903= 2405sim2415)

7629sim7471 (119862119903= 242sim2425)

7709sim7533 (119862119903= 243sim2435)

7582 (119862119903= 244sim2445)

7834 (119862119903= 245)

6762 (119862119903= 240)

6940sim6814 (119862119903= 2405sim241)

6998sim6860 (119862119903= 2415sim242)

7048 (119862119903= 2425)

7252sim7088 (119862119903= 243sim2435)

7296sim7113 (119862119903= 244sim2445)

7323 (119862119903= 245)

EJTP3 3823707sim4477773 3503787sim4159040 3178546sim3834324

119862119903 2455sim250 2455sim250 2455sim250

1198994

2 (119862119903= 2455)

1 (119862119903= 246sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

1198764

62723 (119862119903= 2455)

61837sim68135 (119862119903= 246sim250)

62565sim61676 (119862119903= 2455)

2465sim250 (119862119903= 246sim250)

62400sim61508 (119862119903= 2455sim246)

81257sim67924 (119862119903= 246sim250)

EJTP4 2116836sim3088182 1785043sim2763725 1446121sim2432425












References













































1 Introduction














2

1

Product 1

Product 2

Stock

Stock

Stock

Machine

Randomdemand 1

Random

Random

demand 2

demand n

Product n

n







3 Notations







Decision Variables


4 Production Policy


lowast(119880lowast

= 119880lowast

119894119895119896 119910lowast

119894119895119896and 120575

lowast



Period 1

Δt Δt

j = 1 j = 2 j = 3

1205751 1205752 1205753

Period k


Subperiod j












Formally


Production cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + Cp (119894) times 119880119894119895119896)

Holding cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1



(2)





minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(3)





(4)



119896

sum

119904=1

119863 (119894 119904) + Stock min (119894 119896)

le

119896

sum

119904=1

119901

sum

119895=1

(119910(119894119895119904) times 119880119894119895119904) + stock init (119894 119904 = 0)

forall 119894 = 1 119899

(5)





Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(6)






Min[Cs times 119864 [119878 (119867)2]

+

119867minus1

sum

119896=0

(Cs times 119864 [119878 (119896)2] + Cp times 119864 [119906 (119896)

2])]

(7)



Min (Ζ (119880))

119880 = 119880119894119895119896 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(8)with119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]




minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(10)


(11)


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(12)

119901

sum

119895=1





119899

sum

119894=1

119910119894119895119896 = 1 forall 119895 = 1 119901 for 119896 = 1 119867

119901

sum

119895=1

119910119894119895119896 = 1 forall 119894 = 1 119899 for 119896 = 1 119867

(14)

119910119894119895119896 isin0 1 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(15)










minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(17)



119864 [119889119894 (119896)] = 119889119894 (119896) Var [119889119894 (119896)] = 1205902(119889119894 (119896))

forall 119894 = 1 119899 119896 = 1 119867

(18)



forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(19)

We note that

119864 [119880119894119895119896] = 119894119895119896 = 119880119894119895119896 (20)


Var (119880119894119895119896) = 0

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(21)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(22)




sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(23)


Proof We know that


(24)


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896) ge 0) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) ge 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

ge 119889119894 (119896) minus 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))

ge119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896))) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(25)

Noting that

119883 =119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896)) (26)



119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))ge 119883) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(27)



Hence

119878119894(119896minus1)times119901 +

119901

sum

119895=1


(120579119894)

forall 119894 = 1 119899 119896 = 1 119867

(28)

Therefore

119901

sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(29)




119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(30)





119880lowast= Min

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]]

(31)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(120579119894)


forall 119894 = 1 119899 119896 = 1 119867


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(32)



q = 1 q = 2

j = 1j = 2 j = p

Deg

rada

tion

rate

k = 1 k = 2 k = 3

T t2T

1205822 1205822p1205821

120582p+1

120575p+1
















(q minus 1) times T



T


1

2

3

Δt

120575((k+m)timesp)






119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896


times1

119880119894 nomΔ119905times 120582119899 (119905)]


(34)



Therefore




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(35)


Note that


119879] + 1 (36)




= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(37)




119879 = Round [119867 times 119901

119873] (38)








120601119879

(119880119873)= [

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(39)



=

119873+1

sum

119902=1

120601119879

(119880119873) (40)



120601(119880119873)

=

119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(41)

Note that119873 = 1 2




Γ (119873) = [

[

Mc times119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

+Mp times 119873]

]

(43)







120597Γ (119873)

120597119873= 0 (44)



lim119873rarr1

Γ (119880119873) = lim119873rarr1

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


rarrconstant

)

= 119872119888 times 120601 (119880 1) + 119872119901

lim119873rarr+infin

Γ (119880119873) = lim119873rarr+infin

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr0

+ 119872119901 times 119873⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr+infin

)

= +infin

(45)


Moreover

Γ (119880119873 + 1) minus Γ (119880119873) ge 0

997904rArr [119872119888 times 120601 (119880 (119873 + 1)) + 119872119901 times (119873 + 1)]



997904rArr 120601 (119880 (119873 + 1)) minus 120601 (119880119873) le119872119901

119872119888

(46)

In addition

Γ (119880119873) minus Γ (119880119873 minus 1) le 0

997904rArr [119872119888 times 120601 (119880119873) + 119872119901 times (119873)]




119872119888

(47)




exist119873lowast119904119894 120585119873 le

119872119901

119872119888

le 120585119873minus1 (48)

with

120585119873 = 120601 (119880119873) minus 120601 (119880 (119873 + 1)) (49)



119873lowastexist119904119894

120601 (119880 (119873 + 1)) minus 120601 (119880119873) ge 0

120601 (119880119873) minus 120601 (119880 (119873 minus 1)) le 0

lim119873rarr1

Γ (119880119873) = 119862119900119899119904119905119886119899119905

lim119873rarr+infin

Γ (119880119873) = +infin

(50)



6 Numerical Example




1198891 = 200 120590 (1198891) = 15

1198892 = 110 120590 (1198892) = 09

1198893 = 320 120590 (1198893) = 12

(51)





Table 1



Table 2









1205752

1205753

1205754

1205755

1205756

1205757

1205758

1205759

12057510

12057511

12057512



12057514

12057515

12057516

12057517

12057518

12057519

12057520

12057521

12057522

12057523

12057524






120582119899 (119905) =120572

120573times (

119905

120573)

120572minus1

=2

12times (

119905

12) (52)















Nlowast

d = di k ( )

Ulowast= Uijk ( )

k =

i =

k =i =j =

1 H1 n

1 p1 H1 n


0 2 4 6 8 10

4000

5000

6000

7000

8000


The t

otal

cost

of m

aint

enan

ceΓ

(N)



Γlowast(119873) (um) 119873




=





8 Conclusion






Appendices


We have119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A1)

Also


minus Ent [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A2)

Therefore


]


minus Ent [119895

119901] times 119889119894 (119896)]


minusEnt [119895

119901] times 119889119894 (119896) ])

2

]


]


minus [Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896))])

2

]


]



times (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]


]


]


times(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ 119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A3)



]


]


times 119864 [(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A4)




(A5)

Therefore


]


]

+(Ent [119895

119901])

2

times 119864 [(119889119894 (119896) minus 119889119894 (119896))2

]]

(A6)

We know that

119864 [(119909119896 minus 119909119896)2] = Var (119909119896)

(Int [119895

119901])

2

= Int [119895


119895

119901le 1

(A7)

Therefore


]


119901] times Var (119889119894 (119896))

(A8)


Finally


+ Ent [119895

119901] times Var (119889119894 (119896))

(A9)

Consequently

(i) for 119896 = 1

(a) 119895 = 1

Var (1198781198941) = Var (1198781198940) + (Ent [ 1

119901]) times Var (119889119894 (1))

(A10)

(b) 119895 = 2

Var (1198781198942) = Var (1198781198940) +2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A11)

(c) 119895 = 119901

Var (119878119894119901) = Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A12)

(ii) for 119896 = 2

(a) 119895 = 1

Var (119878119894119901+1) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+ Ent [ 1

119901] times Var (119889119894 (2))]

(A13)

(b) 119895 = 2

Var (119878119894119901+2) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (1)) + Ent [ 1

119901]

times Var (119889119894 (2)) + Ent [ 2

119901] times Var (119889119894 (2))]

(A14)

(c) 119895 = 119901

Var (119878119894(2times119901)) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+

119875

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119901))]

(A15)


(a) 119895 = 1


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

1

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A16)

(b) 119895 = 2


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A17)



= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A18)

On the other hand


]




]


]


]



]]

(A19)

We know that



(A20)


Hence


]




]



]





]

(A21)

Consequently


]



(A22)

Noting that


]





(A23)




= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]


]

= [ Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A24)

Consequently


]

= [ 1205902(1198781198940) +

119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times 1205902(119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A25)


119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(A26)



(i) for 119896 = 1

(a) 119895 = 1

1205821 (119905) = (1205820) times (1 minus In [0

119902 times 119879]) +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (119905)

(B1)

(b) 119895 = 2

1205822 (119905) = 1205821 (1205751) times (1 minus In [1

119902 times 119879])

+

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)


1205822 (119905) = (1205820 +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (120575(1)))


119902 times 119879]) +

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)

(B2)

(c) 119895 = 119901


119902 times 119879])

+

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)

120582119901 (119905) = [(1205820 +

119901minus1

sum

119897=1

119899

sum

119894=1

1198801198941198971 times Δ119905

119880119894 nom times 120575119897

times 120582119899 (120575(119897)))


119902 times 119879]) +

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)]

(B3)


(a) 119895 = 1

120582((119896minus1)times119901)+1 (119905)


119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

120582((119896minus1)times119901)+1 (119905)

= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

(B4)



= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(B5)



References
























1 Introduction



















16

14

12

10

08

06

Ship

men

t cos

t (lowast$100)



R2= 0926



119864 (119876) = 119901119864 (min 119876119863) + 119904119864(119876 minus 119863)+ minus 119888119876

minus 119886 + 119887 ln [119864 (min 119876119863)]

minus 119886 + 119887 ln [119864(119876 minus 119863)+]



(1)









times[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2



(3)


119864(119863 minus 119876)+le[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2

(4)


119889119864 (119876)

119889119876=119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876 minus 120583)

2[1205902 + (119876 minus 120583)2]12

minus 1198872119876 + 2120583 (119876 minus 120583) [120590

2+ (119876 minus 120583)

2]minus12


(5)

1198892119864 (119876)

1198891198762= minus

(119901 minus 119904) 1205902

2[1205902 + (119876 minus 120583)2]32

minus 119887

minus 2 + 21205832minus 21198762

+ 4120583[1205902+ (119876 minus 120583)

2]12

+2120583 (minus1 minus 120583

2+ 4120583119876 minus 3119876

2)

[1205902 + (119876 minus 120583)2]12

minus81205832(119876 minus 120583)

2

1205902 + (119876 minus 120583)2

+(119876 minus 120583)

2(21205833+ 2120583 minus 2120583119876

2)

[1205902 + (119876 minus 120583)2]32

sdot minus1 minus 1205832+ 1198762+ 4120583[120590

2+ (119876 minus 120583)

2]12

minus2

(6)




Iteration 119894 119876119894

1198911015840(119876119894) 119891

10158401015840(119876119894) 119891

1015840(119876119894)11989110158401015840(119876119894) 119876

119894+1





Let119876119894+1 = 119876119894 minus (1198911015840(119876119894)119891


1198911015840(119876119894) =

119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876119894 minus 120583)

2[1205902 + (119876119894 minus 120583)2]12

minus 1198872119876119894 + 2120583 (119876119894 minus 120583) [120590

2+ (119876119894 minus 120583)

2]minus12


2]12

(7)

From (6) we set

11989110158401015840(119876119894) = minus

(119901 minus 119904) 1205902

2[1205902 + (119876119894 minus 120583)2]32

minus 119887 minus 21198762

119894minus 2 + 2120583

2+ 4120583[120590

2+ (119876119894 minus 120583)

2]12

+2120583 (4120583119876119894 minus 3119876

2

119894minus 1205832minus 1)

[1205902 + (119876119894 minus 120583)2]12

+(119876119894 minus 120583)

2(21205833+ 2120583 minus 2120583119876

3

119894)

[1205902 + (119876119894 minus 120583)2]32

minus81205832(119876119894 minus 120583)

2

1205902 + (119876119894 minus 120583)2

(8)






0 119876lowast


0= exp(1minus


0= 119876lowast



is119876lowastlowast119894= (119876119904

119894+119876lowast



119894= (119876119904

119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887








10158401015840(1198760)) =








lowast) 119864(119876

lowast) ()


Average 405

12

10

8

6

4

2

0

5 10 15 20

Order quantity Q

Expe

cted

pro

fitE

(Q)














119864 (119878) = 119901119864 (min 119878 119863) + 119904119864(119878 minus 119863)+


minus 119886 + 119887 ln [119864(119878 minus 119863)+]







2]12

minus (119878 minus 120583)

2



(10)


119901 minus 119904

2[1205902+ (119903 minus 120583)

2]12

minus 119903 + (119888 minus 119904) 119903



2[1205902+ (119878 minus 120583)

2]12



= 0

(11)





1015840(119903119894)) where

119891 (119903119894) =119901 minus 119904

2[1205902+ (119903119894 minus 120583)

2]12

minus 119903119894 + (119888 minus 119904) 119903119894

+ 119887 ln minus1205832 minus 1 + 1199032119894+ 2120583[120590

2+ (119903119894 minus 120583)

2]12


2[1205902+ (119878 minus 120583)

2]12



1198911015840(119903119894) =

2119888 minus 119901 minus 119904

2+

(119901 minus 119904) (119903119894 minus 120583)

2[1205902 + (119903119894 minus 120583)2]12

+ 1198872119903119894 + 2120583 (119903119894 minus 120583) [120590

2+ (119903119894 minus 120583)

2]minus12


2]12

(12)










=119888

120588(119898 + 119889) 120583 minus (119898 + 119889) 119864[119863 minus 119866 (119876)]

+




(13)


119864[119863 minus 119866 (119876)]+le[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

(14)


119864 (119876) ge119888

120588

(119898 + 119889) 120583 minus (119898 + 119889)

times[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

minus (120588 + 119889 minus 1)119876

minus 2119886 + 2119887 ln 2


+ 2 (120583 + 120588119876 minus 119876)

times [1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

(15)


119889119876

= minus119888 (119898 + 119889)

2[1


119888

120588(120588 + 119889 minus 1)





(16)

where119883 = 1205902 + 120588119902119876 + (120588119876 minus 120583)2

1198892119864 (119876)

1198891198762= minus119888 (119898 + 119889)

2[minus120588

4(119902 minus 2120583 + 2120588119876)

2119883minus32

+ 120588119883minus12]

minus 1198871198841015840119885 minus 119884119885

1015840

1198852


12



+ 2 (120583 + 120588119876 minus 119876)11988312

1198841015840= 4120588 (1 minus 120588) minus 2120588


minus1205882

2(120583 + 120588119876 minus 119876) (119902 minus 2120583 + 2120588119876)

2119883minus32

1198851015840= 2 (1 minus 120588) (2120588119876 + 120583) minus 120588119902

minus 2 (1 minus 120588)11988312+120588

4(120583 + 120588119876 minus 119876)


(18)





1015840(119876119894)119891

10158401015840(119876119894)) where 119891

1015840(119876119894) and




2119864(119876119894+1)119889119876

2




0 119876lowast


0= exp(1minus


0= 119876lowast



is 119876lowastlowast119894= (119876119904

119894+ 119876lowast




119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887


119894 = 119894 + 1



2





2

119895 respectively



1 119876lowast2 and

119876lowast

119873

max1198761 119876119873

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583[1205902

119895+ (119876119895 minus 120583119895)

2

]12

(19)

Subject to119873

sum

119895=1

119888119895119876119895 le 119861 (20)


119871 (1198761 119876119873 120582)

=

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583119895[1205902

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582(

119873

sum

119895=1

119888119895119876119895 minus 119861)

(21)


120597119871 (1198761 119876119873 120582)

120597119876119895

=119901119895 + 119904119895 minus 2119888119895


2[1205902119895+ (119876119895 minus 120583119895)

2

]12

minus 119887

2119876119895 + 2120583119895 (119876119895 minus 120583119895) [1205902

119895+ (119876119895 minus 120583119895)

2

]minus12

minus1 minus 1205832 + 1198762119895+ 4120583119895[120590

2

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582119888119895

(22)


119873with

















Step 1 Let 119895 = 1 2 3 4


1= 8731 119876lowast

2= 7762 119876lowast

3= 11072 and 119876lowast

4

= 21243 Check sum4119895=1119888119895119876lowast

119895= $92 gt $80 where (20) is not



3is set to 0


119895= $61 lt $80 (20) is satisfied










Highlights









References



































1 Introduction













max 120601

st 120582119883 le 119909119863

119900

120582119884119863ge 120601119910119863

119900

120582119884119880

ge 120601119910119900119880

119890120582 = 1

120582 ge 0

(1)

in which 119910119900119880= minus119884



max 120601

st 120582119883119863le (1 minus 120601) 119909

119863

119900

120582119883119880

le (1 minus 120601) 119909119900119880

120582119884119863ge (1 + 120601) 119910

119863

119900

120582119884119880

ge (1 + 120601) 119910119900119880

119890120582 = 1

120582 ge 0

(2)

in which 119910119900119880= minus119884

119880+ V gt 0 and 119883

119880

= minus119883119880+ 119908 gt 0




min 1 minus [

[

119908119900 +1

119898 + 119904(sum

119894isin119868119863

119905minus119863

119894+ sum

119903isin119874119863

119905+119863

119903)]

]

st119899

sum

119895=1

120582119895119909119863

119894119895+ 119905minus119863

119894= 119909119863

119894119900minus 119908119900 119894 isin 119868119863

119899

sum

119895=1

120582119895119909119880

119894119895+ 119905minus119880

119894= 119909119880

119894119900+ 119908119900 119894 isin 119868119880

119899

sum

119895=1

120582119895119910119863

119903119895minus 119905+119863

119903= 119910119863

119903119900+ 119908119900 119903 isin 119874119863

119899

sum

119895=1

120582119895119910119880

119903119895minus 119905+119880

119903= 119910119880

119903119900minus 119908119900 119903 isin 119874119880

119899

sum

119895=1

120582119895 = 1

(3)


min 120579

st119899

sum

119895=1

120582119895119909119894119895 le 120579119909119894119900 forall119894 isin 119868

119899

sum

119895=1

120582119895119910119863

119903119895ge 119910119863

119903119900forall119903 isin 119874

119863

119899

sum

119895=1

120582119895119910119880

119903119895le 120579119910119880

119903119900forall119903 isin 119874

119880

120582119895 ge 0 forall119895 isin 119873

(4)




min 120579

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le 120579

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 ge

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(5)


max119898

sum

119894=1

119904minus

119894+

119904

sum

119903=1

119905+

119903

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 + 119904minus

119894= 120579lowast

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 minus 119905+

119903=

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

119904minus

119894ge 0 119905

+

119903ge 0 119894 = 1 119898 119903 = 1 119904

120582119895119896 ge 0 119896 119895 = 1 119899

(6)




119909119894119900 =

119899

sum

119895=1

120582119900lowast

119895119909119894119895 = 120579

lowast119909119894119900 minus 119904

minuslowast

119894119894 = 1 119898

119910119903119900 =

119899

sum

119895=1

120582119900lowast

119895119910119903119895 = 119910119903119900 + 119905

+lowast

119903119903 = 1 119904

(7)


119878119894 =

119899

sum

119895=1

119909119894119895 minus

119899

sum

119895=1

119909119894119895 ge 0 119894 = 1 119898

119879119903 =

119899

sum

119895=1

119910119903119895 minus

119899

sum

119895=1

119910119903119895 ge 0 119903 = 1 119904

(8)


max119899

sum

119895=1

119904

sum

119903=1

119906119903119910119903119895 +

119899

sum

119896=1

120577119896

st119899

sum

119895=1

119898

sum

119894=1

V119894119909119894119895 = 1

119904

sum

119903=1

119906119903119910119903119895 minus

119898

sum

119894=1

V119894119909119894119895 + 120577119896 le 0 119895 119896 = 1 119899

119906119903 ge 0 119903 = 1 119904

V119894 ge 0 119894 = 1 119898

(9)



5 Proposed Method



119883 = 119909119894119895 isin 119877119898times119899

+ 119884

119863= 119910119863

119903119895 isin 119877119904119863times119899

+

119884119880= 119910119880

119905119895 isin 119877119904119880times119899

+

(10)


max 120593

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le

119899

sum

119895=1

119909119894119895 minus 120593119877119909119894 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119863

119903119895ge

119899

sum

119895=1

119910119863

119903119895+ 120593119877119910

119863

119903119903 = 1 119904

119863

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119880

119905119895le

119899

sum

119895=1

119910119880

119903119895minus 120593119877119910

119880

119905119905 = 1 119904

119880

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(11)

where119877119909119894119877119910119863

119903 and119877119910119880



119863

119903 119877119910119880



119903 119877119910119880

119905) vector as

follows

119877119909119894 =

119899

sum

119895=1

119909119894119895 minus 119899 (min 119909119894119895119895=1119899) 119894 = 1 119898

119877119910119863

119903=

119899

sum

119895=1

119910119863

119903119895minus 119899 (max 119910119863

119903119895119895=1119899

) 119903 = 1 119904119863

119877119910119880

119905=

119899

sum

119895=1

119910119880

119905119895minus 119899 (min 119910119880

119905119895119895=1119899

) 119905 = 1 119904119880

(12)


119863

119903 119877119910119880

119905) can be

chosen as follows

(119877119909119894 119877119910119863

119903 119877119910119880

119905) = (

119899

sum

119895=1

119909119894119895

119899

sum

119895=1

119910119863

119903119895

119899

sum

119895=1

119910119880

119905119895) (13)











119863

119903 119877119910119880



6 Numerical Example





7 Conclusion







References


































1 Introduction







20 40 60 80 100Utilization ()

Cycle

tim

eIn

crea

sing


20 40 60 80 100

Low variability

Utilization ()

High variability

Cycle

tim

eIn

crea

sing

te

to




CT119902 = 119881119880119879 (1)



2 Literature Review














119886= 1205902

1198861199052

119886and

1198622

119890= 1205902

1198901199052



2



2

119890gt 1198622



CT119902 = (1198622

119886+ 1198622

119890

2)(

119906

1 minus 119906) 119905119890 (2)

1198622

119889= 11990621198622

119890+ (1 minus 119906

2) 1198622

119886 (3)










Patient



Home

10 min24 min

20 min11 min11 min

98 min0 min

10 min0 min

3 min3 min0 min

10 min




Clean room




ORemergence

time

Patient out of room

Inpatient room


(Nj)

WW W W W W


1205902


1198882




arrivals






2


as 083

119860 =119898119891

119898119891 + 119898119903

(4)

119905119890 =119905119900

119860 (5)

119906 =119903119886

119903119890

=119903119886119905119890

119898 (6)

1198622

119890= 1198622

119900+ 2119860 (1 minus 119860)

119898119903

119905119900

(7)






Specialty 1

FCFS

Specialty 2

Specialty 15


Operating room

Recoverroom

First stage (W1)

Second stage (W2)

(Mj)

(Nj)

M1

M2

M12

N1

N2

N3

N4

N5

N24





119905119890 = 119905119900 +119905119904

119873119904

(8)

1205902

119890= 1205902

119900+1205902

119904

119873119904

+119873119904 minus 1

1198732119904

1199052

119904

1198882

119890=1205902

119890

1199052119890

(9)




119890= 749







CT119902 =1198622

119886+ 1198622

119890

2

119906

1 minus 119906119905119890

=(099 + 749)

2(

089

1 minus 089) 14543

= (424) (809) (14543)

(10)














120583 (11)




(12)



5 Empirical Results








1 10800 813 10800

813≒ 13

2 10800 159 10800

159≒ 68

3 10800 134 10800

134≒ 81

4 10800 346 10800

346≒ 31

5 10800 185 10800

185≒ 58

Input

Output


Simulationmodel






1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

0020018001601400120010008000600040002

0

CV


WIP




Parameters



Patient



suture and finish

Home

Leveling

Billing and coding

Waiting time= 5 min




Inpatient room

W W W W





10 min0 min

98 min0 min5 min




Decision variables



Minimize24

sum

119895=1

119874119895 (13)


5

sum

119894=1

119860 119894119895 = 1 forall119895 (14)

119862119894 ge 1 forall119894 (15)

5

sum

119894=1

119862119894 = 24 (16)

119860 119894119895 isin 0 1 forall119894119895 (17)






















Acknowledgments


References


















































Editorial Board










Contents




























Acknowledgments




Chia-Yu Hsu







1 Introduction














2 Related Work




(a) (b) (c)

(d) (e) (f)



3 Proposed Approach






119898119896119873119892]



]


120579119897119873119892]


119881119897119873119892]


119897= [1205951198971

1205951198972 120595119897119873119892]


119892= [1205951198921 1205951198922

120595119892119873119892]


Consensuspartition


WBMs

1 clustering

q clustering

2 clustering



Labels120587 q










1205871

1205872

1205873

x1

1 1 1x2

1 1 1x3

1 1 1x4

2 2 1x5

2 2 2x6

3 1 2x7

3 1 2x8

3 1 2



V2

V3

V4

V5

V6

V7

V8

119867(1)

ℎ11

1 1 1 0 0 0 0 0ℎ12

0 0 0 1 1 0 0 0ℎ13

0 0 0 0 0 1 1 1

119867(2) ℎ

211 1 1 0 0 1 1 1

ℎ22

0 0 0 1 1 0 0 0

119867(3) ℎ

311 1 1 1 0 0 0 0

ℎ32

0 0 0 0 1 1 1 1


119872(119910119903) =

119873119887

sum

119903=1

119890minus119898120573119889(119910119903 119910119904) 119903 = 1 2 3 119873119887 (1)

where

120573 = (119889 (119910119903 minus 119910wc)

119873119887

)

minus1

(2)







(a) Binary value

51015202530






SSE =

119873119888

sum

119896=1

sum

x119894isin119862119896(x119894 minusm119896)

2 (3)






119869119890 =sum119873119888


119870 (5)

In addition

119889max (p119894Z119897) = max119896=12119873119888

[[

[

sum

forallx119894isin119862119897119896

119889 (x119894m119897119896)119899119897119896

]]

]

(6)



[119889 (m119897119906m119897V)] (7)





119894


119896


119894Z119897)


119901119906(120595119897(119905) minusΘ

119897(119905)) + 119888

119892119906(120595119892(119905) minusΘ

119897(119905)) (i)

Θ119897(119905 + 1) = Θ

119897(119905) + V

119897(119905 + 1) (ii)










1012

1518

2315221370

1184 1098945

02004006008001000120014001600

0

5

10

15

20

25

03 04 05 06 07

SSE

Clus

ter n

umbe

r









max 1 1198861

04119888119901

2 1198862

03119888119892

2 1198863

03Iteration 500



Hsu and Chien [8] 1184 1192 1203 1248 1322


KB 2889 3092 3003 4083 3570KC 3331 2490 2603 3169 2603PB 5893 3601 6566 5839 6308PC 4627 4873 3330 3787 6112

Clusteringensemble






Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9

Group 10

Group 11

Group 12

Group 13

Group 14

Group 15





119879119890

(8)



(a) (b) (c)

(d) (e) (f)




Hsu and Chien [8] 04 04 04 04 04


KB 10 10 10 10 10KC 10 10 10 10 10PB 10 10 10 10 10PC 10 10 10 10 10

Clusteringensemble





precision =TP

TP + FP

recall = TPTP + FN

(9)




=2TP

FP + FN + 2TP (10)






Precision

A 070 084 092 092 092 098B 050 066 096 092 062 096C 060 064 100 100 060 100D 070 098 092 092 098 100E 060 094 082 082 098 098F 080 098 076 076 098 098

Avg 065 084 090 089 085 098



Recall

A 100 100 100 093 100 100B 100 097 07 078 083 100C 100 094 067 084 067 097D 100 081 100 100 100 100E 100 079 100 100 100 100F 100 100 100 100 100 100

Avg 100 092 090 093 092 100



119865-measure

A 082 091 096 092 096 099B 067 079 081 084 071 098C 075 076 08 091 063 098D 082 089 096 096 099 100E 075 086 090 090 099 099F 089 099 086 086 099 099

Avg 078 087 088 090 088 099



5 Conclusion









Acknowledgment


References







































1 Introduction





























119903119894119895 = 05 (1 + log9119886119894119895) (1)









119903119894119895 =119894 minus 119895 + 1


(2)

119903119894119895 = 1 minus 119903119895119894 forall119894 lt 119895 (3)


According to (2)

11990321 = 1 minus 03 = 07

11990331 = 15 minus 03 minus 06 = 06


11990332 = 1 minus 06 = 04

11990342 = 15 minus 06 minus 08 = 01

11990343 = 1 minus 08 = 02

(4)

According to (3)

11990313 = 1 minus 06 = 04

11990314 = 1 minus 03 = 07

11990324 = 1 minus 01 = 09

(5)


PR =[[[

[

05 03 04 07

07 05 06 09

06 04 05 08

03 01 02 05

]]]

]

(6)





119878119886119887 = 1 minus

10038161003816100381610038161198861 minus 11988711003816100381610038161003816 +

10038161003816100381610038161198862 minus 11988721003816100381610038161003816 +

10038161003816100381610038161198863 minus 11988731003816100381610038161003816 +

10038161003816100381610038161198864 minus 11988741003816100381610038161003816

4 (7)


AM =

[[[[

[


119878119886119887

1198781198641 1198781198642 sdot sdot sdot 1

]]]]

]

(8)



AA119886 =1

119864 minus 1

119864

sum

119887=1119886 =119887

119878119886119887 forall119886 (9)


RA119886 =AA119886

sum119864

119886=1AA119886

forall119886 (10)










(12)


1

119894119895 1199032

119894119895 1199033

119894119895 1199034

119894119895)





119903119894119895 =1199031

119894119895+ 2 (119903

2

119894119895+ 1199033

119894119895) + 1199034

119894119895

6 (13)


PR119896 =[[[[

[


1

1199031198731 1199031198732 sdot sdot sdot 1

]]]]

]

(14)






120575119894119896 =

119873

sum

119895=1119895 =119894

119903119894119895 minus

119873

sum

119895=1119895 =119894

119903119895119894 forall119894 119896 (15)


DM =

[[[[

[


1205751198731 1205751198732 sdot sdot sdot 120575119873119872

]]]]

]

(16)




aw119896 =(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119896 minus 120575119895119896

10038161003816100381610038161003816)1(119901minus1)

sum119872

119897=1(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119897 minus 120575119895119897

10038161003816100381610038161003816)1(119901minus1)

forall119896 119901 gt 1 (17)

aw119896 =(sum119873

119894=11205752

119894119896)12(119901minus1)

sum119872

119897=1(sum119873

119894=11205752119894119897)12(119901minus1)

forall119896 119901 gt 1 (18)



119872



119862119894 =

119872

sum

119896=1

aw119896120575119894119896 119894 = 1 2 119873 (19)





transformation


relations










4 Numerical Example





11987711 =[[[

[




]]]

]

11987721 =[[[

[




]]]

]

11987731 =

[[[[[[

[

minus1

3minus minus

minus minus1

4minus

minus minus minus 1


]]]]]]

]

(20)


11987711 =[[[

[





]]]

]

11987721 =[[[

[





]]]

]

11987731 =

[[[[[[

[

minus1

31

31

31

3minus minus

minus minus1

41

41

41

4minus



]]]]]]

]

(21)


11987712 =[[[

[





]]]

]

11987722 =[[[

[





]]]

]


11987732 =

[[[[[[

[

minus1

41

41

41

4minus minus


minus minus minus1

31

31

31

3


]]]]]]

]

(22)


then become 119877101584031

and 1198771015840


replaced by 119877101584031and 1198771015840


1198771015840

31=

[[[

[





]]]

]

1198771015840

32=

[[[

[





]]]

]

(23)



PR1 =[[[

[





]]]

]

PR2 =[[[

[





]]]

]

(24)



2are

PR10158401=

[[[

[

0500 0290 0100 0100

0710 0500 0311 0311

0900 0689 0500 0500

0900 0689 0500 0500

]]]

]

PR10158402=

[[[

[

0500 0218 0265 0055

0782 0500 0547 0337

0735 0453 0500 0290

0945 0663 0710 0500

]]]

]

(25)


1and PR1015840

2are consistent


DM =[[[

[

minus2019 minus1923

minus0336 0331

1178 minus0045

1178 1637

]]]

]

(26)




11990312

11990323

11990334


119886119887)

11987812

0925 0775 100011987813

0900 0880 090011987823


119886)

AA1 0913 0828 0950AA2 0900 0730 0950AA3 0888 0783 0900


RA1 0338 0354 0339RA2 0333 0312 0339RA3 0329 0334 0321


120573 = 05

CC1 0319 0327 0320CC2 0317 0306 0320CC3 0364 0367 0361


23= (022 028 034 041) 119903

34= (040 047 053 060)


23= 0311 119903

34= 0500



Very low

02 04 06 1008

02

04

06

08

10

Mem

bers

hip

valu

e



Very low

More highMore

low

02

04

06

08

10

Mem

bers

hip

valu

e




Alternative 119894 120575119894119896

119862119894(ADM) 119862



aw119896by SDM 0509 0491





5 Conclusion











Acknowledgments


References


















































1 Introduction

















(i) Supplier Side





(iii) Retailer Side



22 Assumptions












3 Model Formulation




Repaired products

Defective products

Normal products











=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]

(1)




nQ


nQD

nQD

nQD

nQ

nQ


Z(1 minus R)nQ


nZQ


nQ


P

Q

t

t

t

Q

Q

Q

Q

P

QD

QD (n minus 1)Q

D

nRZQ




119885(1 minus 119877)]119876119894




=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)


2

2119899 [1 minus 119885 (1 minus 119877)]119876119894

(2)




nQ


Lm =nQ

D

X

Q

t

Interest income

Figure 3 119871119898lt 119883

Lm =nQ

D

nQ


X

Q

Interest income

Opportunity cost

t

Figure 4 119871119898ge 119883



(ii) Ordering cost = 119860119898119863119899119894119876119894










TP1198981





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


119876119894

2)

(3)



2 [1 minus 119885 (1 minus 119877)] (4)





TP1198982





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)


minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(5)




22[1 minus 119885(1 minus 119877)]119899119894119876119894



TP1198983





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119894119876119894 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus 119862119898119868mp (119863119884 minus119876119894

2)

(6)



119877)]119899119894119876119894





TP1198984





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119876 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus119862119898119868mp (119863119884)

2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(7)




(ii) Ordering cost = 119860119903119863119899119894119876119894






TP1199031



= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

+ 119875119903119868re (119863119884 minus119876119894

2)





TP1199032



+ interest income

= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

minus119862119903119868rp (119876119894 minus 119863119884)

2

2119876119894

+119875119903119868re (119863119884)

2

2119876119894

(9)


EJTP119894 (119876119894 119899119894)

=





(10)

whereEJTP1 (1198761 1198991)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198761 [1198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198991


ℎ1199031198761

2minus

ℎ11990411986311989911198761

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198761

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989911198761)

2 [1 minus 119885 (1 minus 119877)]

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198761

2)

EJTP2 (1198762 1198992)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198762 [1198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198992


ℎ1199031198762

2minus

ℎ11990411986311989921198762

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989921198762)

2 [1 minus 119885 (1 minus 119877)]

+(119862119903119868me minus 119862119903119868rp) (1198762 minus 119863119884)

2

21198762

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198762

EJTP3 (1198763 1198993)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198763 [1198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198993


ℎ1199031198763

2minus

ℎ11990411986311989931198763

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198763

2)

EJTP4 (1198764 1198994)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]


minus ℎ1198981198631198764 [1198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198994


ℎ1199031198764

2minus

ℎ11990411986311989941198764

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198764

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119898119868se minus 119862119898119868mp) (11989941198764 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119903119868me minus 119862119903119868rp) (1198764 minus 119863119884)

2

21198764

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198764

(11)



120597EJTP1 (1198761 1198991)1205971198761

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198762

1

minus ℎ1198981198631198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198991


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp)

2

(12)

120597EJTP2 (1198762 1198992)1205971198762

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

2

minus ℎ1198981198631198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198992


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987622

+(119862119903119868me minus 119862119903119868rp) [119876

2

2minus (119863119884)

2]

211987622

(13)

120597EJTP3 (1198763 1198993)1205971198763

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198762

3

minus ℎ1198981198631198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198993

2119875 [1 minus 119885 (1 minus 119877)]2

minus(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

+(119862119898119868se minus 119862119898119868mp) [(11989931198763)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

minus(119875119903119868re minus 119862119898119868mp)

2

(14)

120597EJTP4 (1198764 1198994)1205971198764

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198762

4

minus ℎ1198981198631198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198994


2

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987624

+(119862119898119868se minus 119862119898119868mp) [(11989941198764)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

+(119862119903119868me minus 119862119903119868rp) [119876

2

4minus (119863119884)

2]

211987624

(15)


102

1025

103

1035

104

1045

105

2295 2305 2315 2325 2335 2345 2355

Q1



0

100

200

300

400

500

600

700


Q2


777879808182838485

235 236 237 238 239 240

Q3

P = 1100

P = 1200

P = 1300



0

200

400

600

800

1000

1200

235 236 237 238 239 240

Q4

P = 1100

P = 1200

P = 1300




1205972EJTP1 (1198761 1198991)

12059711987621

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198763

1

lt 0

(16)

1205972EJTP2 (1198762 1198992)

12059711987622

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198763

2

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987632

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987632

lt 0

(17)

1205972EJTP3 (1198763 1198993)

12059711987623

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

3

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

lt 0

(18)

1205972EJTP4 (1198764 1198994)

12059711987624

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198763

4

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987634

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987634

lt 0

(19)



60000

60500

61000

61500

62000

62500

63000

63500

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



30000

35000

40000

45000

50000

55000

60000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



43000432004340043600438004400044200444004460044800

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

3



60008000

10000120001400016000180002000022000

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4



119876lowast

1= ((2119863119875119880

2(119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991))

times (1198992 119875119880 [119880 (ℎ119898 (1198991 minus 1) + ℎ119903 +119882) +1198721198991]

+119863 [1198991 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(20)

119876lowast

2= ((119875119880

2[2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

minus 1198992 (119861 +119882) (119863119884)2])


+119863 [1198992 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(21)

119876lowast

3= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

minus (119872 + 119864) (119863119883)2])


+119863 [1198993 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(22)

119876lowast

4= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

minus (119872 + 119864) (119863119883)2minus 1198801198994 (119861 +119882) (119863119884)

2])


+119863 [1198994 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(23)




102

1025

103

1035

104

1045

105

239 240 241 242 243 244 245 246

Q1



560565570575580585590595600605610

245 246 247 248 249 250 251

Q2



200210220230240250260270280290300

239 240 241 242 243 244 245 246

Q3

P = 1100

P = 1200

P = 1300



500550600650700750800850900

245 246 247 248 249 250 251

Q4

P = 1100

P = 1200

P = 1300









5 Numerical Example









119875 = 1100 119875 = 1200 119875 = 1300

119862119898

2300sim2350 2300sim2350 2300sim23501198991

2 2 21198761

10219sim10229 10339sim10349 10444sim10454EJTP1


2300sim2350 2300sim2350 2300sim23501198992

1 1 11198762

18902sim57786 19404sim59321 19862sim60721EJTP2

5846523sim3350315 5877907sim3446259 5905128sim3529477119862119898

2355sim240 2355sim240 2355sim2401198993

14 13 131198763

7873sim7772 8404sim8295 8415sim8306EJTP3

4463785sim4357297 4466066sim4359922 4468691sim4362513119862119898

2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355sim2365) 2 (119862

119898= 2355) 1

1 (119862119898= 2355sim2365) 1 (119862

119898= 2360sim2365)

1198764

64172sim67519 (119862119898= 2355sim2365) 65370 (119862

119898= 2355) 90178sim104684

90662sim99636 (119862119898= 2370sim2400) 89788sim102277 (119862

119898= 2355sim2365)

EJTP4


59500

60000

60500

61000

61500

62000

62500

63000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



33000

34000

35000

36000

37000

38000

39000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



34000

36000

38000

40000

42000

44000

46000

240 2405 241 2415 242 2425 243 2435 244 2445 245

P = 1100

P = 1200

P = 1300

EJTP

3



200002200024000260002800030000320003400036000

2455 246 2465 247 2475 248 2485 249 2495 250

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4


1018102

1022102410261028

103103210341036

229 230 231 232 233 234 235 236

Q1



100150200250300350400450500550600

229 230 231 232 233 234 235 236

Q2



707274767880828486

235 236 237 238 239 240 241

Q3

Z = 01

Z = 02

Z = 03



0

200

400

600

800

1000

1200

235 236 237 238 239 240 241

Q4

Z = 01

Z = 02

Z = 03















119875 = 1100 119875 = 1200 119875 = 1300

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3370094sim3546836 3465934sim3640884 3548895sim3722454


EJTP3 3425530sim4221153 3464154sim4251420 3497160sim4277287

119862119903 2455sim2500 2455sim2500 2455sim2500

1198994

2 (119862119903= 2455sim2465)

1 (119862119903= 2470sim2500)

2 (119862119903= 2455sim2460)

1 (119862119903= 2355sim2500)

2 (119862119903= 2455)

1 (119862119903= 2360sim2500)

1198764

61574sim59822 (119862119903= 2455sim2465)

77530sim66375 (119862119903= 2470sim2500)

62723sim61837(119862119903= 2455sim2460)

81338sim68135 (119862119903= 2465sim250)

63748 (119862119903= 2455)

85010sim69738 (119862119903= 246sim250)

EJTP4 2021540sim2977979 2116825sim3088182 2198873sim3183760


52000

54000

56000

58000

60000

62000

64000

66000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



0

10000

20000

30000

40000

50000

60000

70000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



3000032000340003600038000400004200044000460004800050000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

3



02000400060008000

100001200014000160001800020000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4


1021022102410261028

1031032103410361038

104

239 240 241 242 243 244 245 246

Q1



576578580582584586588590592594

245 246 247 248 249 250 251

Q2



6065707580859095

100

239 240 241 242 243 244 245 246

Q3

Z = 01

Z = 02

Z = 03



500

550

600

650

700

750

800

850

245 246 247 248 249 250 251

Q4

Z = 01

Z = 02

Z = 03





119885 = 01 119885 = 02 119885 = 03


EJTP1


EJTP2 5877907sim3446259 5561648sim3126491 5238827sim2800005


EJTP3 4466066sim4359922 4147330sim4041153 3822615sim3716446

119862119898 2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

1198764

65370 (119862119898= 2355)

89788sim102277 (119862119898= 236sim240)

65203 (119862119898= 2355)

89751sim102171 (119862119898= 236sim240)

65029 (119862119898= 2355)

89708sim102058 (119862119898= 236sim240)

EJTP4 1900021sim1000745 1567517sim667383 1227857sim362920



4600048000500005200054000560005800060000620006400066000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



2500027000290003100033000350003700039000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



3000032000340003600038000400004200044000460004800050000

240 2405 241 2415 242 2425 243 2435 244 2445 245

Z = 01

Z = 02

Z = 03

EJTP

3



10000

15000

20000

25000

30000

35000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4










119885 = 01 119885 = 02 119885 = 03

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3465846sim3640884 3146216sim3322506 2819872sim2997440

119862119903 2400sim245 2400sim245 2400sim245

1198993

17 (119862119903= 240sim241)

16 (119862119903= 2415sim2425)

15 (119862119903= 243sim2435)

14 (119862119903= 244sim2445)

13 (119862119903= 245)

19 (119862119903= 240)

18 (119862119903= 2405sim2415)

17 (119862119903= 242sim2425)

16 (119862119903= 243sim2435)

15 (119862119903= 244sim2445)

14 (119862119903= 245)

21 (119862119903= 240)

20 (119862119903= 2405sim241)

19 (119862119903= 2415sim242)

18 (119862119903= 2425)

17 (119862119903= 243sim2435)

16 (119862119903= 244sim2445)

15 (119862119903= 245)

1198763

8295sim7997 (119862119903= 240sim241)

8277sim811 (119862119903= 2415sim2425)

8221sim8032 (119862119903= 243sim2435)

8325sim8112 (119862119903= 244sim2445)

8417 (119862119903= 245)

7453 (119862119903= 240)

7684sim7399 (119862119903= 2405sim2415)

7629sim7471 (119862119903= 242sim2425)

7709sim7533 (119862119903= 243sim2435)

7582 (119862119903= 244sim2445)

7834 (119862119903= 245)

6762 (119862119903= 240)

6940sim6814 (119862119903= 2405sim241)

6998sim6860 (119862119903= 2415sim242)

7048 (119862119903= 2425)

7252sim7088 (119862119903= 243sim2435)

7296sim7113 (119862119903= 244sim2445)

7323 (119862119903= 245)

EJTP3 3823707sim4477773 3503787sim4159040 3178546sim3834324

119862119903 2455sim250 2455sim250 2455sim250

1198994

2 (119862119903= 2455)

1 (119862119903= 246sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

1198764

62723 (119862119903= 2455)

61837sim68135 (119862119903= 246sim250)

62565sim61676 (119862119903= 2455)

2465sim250 (119862119903= 246sim250)

62400sim61508 (119862119903= 2455sim246)

81257sim67924 (119862119903= 246sim250)

EJTP4 2116836sim3088182 1785043sim2763725 1446121sim2432425












References













































1 Introduction














2

1

Product 1

Product 2

Stock

Stock

Stock

Machine

Randomdemand 1

Random

Random

demand 2

demand n

Product n

n







3 Notations







Decision Variables


4 Production Policy


lowast(119880lowast

= 119880lowast

119894119895119896 119910lowast

119894119895119896and 120575

lowast



Period 1

Δt Δt

j = 1 j = 2 j = 3

1205751 1205752 1205753

Period k


Subperiod j












Formally


Production cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + Cp (119894) times 119880119894119895119896)

Holding cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1



(2)





minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(3)





(4)



119896

sum

119904=1

119863 (119894 119904) + Stock min (119894 119896)

le

119896

sum

119904=1

119901

sum

119895=1

(119910(119894119895119904) times 119880119894119895119904) + stock init (119894 119904 = 0)

forall 119894 = 1 119899

(5)





Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(6)






Min[Cs times 119864 [119878 (119867)2]

+

119867minus1

sum

119896=0

(Cs times 119864 [119878 (119896)2] + Cp times 119864 [119906 (119896)

2])]

(7)



Min (Ζ (119880))

119880 = 119880119894119895119896 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(8)with119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]




minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(10)


(11)


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(12)

119901

sum

119895=1





119899

sum

119894=1

119910119894119895119896 = 1 forall 119895 = 1 119901 for 119896 = 1 119867

119901

sum

119895=1

119910119894119895119896 = 1 forall 119894 = 1 119899 for 119896 = 1 119867

(14)

119910119894119895119896 isin0 1 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(15)










minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(17)



119864 [119889119894 (119896)] = 119889119894 (119896) Var [119889119894 (119896)] = 1205902(119889119894 (119896))

forall 119894 = 1 119899 119896 = 1 119867

(18)



forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(19)

We note that

119864 [119880119894119895119896] = 119894119895119896 = 119880119894119895119896 (20)


Var (119880119894119895119896) = 0

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(21)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(22)




sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(23)


Proof We know that


(24)


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896) ge 0) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) ge 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

ge 119889119894 (119896) minus 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))

ge119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896))) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(25)

Noting that

119883 =119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896)) (26)



119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))ge 119883) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(27)



Hence

119878119894(119896minus1)times119901 +

119901

sum

119895=1


(120579119894)

forall 119894 = 1 119899 119896 = 1 119867

(28)

Therefore

119901

sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(29)




119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(30)





119880lowast= Min

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]]

(31)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(120579119894)


forall 119894 = 1 119899 119896 = 1 119867


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(32)



q = 1 q = 2

j = 1j = 2 j = p

Deg

rada

tion

rate

k = 1 k = 2 k = 3

T t2T

1205822 1205822p1205821

120582p+1

120575p+1
















(q minus 1) times T



T


1

2

3

Δt

120575((k+m)timesp)






119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896


times1

119880119894 nomΔ119905times 120582119899 (119905)]


(34)



Therefore




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(35)


Note that


119879] + 1 (36)




= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(37)




119879 = Round [119867 times 119901

119873] (38)








120601119879

(119880119873)= [

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(39)



=

119873+1

sum

119902=1

120601119879

(119880119873) (40)



120601(119880119873)

=

119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(41)

Note that119873 = 1 2




Γ (119873) = [

[

Mc times119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

+Mp times 119873]

]

(43)







120597Γ (119873)

120597119873= 0 (44)



lim119873rarr1

Γ (119880119873) = lim119873rarr1

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


rarrconstant

)

= 119872119888 times 120601 (119880 1) + 119872119901

lim119873rarr+infin

Γ (119880119873) = lim119873rarr+infin

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr0

+ 119872119901 times 119873⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr+infin

)

= +infin

(45)


Moreover

Γ (119880119873 + 1) minus Γ (119880119873) ge 0

997904rArr [119872119888 times 120601 (119880 (119873 + 1)) + 119872119901 times (119873 + 1)]



997904rArr 120601 (119880 (119873 + 1)) minus 120601 (119880119873) le119872119901

119872119888

(46)

In addition

Γ (119880119873) minus Γ (119880119873 minus 1) le 0

997904rArr [119872119888 times 120601 (119880119873) + 119872119901 times (119873)]




119872119888

(47)




exist119873lowast119904119894 120585119873 le

119872119901

119872119888

le 120585119873minus1 (48)

with

120585119873 = 120601 (119880119873) minus 120601 (119880 (119873 + 1)) (49)



119873lowastexist119904119894

120601 (119880 (119873 + 1)) minus 120601 (119880119873) ge 0

120601 (119880119873) minus 120601 (119880 (119873 minus 1)) le 0

lim119873rarr1

Γ (119880119873) = 119862119900119899119904119905119886119899119905

lim119873rarr+infin

Γ (119880119873) = +infin

(50)



6 Numerical Example




1198891 = 200 120590 (1198891) = 15

1198892 = 110 120590 (1198892) = 09

1198893 = 320 120590 (1198893) = 12

(51)





Table 1



Table 2









1205752

1205753

1205754

1205755

1205756

1205757

1205758

1205759

12057510

12057511

12057512



12057514

12057515

12057516

12057517

12057518

12057519

12057520

12057521

12057522

12057523

12057524






120582119899 (119905) =120572

120573times (

119905

120573)

120572minus1

=2

12times (

119905

12) (52)















Nlowast

d = di k ( )

Ulowast= Uijk ( )

k =

i =

k =i =j =

1 H1 n

1 p1 H1 n


0 2 4 6 8 10

4000

5000

6000

7000

8000


The t

otal

cost

of m

aint

enan

ceΓ

(N)



Γlowast(119873) (um) 119873




=





8 Conclusion






Appendices


We have119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A1)

Also


minus Ent [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A2)

Therefore


]


minus Ent [119895

119901] times 119889119894 (119896)]


minusEnt [119895

119901] times 119889119894 (119896) ])

2

]


]


minus [Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896))])

2

]


]



times (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]


]


]


times(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ 119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A3)



]


]


times 119864 [(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A4)




(A5)

Therefore


]


]

+(Ent [119895

119901])

2

times 119864 [(119889119894 (119896) minus 119889119894 (119896))2

]]

(A6)

We know that

119864 [(119909119896 minus 119909119896)2] = Var (119909119896)

(Int [119895

119901])

2

= Int [119895


119895

119901le 1

(A7)

Therefore


]


119901] times Var (119889119894 (119896))

(A8)


Finally


+ Ent [119895

119901] times Var (119889119894 (119896))

(A9)

Consequently

(i) for 119896 = 1

(a) 119895 = 1

Var (1198781198941) = Var (1198781198940) + (Ent [ 1

119901]) times Var (119889119894 (1))

(A10)

(b) 119895 = 2

Var (1198781198942) = Var (1198781198940) +2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A11)

(c) 119895 = 119901

Var (119878119894119901) = Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A12)

(ii) for 119896 = 2

(a) 119895 = 1

Var (119878119894119901+1) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+ Ent [ 1

119901] times Var (119889119894 (2))]

(A13)

(b) 119895 = 2

Var (119878119894119901+2) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (1)) + Ent [ 1

119901]

times Var (119889119894 (2)) + Ent [ 2

119901] times Var (119889119894 (2))]

(A14)

(c) 119895 = 119901

Var (119878119894(2times119901)) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+

119875

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119901))]

(A15)


(a) 119895 = 1


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

1

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A16)

(b) 119895 = 2


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A17)



= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A18)

On the other hand


]




]


]


]



]]

(A19)

We know that



(A20)


Hence


]




]



]





]

(A21)

Consequently


]



(A22)

Noting that


]





(A23)




= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]


]

= [ Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A24)

Consequently


]

= [ 1205902(1198781198940) +

119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times 1205902(119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A25)


119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(A26)



(i) for 119896 = 1

(a) 119895 = 1

1205821 (119905) = (1205820) times (1 minus In [0

119902 times 119879]) +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (119905)

(B1)

(b) 119895 = 2

1205822 (119905) = 1205821 (1205751) times (1 minus In [1

119902 times 119879])

+

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)


1205822 (119905) = (1205820 +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (120575(1)))


119902 times 119879]) +

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)

(B2)

(c) 119895 = 119901


119902 times 119879])

+

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)

120582119901 (119905) = [(1205820 +

119901minus1

sum

119897=1

119899

sum

119894=1

1198801198941198971 times Δ119905

119880119894 nom times 120575119897

times 120582119899 (120575(119897)))


119902 times 119879]) +

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)]

(B3)


(a) 119895 = 1

120582((119896minus1)times119901)+1 (119905)


119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

120582((119896minus1)times119901)+1 (119905)

= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

(B4)



= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(B5)



References
























1 Introduction



















16

14

12

10

08

06

Ship

men

t cos

t (lowast$100)



R2= 0926



119864 (119876) = 119901119864 (min 119876119863) + 119904119864(119876 minus 119863)+ minus 119888119876

minus 119886 + 119887 ln [119864 (min 119876119863)]

minus 119886 + 119887 ln [119864(119876 minus 119863)+]



(1)









times[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2



(3)


119864(119863 minus 119876)+le[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2

(4)


119889119864 (119876)

119889119876=119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876 minus 120583)

2[1205902 + (119876 minus 120583)2]12

minus 1198872119876 + 2120583 (119876 minus 120583) [120590

2+ (119876 minus 120583)

2]minus12


(5)

1198892119864 (119876)

1198891198762= minus

(119901 minus 119904) 1205902

2[1205902 + (119876 minus 120583)2]32

minus 119887

minus 2 + 21205832minus 21198762

+ 4120583[1205902+ (119876 minus 120583)

2]12

+2120583 (minus1 minus 120583

2+ 4120583119876 minus 3119876

2)

[1205902 + (119876 minus 120583)2]12

minus81205832(119876 minus 120583)

2

1205902 + (119876 minus 120583)2

+(119876 minus 120583)

2(21205833+ 2120583 minus 2120583119876

2)

[1205902 + (119876 minus 120583)2]32

sdot minus1 minus 1205832+ 1198762+ 4120583[120590

2+ (119876 minus 120583)

2]12

minus2

(6)




Iteration 119894 119876119894

1198911015840(119876119894) 119891

10158401015840(119876119894) 119891

1015840(119876119894)11989110158401015840(119876119894) 119876

119894+1





Let119876119894+1 = 119876119894 minus (1198911015840(119876119894)119891


1198911015840(119876119894) =

119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876119894 minus 120583)

2[1205902 + (119876119894 minus 120583)2]12

minus 1198872119876119894 + 2120583 (119876119894 minus 120583) [120590

2+ (119876119894 minus 120583)

2]minus12


2]12

(7)

From (6) we set

11989110158401015840(119876119894) = minus

(119901 minus 119904) 1205902

2[1205902 + (119876119894 minus 120583)2]32

minus 119887 minus 21198762

119894minus 2 + 2120583

2+ 4120583[120590

2+ (119876119894 minus 120583)

2]12

+2120583 (4120583119876119894 minus 3119876

2

119894minus 1205832minus 1)

[1205902 + (119876119894 minus 120583)2]12

+(119876119894 minus 120583)

2(21205833+ 2120583 minus 2120583119876

3

119894)

[1205902 + (119876119894 minus 120583)2]32

minus81205832(119876119894 minus 120583)

2

1205902 + (119876119894 minus 120583)2

(8)






0 119876lowast


0= exp(1minus


0= 119876lowast



is119876lowastlowast119894= (119876119904

119894+119876lowast



119894= (119876119904

119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887








10158401015840(1198760)) =








lowast) 119864(119876

lowast) ()


Average 405

12

10

8

6

4

2

0

5 10 15 20

Order quantity Q

Expe

cted

pro

fitE

(Q)














119864 (119878) = 119901119864 (min 119878 119863) + 119904119864(119878 minus 119863)+


minus 119886 + 119887 ln [119864(119878 minus 119863)+]







2]12

minus (119878 minus 120583)

2



(10)


119901 minus 119904

2[1205902+ (119903 minus 120583)

2]12

minus 119903 + (119888 minus 119904) 119903



2[1205902+ (119878 minus 120583)

2]12



= 0

(11)





1015840(119903119894)) where

119891 (119903119894) =119901 minus 119904

2[1205902+ (119903119894 minus 120583)

2]12

minus 119903119894 + (119888 minus 119904) 119903119894

+ 119887 ln minus1205832 minus 1 + 1199032119894+ 2120583[120590

2+ (119903119894 minus 120583)

2]12


2[1205902+ (119878 minus 120583)

2]12



1198911015840(119903119894) =

2119888 minus 119901 minus 119904

2+

(119901 minus 119904) (119903119894 minus 120583)

2[1205902 + (119903119894 minus 120583)2]12

+ 1198872119903119894 + 2120583 (119903119894 minus 120583) [120590

2+ (119903119894 minus 120583)

2]minus12


2]12

(12)










=119888

120588(119898 + 119889) 120583 minus (119898 + 119889) 119864[119863 minus 119866 (119876)]

+




(13)


119864[119863 minus 119866 (119876)]+le[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

(14)


119864 (119876) ge119888

120588

(119898 + 119889) 120583 minus (119898 + 119889)

times[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

minus (120588 + 119889 minus 1)119876

minus 2119886 + 2119887 ln 2


+ 2 (120583 + 120588119876 minus 119876)

times [1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

(15)


119889119876

= minus119888 (119898 + 119889)

2[1


119888

120588(120588 + 119889 minus 1)





(16)

where119883 = 1205902 + 120588119902119876 + (120588119876 minus 120583)2

1198892119864 (119876)

1198891198762= minus119888 (119898 + 119889)

2[minus120588

4(119902 minus 2120583 + 2120588119876)

2119883minus32

+ 120588119883minus12]

minus 1198871198841015840119885 minus 119884119885

1015840

1198852


12



+ 2 (120583 + 120588119876 minus 119876)11988312

1198841015840= 4120588 (1 minus 120588) minus 2120588


minus1205882

2(120583 + 120588119876 minus 119876) (119902 minus 2120583 + 2120588119876)

2119883minus32

1198851015840= 2 (1 minus 120588) (2120588119876 + 120583) minus 120588119902

minus 2 (1 minus 120588)11988312+120588

4(120583 + 120588119876 minus 119876)


(18)





1015840(119876119894)119891

10158401015840(119876119894)) where 119891

1015840(119876119894) and




2119864(119876119894+1)119889119876

2




0 119876lowast


0= exp(1minus


0= 119876lowast



is 119876lowastlowast119894= (119876119904

119894+ 119876lowast




119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887


119894 = 119894 + 1



2





2

119895 respectively



1 119876lowast2 and

119876lowast

119873

max1198761 119876119873

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583[1205902

119895+ (119876119895 minus 120583119895)

2

]12

(19)

Subject to119873

sum

119895=1

119888119895119876119895 le 119861 (20)


119871 (1198761 119876119873 120582)

=

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583119895[1205902

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582(

119873

sum

119895=1

119888119895119876119895 minus 119861)

(21)


120597119871 (1198761 119876119873 120582)

120597119876119895

=119901119895 + 119904119895 minus 2119888119895


2[1205902119895+ (119876119895 minus 120583119895)

2

]12

minus 119887

2119876119895 + 2120583119895 (119876119895 minus 120583119895) [1205902

119895+ (119876119895 minus 120583119895)

2

]minus12

minus1 minus 1205832 + 1198762119895+ 4120583119895[120590

2

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582119888119895

(22)


119873with

















Step 1 Let 119895 = 1 2 3 4


1= 8731 119876lowast

2= 7762 119876lowast

3= 11072 and 119876lowast

4

= 21243 Check sum4119895=1119888119895119876lowast

119895= $92 gt $80 where (20) is not



3is set to 0


119895= $61 lt $80 (20) is satisfied










Highlights









References



































1 Introduction













max 120601

st 120582119883 le 119909119863

119900

120582119884119863ge 120601119910119863

119900

120582119884119880

ge 120601119910119900119880

119890120582 = 1

120582 ge 0

(1)

in which 119910119900119880= minus119884



max 120601

st 120582119883119863le (1 minus 120601) 119909

119863

119900

120582119883119880

le (1 minus 120601) 119909119900119880

120582119884119863ge (1 + 120601) 119910

119863

119900

120582119884119880

ge (1 + 120601) 119910119900119880

119890120582 = 1

120582 ge 0

(2)

in which 119910119900119880= minus119884

119880+ V gt 0 and 119883

119880

= minus119883119880+ 119908 gt 0




min 1 minus [

[

119908119900 +1

119898 + 119904(sum

119894isin119868119863

119905minus119863

119894+ sum

119903isin119874119863

119905+119863

119903)]

]

st119899

sum

119895=1

120582119895119909119863

119894119895+ 119905minus119863

119894= 119909119863

119894119900minus 119908119900 119894 isin 119868119863

119899

sum

119895=1

120582119895119909119880

119894119895+ 119905minus119880

119894= 119909119880

119894119900+ 119908119900 119894 isin 119868119880

119899

sum

119895=1

120582119895119910119863

119903119895minus 119905+119863

119903= 119910119863

119903119900+ 119908119900 119903 isin 119874119863

119899

sum

119895=1

120582119895119910119880

119903119895minus 119905+119880

119903= 119910119880

119903119900minus 119908119900 119903 isin 119874119880

119899

sum

119895=1

120582119895 = 1

(3)


min 120579

st119899

sum

119895=1

120582119895119909119894119895 le 120579119909119894119900 forall119894 isin 119868

119899

sum

119895=1

120582119895119910119863

119903119895ge 119910119863

119903119900forall119903 isin 119874

119863

119899

sum

119895=1

120582119895119910119880

119903119895le 120579119910119880

119903119900forall119903 isin 119874

119880

120582119895 ge 0 forall119895 isin 119873

(4)




min 120579

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le 120579

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 ge

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(5)


max119898

sum

119894=1

119904minus

119894+

119904

sum

119903=1

119905+

119903

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 + 119904minus

119894= 120579lowast

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 minus 119905+

119903=

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

119904minus

119894ge 0 119905

+

119903ge 0 119894 = 1 119898 119903 = 1 119904

120582119895119896 ge 0 119896 119895 = 1 119899

(6)




119909119894119900 =

119899

sum

119895=1

120582119900lowast

119895119909119894119895 = 120579

lowast119909119894119900 minus 119904

minuslowast

119894119894 = 1 119898

119910119903119900 =

119899

sum

119895=1

120582119900lowast

119895119910119903119895 = 119910119903119900 + 119905

+lowast

119903119903 = 1 119904

(7)


119878119894 =

119899

sum

119895=1

119909119894119895 minus

119899

sum

119895=1

119909119894119895 ge 0 119894 = 1 119898

119879119903 =

119899

sum

119895=1

119910119903119895 minus

119899

sum

119895=1

119910119903119895 ge 0 119903 = 1 119904

(8)


max119899

sum

119895=1

119904

sum

119903=1

119906119903119910119903119895 +

119899

sum

119896=1

120577119896

st119899

sum

119895=1

119898

sum

119894=1

V119894119909119894119895 = 1

119904

sum

119903=1

119906119903119910119903119895 minus

119898

sum

119894=1

V119894119909119894119895 + 120577119896 le 0 119895 119896 = 1 119899

119906119903 ge 0 119903 = 1 119904

V119894 ge 0 119894 = 1 119898

(9)



5 Proposed Method



119883 = 119909119894119895 isin 119877119898times119899

+ 119884

119863= 119910119863

119903119895 isin 119877119904119863times119899

+

119884119880= 119910119880

119905119895 isin 119877119904119880times119899

+

(10)


max 120593

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le

119899

sum

119895=1

119909119894119895 minus 120593119877119909119894 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119863

119903119895ge

119899

sum

119895=1

119910119863

119903119895+ 120593119877119910

119863

119903119903 = 1 119904

119863

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119880

119905119895le

119899

sum

119895=1

119910119880

119903119895minus 120593119877119910

119880

119905119905 = 1 119904

119880

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(11)

where119877119909119894119877119910119863

119903 and119877119910119880



119863

119903 119877119910119880



119903 119877119910119880

119905) vector as

follows

119877119909119894 =

119899

sum

119895=1

119909119894119895 minus 119899 (min 119909119894119895119895=1119899) 119894 = 1 119898

119877119910119863

119903=

119899

sum

119895=1

119910119863

119903119895minus 119899 (max 119910119863

119903119895119895=1119899

) 119903 = 1 119904119863

119877119910119880

119905=

119899

sum

119895=1

119910119880

119905119895minus 119899 (min 119910119880

119905119895119895=1119899

) 119905 = 1 119904119880

(12)


119863

119903 119877119910119880

119905) can be

chosen as follows

(119877119909119894 119877119910119863

119903 119877119910119880

119905) = (

119899

sum

119895=1

119909119894119895

119899

sum

119895=1

119910119863

119903119895

119899

sum

119895=1

119910119880

119905119895) (13)











119863

119903 119877119910119880



6 Numerical Example





7 Conclusion







References


































1 Introduction







20 40 60 80 100Utilization ()

Cycle

tim

eIn

crea

sing


20 40 60 80 100

Low variability

Utilization ()

High variability

Cycle

tim

eIn

crea

sing

te

to




CT119902 = 119881119880119879 (1)



2 Literature Review














119886= 1205902

1198861199052

119886and

1198622

119890= 1205902

1198901199052



2



2

119890gt 1198622



CT119902 = (1198622

119886+ 1198622

119890

2)(

119906

1 minus 119906) 119905119890 (2)

1198622

119889= 11990621198622

119890+ (1 minus 119906

2) 1198622

119886 (3)










Patient



Home

10 min24 min

20 min11 min11 min

98 min0 min

10 min0 min

3 min3 min0 min

10 min




Clean room




ORemergence

time

Patient out of room

Inpatient room


(Nj)

WW W W W W


1205902


1198882




arrivals






2


as 083

119860 =119898119891

119898119891 + 119898119903

(4)

119905119890 =119905119900

119860 (5)

119906 =119903119886

119903119890

=119903119886119905119890

119898 (6)

1198622

119890= 1198622

119900+ 2119860 (1 minus 119860)

119898119903

119905119900

(7)






Specialty 1

FCFS

Specialty 2

Specialty 15


Operating room

Recoverroom

First stage (W1)

Second stage (W2)

(Mj)

(Nj)

M1

M2

M12

N1

N2

N3

N4

N5

N24





119905119890 = 119905119900 +119905119904

119873119904

(8)

1205902

119890= 1205902

119900+1205902

119904

119873119904

+119873119904 minus 1

1198732119904

1199052

119904

1198882

119890=1205902

119890

1199052119890

(9)




119890= 749







CT119902 =1198622

119886+ 1198622

119890

2

119906

1 minus 119906119905119890

=(099 + 749)

2(

089

1 minus 089) 14543

= (424) (809) (14543)

(10)














120583 (11)




(12)



5 Empirical Results








1 10800 813 10800

813≒ 13

2 10800 159 10800

159≒ 68

3 10800 134 10800

134≒ 81

4 10800 346 10800

346≒ 31

5 10800 185 10800

185≒ 58

Input

Output


Simulationmodel






1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

0020018001601400120010008000600040002

0

CV


WIP




Parameters



Patient



suture and finish

Home

Leveling

Billing and coding

Waiting time= 5 min




Inpatient room

W W W W





10 min0 min

98 min0 min5 min




Decision variables



Minimize24

sum

119895=1

119874119895 (13)


5

sum

119894=1

119860 119894119895 = 1 forall119895 (14)

119862119894 ge 1 forall119894 (15)

5

sum

119894=1

119862119894 = 24 (16)

119860 119894119895 isin 0 1 forall119894119895 (17)






















Acknowledgments


References















































Editorial Board










Contents




























Acknowledgments




Chia-Yu Hsu







1 Introduction














2 Related Work




(a) (b) (c)

(d) (e) (f)



3 Proposed Approach






119898119896119873119892]



]


120579119897119873119892]


119881119897119873119892]


119897= [1205951198971

1205951198972 120595119897119873119892]


119892= [1205951198921 1205951198922

120595119892119873119892]


Consensuspartition


WBMs

1 clustering

q clustering

2 clustering



Labels120587 q










1205871

1205872

1205873

x1

1 1 1x2

1 1 1x3

1 1 1x4

2 2 1x5

2 2 2x6

3 1 2x7

3 1 2x8

3 1 2



V2

V3

V4

V5

V6

V7

V8

119867(1)

ℎ11

1 1 1 0 0 0 0 0ℎ12

0 0 0 1 1 0 0 0ℎ13

0 0 0 0 0 1 1 1

119867(2) ℎ

211 1 1 0 0 1 1 1

ℎ22

0 0 0 1 1 0 0 0

119867(3) ℎ

311 1 1 1 0 0 0 0

ℎ32

0 0 0 0 1 1 1 1


119872(119910119903) =

119873119887

sum

119903=1

119890minus119898120573119889(119910119903 119910119904) 119903 = 1 2 3 119873119887 (1)

where

120573 = (119889 (119910119903 minus 119910wc)

119873119887

)

minus1

(2)







(a) Binary value

51015202530






SSE =

119873119888

sum

119896=1

sum

x119894isin119862119896(x119894 minusm119896)

2 (3)






119869119890 =sum119873119888


119870 (5)

In addition

119889max (p119894Z119897) = max119896=12119873119888

[[

[

sum

forallx119894isin119862119897119896

119889 (x119894m119897119896)119899119897119896

]]

]

(6)



[119889 (m119897119906m119897V)] (7)





119894


119896


119894Z119897)


119901119906(120595119897(119905) minusΘ

119897(119905)) + 119888

119892119906(120595119892(119905) minusΘ

119897(119905)) (i)

Θ119897(119905 + 1) = Θ

119897(119905) + V

119897(119905 + 1) (ii)










1012

1518

2315221370

1184 1098945

02004006008001000120014001600

0

5

10

15

20

25

03 04 05 06 07

SSE

Clus

ter n

umbe

r









max 1 1198861

04119888119901

2 1198862

03119888119892

2 1198863

03Iteration 500



Hsu and Chien [8] 1184 1192 1203 1248 1322


KB 2889 3092 3003 4083 3570KC 3331 2490 2603 3169 2603PB 5893 3601 6566 5839 6308PC 4627 4873 3330 3787 6112

Clusteringensemble






Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9

Group 10

Group 11

Group 12

Group 13

Group 14

Group 15





119879119890

(8)



(a) (b) (c)

(d) (e) (f)




Hsu and Chien [8] 04 04 04 04 04


KB 10 10 10 10 10KC 10 10 10 10 10PB 10 10 10 10 10PC 10 10 10 10 10

Clusteringensemble





precision =TP

TP + FP

recall = TPTP + FN

(9)




=2TP

FP + FN + 2TP (10)






Precision

A 070 084 092 092 092 098B 050 066 096 092 062 096C 060 064 100 100 060 100D 070 098 092 092 098 100E 060 094 082 082 098 098F 080 098 076 076 098 098

Avg 065 084 090 089 085 098



Recall

A 100 100 100 093 100 100B 100 097 07 078 083 100C 100 094 067 084 067 097D 100 081 100 100 100 100E 100 079 100 100 100 100F 100 100 100 100 100 100

Avg 100 092 090 093 092 100



119865-measure

A 082 091 096 092 096 099B 067 079 081 084 071 098C 075 076 08 091 063 098D 082 089 096 096 099 100E 075 086 090 090 099 099F 089 099 086 086 099 099

Avg 078 087 088 090 088 099



5 Conclusion









Acknowledgment


References







































1 Introduction





























119903119894119895 = 05 (1 + log9119886119894119895) (1)









119903119894119895 =119894 minus 119895 + 1


(2)

119903119894119895 = 1 minus 119903119895119894 forall119894 lt 119895 (3)


According to (2)

11990321 = 1 minus 03 = 07

11990331 = 15 minus 03 minus 06 = 06


11990332 = 1 minus 06 = 04

11990342 = 15 minus 06 minus 08 = 01

11990343 = 1 minus 08 = 02

(4)

According to (3)

11990313 = 1 minus 06 = 04

11990314 = 1 minus 03 = 07

11990324 = 1 minus 01 = 09

(5)


PR =[[[

[

05 03 04 07

07 05 06 09

06 04 05 08

03 01 02 05

]]]

]

(6)





119878119886119887 = 1 minus

10038161003816100381610038161198861 minus 11988711003816100381610038161003816 +

10038161003816100381610038161198862 minus 11988721003816100381610038161003816 +

10038161003816100381610038161198863 minus 11988731003816100381610038161003816 +

10038161003816100381610038161198864 minus 11988741003816100381610038161003816

4 (7)


AM =

[[[[

[


119878119886119887

1198781198641 1198781198642 sdot sdot sdot 1

]]]]

]

(8)



AA119886 =1

119864 minus 1

119864

sum

119887=1119886 =119887

119878119886119887 forall119886 (9)


RA119886 =AA119886

sum119864

119886=1AA119886

forall119886 (10)










(12)


1

119894119895 1199032

119894119895 1199033

119894119895 1199034

119894119895)





119903119894119895 =1199031

119894119895+ 2 (119903

2

119894119895+ 1199033

119894119895) + 1199034

119894119895

6 (13)


PR119896 =[[[[

[


1

1199031198731 1199031198732 sdot sdot sdot 1

]]]]

]

(14)






120575119894119896 =

119873

sum

119895=1119895 =119894

119903119894119895 minus

119873

sum

119895=1119895 =119894

119903119895119894 forall119894 119896 (15)


DM =

[[[[

[


1205751198731 1205751198732 sdot sdot sdot 120575119873119872

]]]]

]

(16)




aw119896 =(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119896 minus 120575119895119896

10038161003816100381610038161003816)1(119901minus1)

sum119872

119897=1(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119897 minus 120575119895119897

10038161003816100381610038161003816)1(119901minus1)

forall119896 119901 gt 1 (17)

aw119896 =(sum119873

119894=11205752

119894119896)12(119901minus1)

sum119872

119897=1(sum119873

119894=11205752119894119897)12(119901minus1)

forall119896 119901 gt 1 (18)



119872



119862119894 =

119872

sum

119896=1

aw119896120575119894119896 119894 = 1 2 119873 (19)





transformation


relations










4 Numerical Example





11987711 =[[[

[




]]]

]

11987721 =[[[

[




]]]

]

11987731 =

[[[[[[

[

minus1

3minus minus

minus minus1

4minus

minus minus minus 1


]]]]]]

]

(20)


11987711 =[[[

[





]]]

]

11987721 =[[[

[





]]]

]

11987731 =

[[[[[[

[

minus1

31

31

31

3minus minus

minus minus1

41

41

41

4minus



]]]]]]

]

(21)


11987712 =[[[

[





]]]

]

11987722 =[[[

[





]]]

]


11987732 =

[[[[[[

[

minus1

41

41

41

4minus minus


minus minus minus1

31

31

31

3


]]]]]]

]

(22)


then become 119877101584031

and 1198771015840


replaced by 119877101584031and 1198771015840


1198771015840

31=

[[[

[





]]]

]

1198771015840

32=

[[[

[





]]]

]

(23)



PR1 =[[[

[





]]]

]

PR2 =[[[

[





]]]

]

(24)



2are

PR10158401=

[[[

[

0500 0290 0100 0100

0710 0500 0311 0311

0900 0689 0500 0500

0900 0689 0500 0500

]]]

]

PR10158402=

[[[

[

0500 0218 0265 0055

0782 0500 0547 0337

0735 0453 0500 0290

0945 0663 0710 0500

]]]

]

(25)


1and PR1015840

2are consistent


DM =[[[

[

minus2019 minus1923

minus0336 0331

1178 minus0045

1178 1637

]]]

]

(26)




11990312

11990323

11990334


119886119887)

11987812

0925 0775 100011987813

0900 0880 090011987823


119886)

AA1 0913 0828 0950AA2 0900 0730 0950AA3 0888 0783 0900


RA1 0338 0354 0339RA2 0333 0312 0339RA3 0329 0334 0321


120573 = 05

CC1 0319 0327 0320CC2 0317 0306 0320CC3 0364 0367 0361


23= (022 028 034 041) 119903

34= (040 047 053 060)


23= 0311 119903

34= 0500



Very low

02 04 06 1008

02

04

06

08

10

Mem

bers

hip

valu

e



Very low

More highMore

low

02

04

06

08

10

Mem

bers

hip

valu

e




Alternative 119894 120575119894119896

119862119894(ADM) 119862



aw119896by SDM 0509 0491





5 Conclusion











Acknowledgments


References


















































1 Introduction

















(i) Supplier Side





(iii) Retailer Side



22 Assumptions












3 Model Formulation




Repaired products

Defective products

Normal products











=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]

(1)




nQ


nQD

nQD

nQD

nQ

nQ


Z(1 minus R)nQ


nZQ


nQ


P

Q

t

t

t

Q

Q

Q

Q

P

QD

QD (n minus 1)Q

D

nRZQ




119885(1 minus 119877)]119876119894




=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)


2

2119899 [1 minus 119885 (1 minus 119877)]119876119894

(2)




nQ


Lm =nQ

D

X

Q

t

Interest income

Figure 3 119871119898lt 119883

Lm =nQ

D

nQ


X

Q

Interest income

Opportunity cost

t

Figure 4 119871119898ge 119883



(ii) Ordering cost = 119860119898119863119899119894119876119894










TP1198981





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


119876119894

2)

(3)



2 [1 minus 119885 (1 minus 119877)] (4)





TP1198982





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)


minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(5)




22[1 minus 119885(1 minus 119877)]119899119894119876119894



TP1198983





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119894119876119894 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus 119862119898119868mp (119863119884 minus119876119894

2)

(6)



119877)]119899119894119876119894





TP1198984





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119876 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus119862119898119868mp (119863119884)

2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(7)




(ii) Ordering cost = 119860119903119863119899119894119876119894






TP1199031



= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

+ 119875119903119868re (119863119884 minus119876119894

2)





TP1199032



+ interest income

= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

minus119862119903119868rp (119876119894 minus 119863119884)

2

2119876119894

+119875119903119868re (119863119884)

2

2119876119894

(9)


EJTP119894 (119876119894 119899119894)

=





(10)

whereEJTP1 (1198761 1198991)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198761 [1198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198991


ℎ1199031198761

2minus

ℎ11990411986311989911198761

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198761

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989911198761)

2 [1 minus 119885 (1 minus 119877)]

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198761

2)

EJTP2 (1198762 1198992)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198762 [1198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198992


ℎ1199031198762

2minus

ℎ11990411986311989921198762

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989921198762)

2 [1 minus 119885 (1 minus 119877)]

+(119862119903119868me minus 119862119903119868rp) (1198762 minus 119863119884)

2

21198762

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198762

EJTP3 (1198763 1198993)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198763 [1198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198993


ℎ1199031198763

2minus

ℎ11990411986311989931198763

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198763

2)

EJTP4 (1198764 1198994)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]


minus ℎ1198981198631198764 [1198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198994


ℎ1199031198764

2minus

ℎ11990411986311989941198764

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198764

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119898119868se minus 119862119898119868mp) (11989941198764 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119903119868me minus 119862119903119868rp) (1198764 minus 119863119884)

2

21198764

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198764

(11)



120597EJTP1 (1198761 1198991)1205971198761

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198762

1

minus ℎ1198981198631198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198991


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp)

2

(12)

120597EJTP2 (1198762 1198992)1205971198762

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

2

minus ℎ1198981198631198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198992


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987622

+(119862119903119868me minus 119862119903119868rp) [119876

2

2minus (119863119884)

2]

211987622

(13)

120597EJTP3 (1198763 1198993)1205971198763

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198762

3

minus ℎ1198981198631198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198993

2119875 [1 minus 119885 (1 minus 119877)]2

minus(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

+(119862119898119868se minus 119862119898119868mp) [(11989931198763)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

minus(119875119903119868re minus 119862119898119868mp)

2

(14)

120597EJTP4 (1198764 1198994)1205971198764

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198762

4

minus ℎ1198981198631198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198994


2

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987624

+(119862119898119868se minus 119862119898119868mp) [(11989941198764)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

+(119862119903119868me minus 119862119903119868rp) [119876

2

4minus (119863119884)

2]

211987624

(15)


102

1025

103

1035

104

1045

105

2295 2305 2315 2325 2335 2345 2355

Q1



0

100

200

300

400

500

600

700


Q2


777879808182838485

235 236 237 238 239 240

Q3

P = 1100

P = 1200

P = 1300



0

200

400

600

800

1000

1200

235 236 237 238 239 240

Q4

P = 1100

P = 1200

P = 1300




1205972EJTP1 (1198761 1198991)

12059711987621

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198763

1

lt 0

(16)

1205972EJTP2 (1198762 1198992)

12059711987622

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198763

2

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987632

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987632

lt 0

(17)

1205972EJTP3 (1198763 1198993)

12059711987623

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

3

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

lt 0

(18)

1205972EJTP4 (1198764 1198994)

12059711987624

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198763

4

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987634

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987634

lt 0

(19)



60000

60500

61000

61500

62000

62500

63000

63500

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



30000

35000

40000

45000

50000

55000

60000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



43000432004340043600438004400044200444004460044800

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

3



60008000

10000120001400016000180002000022000

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4



119876lowast

1= ((2119863119875119880

2(119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991))

times (1198992 119875119880 [119880 (ℎ119898 (1198991 minus 1) + ℎ119903 +119882) +1198721198991]

+119863 [1198991 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(20)

119876lowast

2= ((119875119880

2[2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

minus 1198992 (119861 +119882) (119863119884)2])


+119863 [1198992 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(21)

119876lowast

3= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

minus (119872 + 119864) (119863119883)2])


+119863 [1198993 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(22)

119876lowast

4= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

minus (119872 + 119864) (119863119883)2minus 1198801198994 (119861 +119882) (119863119884)

2])


+119863 [1198994 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(23)




102

1025

103

1035

104

1045

105

239 240 241 242 243 244 245 246

Q1



560565570575580585590595600605610

245 246 247 248 249 250 251

Q2



200210220230240250260270280290300

239 240 241 242 243 244 245 246

Q3

P = 1100

P = 1200

P = 1300



500550600650700750800850900

245 246 247 248 249 250 251

Q4

P = 1100

P = 1200

P = 1300









5 Numerical Example









119875 = 1100 119875 = 1200 119875 = 1300

119862119898

2300sim2350 2300sim2350 2300sim23501198991

2 2 21198761

10219sim10229 10339sim10349 10444sim10454EJTP1


2300sim2350 2300sim2350 2300sim23501198992

1 1 11198762

18902sim57786 19404sim59321 19862sim60721EJTP2

5846523sim3350315 5877907sim3446259 5905128sim3529477119862119898

2355sim240 2355sim240 2355sim2401198993

14 13 131198763

7873sim7772 8404sim8295 8415sim8306EJTP3

4463785sim4357297 4466066sim4359922 4468691sim4362513119862119898

2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355sim2365) 2 (119862

119898= 2355) 1

1 (119862119898= 2355sim2365) 1 (119862

119898= 2360sim2365)

1198764

64172sim67519 (119862119898= 2355sim2365) 65370 (119862

119898= 2355) 90178sim104684

90662sim99636 (119862119898= 2370sim2400) 89788sim102277 (119862

119898= 2355sim2365)

EJTP4


59500

60000

60500

61000

61500

62000

62500

63000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



33000

34000

35000

36000

37000

38000

39000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



34000

36000

38000

40000

42000

44000

46000

240 2405 241 2415 242 2425 243 2435 244 2445 245

P = 1100

P = 1200

P = 1300

EJTP

3



200002200024000260002800030000320003400036000

2455 246 2465 247 2475 248 2485 249 2495 250

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4


1018102

1022102410261028

103103210341036

229 230 231 232 233 234 235 236

Q1



100150200250300350400450500550600

229 230 231 232 233 234 235 236

Q2



707274767880828486

235 236 237 238 239 240 241

Q3

Z = 01

Z = 02

Z = 03



0

200

400

600

800

1000

1200

235 236 237 238 239 240 241

Q4

Z = 01

Z = 02

Z = 03















119875 = 1100 119875 = 1200 119875 = 1300

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3370094sim3546836 3465934sim3640884 3548895sim3722454


EJTP3 3425530sim4221153 3464154sim4251420 3497160sim4277287

119862119903 2455sim2500 2455sim2500 2455sim2500

1198994

2 (119862119903= 2455sim2465)

1 (119862119903= 2470sim2500)

2 (119862119903= 2455sim2460)

1 (119862119903= 2355sim2500)

2 (119862119903= 2455)

1 (119862119903= 2360sim2500)

1198764

61574sim59822 (119862119903= 2455sim2465)

77530sim66375 (119862119903= 2470sim2500)

62723sim61837(119862119903= 2455sim2460)

81338sim68135 (119862119903= 2465sim250)

63748 (119862119903= 2455)

85010sim69738 (119862119903= 246sim250)

EJTP4 2021540sim2977979 2116825sim3088182 2198873sim3183760


52000

54000

56000

58000

60000

62000

64000

66000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



0

10000

20000

30000

40000

50000

60000

70000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



3000032000340003600038000400004200044000460004800050000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

3



02000400060008000

100001200014000160001800020000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4


1021022102410261028

1031032103410361038

104

239 240 241 242 243 244 245 246

Q1



576578580582584586588590592594

245 246 247 248 249 250 251

Q2



6065707580859095

100

239 240 241 242 243 244 245 246

Q3

Z = 01

Z = 02

Z = 03



500

550

600

650

700

750

800

850

245 246 247 248 249 250 251

Q4

Z = 01

Z = 02

Z = 03





119885 = 01 119885 = 02 119885 = 03


EJTP1


EJTP2 5877907sim3446259 5561648sim3126491 5238827sim2800005


EJTP3 4466066sim4359922 4147330sim4041153 3822615sim3716446

119862119898 2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

1198764

65370 (119862119898= 2355)

89788sim102277 (119862119898= 236sim240)

65203 (119862119898= 2355)

89751sim102171 (119862119898= 236sim240)

65029 (119862119898= 2355)

89708sim102058 (119862119898= 236sim240)

EJTP4 1900021sim1000745 1567517sim667383 1227857sim362920



4600048000500005200054000560005800060000620006400066000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



2500027000290003100033000350003700039000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



3000032000340003600038000400004200044000460004800050000

240 2405 241 2415 242 2425 243 2435 244 2445 245

Z = 01

Z = 02

Z = 03

EJTP

3



10000

15000

20000

25000

30000

35000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4










119885 = 01 119885 = 02 119885 = 03

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3465846sim3640884 3146216sim3322506 2819872sim2997440

119862119903 2400sim245 2400sim245 2400sim245

1198993

17 (119862119903= 240sim241)

16 (119862119903= 2415sim2425)

15 (119862119903= 243sim2435)

14 (119862119903= 244sim2445)

13 (119862119903= 245)

19 (119862119903= 240)

18 (119862119903= 2405sim2415)

17 (119862119903= 242sim2425)

16 (119862119903= 243sim2435)

15 (119862119903= 244sim2445)

14 (119862119903= 245)

21 (119862119903= 240)

20 (119862119903= 2405sim241)

19 (119862119903= 2415sim242)

18 (119862119903= 2425)

17 (119862119903= 243sim2435)

16 (119862119903= 244sim2445)

15 (119862119903= 245)

1198763

8295sim7997 (119862119903= 240sim241)

8277sim811 (119862119903= 2415sim2425)

8221sim8032 (119862119903= 243sim2435)

8325sim8112 (119862119903= 244sim2445)

8417 (119862119903= 245)

7453 (119862119903= 240)

7684sim7399 (119862119903= 2405sim2415)

7629sim7471 (119862119903= 242sim2425)

7709sim7533 (119862119903= 243sim2435)

7582 (119862119903= 244sim2445)

7834 (119862119903= 245)

6762 (119862119903= 240)

6940sim6814 (119862119903= 2405sim241)

6998sim6860 (119862119903= 2415sim242)

7048 (119862119903= 2425)

7252sim7088 (119862119903= 243sim2435)

7296sim7113 (119862119903= 244sim2445)

7323 (119862119903= 245)

EJTP3 3823707sim4477773 3503787sim4159040 3178546sim3834324

119862119903 2455sim250 2455sim250 2455sim250

1198994

2 (119862119903= 2455)

1 (119862119903= 246sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

1198764

62723 (119862119903= 2455)

61837sim68135 (119862119903= 246sim250)

62565sim61676 (119862119903= 2455)

2465sim250 (119862119903= 246sim250)

62400sim61508 (119862119903= 2455sim246)

81257sim67924 (119862119903= 246sim250)

EJTP4 2116836sim3088182 1785043sim2763725 1446121sim2432425












References













































1 Introduction














2

1

Product 1

Product 2

Stock

Stock

Stock

Machine

Randomdemand 1

Random

Random

demand 2

demand n

Product n

n







3 Notations







Decision Variables


4 Production Policy


lowast(119880lowast

= 119880lowast

119894119895119896 119910lowast

119894119895119896and 120575

lowast



Period 1

Δt Δt

j = 1 j = 2 j = 3

1205751 1205752 1205753

Period k


Subperiod j












Formally


Production cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + Cp (119894) times 119880119894119895119896)

Holding cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1



(2)





minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(3)





(4)



119896

sum

119904=1

119863 (119894 119904) + Stock min (119894 119896)

le

119896

sum

119904=1

119901

sum

119895=1

(119910(119894119895119904) times 119880119894119895119904) + stock init (119894 119904 = 0)

forall 119894 = 1 119899

(5)





Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(6)






Min[Cs times 119864 [119878 (119867)2]

+

119867minus1

sum

119896=0

(Cs times 119864 [119878 (119896)2] + Cp times 119864 [119906 (119896)

2])]

(7)



Min (Ζ (119880))

119880 = 119880119894119895119896 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(8)with119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]




minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(10)


(11)


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(12)

119901

sum

119895=1





119899

sum

119894=1

119910119894119895119896 = 1 forall 119895 = 1 119901 for 119896 = 1 119867

119901

sum

119895=1

119910119894119895119896 = 1 forall 119894 = 1 119899 for 119896 = 1 119867

(14)

119910119894119895119896 isin0 1 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(15)










minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(17)



119864 [119889119894 (119896)] = 119889119894 (119896) Var [119889119894 (119896)] = 1205902(119889119894 (119896))

forall 119894 = 1 119899 119896 = 1 119867

(18)



forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(19)

We note that

119864 [119880119894119895119896] = 119894119895119896 = 119880119894119895119896 (20)


Var (119880119894119895119896) = 0

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(21)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(22)




sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(23)


Proof We know that


(24)


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896) ge 0) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) ge 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

ge 119889119894 (119896) minus 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))

ge119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896))) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(25)

Noting that

119883 =119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896)) (26)



119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))ge 119883) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(27)



Hence

119878119894(119896minus1)times119901 +

119901

sum

119895=1


(120579119894)

forall 119894 = 1 119899 119896 = 1 119867

(28)

Therefore

119901

sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(29)




119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(30)





119880lowast= Min

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]]

(31)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(120579119894)


forall 119894 = 1 119899 119896 = 1 119867


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(32)



q = 1 q = 2

j = 1j = 2 j = p

Deg

rada

tion

rate

k = 1 k = 2 k = 3

T t2T

1205822 1205822p1205821

120582p+1

120575p+1
















(q minus 1) times T



T


1

2

3

Δt

120575((k+m)timesp)






119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896


times1

119880119894 nomΔ119905times 120582119899 (119905)]


(34)



Therefore




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(35)


Note that


119879] + 1 (36)




= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(37)




119879 = Round [119867 times 119901

119873] (38)








120601119879

(119880119873)= [

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(39)



=

119873+1

sum

119902=1

120601119879

(119880119873) (40)



120601(119880119873)

=

119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(41)

Note that119873 = 1 2




Γ (119873) = [

[

Mc times119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

+Mp times 119873]

]

(43)







120597Γ (119873)

120597119873= 0 (44)



lim119873rarr1

Γ (119880119873) = lim119873rarr1

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


rarrconstant

)

= 119872119888 times 120601 (119880 1) + 119872119901

lim119873rarr+infin

Γ (119880119873) = lim119873rarr+infin

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr0

+ 119872119901 times 119873⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr+infin

)

= +infin

(45)


Moreover

Γ (119880119873 + 1) minus Γ (119880119873) ge 0

997904rArr [119872119888 times 120601 (119880 (119873 + 1)) + 119872119901 times (119873 + 1)]



997904rArr 120601 (119880 (119873 + 1)) minus 120601 (119880119873) le119872119901

119872119888

(46)

In addition

Γ (119880119873) minus Γ (119880119873 minus 1) le 0

997904rArr [119872119888 times 120601 (119880119873) + 119872119901 times (119873)]




119872119888

(47)




exist119873lowast119904119894 120585119873 le

119872119901

119872119888

le 120585119873minus1 (48)

with

120585119873 = 120601 (119880119873) minus 120601 (119880 (119873 + 1)) (49)



119873lowastexist119904119894

120601 (119880 (119873 + 1)) minus 120601 (119880119873) ge 0

120601 (119880119873) minus 120601 (119880 (119873 minus 1)) le 0

lim119873rarr1

Γ (119880119873) = 119862119900119899119904119905119886119899119905

lim119873rarr+infin

Γ (119880119873) = +infin

(50)



6 Numerical Example




1198891 = 200 120590 (1198891) = 15

1198892 = 110 120590 (1198892) = 09

1198893 = 320 120590 (1198893) = 12

(51)





Table 1



Table 2









1205752

1205753

1205754

1205755

1205756

1205757

1205758

1205759

12057510

12057511

12057512



12057514

12057515

12057516

12057517

12057518

12057519

12057520

12057521

12057522

12057523

12057524






120582119899 (119905) =120572

120573times (

119905

120573)

120572minus1

=2

12times (

119905

12) (52)















Nlowast

d = di k ( )

Ulowast= Uijk ( )

k =

i =

k =i =j =

1 H1 n

1 p1 H1 n


0 2 4 6 8 10

4000

5000

6000

7000

8000


The t

otal

cost

of m

aint

enan

ceΓ

(N)



Γlowast(119873) (um) 119873




=





8 Conclusion






Appendices


We have119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A1)

Also


minus Ent [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A2)

Therefore


]


minus Ent [119895

119901] times 119889119894 (119896)]


minusEnt [119895

119901] times 119889119894 (119896) ])

2

]


]


minus [Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896))])

2

]


]



times (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]


]


]


times(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ 119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A3)



]


]


times 119864 [(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A4)




(A5)

Therefore


]


]

+(Ent [119895

119901])

2

times 119864 [(119889119894 (119896) minus 119889119894 (119896))2

]]

(A6)

We know that

119864 [(119909119896 minus 119909119896)2] = Var (119909119896)

(Int [119895

119901])

2

= Int [119895


119895

119901le 1

(A7)

Therefore


]


119901] times Var (119889119894 (119896))

(A8)


Finally


+ Ent [119895

119901] times Var (119889119894 (119896))

(A9)

Consequently

(i) for 119896 = 1

(a) 119895 = 1

Var (1198781198941) = Var (1198781198940) + (Ent [ 1

119901]) times Var (119889119894 (1))

(A10)

(b) 119895 = 2

Var (1198781198942) = Var (1198781198940) +2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A11)

(c) 119895 = 119901

Var (119878119894119901) = Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A12)

(ii) for 119896 = 2

(a) 119895 = 1

Var (119878119894119901+1) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+ Ent [ 1

119901] times Var (119889119894 (2))]

(A13)

(b) 119895 = 2

Var (119878119894119901+2) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (1)) + Ent [ 1

119901]

times Var (119889119894 (2)) + Ent [ 2

119901] times Var (119889119894 (2))]

(A14)

(c) 119895 = 119901

Var (119878119894(2times119901)) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+

119875

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119901))]

(A15)


(a) 119895 = 1


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

1

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A16)

(b) 119895 = 2


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A17)



= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A18)

On the other hand


]




]


]


]



]]

(A19)

We know that



(A20)


Hence


]




]



]





]

(A21)

Consequently


]



(A22)

Noting that


]





(A23)




= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]


]

= [ Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A24)

Consequently


]

= [ 1205902(1198781198940) +

119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times 1205902(119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A25)


119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(A26)



(i) for 119896 = 1

(a) 119895 = 1

1205821 (119905) = (1205820) times (1 minus In [0

119902 times 119879]) +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (119905)

(B1)

(b) 119895 = 2

1205822 (119905) = 1205821 (1205751) times (1 minus In [1

119902 times 119879])

+

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)


1205822 (119905) = (1205820 +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (120575(1)))


119902 times 119879]) +

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)

(B2)

(c) 119895 = 119901


119902 times 119879])

+

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)

120582119901 (119905) = [(1205820 +

119901minus1

sum

119897=1

119899

sum

119894=1

1198801198941198971 times Δ119905

119880119894 nom times 120575119897

times 120582119899 (120575(119897)))


119902 times 119879]) +

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)]

(B3)


(a) 119895 = 1

120582((119896minus1)times119901)+1 (119905)


119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

120582((119896minus1)times119901)+1 (119905)

= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

(B4)



= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(B5)



References
























1 Introduction



















16

14

12

10

08

06

Ship

men

t cos

t (lowast$100)



R2= 0926



119864 (119876) = 119901119864 (min 119876119863) + 119904119864(119876 minus 119863)+ minus 119888119876

minus 119886 + 119887 ln [119864 (min 119876119863)]

minus 119886 + 119887 ln [119864(119876 minus 119863)+]



(1)









times[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2



(3)


119864(119863 minus 119876)+le[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2

(4)


119889119864 (119876)

119889119876=119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876 minus 120583)

2[1205902 + (119876 minus 120583)2]12

minus 1198872119876 + 2120583 (119876 minus 120583) [120590

2+ (119876 minus 120583)

2]minus12


(5)

1198892119864 (119876)

1198891198762= minus

(119901 minus 119904) 1205902

2[1205902 + (119876 minus 120583)2]32

minus 119887

minus 2 + 21205832minus 21198762

+ 4120583[1205902+ (119876 minus 120583)

2]12

+2120583 (minus1 minus 120583

2+ 4120583119876 minus 3119876

2)

[1205902 + (119876 minus 120583)2]12

minus81205832(119876 minus 120583)

2

1205902 + (119876 minus 120583)2

+(119876 minus 120583)

2(21205833+ 2120583 minus 2120583119876

2)

[1205902 + (119876 minus 120583)2]32

sdot minus1 minus 1205832+ 1198762+ 4120583[120590

2+ (119876 minus 120583)

2]12

minus2

(6)




Iteration 119894 119876119894

1198911015840(119876119894) 119891

10158401015840(119876119894) 119891

1015840(119876119894)11989110158401015840(119876119894) 119876

119894+1





Let119876119894+1 = 119876119894 minus (1198911015840(119876119894)119891


1198911015840(119876119894) =

119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876119894 minus 120583)

2[1205902 + (119876119894 minus 120583)2]12

minus 1198872119876119894 + 2120583 (119876119894 minus 120583) [120590

2+ (119876119894 minus 120583)

2]minus12


2]12

(7)

From (6) we set

11989110158401015840(119876119894) = minus

(119901 minus 119904) 1205902

2[1205902 + (119876119894 minus 120583)2]32

minus 119887 minus 21198762

119894minus 2 + 2120583

2+ 4120583[120590

2+ (119876119894 minus 120583)

2]12

+2120583 (4120583119876119894 minus 3119876

2

119894minus 1205832minus 1)

[1205902 + (119876119894 minus 120583)2]12

+(119876119894 minus 120583)

2(21205833+ 2120583 minus 2120583119876

3

119894)

[1205902 + (119876119894 minus 120583)2]32

minus81205832(119876119894 minus 120583)

2

1205902 + (119876119894 minus 120583)2

(8)






0 119876lowast


0= exp(1minus


0= 119876lowast



is119876lowastlowast119894= (119876119904

119894+119876lowast



119894= (119876119904

119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887








10158401015840(1198760)) =








lowast) 119864(119876

lowast) ()


Average 405

12

10

8

6

4

2

0

5 10 15 20

Order quantity Q

Expe

cted

pro

fitE

(Q)














119864 (119878) = 119901119864 (min 119878 119863) + 119904119864(119878 minus 119863)+


minus 119886 + 119887 ln [119864(119878 minus 119863)+]







2]12

minus (119878 minus 120583)

2



(10)


119901 minus 119904

2[1205902+ (119903 minus 120583)

2]12

minus 119903 + (119888 minus 119904) 119903



2[1205902+ (119878 minus 120583)

2]12



= 0

(11)





1015840(119903119894)) where

119891 (119903119894) =119901 minus 119904

2[1205902+ (119903119894 minus 120583)

2]12

minus 119903119894 + (119888 minus 119904) 119903119894

+ 119887 ln minus1205832 minus 1 + 1199032119894+ 2120583[120590

2+ (119903119894 minus 120583)

2]12


2[1205902+ (119878 minus 120583)

2]12



1198911015840(119903119894) =

2119888 minus 119901 minus 119904

2+

(119901 minus 119904) (119903119894 minus 120583)

2[1205902 + (119903119894 minus 120583)2]12

+ 1198872119903119894 + 2120583 (119903119894 minus 120583) [120590

2+ (119903119894 minus 120583)

2]minus12


2]12

(12)










=119888

120588(119898 + 119889) 120583 minus (119898 + 119889) 119864[119863 minus 119866 (119876)]

+




(13)


119864[119863 minus 119866 (119876)]+le[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

(14)


119864 (119876) ge119888

120588

(119898 + 119889) 120583 minus (119898 + 119889)

times[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

minus (120588 + 119889 minus 1)119876

minus 2119886 + 2119887 ln 2


+ 2 (120583 + 120588119876 minus 119876)

times [1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

(15)


119889119876

= minus119888 (119898 + 119889)

2[1


119888

120588(120588 + 119889 minus 1)





(16)

where119883 = 1205902 + 120588119902119876 + (120588119876 minus 120583)2

1198892119864 (119876)

1198891198762= minus119888 (119898 + 119889)

2[minus120588

4(119902 minus 2120583 + 2120588119876)

2119883minus32

+ 120588119883minus12]

minus 1198871198841015840119885 minus 119884119885

1015840

1198852


12



+ 2 (120583 + 120588119876 minus 119876)11988312

1198841015840= 4120588 (1 minus 120588) minus 2120588


minus1205882

2(120583 + 120588119876 minus 119876) (119902 minus 2120583 + 2120588119876)

2119883minus32

1198851015840= 2 (1 minus 120588) (2120588119876 + 120583) minus 120588119902

minus 2 (1 minus 120588)11988312+120588

4(120583 + 120588119876 minus 119876)


(18)





1015840(119876119894)119891

10158401015840(119876119894)) where 119891

1015840(119876119894) and




2119864(119876119894+1)119889119876

2




0 119876lowast


0= exp(1minus


0= 119876lowast



is 119876lowastlowast119894= (119876119904

119894+ 119876lowast




119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887


119894 = 119894 + 1



2





2

119895 respectively



1 119876lowast2 and

119876lowast

119873

max1198761 119876119873

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583[1205902

119895+ (119876119895 minus 120583119895)

2

]12

(19)

Subject to119873

sum

119895=1

119888119895119876119895 le 119861 (20)


119871 (1198761 119876119873 120582)

=

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583119895[1205902

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582(

119873

sum

119895=1

119888119895119876119895 minus 119861)

(21)


120597119871 (1198761 119876119873 120582)

120597119876119895

=119901119895 + 119904119895 minus 2119888119895


2[1205902119895+ (119876119895 minus 120583119895)

2

]12

minus 119887

2119876119895 + 2120583119895 (119876119895 minus 120583119895) [1205902

119895+ (119876119895 minus 120583119895)

2

]minus12

minus1 minus 1205832 + 1198762119895+ 4120583119895[120590

2

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582119888119895

(22)


119873with

















Step 1 Let 119895 = 1 2 3 4


1= 8731 119876lowast

2= 7762 119876lowast

3= 11072 and 119876lowast

4

= 21243 Check sum4119895=1119888119895119876lowast

119895= $92 gt $80 where (20) is not



3is set to 0


119895= $61 lt $80 (20) is satisfied










Highlights









References



































1 Introduction













max 120601

st 120582119883 le 119909119863

119900

120582119884119863ge 120601119910119863

119900

120582119884119880

ge 120601119910119900119880

119890120582 = 1

120582 ge 0

(1)

in which 119910119900119880= minus119884



max 120601

st 120582119883119863le (1 minus 120601) 119909

119863

119900

120582119883119880

le (1 minus 120601) 119909119900119880

120582119884119863ge (1 + 120601) 119910

119863

119900

120582119884119880

ge (1 + 120601) 119910119900119880

119890120582 = 1

120582 ge 0

(2)

in which 119910119900119880= minus119884

119880+ V gt 0 and 119883

119880

= minus119883119880+ 119908 gt 0




min 1 minus [

[

119908119900 +1

119898 + 119904(sum

119894isin119868119863

119905minus119863

119894+ sum

119903isin119874119863

119905+119863

119903)]

]

st119899

sum

119895=1

120582119895119909119863

119894119895+ 119905minus119863

119894= 119909119863

119894119900minus 119908119900 119894 isin 119868119863

119899

sum

119895=1

120582119895119909119880

119894119895+ 119905minus119880

119894= 119909119880

119894119900+ 119908119900 119894 isin 119868119880

119899

sum

119895=1

120582119895119910119863

119903119895minus 119905+119863

119903= 119910119863

119903119900+ 119908119900 119903 isin 119874119863

119899

sum

119895=1

120582119895119910119880

119903119895minus 119905+119880

119903= 119910119880

119903119900minus 119908119900 119903 isin 119874119880

119899

sum

119895=1

120582119895 = 1

(3)


min 120579

st119899

sum

119895=1

120582119895119909119894119895 le 120579119909119894119900 forall119894 isin 119868

119899

sum

119895=1

120582119895119910119863

119903119895ge 119910119863

119903119900forall119903 isin 119874

119863

119899

sum

119895=1

120582119895119910119880

119903119895le 120579119910119880

119903119900forall119903 isin 119874

119880

120582119895 ge 0 forall119895 isin 119873

(4)




min 120579

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le 120579

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 ge

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(5)


max119898

sum

119894=1

119904minus

119894+

119904

sum

119903=1

119905+

119903

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 + 119904minus

119894= 120579lowast

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 minus 119905+

119903=

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

119904minus

119894ge 0 119905

+

119903ge 0 119894 = 1 119898 119903 = 1 119904

120582119895119896 ge 0 119896 119895 = 1 119899

(6)




119909119894119900 =

119899

sum

119895=1

120582119900lowast

119895119909119894119895 = 120579

lowast119909119894119900 minus 119904

minuslowast

119894119894 = 1 119898

119910119903119900 =

119899

sum

119895=1

120582119900lowast

119895119910119903119895 = 119910119903119900 + 119905

+lowast

119903119903 = 1 119904

(7)


119878119894 =

119899

sum

119895=1

119909119894119895 minus

119899

sum

119895=1

119909119894119895 ge 0 119894 = 1 119898

119879119903 =

119899

sum

119895=1

119910119903119895 minus

119899

sum

119895=1

119910119903119895 ge 0 119903 = 1 119904

(8)


max119899

sum

119895=1

119904

sum

119903=1

119906119903119910119903119895 +

119899

sum

119896=1

120577119896

st119899

sum

119895=1

119898

sum

119894=1

V119894119909119894119895 = 1

119904

sum

119903=1

119906119903119910119903119895 minus

119898

sum

119894=1

V119894119909119894119895 + 120577119896 le 0 119895 119896 = 1 119899

119906119903 ge 0 119903 = 1 119904

V119894 ge 0 119894 = 1 119898

(9)



5 Proposed Method



119883 = 119909119894119895 isin 119877119898times119899

+ 119884

119863= 119910119863

119903119895 isin 119877119904119863times119899

+

119884119880= 119910119880

119905119895 isin 119877119904119880times119899

+

(10)


max 120593

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le

119899

sum

119895=1

119909119894119895 minus 120593119877119909119894 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119863

119903119895ge

119899

sum

119895=1

119910119863

119903119895+ 120593119877119910

119863

119903119903 = 1 119904

119863

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119880

119905119895le

119899

sum

119895=1

119910119880

119903119895minus 120593119877119910

119880

119905119905 = 1 119904

119880

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(11)

where119877119909119894119877119910119863

119903 and119877119910119880



119863

119903 119877119910119880



119903 119877119910119880

119905) vector as

follows

119877119909119894 =

119899

sum

119895=1

119909119894119895 minus 119899 (min 119909119894119895119895=1119899) 119894 = 1 119898

119877119910119863

119903=

119899

sum

119895=1

119910119863

119903119895minus 119899 (max 119910119863

119903119895119895=1119899

) 119903 = 1 119904119863

119877119910119880

119905=

119899

sum

119895=1

119910119880

119905119895minus 119899 (min 119910119880

119905119895119895=1119899

) 119905 = 1 119904119880

(12)


119863

119903 119877119910119880

119905) can be

chosen as follows

(119877119909119894 119877119910119863

119903 119877119910119880

119905) = (

119899

sum

119895=1

119909119894119895

119899

sum

119895=1

119910119863

119903119895

119899

sum

119895=1

119910119880

119905119895) (13)











119863

119903 119877119910119880



6 Numerical Example





7 Conclusion







References


































1 Introduction







20 40 60 80 100Utilization ()

Cycle

tim

eIn

crea

sing


20 40 60 80 100

Low variability

Utilization ()

High variability

Cycle

tim

eIn

crea

sing

te

to




CT119902 = 119881119880119879 (1)



2 Literature Review














119886= 1205902

1198861199052

119886and

1198622

119890= 1205902

1198901199052



2



2

119890gt 1198622



CT119902 = (1198622

119886+ 1198622

119890

2)(

119906

1 minus 119906) 119905119890 (2)

1198622

119889= 11990621198622

119890+ (1 minus 119906

2) 1198622

119886 (3)










Patient



Home

10 min24 min

20 min11 min11 min

98 min0 min

10 min0 min

3 min3 min0 min

10 min




Clean room




ORemergence

time

Patient out of room

Inpatient room


(Nj)

WW W W W W


1205902


1198882




arrivals






2


as 083

119860 =119898119891

119898119891 + 119898119903

(4)

119905119890 =119905119900

119860 (5)

119906 =119903119886

119903119890

=119903119886119905119890

119898 (6)

1198622

119890= 1198622

119900+ 2119860 (1 minus 119860)

119898119903

119905119900

(7)






Specialty 1

FCFS

Specialty 2

Specialty 15


Operating room

Recoverroom

First stage (W1)

Second stage (W2)

(Mj)

(Nj)

M1

M2

M12

N1

N2

N3

N4

N5

N24





119905119890 = 119905119900 +119905119904

119873119904

(8)

1205902

119890= 1205902

119900+1205902

119904

119873119904

+119873119904 minus 1

1198732119904

1199052

119904

1198882

119890=1205902

119890

1199052119890

(9)




119890= 749







CT119902 =1198622

119886+ 1198622

119890

2

119906

1 minus 119906119905119890

=(099 + 749)

2(

089

1 minus 089) 14543

= (424) (809) (14543)

(10)














120583 (11)




(12)



5 Empirical Results








1 10800 813 10800

813≒ 13

2 10800 159 10800

159≒ 68

3 10800 134 10800

134≒ 81

4 10800 346 10800

346≒ 31

5 10800 185 10800

185≒ 58

Input

Output


Simulationmodel






1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

0020018001601400120010008000600040002

0

CV


WIP




Parameters



Patient



suture and finish

Home

Leveling

Billing and coding

Waiting time= 5 min




Inpatient room

W W W W





10 min0 min

98 min0 min5 min




Decision variables



Minimize24

sum

119895=1

119874119895 (13)


5

sum

119894=1

119860 119894119895 = 1 forall119895 (14)

119862119894 ge 1 forall119894 (15)

5

sum

119894=1

119862119894 = 24 (16)

119860 119894119895 isin 0 1 forall119894119895 (17)






















Acknowledgments


References













































Editorial Board










Contents




























Acknowledgments




Chia-Yu Hsu







1 Introduction














2 Related Work




(a) (b) (c)

(d) (e) (f)



3 Proposed Approach






119898119896119873119892]



]


120579119897119873119892]


119881119897119873119892]


119897= [1205951198971

1205951198972 120595119897119873119892]


119892= [1205951198921 1205951198922

120595119892119873119892]


Consensuspartition


WBMs

1 clustering

q clustering

2 clustering



Labels120587 q










1205871

1205872

1205873

x1

1 1 1x2

1 1 1x3

1 1 1x4

2 2 1x5

2 2 2x6

3 1 2x7

3 1 2x8

3 1 2



V2

V3

V4

V5

V6

V7

V8

119867(1)

ℎ11

1 1 1 0 0 0 0 0ℎ12

0 0 0 1 1 0 0 0ℎ13

0 0 0 0 0 1 1 1

119867(2) ℎ

211 1 1 0 0 1 1 1

ℎ22

0 0 0 1 1 0 0 0

119867(3) ℎ

311 1 1 1 0 0 0 0

ℎ32

0 0 0 0 1 1 1 1


119872(119910119903) =

119873119887

sum

119903=1

119890minus119898120573119889(119910119903 119910119904) 119903 = 1 2 3 119873119887 (1)

where

120573 = (119889 (119910119903 minus 119910wc)

119873119887

)

minus1

(2)







(a) Binary value

51015202530






SSE =

119873119888

sum

119896=1

sum

x119894isin119862119896(x119894 minusm119896)

2 (3)






119869119890 =sum119873119888


119870 (5)

In addition

119889max (p119894Z119897) = max119896=12119873119888

[[

[

sum

forallx119894isin119862119897119896

119889 (x119894m119897119896)119899119897119896

]]

]

(6)



[119889 (m119897119906m119897V)] (7)





119894


119896


119894Z119897)


119901119906(120595119897(119905) minusΘ

119897(119905)) + 119888

119892119906(120595119892(119905) minusΘ

119897(119905)) (i)

Θ119897(119905 + 1) = Θ

119897(119905) + V

119897(119905 + 1) (ii)










1012

1518

2315221370

1184 1098945

02004006008001000120014001600

0

5

10

15

20

25

03 04 05 06 07

SSE

Clus

ter n

umbe

r









max 1 1198861

04119888119901

2 1198862

03119888119892

2 1198863

03Iteration 500



Hsu and Chien [8] 1184 1192 1203 1248 1322


KB 2889 3092 3003 4083 3570KC 3331 2490 2603 3169 2603PB 5893 3601 6566 5839 6308PC 4627 4873 3330 3787 6112

Clusteringensemble






Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9

Group 10

Group 11

Group 12

Group 13

Group 14

Group 15





119879119890

(8)



(a) (b) (c)

(d) (e) (f)




Hsu and Chien [8] 04 04 04 04 04


KB 10 10 10 10 10KC 10 10 10 10 10PB 10 10 10 10 10PC 10 10 10 10 10

Clusteringensemble





precision =TP

TP + FP

recall = TPTP + FN

(9)




=2TP

FP + FN + 2TP (10)






Precision

A 070 084 092 092 092 098B 050 066 096 092 062 096C 060 064 100 100 060 100D 070 098 092 092 098 100E 060 094 082 082 098 098F 080 098 076 076 098 098

Avg 065 084 090 089 085 098



Recall

A 100 100 100 093 100 100B 100 097 07 078 083 100C 100 094 067 084 067 097D 100 081 100 100 100 100E 100 079 100 100 100 100F 100 100 100 100 100 100

Avg 100 092 090 093 092 100



119865-measure

A 082 091 096 092 096 099B 067 079 081 084 071 098C 075 076 08 091 063 098D 082 089 096 096 099 100E 075 086 090 090 099 099F 089 099 086 086 099 099

Avg 078 087 088 090 088 099



5 Conclusion









Acknowledgment


References







































1 Introduction





























119903119894119895 = 05 (1 + log9119886119894119895) (1)









119903119894119895 =119894 minus 119895 + 1


(2)

119903119894119895 = 1 minus 119903119895119894 forall119894 lt 119895 (3)


According to (2)

11990321 = 1 minus 03 = 07

11990331 = 15 minus 03 minus 06 = 06


11990332 = 1 minus 06 = 04

11990342 = 15 minus 06 minus 08 = 01

11990343 = 1 minus 08 = 02

(4)

According to (3)

11990313 = 1 minus 06 = 04

11990314 = 1 minus 03 = 07

11990324 = 1 minus 01 = 09

(5)


PR =[[[

[

05 03 04 07

07 05 06 09

06 04 05 08

03 01 02 05

]]]

]

(6)





119878119886119887 = 1 minus

10038161003816100381610038161198861 minus 11988711003816100381610038161003816 +

10038161003816100381610038161198862 minus 11988721003816100381610038161003816 +

10038161003816100381610038161198863 minus 11988731003816100381610038161003816 +

10038161003816100381610038161198864 minus 11988741003816100381610038161003816

4 (7)


AM =

[[[[

[


119878119886119887

1198781198641 1198781198642 sdot sdot sdot 1

]]]]

]

(8)



AA119886 =1

119864 minus 1

119864

sum

119887=1119886 =119887

119878119886119887 forall119886 (9)


RA119886 =AA119886

sum119864

119886=1AA119886

forall119886 (10)










(12)


1

119894119895 1199032

119894119895 1199033

119894119895 1199034

119894119895)





119903119894119895 =1199031

119894119895+ 2 (119903

2

119894119895+ 1199033

119894119895) + 1199034

119894119895

6 (13)


PR119896 =[[[[

[


1

1199031198731 1199031198732 sdot sdot sdot 1

]]]]

]

(14)






120575119894119896 =

119873

sum

119895=1119895 =119894

119903119894119895 minus

119873

sum

119895=1119895 =119894

119903119895119894 forall119894 119896 (15)


DM =

[[[[

[


1205751198731 1205751198732 sdot sdot sdot 120575119873119872

]]]]

]

(16)




aw119896 =(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119896 minus 120575119895119896

10038161003816100381610038161003816)1(119901minus1)

sum119872

119897=1(sum119873

119894=1sum119873

119895=1

10038161003816100381610038161003816120575119894119897 minus 120575119895119897

10038161003816100381610038161003816)1(119901minus1)

forall119896 119901 gt 1 (17)

aw119896 =(sum119873

119894=11205752

119894119896)12(119901minus1)

sum119872

119897=1(sum119873

119894=11205752119894119897)12(119901minus1)

forall119896 119901 gt 1 (18)



119872



119862119894 =

119872

sum

119896=1

aw119896120575119894119896 119894 = 1 2 119873 (19)





transformation


relations










4 Numerical Example





11987711 =[[[

[




]]]

]

11987721 =[[[

[




]]]

]

11987731 =

[[[[[[

[

minus1

3minus minus

minus minus1

4minus

minus minus minus 1


]]]]]]

]

(20)


11987711 =[[[

[





]]]

]

11987721 =[[[

[





]]]

]

11987731 =

[[[[[[

[

minus1

31

31

31

3minus minus

minus minus1

41

41

41

4minus



]]]]]]

]

(21)


11987712 =[[[

[





]]]

]

11987722 =[[[

[





]]]

]


11987732 =

[[[[[[

[

minus1

41

41

41

4minus minus


minus minus minus1

31

31

31

3


]]]]]]

]

(22)


then become 119877101584031

and 1198771015840


replaced by 119877101584031and 1198771015840


1198771015840

31=

[[[

[





]]]

]

1198771015840

32=

[[[

[





]]]

]

(23)



PR1 =[[[

[





]]]

]

PR2 =[[[

[





]]]

]

(24)



2are

PR10158401=

[[[

[

0500 0290 0100 0100

0710 0500 0311 0311

0900 0689 0500 0500

0900 0689 0500 0500

]]]

]

PR10158402=

[[[

[

0500 0218 0265 0055

0782 0500 0547 0337

0735 0453 0500 0290

0945 0663 0710 0500

]]]

]

(25)


1and PR1015840

2are consistent


DM =[[[

[

minus2019 minus1923

minus0336 0331

1178 minus0045

1178 1637

]]]

]

(26)




11990312

11990323

11990334


119886119887)

11987812

0925 0775 100011987813

0900 0880 090011987823


119886)

AA1 0913 0828 0950AA2 0900 0730 0950AA3 0888 0783 0900


RA1 0338 0354 0339RA2 0333 0312 0339RA3 0329 0334 0321


120573 = 05

CC1 0319 0327 0320CC2 0317 0306 0320CC3 0364 0367 0361


23= (022 028 034 041) 119903

34= (040 047 053 060)


23= 0311 119903

34= 0500



Very low

02 04 06 1008

02

04

06

08

10

Mem

bers

hip

valu

e



Very low

More highMore

low

02

04

06

08

10

Mem

bers

hip

valu

e




Alternative 119894 120575119894119896

119862119894(ADM) 119862



aw119896by SDM 0509 0491





5 Conclusion











Acknowledgments


References


















































1 Introduction

















(i) Supplier Side





(iii) Retailer Side



22 Assumptions












3 Model Formulation




Repaired products

Defective products

Normal products











=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)

2 [1 minus 119885 (1 minus 119877)]

(1)




nQ


nQD

nQD

nQD

nQ

nQ


Z(1 minus R)nQ


nZQ


nQ


P

Q

t

t

t

Q

Q

Q

Q

P

QD

QD (n minus 1)Q

D

nRZQ




119885(1 minus 119877)]119876119894




=119863 (119862119898 minus 119862119904)


119899119894119876119894

minusℎ119904119863119899119894119876119894

2119875 [1 minus 119885 (1 minus 119877)]2

minus119862119904119868sp (2119863119883 minus 119899119894119876119894)


2

2119899 [1 minus 119885 (1 minus 119877)]119876119894

(2)




nQ


Lm =nQ

D

X

Q

t

Interest income

Figure 3 119871119898lt 119883

Lm =nQ

D

nQ


X

Q

Interest income

Opportunity cost

t

Figure 4 119871119898ge 119883



(ii) Ordering cost = 119860119898119863119899119894119876119894










TP1198981





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


119876119894

2)

(3)



2 [1 minus 119885 (1 minus 119877)] (4)





TP1198982





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)


minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

+119862119903119868me (2119863119883 minus 119899119894119876119894)


2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(5)




22[1 minus 119885(1 minus 119877)]119899119894119876119894



TP1198983





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119894119876119894 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus 119862119898119868mp (119863119884 minus119876119894

2)

(6)



119877)]119899119894119876119894





TP1198984





= 119863[119862119903 minus119862119898


119860119898119863

119899119894119876119894

minus ℎ119898119863119876119894 [119899119894 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus119905119904119885119877119899119894

1 minus 119885 (1 minus 119877)

minus119865119898119863

119899119894119876119894

minus119863 [119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)]

1 minus 119885 (1 minus 119877)

minus119862119898119868mp (119899119876 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

+119862119903119868me (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 119899119894119876119894

minus119862119898119868mp (119863119884)

2

2119876119894

+119862119903119868me (119876119894 minus 119863119884)

2

2119876119894

(7)




(ii) Ordering cost = 119860119903119863119899119894119876119894






TP1199031



= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

+ 119875119903119868re (119863119884 minus119876119894

2)





TP1199032



+ interest income

= 119863 (119875119903 minus 119862119903) minus119860119903119863

119899119894119876119894

minusℎ119903119876119894

2minus119865119903119863

119876119894

minus119862119903119868rp (119876119894 minus 119863119884)

2

2119876119894

+119875119903119868re (119863119884)

2

2119876119894

(9)


EJTP119894 (119876119894 119899119894)

=





(10)

whereEJTP1 (1198761 1198991)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198761 [1198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198991


ℎ1199031198761

2minus

ℎ11990411986311989911198761

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198761

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989911198761)

2 [1 minus 119885 (1 minus 119877)]

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198761

2)

EJTP2 (1198762 1198992)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198762 [1198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198992


ℎ1199031198762

2minus

ℎ11990411986311989921198762

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

+(119862119903119868me minus 119862119904119868sp) (2119863119883 minus 11989921198762)

2 [1 minus 119885 (1 minus 119877)]

+(119862119903119868me minus 119862119903119868rp) (1198762 minus 119863119884)

2

21198762

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198762

EJTP3 (1198763 1198993)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]

minus ℎ1198981198631198763 [1198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198993


ℎ1199031198763

2minus

ℎ11990411986311989931198763

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+(119862119898119868se minus 119862119898119868mp) (11989931198763 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198763

+ (119875119903119868re minus 119862119898119868mp) (119863119884 minus1198763

2)

EJTP4 (1198764 1198994)

= 119863[119875119903 minus119862119904 +119882 + 119862rm119885119877 + 119866119885 (1 minus 119877)

1 minus 119885 (1 minus 119877)]


minus ℎ1198981198631198764 [1198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875]

minus1199051199041198851198771198994


ℎ1199031198764

2minus

ℎ11990411986311989941198764

2119875 [1 minus 119885 (1 minus 119877)]2

minus119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198764

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119898119868se minus 119862119898119868mp) (11989941198764 minus 119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989941198764

+(119862119903119868me minus 119862119903119868rp) (1198764 minus 119863119884)

2

21198764

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

21198764

(11)



120597EJTP1 (1198761 1198991)1205971198761

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198762

1

minus ℎ1198981198631198991 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198991


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp)

2

(12)

120597EJTP2 (1198762 1198992)1205971198762

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198762

2

minus ℎ1198981198631198992 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198992


2 [1 minus 119885 (1 minus 119877)]

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987622

+(119862119903119868me minus 119862119903119868rp) [119876

2

2minus (119863119884)

2]

211987622

(13)

120597EJTP3 (1198763 1198993)1205971198763

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198762

3

minus ℎ1198981198631198993 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198993

2119875 [1 minus 119885 (1 minus 119877)]2

minus(119862119903119868me minus 119862119904119868sp) (119863119883)

2

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

+(119862119898119868se minus 119862119898119868mp) [(11989931198763)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989931198762

3

minus(119875119903119868re minus 119862119898119868mp)

2

(14)

120597EJTP4 (1198764 1198994)1205971198764

=119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198762

4

minus ℎ1198981198631198994 minus 1


2119875 [1 minus 119885 (1 minus 119877)]2

+1

119875

minusℎ119903

2minus

ℎ1199041198631198994


2

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

minus(119875119903119868re minus 119862119898119868mp) (119863119884)

2

211987624

+(119862119898119868se minus 119862119898119868mp) [(11989941198764)

2minus (119863119883)

2]

2 [1 minus 119885 (1 minus 119877)] 11989941198762

4

+(119862119903119868me minus 119862119903119868rp) [119876

2

4minus (119863119884)

2]

211987624

(15)


102

1025

103

1035

104

1045

105

2295 2305 2315 2325 2335 2345 2355

Q1



0

100

200

300

400

500

600

700


Q2


777879808182838485

235 236 237 238 239 240

Q3

P = 1100

P = 1200

P = 1300



0

200

400

600

800

1000

1200

235 236 237 238 239 240

Q4

P = 1100

P = 1200

P = 1300




1205972EJTP1 (1198761 1198991)

12059711987621

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991)

11989911198763

1

lt 0

(16)

1205972EJTP2 (1198762 1198992)

12059711987622

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

11989921198763

2

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987632

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987632

lt 0

(17)

1205972EJTP3 (1198763 1198993)

12059711987623

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

11989931198763

3

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989931198763

3

lt 0

(18)

1205972EJTP4 (1198764 1198994)

12059711987624

= minus2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

11989941198763

4

+(119862119903119868me minus 119862119904119868sp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119898119868se minus 119862119898119868mp) (119863119883)

2

[1 minus 119885 (1 minus 119877)] 11989941198763

4

+(119862119903119868me minus 119862119903119868rp) (119863119884)

2

11987634

+(119875119903119868re minus 119862119898119868mp) (119863119884)

2

11987634

lt 0

(19)



60000

60500

61000

61500

62000

62500

63000

63500

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



30000

35000

40000

45000

50000

55000

60000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



43000432004340043600438004400044200444004460044800

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

3



60008000

10000120001400016000180002000022000

2355 236 2365 237 2375 238 2385 239 2395 240

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4



119876lowast

1= ((2119863119875119880

2(119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198991))

times (1198992 119875119880 [119880 (ℎ119898 (1198991 minus 1) + ℎ119903 +119882) +1198721198991]

+119863 [1198991 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(20)

119876lowast

2= ((119875119880

2[2119863 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198992)

minus 1198992 (119861 +119882) (119863119884)2])


+119863 [1198992 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(21)

119876lowast

3= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198993)

minus (119872 + 119864) (119863119883)2])


+119863 [1198993 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(22)

119876lowast

4= ((119875119880 [2119863119880 (119860 119904 + 119860119898 + 119865119898 + 119860119903 + 1198651199031198994)

minus (119872 + 119864) (119863119883)2minus 1198801198994 (119861 +119882) (119863119884)

2])


+119863 [1198994 (ℎ119904 + ℎ119898 (1 minus 2119880)) + 2ℎ1198981198802])minus1

)12

(23)




102

1025

103

1035

104

1045

105

239 240 241 242 243 244 245 246

Q1



560565570575580585590595600605610

245 246 247 248 249 250 251

Q2



200210220230240250260270280290300

239 240 241 242 243 244 245 246

Q3

P = 1100

P = 1200

P = 1300



500550600650700750800850900

245 246 247 248 249 250 251

Q4

P = 1100

P = 1200

P = 1300









5 Numerical Example









119875 = 1100 119875 = 1200 119875 = 1300

119862119898

2300sim2350 2300sim2350 2300sim23501198991

2 2 21198761

10219sim10229 10339sim10349 10444sim10454EJTP1


2300sim2350 2300sim2350 2300sim23501198992

1 1 11198762

18902sim57786 19404sim59321 19862sim60721EJTP2

5846523sim3350315 5877907sim3446259 5905128sim3529477119862119898

2355sim240 2355sim240 2355sim2401198993

14 13 131198763

7873sim7772 8404sim8295 8415sim8306EJTP3

4463785sim4357297 4466066sim4359922 4468691sim4362513119862119898

2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355sim2365) 2 (119862

119898= 2355) 1

1 (119862119898= 2355sim2365) 1 (119862

119898= 2360sim2365)

1198764

64172sim67519 (119862119898= 2355sim2365) 65370 (119862

119898= 2355) 90178sim104684

90662sim99636 (119862119898= 2370sim2400) 89788sim102277 (119862

119898= 2355sim2365)

EJTP4


59500

60000

60500

61000

61500

62000

62500

63000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



33000

34000

35000

36000

37000

38000

39000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



34000

36000

38000

40000

42000

44000

46000

240 2405 241 2415 242 2425 243 2435 244 2445 245

P = 1100

P = 1200

P = 1300

EJTP

3



200002200024000260002800030000320003400036000

2455 246 2465 247 2475 248 2485 249 2495 250

P = 1100

P = 1200

P = 1300

EJTP

4




119894 for 119894 = 1 2 3 4


1018102

1022102410261028

103103210341036

229 230 231 232 233 234 235 236

Q1



100150200250300350400450500550600

229 230 231 232 233 234 235 236

Q2



707274767880828486

235 236 237 238 239 240 241

Q3

Z = 01

Z = 02

Z = 03



0

200

400

600

800

1000

1200

235 236 237 238 239 240 241

Q4

Z = 01

Z = 02

Z = 03















119875 = 1100 119875 = 1200 119875 = 1300

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3370094sim3546836 3465934sim3640884 3548895sim3722454


EJTP3 3425530sim4221153 3464154sim4251420 3497160sim4277287

119862119903 2455sim2500 2455sim2500 2455sim2500

1198994

2 (119862119903= 2455sim2465)

1 (119862119903= 2470sim2500)

2 (119862119903= 2455sim2460)

1 (119862119903= 2355sim2500)

2 (119862119903= 2455)

1 (119862119903= 2360sim2500)

1198764

61574sim59822 (119862119903= 2455sim2465)

77530sim66375 (119862119903= 2470sim2500)

62723sim61837(119862119903= 2455sim2460)

81338sim68135 (119862119903= 2465sim250)

63748 (119862119903= 2455)

85010sim69738 (119862119903= 246sim250)

EJTP4 2021540sim2977979 2116825sim3088182 2198873sim3183760


52000

54000

56000

58000

60000

62000

64000

66000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

1



0

10000

20000

30000

40000

50000

60000

70000

230 2305 231 2315 232 2325 233 2335 234 2345 235

EJTP

2



3000032000340003600038000400004200044000460004800050000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

3



02000400060008000

100001200014000160001800020000

2355 236 2365 237 2375 238 2385 239 2395 240

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4


1021022102410261028

1031032103410361038

104

239 240 241 242 243 244 245 246

Q1



576578580582584586588590592594

245 246 247 248 249 250 251

Q2



6065707580859095

100

239 240 241 242 243 244 245 246

Q3

Z = 01

Z = 02

Z = 03



500

550

600

650

700

750

800

850

245 246 247 248 249 250 251

Q4

Z = 01

Z = 02

Z = 03





119885 = 01 119885 = 02 119885 = 03


EJTP1


EJTP2 5877907sim3446259 5561648sim3126491 5238827sim2800005


EJTP3 4466066sim4359922 4147330sim4041153 3822615sim3716446

119862119898 2355sim240 2355sim240 2355sim240

1198994

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

2 (119862119898= 2355)

1 (119862119898= 236sim240)

1198764

65370 (119862119898= 2355)

89788sim102277 (119862119898= 236sim240)

65203 (119862119898= 2355)

89751sim102171 (119862119898= 236sim240)

65029 (119862119898= 2355)

89708sim102058 (119862119898= 236sim240)

EJTP4 1900021sim1000745 1567517sim667383 1227857sim362920



4600048000500005200054000560005800060000620006400066000

240 2405 241 2415 242 2425 243 2435 244 2445 245

EJTP

1



2500027000290003100033000350003700039000

2455 246 2465 247 2475 248 2485 249 2495 250

EJTP

2



3000032000340003600038000400004200044000460004800050000

240 2405 241 2415 242 2425 243 2435 244 2445 245

Z = 01

Z = 02

Z = 03

EJTP

3



10000

15000

20000

25000

30000

35000

40000

2455 246 2465 247 2475 248 2485 249 2495 250

Z = 01

Z = 02

Z = 03

EJTP

4




119894 for 119894 = 1 2 3 4










119885 = 01 119885 = 02 119885 = 03

119862119903 2400sim2450 2400sim2450 2400sim24501198991 2 2 21198761



EJTP2 3465846sim3640884 3146216sim3322506 2819872sim2997440

119862119903 2400sim245 2400sim245 2400sim245

1198993

17 (119862119903= 240sim241)

16 (119862119903= 2415sim2425)

15 (119862119903= 243sim2435)

14 (119862119903= 244sim2445)

13 (119862119903= 245)

19 (119862119903= 240)

18 (119862119903= 2405sim2415)

17 (119862119903= 242sim2425)

16 (119862119903= 243sim2435)

15 (119862119903= 244sim2445)

14 (119862119903= 245)

21 (119862119903= 240)

20 (119862119903= 2405sim241)

19 (119862119903= 2415sim242)

18 (119862119903= 2425)

17 (119862119903= 243sim2435)

16 (119862119903= 244sim2445)

15 (119862119903= 245)

1198763

8295sim7997 (119862119903= 240sim241)

8277sim811 (119862119903= 2415sim2425)

8221sim8032 (119862119903= 243sim2435)

8325sim8112 (119862119903= 244sim2445)

8417 (119862119903= 245)

7453 (119862119903= 240)

7684sim7399 (119862119903= 2405sim2415)

7629sim7471 (119862119903= 242sim2425)

7709sim7533 (119862119903= 243sim2435)

7582 (119862119903= 244sim2445)

7834 (119862119903= 245)

6762 (119862119903= 240)

6940sim6814 (119862119903= 2405sim241)

6998sim6860 (119862119903= 2415sim242)

7048 (119862119903= 2425)

7252sim7088 (119862119903= 243sim2435)

7296sim7113 (119862119903= 244sim2445)

7323 (119862119903= 245)

EJTP3 3823707sim4477773 3503787sim4159040 3178546sim3834324

119862119903 2455sim250 2455sim250 2455sim250

1198994

2 (119862119903= 2455)

1 (119862119903= 246sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

2 (119862119903= 2455sim246)

1 (119862119903= 2465sim250)

1198764

62723 (119862119903= 2455)

61837sim68135 (119862119903= 246sim250)

62565sim61676 (119862119903= 2455)

2465sim250 (119862119903= 246sim250)

62400sim61508 (119862119903= 2455sim246)

81257sim67924 (119862119903= 246sim250)

EJTP4 2116836sim3088182 1785043sim2763725 1446121sim2432425












References













































1 Introduction














2

1

Product 1

Product 2

Stock

Stock

Stock

Machine

Randomdemand 1

Random

Random

demand 2

demand n

Product n

n







3 Notations







Decision Variables


4 Production Policy


lowast(119880lowast

= 119880lowast

119894119895119896 119910lowast

119894119895119896and 120575

lowast



Period 1

Δt Δt

j = 1 j = 2 j = 3

1205751 1205752 1205753

Period k


Subperiod j












Formally


Production cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + Cp (119894) times 119880119894119895119896)

Holding cost

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1



(2)





minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(3)





(4)



119896

sum

119904=1

119863 (119894 119904) + Stock min (119894 119896)

le

119896

sum

119904=1

119901

sum

119895=1

(119910(119894119895119904) times 119880119894119895119904) + stock init (119894 119904 = 0)

forall 119894 = 1 119899

(5)





Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(6)






Min[Cs times 119864 [119878 (119867)2]

+

119867minus1

sum

119896=0

(Cs times 119864 [119878 (119896)2] + Cp times 119864 [119906 (119896)

2])]

(7)



Min (Ζ (119880))

119880 = 119880119894119895119896 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(8)with119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]




minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(10)


(11)


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(12)

119901

sum

119895=1





119899

sum

119894=1

119910119894119895119896 = 1 forall 119895 = 1 119901 for 119896 = 1 119867

119901

sum

119895=1

119910119894119895119896 = 1 forall 119894 = 1 119899 for 119896 = 1 119867

(14)

119910119894119895119896 isin0 1 forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(15)










minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(17)



119864 [119889119894 (119896)] = 119889119894 (119896) Var [119889119894 (119896)] = 1205902(119889119894 (119896))

forall 119894 = 1 119899 119896 = 1 119867

(18)



forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(19)

We note that

119864 [119880119894119895119896] = 119894119895119896 = 119880119894119895119896 (20)


Var (119880119894119895119896) = 0

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(21)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(22)




sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(23)


Proof We know that


(24)


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896) ge 0) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


+

119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) ge 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

sum

119895=1

(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

ge 119889119894 (119896) minus 119889119894 (119896)) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867


119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))

ge119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896))) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(25)

Noting that

119883 =119889119894 (119896) minus 119889119894 (119896)

Var (119889119894 (119896)) (26)



119901

119895=1(119910119894119895119896 times 119880119894119895119896) minus 119889119894 (119896)

Var (119889119894 (119896))ge 119883) ge 120579119894

forall 119894 = 1 119899 119896 = 1 119867

(27)



Hence

119878119894(119896minus1)times119901 +

119901

sum

119895=1


(120579119894)

forall 119894 = 1 119899 119896 = 1 119867

(28)

Therefore

119901

sum

119895=1



forall 119894 = 1 119899 119896 = 1 119867

(29)




119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(30)





119880lowast= Min

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]]

(31)



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(120579119894)


forall 119894 = 1 119899 119896 = 1 119867


Δ119905times 119880119894 nom

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

119901

sum

119895=1


(32)



q = 1 q = 2

j = 1j = 2 j = p

Deg

rada

tion

rate

k = 1 k = 2 k = 3

T t2T

1205822 1205822p1205821

120582p+1

120575p+1
















(q minus 1) times T



T


1

2

3

Δt

120575((k+m)timesp)






119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896


times1

119880119894 nomΔ119905times 120582119899 (119905)]


(34)



Therefore




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(35)


Note that


119879] + 1 (36)




= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(37)




119879 = Round [119867 times 119901

119873] (38)








120601119879

(119880119873)= [

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(39)



=

119873+1

sum

119902=1

120601119879

(119880119873) (40)



120601(119880119873)

=

119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

(41)

Note that119873 = 1 2




Γ (119873) = [

[

Mc times119873+1

sum

119902=1

[

[

119901

sum



0


+

In[(119902times119879)Δ119905]

sum


119901

sum

119895=1

int


0


+


sum

119895=1

int


0


]

+Mp times 119873]

]

(43)







120597Γ (119873)

120597119873= 0 (44)



lim119873rarr1

Γ (119880119873) = lim119873rarr1

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


rarrconstant

)

= 119872119888 times 120601 (119880 1) + 119872119901

lim119873rarr+infin

Γ (119880119873) = lim119873rarr+infin

(119872119888 times 120601 (119880119873)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr0

+ 119872119901 times 119873⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr+infin

)

= +infin

(45)


Moreover

Γ (119880119873 + 1) minus Γ (119880119873) ge 0

997904rArr [119872119888 times 120601 (119880 (119873 + 1)) + 119872119901 times (119873 + 1)]



997904rArr 120601 (119880 (119873 + 1)) minus 120601 (119880119873) le119872119901

119872119888

(46)

In addition

Γ (119880119873) minus Γ (119880119873 minus 1) le 0

997904rArr [119872119888 times 120601 (119880119873) + 119872119901 times (119873)]




119872119888

(47)




exist119873lowast119904119894 120585119873 le

119872119901

119872119888

le 120585119873minus1 (48)

with

120585119873 = 120601 (119880119873) minus 120601 (119880 (119873 + 1)) (49)



119873lowastexist119904119894

120601 (119880 (119873 + 1)) minus 120601 (119880119873) ge 0

120601 (119880119873) minus 120601 (119880 (119873 minus 1)) le 0

lim119873rarr1

Γ (119880119873) = 119862119900119899119904119905119886119899119905

lim119873rarr+infin

Γ (119880119873) = +infin

(50)



6 Numerical Example




1198891 = 200 120590 (1198891) = 15

1198892 = 110 120590 (1198892) = 09

1198893 = 320 120590 (1198893) = 12

(51)





Table 1



Table 2









1205752

1205753

1205754

1205755

1205756

1205757

1205758

1205759

12057510

12057511

12057512



12057514

12057515

12057516

12057517

12057518

12057519

12057520

12057521

12057522

12057523

12057524






120582119899 (119905) =120572

120573times (

119905

120573)

120572minus1

=2

12times (

119905

12) (52)















Nlowast

d = di k ( )

Ulowast= Uijk ( )

k =

i =

k =i =j =

1 H1 n

1 p1 H1 n


0 2 4 6 8 10

4000

5000

6000

7000

8000


The t

otal

cost

of m

aint

enan

ceΓ

(N)



Γlowast(119873) (um) 119873




=





8 Conclusion






Appendices


We have119885 (119880)

=

119867

sum

119896=1

119901

sum

119895=1

119899

sum

119894=1

[119910119894119895119896 times (St (119894) + (Cp (119894) times 119864 [(119880119894119895119896)2

]))


Δ119905


])]



minus Int [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A1)

Also


minus Ent [119895

119901] times 119889119894 (119896)

forall 119894 = 1 119899 119895 = 1 119901 119896 = 1 119867

(A2)

Therefore


]


minus Ent [119895

119901] times 119889119894 (119896)]


minusEnt [119895

119901] times 119889119894 (119896) ])

2

]


]


minus [Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896))])

2

]


]



times (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ (Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]


]


]


times(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+ 119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A3)



]


]


times 119864 [(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))]

+119864[(Ent [119895

119901] times (119889119894 (119896) minus 119889119894 (119896)))

2

]]

(A4)




(A5)

Therefore


]


]

+(Ent [119895

119901])

2

times 119864 [(119889119894 (119896) minus 119889119894 (119896))2

]]

(A6)

We know that

119864 [(119909119896 minus 119909119896)2] = Var (119909119896)

(Int [119895

119901])

2

= Int [119895


119895

119901le 1

(A7)

Therefore


]


119901] times Var (119889119894 (119896))

(A8)


Finally


+ Ent [119895

119901] times Var (119889119894 (119896))

(A9)

Consequently

(i) for 119896 = 1

(a) 119895 = 1

Var (1198781198941) = Var (1198781198940) + (Ent [ 1

119901]) times Var (119889119894 (1))

(A10)

(b) 119895 = 2

Var (1198781198942) = Var (1198781198940) +2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A11)

(c) 119895 = 119901

Var (119878119894119901) = Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

(A12)

(ii) for 119896 = 2

(a) 119895 = 1

Var (119878119894119901+1) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+ Ent [ 1

119901] times Var (119889119894 (2))]

(A13)

(b) 119895 = 2

Var (119878119894119901+2) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (1)) + Ent [ 1

119901]

times Var (119889119894 (2)) + Ent [ 2

119901] times Var (119889119894 (2))]

(A14)

(c) 119895 = 119901

Var (119878119894(2times119901)) = [Var (1198781198940) +119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (1))

+

119875

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119901))]

(A15)


(a) 119895 = 1


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

1

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A16)

(b) 119895 = 2


sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901]

times Var (119889119894 (119876))

+

2

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A17)



= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]

(A18)

On the other hand


]




]


]


]



]]

(A19)

We know that



(A20)


Hence


]




]



]





]

(A21)

Consequently


]



(A22)

Noting that


]





(A23)




= [Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119896))]


]

= [ Var (1198781198940) +119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times Var (119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A24)

Consequently


]

= [ 1205902(1198781198940) +

119896minus1

sum

119876=1

119901

sum

119897=1

Ent [ 119897

119901] times 1205902(119889119894 (119876))

+

119895

sum

119897=1

Ent [ 119897


2

]

(A25)


119885 (119880) =

119867

sum

119896=1

119875

sum

119895=1

119899

sum

119894=1

119910119894119895119896 times (St (119894) + (Cp (119894) times 1198802

119894119895119896))


Δ119905

times [1205902(1198781198940)

+ (

119896minus1

sum

119876=1

119901

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119876)))

+ (

119895

sum

119897=1

Int( 119897

119901) times 1205902(119889119894 (119896)))


]

(A26)



(i) for 119896 = 1

(a) 119895 = 1

1205821 (119905) = (1205820) times (1 minus In [0

119902 times 119879]) +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (119905)

(B1)

(b) 119895 = 2

1205822 (119905) = 1205821 (1205751) times (1 minus In [1

119902 times 119879])

+

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)


1205822 (119905) = (1205820 +

119899

sum

119894=1

11988011989411 times Δ119905

119880119894 nom times 1205751

times 120582119899 (120575(1)))


119902 times 119879]) +

119899

sum

119894=1

11988011989421 times Δ119905

119880119894 nom times 1205752

times 120582119899 (119905)

(B2)

(c) 119895 = 119901


119902 times 119879])

+

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)

120582119901 (119905) = [(1205820 +

119901minus1

sum

119897=1

119899

sum

119894=1

1198801198941198971 times Δ119905

119880119894 nom times 120575119897

times 120582119899 (120575(119897)))


119902 times 119879]) +

119899

sum

119894=1

1198801198941199011 times Δ119905

119880119894 nom times 120575119901

times 120582119899 (119905)]

(B3)


(a) 119895 = 1

120582((119896minus1)times119901)+1 (119905)


119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

120582((119896minus1)times119901)+1 (119905)

= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

1198801198941119896 times Δ119905


times 120582119899 (119905)]

(B4)



= [(1205820 +

119896minus1

sum

119876=1

119901

sum

119897=1

119899

sum

119894=1

119880119894119897119876 times Δ119905



+

119895minus1

sum

119897=1

119899

sum

119894=1

119880119894119897119896 times Δ119905




119902 times 119879])

+

119899

sum

119894=1

119880119894119895119896 times Δ119905


times 120582119899 (119905)]


(B5)



References
























1 Introduction



















16

14

12

10

08

06

Ship

men

t cos

t (lowast$100)



R2= 0926



119864 (119876) = 119901119864 (min 119876119863) + 119904119864(119876 minus 119863)+ minus 119888119876

minus 119886 + 119887 ln [119864 (min 119876119863)]

minus 119886 + 119887 ln [119864(119876 minus 119863)+]



(1)









times[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2



(3)


119864(119863 minus 119876)+le[1205902+ (119876 minus 120583)

2]12

minus (119876 minus 120583)

2

(4)


119889119864 (119876)

119889119876=119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876 minus 120583)

2[1205902 + (119876 minus 120583)2]12

minus 1198872119876 + 2120583 (119876 minus 120583) [120590

2+ (119876 minus 120583)

2]minus12


(5)

1198892119864 (119876)

1198891198762= minus

(119901 minus 119904) 1205902

2[1205902 + (119876 minus 120583)2]32

minus 119887

minus 2 + 21205832minus 21198762

+ 4120583[1205902+ (119876 minus 120583)

2]12

+2120583 (minus1 minus 120583

2+ 4120583119876 minus 3119876

2)

[1205902 + (119876 minus 120583)2]12

minus81205832(119876 minus 120583)

2

1205902 + (119876 minus 120583)2

+(119876 minus 120583)

2(21205833+ 2120583 minus 2120583119876

2)

[1205902 + (119876 minus 120583)2]32

sdot minus1 minus 1205832+ 1198762+ 4120583[120590

2+ (119876 minus 120583)

2]12

minus2

(6)




Iteration 119894 119876119894

1198911015840(119876119894) 119891

10158401015840(119876119894) 119891

1015840(119876119894)11989110158401015840(119876119894) 119876

119894+1





Let119876119894+1 = 119876119894 minus (1198911015840(119876119894)119891


1198911015840(119876119894) =

119901 + 119904 minus 2119888

2minus

(119901 minus 119904) (119876119894 minus 120583)

2[1205902 + (119876119894 minus 120583)2]12

minus 1198872119876119894 + 2120583 (119876119894 minus 120583) [120590

2+ (119876119894 minus 120583)

2]minus12


2]12

(7)

From (6) we set

11989110158401015840(119876119894) = minus

(119901 minus 119904) 1205902

2[1205902 + (119876119894 minus 120583)2]32

minus 119887 minus 21198762

119894minus 2 + 2120583

2+ 4120583[120590

2+ (119876119894 minus 120583)

2]12

+2120583 (4120583119876119894 minus 3119876

2

119894minus 1205832minus 1)

[1205902 + (119876119894 minus 120583)2]12

+(119876119894 minus 120583)

2(21205833+ 2120583 minus 2120583119876

3

119894)

[1205902 + (119876119894 minus 120583)2]32

minus81205832(119876119894 minus 120583)

2

1205902 + (119876119894 minus 120583)2

(8)






0 119876lowast


0= exp(1minus


0= 119876lowast



is119876lowastlowast119894= (119876119904

119894+119876lowast



119894= (119876119904

119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887








10158401015840(1198760)) =








lowast) 119864(119876

lowast) ()


Average 405

12

10

8

6

4

2

0

5 10 15 20

Order quantity Q

Expe

cted

pro

fitE

(Q)














119864 (119878) = 119901119864 (min 119878 119863) + 119904119864(119878 minus 119863)+


minus 119886 + 119887 ln [119864(119878 minus 119863)+]







2]12

minus (119878 minus 120583)

2



(10)


119901 minus 119904

2[1205902+ (119903 minus 120583)

2]12

minus 119903 + (119888 minus 119904) 119903



2[1205902+ (119878 minus 120583)

2]12



= 0

(11)





1015840(119903119894)) where

119891 (119903119894) =119901 minus 119904

2[1205902+ (119903119894 minus 120583)

2]12

minus 119903119894 + (119888 minus 119904) 119903119894

+ 119887 ln minus1205832 minus 1 + 1199032119894+ 2120583[120590

2+ (119903119894 minus 120583)

2]12


2[1205902+ (119878 minus 120583)

2]12



1198911015840(119903119894) =

2119888 minus 119901 minus 119904

2+

(119901 minus 119904) (119903119894 minus 120583)

2[1205902 + (119903119894 minus 120583)2]12

+ 1198872119903119894 + 2120583 (119903119894 minus 120583) [120590

2+ (119903119894 minus 120583)

2]minus12


2]12

(12)










=119888

120588(119898 + 119889) 120583 minus (119898 + 119889) 119864[119863 minus 119866 (119876)]

+




(13)


119864[119863 minus 119866 (119876)]+le[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

(14)


119864 (119876) ge119888

120588

(119898 + 119889) 120583 minus (119898 + 119889)

times[1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

minus (120588119876 minus 120583)

2

minus (120588 + 119889 minus 1)119876

minus 2119886 + 2119887 ln 2


+ 2 (120583 + 120588119876 minus 119876)

times [1205902+ 120588119902119876 + (120588119876 minus 120583)

2]12

(15)


119889119876

= minus119888 (119898 + 119889)

2[1


119888

120588(120588 + 119889 minus 1)





(16)

where119883 = 1205902 + 120588119902119876 + (120588119876 minus 120583)2

1198892119864 (119876)

1198891198762= minus119888 (119898 + 119889)

2[minus120588

4(119902 minus 2120583 + 2120588119876)

2119883minus32

+ 120588119883minus12]

minus 1198871198841015840119885 minus 119884119885

1015840

1198852


12



+ 2 (120583 + 120588119876 minus 119876)11988312

1198841015840= 4120588 (1 minus 120588) minus 2120588


minus1205882

2(120583 + 120588119876 minus 119876) (119902 minus 2120583 + 2120588119876)

2119883minus32

1198851015840= 2 (1 minus 120588) (2120588119876 + 120583) minus 120588119902

minus 2 (1 minus 120588)11988312+120588

4(120583 + 120588119876 minus 119876)


(18)





1015840(119876119894)119891

10158401015840(119876119894)) where 119891

1015840(119876119894) and




2119864(119876119894+1)119889119876

2




0 119876lowast


0= exp(1minus


0= 119876lowast



is 119876lowastlowast119894= (119876119904

119894+ 119876lowast




119894+ 119876lowast

119894)2

Step 6 If 119864(119876119887119894) ge 119864(119876

lowast


119894+1= 119876119887

119894and 119876119904

119894+1= 119876lowast

119894


119894and 119876119904

119894+1= 119876119887


119894 = 119894 + 1



2





2

119895 respectively



1 119876lowast2 and

119876lowast

119873

max1198761 119876119873

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583[1205902

119895+ (119876119895 minus 120583119895)

2

]12

(19)

Subject to119873

sum

119895=1

119888119895119876119895 le 119861 (20)


119871 (1198761 119876119873 120582)

=

119873

sum

119895=1

(119901119895 minus 119904119895)

(119876119895 + 120583119895) minus [1205902

119895+ (119876119895 minus 120583119895)

2

]12

2



2

119895

+2120583119895[1205902

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582(

119873

sum

119895=1

119888119895119876119895 minus 119861)

(21)


120597119871 (1198761 119876119873 120582)

120597119876119895

=119901119895 + 119904119895 minus 2119888119895


2[1205902119895+ (119876119895 minus 120583119895)

2

]12

minus 119887

2119876119895 + 2120583119895 (119876119895 minus 120583119895) [1205902

119895+ (119876119895 minus 120583119895)

2

]minus12

minus1 minus 1205832 + 1198762119895+ 4120583119895[120590

2

119895+ (119876119895 minus 120583119895)

2

]12

+ 120582119888119895

(22)


119873with

















Step 1 Let 119895 = 1 2 3 4


1= 8731 119876lowast

2= 7762 119876lowast

3= 11072 and 119876lowast

4

= 21243 Check sum4119895=1119888119895119876lowast

119895= $92 gt $80 where (20) is not



3is set to 0


119895= $61 lt $80 (20) is satisfied










Highlights









References



































1 Introduction













max 120601

st 120582119883 le 119909119863

119900

120582119884119863ge 120601119910119863

119900

120582119884119880

ge 120601119910119900119880

119890120582 = 1

120582 ge 0

(1)

in which 119910119900119880= minus119884



max 120601

st 120582119883119863le (1 minus 120601) 119909

119863

119900

120582119883119880

le (1 minus 120601) 119909119900119880

120582119884119863ge (1 + 120601) 119910

119863

119900

120582119884119880

ge (1 + 120601) 119910119900119880

119890120582 = 1

120582 ge 0

(2)

in which 119910119900119880= minus119884

119880+ V gt 0 and 119883

119880

= minus119883119880+ 119908 gt 0




min 1 minus [

[

119908119900 +1

119898 + 119904(sum

119894isin119868119863

119905minus119863

119894+ sum

119903isin119874119863

119905+119863

119903)]

]

st119899

sum

119895=1

120582119895119909119863

119894119895+ 119905minus119863

119894= 119909119863

119894119900minus 119908119900 119894 isin 119868119863

119899

sum

119895=1

120582119895119909119880

119894119895+ 119905minus119880

119894= 119909119880

119894119900+ 119908119900 119894 isin 119868119880

119899

sum

119895=1

120582119895119910119863

119903119895minus 119905+119863

119903= 119910119863

119903119900+ 119908119900 119903 isin 119874119863

119899

sum

119895=1

120582119895119910119880

119903119895minus 119905+119880

119903= 119910119880

119903119900minus 119908119900 119903 isin 119874119880

119899

sum

119895=1

120582119895 = 1

(3)


min 120579

st119899

sum

119895=1

120582119895119909119894119895 le 120579119909119894119900 forall119894 isin 119868

119899

sum

119895=1

120582119895119910119863

119903119895ge 119910119863

119903119900forall119903 isin 119874

119863

119899

sum

119895=1

120582119895119910119880

119903119895le 120579119910119880

119903119900forall119903 isin 119874

119880

120582119895 ge 0 forall119895 isin 119873

(4)




min 120579

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le 120579

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 ge

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(5)


max119898

sum

119894=1

119904minus

119894+

119904

sum

119903=1

119905+

119903

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 + 119904minus

119894= 120579lowast

119899

sum

119895=1

119909119894119895 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119903119895 minus 119905+

119903=

119899

sum

119895=1

119910119903119895 119903 = 1 119904

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

119904minus

119894ge 0 119905

+

119903ge 0 119894 = 1 119898 119903 = 1 119904

120582119895119896 ge 0 119896 119895 = 1 119899

(6)




119909119894119900 =

119899

sum

119895=1

120582119900lowast

119895119909119894119895 = 120579

lowast119909119894119900 minus 119904

minuslowast

119894119894 = 1 119898

119910119903119900 =

119899

sum

119895=1

120582119900lowast

119895119910119903119895 = 119910119903119900 + 119905

+lowast

119903119903 = 1 119904

(7)


119878119894 =

119899

sum

119895=1

119909119894119895 minus

119899

sum

119895=1

119909119894119895 ge 0 119894 = 1 119898

119879119903 =

119899

sum

119895=1

119910119903119895 minus

119899

sum

119895=1

119910119903119895 ge 0 119903 = 1 119904

(8)


max119899

sum

119895=1

119904

sum

119903=1

119906119903119910119903119895 +

119899

sum

119896=1

120577119896

st119899

sum

119895=1

119898

sum

119894=1

V119894119909119894119895 = 1

119904

sum

119903=1

119906119903119910119903119895 minus

119898

sum

119894=1

V119894119909119894119895 + 120577119896 le 0 119895 119896 = 1 119899

119906119903 ge 0 119903 = 1 119904

V119894 ge 0 119894 = 1 119898

(9)



5 Proposed Method



119883 = 119909119894119895 isin 119877119898times119899

+ 119884

119863= 119910119863

119903119895 isin 119877119904119863times119899

+

119884119880= 119910119880

119905119895 isin 119877119904119880times119899

+

(10)


max 120593

st119899

sum

119896=1

119899

sum

119895=1

120582119895119896119909119894119895 le

119899

sum

119895=1

119909119894119895 minus 120593119877119909119894 119894 = 1 119898

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119863

119903119895ge

119899

sum

119895=1

119910119863

119903119895+ 120593119877119910

119863

119903119903 = 1 119904

119863

119899

sum

119896=1

119899

sum

119895=1

120582119895119896119910119880

119905119895le

119899

sum

119895=1

119910119880

119903119895minus 120593119877119910

119880

119905119905 = 1 119904

119880

119899

sum

119895=1

120582119895119896 = 1 119896 = 1 119899

120582119895119896 ge 0 119896 119895 = 1 119899

(11)

where119877119909119894119877119910119863

119903 and119877119910119880



119863

119903 119877119910119880



119903 119877119910119880

119905) vector as

follows

119877119909119894 =

119899

sum

119895=1

119909119894119895 minus 119899 (min 119909119894119895119895=1119899) 119894 = 1 119898

119877119910119863

119903=

119899

sum

119895=1

119910119863

119903119895minus 119899 (max 119910119863

119903119895119895=1119899

) 119903 = 1 119904119863

119877119910119880

119905=

119899

sum

119895=1

119910119880

119905119895minus 119899 (min 119910119880

119905119895119895=1119899

) 119905 = 1 119904119880

(12)


119863

119903 119877119910119880

119905) can be

chosen as follows

(119877119909119894 119877119910119863

119903 119877119910119880

119905) = (

119899

sum

119895=1

119909119894119895

119899

sum

119895=1

119910119863

119903119895

119899

sum

119895=1

119910119880

119905119895) (13)











119863

119903 119877119910119880



6 Numerical Example





7 Conclusion







References


































1 Introduction







20 40 60 80 100Utilization ()

Cycle

tim

eIn

crea

sing


20 40 60 80 100

Low variability

Utilization ()

High variability

Cycle

tim

eIn

crea

sing

te

to




CT119902 = 119881119880119879 (1)



2 Literature Review














119886= 1205902

1198861199052

119886and

1198622

119890= 1205902

1198901199052



2



2

119890gt 1198622



CT119902 = (1198622

119886+ 1198622

119890

2)(

119906

1 minus 119906) 119905119890 (2)

1198622

119889= 11990621198622

119890+ (1 minus 119906

2) 1198622

119886 (3)










Patient



Home

10 min24 min

20 min11 min11 min

98 min0 min

10 min0 min

3 min3 min0 min

10 min




Clean room




ORemergence

time

Patient out of room

Inpatient room


(Nj)

WW W W W W


1205902


1198882




arrivals






2


as 083

119860 =119898119891

119898119891 + 119898119903

(4)

119905119890 =119905119900

119860 (5)

119906 =119903119886

119903119890

=119903119886119905119890

119898 (6)

1198622

119890= 1198622

119900+ 2119860 (1 minus 119860)

119898119903

119905119900

(7)






Specialty 1

FCFS

Specialty 2

Specialty 15


Operating room

Recoverroom

First stage (W1)

Second stage (W2)

(Mj)

(Nj)

M1

M2

M12

N1

N2

N3

N4

N5

N24





119905119890 = 119905119900 +119905119904

119873119904

(8)

1205902

119890= 1205902

119900+1205902

119904

119873119904

+119873119904 minus 1

1198732119904

1199052

119904

1198882

119890=1205902

119890

1199052119890

(9)




119890= 749







CT119902 =1198622

119886+ 1198622

119890

2

119906

1 minus 119906119905119890

=(099 + 749)

2(

089

1 minus 089) 14543

= (424) (809) (14543)

(10)














120583 (11)




(12)



5 Empirical Results








1 10800 813 10800

813≒ 13

2 10800 159 10800

159≒ 68

3 10800 134 10800

134≒ 81

4 10800 346 10800

346≒ 31

5 10800 185 10800

185≒ 58

Input

Output


Simulationmodel






1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

0020018001601400120010008000600040002

0

CV


WIP




Parameters



Patient



suture and finish

Home

Leveling

Billing and coding

Waiting time= 5 min




Inpatient room

W W W W





10 min0 min

98 min0 min5 min




Decision variables



Minimize24

sum

119895=1

119874119895 (13)


5

sum

119894=1

119860 119894119895 = 1 forall119895 (14)

119862119894 ge 1 forall119894 (15)

5

sum

119894=1

119862119894 = 24 (16)

119860 119894119895 isin 0 1 forall119894119895 (17)






















Acknowledgments


References













































Mathematical Problems in Emerging Manufacturing ... - Hindawi

Documents