Dynamic Drum-Buffer-Rope approach for production planningand
control in capacitated ow-shop manufacturing systemsqPatroklos
Georgiadis, Alexandra PolitouIndustrial Management Division,
Department of Mechanical Engineering, Aristotle University of
Thessaloniki, P.O. Box 461, 541 24 Thessaloniki, Greecearti cle i
nfoArticle history:Received 28 March 2012Received in revised form
22 February 2013Accepted 23 April 2013Available online 3 May
2013Keywords:Drum-Buffer-RopeTime-bufferFlow-shopProduction
planning and controlSimulationSystem
dynamicsabstractDrum-Buffer-Rope-based production planning and
control (PPC) approaches provide productionmanagers with effective
tools to manage production disruptions and improve operational
performance.Thecornerstoneoftheseapproachesistheproperselectionoftime-bufferswhichareconsideredasexogenously
dened constant. However, the majority of real-world manufacturing
systems arecharacterized by the dynamic change of demand and by
stochastic production times. This fact calls fora dynamic approach
in supporting the decision making on time-buffer policies. To this
end, we study acapacitated, single-product, three-operation,
ow-shopmanufacturingsystem. Weproposeadynamictime-buffer control
mechanism for short/medium-term PPC with adaptive response to
demand changesand robustness to sudden disturbances in both
internal and external shop environment. By integratingthecontrol
mechanismintotheow-shopsystem, wedevelopasystemdynamicsmodel
tosupportthedecision-makingontime-bufferpolicies. Usingthemodel,
westudytheeffectof policiesonshopperformance by means of analysis
of variance. Extensive numerical investigation reveals the
insensitivityof time-buffer policies to key factors related to
demand, demand due date and operational characteristicssuch as
protective capacity and production times. 2013 Elsevier Ltd. All
rights reserved.1. IntroductionInsufcient productionplanning in
manufacturing systems
oftenturnsanon-bottleneckresourcetocapacityconstraint
resource(CCR), whichoperates as a bottleneckwithonaverage excess
capacity(Goldratt, 1988). Drum-Buffer-Rope (DBR)-based
productionplanning andcontrol (PPC) approaches focus onthe
synchronizationof resources andmaterial utilizationinCCRs of
manufacturingsystems (Goldratt &Fox, 1986; Sivasubramanian,
Selladurai, &Rajamramasamy, 2000). This synchronization calls
for time-buffersthat protect the production plans of CCR fromthe
effects ofdisruptionsat theprecedingproductionresources.
Bymeansoftime-buffers (i.e. constraint, assembly, shipping
time-buffers),buffermanagementmonitorstheinventoryinfrontofprotectedresources
to effectively manage and improve systems performance(Schragenheim
& Ronen, 1990; Schragenheim & Ronen, 1991).The research
agenda on the efciency of DBR approach in PPC
ofmanufacturingsystemshasreceivedincreasedattentionduringthe last
decade. The basic assumption in all relative studies is
theexogenous determination of time-buffers as a constant
throughoutthe planning horizon. However, the majority of
real-worldmanufacturing systems are characterized by the dynamic
changeof demandandbystochastic productiontimes. Therefore,
thedecision making on time-buffer policies calls for a
dynamicmechanism. This is exactly the purpose of this paper.
Morespecically, weconsideradynamic, capacitated,
single-product,three-operation, ow-shop production system. We dene
asproduction time-buffer (PTB), the total of constraint and
shippingtime-buffers. We propose a dynamic, goal-seeking,
feedbackmechanism to dene PTB for short/medium-term PPC. By
integratingthe proposed mechanism into the ow-shop system, we
develop
asystemdynamics(SD)modeltosupportthedecisionmakingonPTBpolicies.
Westudytheshopresponse(dynamicsof productows, inventories,
performancemeasures)toPTBpoliciesunderstochastic demand and
production times. Since the dynamicbehavior may be used to evaluate
the efciency of a specic PTBpolicy, the SD model can be viewed as a
decision support system(DSS) for PTB-related decisions. In
particular, by continuousmonitoring, the actual level of PTB is
adjusted to demand-drivendesired values. The innovative element of
the control mechanismistheendogenousdenitionof desiredPTBvalues.
Inaddition,the mechanism provides robustness to sudden disturbance
occur-rences in demand and shop operations. This is a positive
propertyto cope with uncertainty issues in both external and
internal shopenvironment. UsingtheSDmodel,
wedeterminePTBincrease/decreasepolicies throughout
agivenplanninghorizonandwestudytheireffectonshopperformancebymeansof
analysisof0360-8352/$ - see front matter 2013 Elsevier Ltd. All
rights reserved.http://dx.doi.org/10.1016/j.cie.2013.04.013qThis
manuscript was processed by Area Editor Manoj Tiwari.Corresponding
author. Tel.: +30 2310 996046; fax: +30 2310 996018.E-mail
addresses: [email protected], [email protected] (P. Georgiadis),
[email protected] (A. Politou).Computers & Industrial
Engineering 65 (2013)
689703ContentslistsavailableatSciVerseScienceDirectComputers &
Industrial Engineeringj our nal homepage: www. el sevi er. com/ l
ocat e/ cai evariance (ANOVA). The examination of results obtained
byextensive numerical investigation reveals the insensitivity of
PTBpoliciestokeyfactorsrelatedtodemand, demandduedateandoperational
characteristics such as protective capacity and produc-tion times.
This is an additional appealing feature of the proposedPTB control
mechanism which provides production managers withexibility on
PTB-related decisions.The rest of the paper is organized as
follows. Section 2presents the literature review on DBR studies and
applications inmanufacturing systems and justies the suitability of
SDmethodology in developing dynamic DBR-based PPC systems.Section 3
contains the ow-shop systemunder study and its perfor-mance
measures, the description of the SDmodel, the
mathematicalformulation and the models validation. Section 4
presents the con-trol parameters under study along with their sets
of values,
whileSection5presentstheadaptabilityandrobustnesspropertiesofthe
dynamic PTB control mechanism. The effect of PTB policies onthe
shops performance obtained by numerical investigation is givenin
Section 6. Finally, in Section 7 we wrap-up with a summary,
thelimitations of our work and directions for model extensions.2.
Literature reviewReview papers present a variety of PPC problems
dealing withow-shop scheduling in manufacturing systems; either
withsequence-independent set-up times or sequence-dependentset-up
times (Hejazi and Saghaan, 2005; Zhu & Wilhelm, 2006).The
proposed scheduling methods include: (i) exact methods suchas
dynamic programming (Held & Karp, 1962),
branch-and-bound(Grabowski, Skubalska, & Smutnicki, 1983),
integer programming(Frieze & Yadegar, 1989) and complete
enumeration; (ii) heuristicmethods such as Palmer algorithm
(Palmer, 1965), Guptaalgorithm(Gupta, 1971), CDS
algorithm(Campbell, Dudek, &Smith, 1970) and NEH algorithm
(Nawaz, Enscore, & Ham,
1983);and(iii)metaheuristicmethodssuchassimulatedannealing-SA(Liu,
1999), genetic algorithms-GA (Reeves, 1995), tabu search-TA(Widmer
& Hertz, 1989), greedy approaches (Carpov, Carlier, Nace,&
Sirdey,2012),variable-depth search approach (Jin,Yang,&
Ito,2006), pilot methods (Vob, Fink, &Duin, 2005), hill
climbingprocedures(Nearchou, 2004),
antcolonysystem-ACS(Rajendran&Ziegler, 2004), articial neural
network-ANN(Lee &Shaw,2000) and hybrid algorithms (Wang &
Zheng, 2003).For the specic make-to-order ow-shop environment,
Stevenson,Hendry, andKingsman(2005)
provideadetailedreviewontheemployed PPC approaches. The commonly
used approaches includeConstant Work In Process-CONWIP (Framinan,
Gonzlez, &Ruiz-Usano, 2003), Workload Control-WLC (Th} urer,
Stevenson,&Silva, 2011), Material
RequirementPlanning-MRP(Bertrand&Muntslag, 1993),
Just-in-Time-JIT (Singh & Brar, 1992), Theory ofConstraints-TOC
(Atwater, Stephens, & Chakravorty, 2004; Goldratt& Cox,
1984; Mabin & Balderstone, 2003), Paired cell
OverlappingLoopsof Cards withAuthorization-POLCA(Riezebos, 2010)
andweb- or e-based Supply Chain Management-SCM (Cagliano,
Caniato,& Spina, 2003; Kehoe & Boughton, 2001). The
comparison of MRP,TOC and JIT approaches justies the TOC to be more
effective for apure ow shop or general ow shop system, when the
bottleneckresources are stationary positionedinthe
productionprocess.The effectiveness of TOC approaches is further
discussed for highlycustomized industries facing difculties in
estimating in advancethe processing times (Stevenson et al., 2005).
This is due to the factthat TOC requires dataaccuracy only in CCR
tocontrol theplantthroughput (Gupta & Snyder, 2009).TOCwas rst
developedinthemid-1980s (Goldratt &Cox,1984; Gupta, 2003). It
uses the DBR production
schedulingapproach;productionprocessisscheduledtoruninaccordancewith
the needs of the CCR, as CCR determines the performance ofthe whole
production system. The advantages of TOC are
discussedinvariousindustrialimplementationsreportingthereductionofinventories
by 49% and the improvement of due date and nancialperformance by
60% (Gupta, 2003; Mabin & Balderstone,2003).
Thestatisticalanalysisofasurveywithquestionnairestomanufacturing
managers performing TOC, JIT and traditionalmethods provides
further insights regardingthe superiorityofTOCapproachesonother
approachesintermsof nancial andoperational performance measures
(Sale & Inman, 2003). TOC hasalso been used for the
determination of optimal, or near optimal,product mix decisions
(Aryanezhad &Komijan, 2004; Souren,Ahn, & Schmitz, 2005).In
certain studies, the usefulness of DBR logic in PPC ofmanufacturing
systems is revealed by conducted simulation exper-imentsunder
different manufacturingsettings.
Inthesestudiestimebuffersarenotoptimizedandremainconstantthroughoutthe
simulation process. In a make-to-stock environment,DBR-based PPC is
combined to manufacturing expediting ofproducts (Schragenheim, Cox,
& Ronen, 1994). In a make-to orderenvironment, DBR approach is
combined with different orderreview/releasepolicies(Russell
&Fry, 1997)anditiscomparedto the previously used approach in
furniture manufacturing rms(Wu, Morris, &Gordon, 1994).
Inaserial productionlinewithexponentially distributed processing
times setting, DBR-basedPPC is compared to CONWIP approach
(Gilland, 2002). The differ-encebetweenthetwoapproachesisthat
inCONWIPapproachmaterial unitsarereleasedintothelineat arateequal
tolinethroughput, while in DBR approach at the rate they are
producedat CCR. The outperformance of DBR approach is proved to
increaseas CCR moves closer to the rst operation of production as
well aswhentherequiredthroughput or service level is
closetothesystemscapacity(Framinanetal., 2003). Finally,
inaow-shopsetting, Sirikri
andYenradee(2006)employDBR-basedPPCandinvestigate how buffer sizes
related to lead time up to CCR affectspecic performance
measures.The analytical approaches to determine time-buffer sizes
basedon queuing theory are limited in simple PPC
manufacturingproblems. In these approaches the constraint resource
is modeledasaM/M/1/Ksystem(Radovilsky, 1998),
whiletheproductionsystem is modeled either as a determination
model, where a treestructure represents the relationship between
the constraintmachine and its feeder machines (Tu & Li, 1998;
Ye & Han, 2008)or as multiproduct open queuing network in which
the productionoperations are modeled as GI/G/m (Louw & Page,
2004).The applicability of DBR in real world case studies is
denoted bya lot of DBRimplementations inmanufacturing rms; e.g.
inOrko-PakinNetherlands that manufactures
packagingmaterialfromcorrugatedcardboard(Riezebos, Korte,
&Land, 2003), inOregon Freeze Dry processing products by
removing water at lowtemperature and pressure (Umble, Umble, &
von Deylen, 2001), inAlamedaNaval AviationDepot that
remanufacturesaircraft, jetturbine engines, engine components and
avionics equipment(Guide & Ghishelli, 1995), in a light
assembly rm for heavy dutytrucks andtrailers (Pegels &Watrous,
2005) andinabearingmanufacturing company (Steele, Philipoom,
Malhotra, & Fry,2005). DBRisevenusedinaghtersquadronof
theIsraeli AirForceforbetterschedulingof
itsmissionsandallocatingcrewsto aircraft (Ronen, Gur, & Pass,
1994).DBR literature suggests several performance measures
inevaluating the efciency of the proposed approaches. Thesemeasures
include: theaveragesystemthroughput, theaveragenished product
inventory, the average number of stockouts(Duclos & Spencer,
1995); the throughput, the utilization ofmachines, the average wait
time of items, the percentage ofmachineblocking(Mahapatra&Sahu,
2006); themeanpercent690 P. Georgiadis, A. Politou/ Computers &
Industrial Engineering 65 (2013) 689703completionto schedule, the
meanwork-in-process (WIP), themeanthroughput rate,
theproductmeantimeinsystem (Guide,1996); the total system output,
the average and standard deviationof ow time (Cook, 1994); and the
average wait time of items, theaverage WIP in queue and the system
throughput (Betterton & Cox,2009).In the above-mentioned DBR
studies, time-buffer is consideredas an exogenously dened constant
throughout the planninghorizon. Further limitation is the inability
to handle the non-linear,non-stationaryanduncertainnatureof
productionprocessthatcharacterizesthemajorityof
ow-shopmanufacturingsystems.Theselimitationscall
foradynamictime-buffercontrol mecha-nism, suitable for monitoring
and adjusting time-buffer values todesired levels. Such a control
mechanism can be provided by theusage of SD methodology. Therefore,
SD is the primary modelingandanalysistoolusedinthispaper.
Forrester(1961)introducedSDintheearly1960sasamodelingandcontinuoussimulationmethodologyfordecision-makingincomplexdynamicindustrialmanagement
problems (GrBler, Thun, & Milling, 2008). Incontrast to the
traditional discrete event simulation-basedDSS, the methodology
provides an understanding of
changesoccurringwithinamanufacturingenvironment, byfocusingonthe
interaction between physical ows, information ows,
delaysandpoliciesthat createthedynamics ofthevariables
ofinterestandthereafter searches for policies
toimprovesystemperfor-mance (Georgiadis &Michaloudis, 2012;
Sterman, 2000). Thestructureof aSDmodel isdescribedbystocksandows.
Thisstructure provides the PPC with a capability to capture
thedynamics of material, product and information ows under
causaleffectsoriginatedfromtheinternal andexternal
shopenviron-ment. The SD discipline acknowledges at the outset that
realisticrepresentations may include non-linear elements,so
closed-formsolutions are bypassed in favor of a simulation
methodology.Although the rich body of SD studies in PPC issues,
DBR-basedapproaches are very limited. In particular, these studies
introducethe potential use of SD theory along with the expected
advantages.WixsonandMills (2003) present anumerical exampleof
DBRproduction process and show how SD may help in
understandingthesystemconstraints.
Theyconsiderthetime-bufferofDBRasanexternal parameter and assume
constant productiontimesand innite raw materials. The potential use
of SD in developingDBR-based PPC systems for a make-to-order,
ow-shop system ispresented by Politou and Georgiadis (2008). They
assumeexponentiallydistributedproductiontimes,
niterawmaterialsandconstant PTBthroughout theplanninghorizon. This
papertakes the last research further by developing an
endogenouscontrol
mechanismforPTBandintegratingintoaSD-basedPPCsystemforow-shopoperations.
Thenewpossibilitiesprovidedby the proposed SD model can be
summarized in its ability to
copewiththechallengesfordynamicDBRapproachesinastochasticow-shop
environment. This ability contributes to the PPCfunctionof
ow-shopmanufacturingsystems
providingstable,controllableandadaptiveproductionplanswhichdeal
withthenon-linear, non-stationary and uncertain nature of
productionprocesses.3. The SD model3.1. The ow-shop system under
studyWe consider a three-operation, capacitated ow-shop
thatproduces a single product and purchases one type of raw
material(referredasmaterialattheremainderofpaper).
TheCCRoftheow-shopliesinitssecondoperationandproductionratesaredened
by DBR approach. The demand follows a normaldistribution.
Eachoperationof theow-shopisconsideredasaqueueing model M/M/1, in
which product arrival is described by aPoisson process with the
parameter k being equal to mean demand.The capacities of the three
operations are described by a
PoissonprocessandtheirmeanvaluesaresetequaltoCapi_M(i = 1, 2,
3).The mean value of the production time at each operation (1/li)
isdened in Eq. (1) by means of the respective mean
capacity:1li1Capi M1Therefore, production times followexponential
distributionwithparameters liequaltoorgreater thank. Consequently,
theshop operation is considered as a series of three queuing
modelsM/M/1(Hillier &Lieberman, 1995). Thus,
themeanproductiontime (MPT) which is the mean value of the total
production timeof shops operations is given in Eq. (2):MPT
X3i11liX3i11Capi M2Thenotationandtherespectiveunits of measure are
giveninTable 1.Table 1Notation list and units of measure.Flow-shop
variablesCapiCapacity of the i operation (i = 1, 2, 3),
items/dayCapi_MMean value of Capi (i = 1, 2, 3),
items/dayCapCCRCapacity of the CCR operation, which is equal to
Cap2, items/dayCapCCR,mMean of CapCCR, items/dayD Demand,
items/dayDmMean value of D, items/dayDSDStandard Deviation of D,
items/dayDB Demand Backlog, itemsDBDR Demand Backlog Decrease Rate,
items/dayDBIR Demand Backlog Increase Rate, items/dayDDD Demand Due
Date, daysED Expected Demand, items/dayFPI Finished Product
Inventory, itemsMCR Material Consumption Rate, kg/dayMF Material
Factor (i.e. for production of one item, MF kg of rawmaterial are
required), kg/itemMFO Material For Order, kgMFODR Material For
Order Decrease Rate, kg/dayMFOIR Material For Order Increase Rate,
kg/dayMI Material Inventory, kgMLT Material Lead Time, daysMO
Material Order, kgMOR Material Order Rope, items/dayMP Material
Procurement, kgMPR Material Procurement Rate, kg/dayMPT Mean
Production Time, daysMRR Material Release Rate, kg/dayMRT Material
Release Time, daysMUR Material Usage Rate, kg/dayOB Orders Backlog,
itemsOR Order Release backlog, itemsORR Order Release Rate,
items/dayORT Orders Rate, items/dayPDF Planned Demand Fulllment,
itemsPORR Planned Order Release Rate, items/dayPRi Production Rate
of the i operation (i = 1, 2, 3), items/dayPTB Production
Time-Buffer, daysr Delay time used in computation of MOR, daysSR
Shipments Rate, items/dayWIP0 MI that has been released in
production process and waits to beprocessed at rst operation,
kgWIPi Work-In-Process inventory at the i operation (i = 1, 2),
items1/c Smoothing factor used in computation of ED, 1/days1/liMean
value of production time of the i operation (i = 1, 2,
3),days/itemP. Georgiadis, A. Politou/ Computers & Industrial
Engineering 65 (2013) 689703 6913.2. Performance measuresThe
efciency of PTB policies in association with themanufacturing
process is obtained at the end of a given planninghorizon using
performance measures suggested by the DBRliterature. Besides, the
efciency of PTB policies in association withtheevaluationprocess is
obtainedat theendof theplanninghorizon using two performance
measures related to PTB. All thesemeasures are shown in Table
2.3.3. Conceptual modelingIn SD discipline, causal-loop diagrams
are maps of the systemsunder study showing the causal links among
the incorporated vari-ables (Sterman, 2000). Thegeneric
causal-loopdiagramof thedeveloped SD model is depicted in Fig.
1.ThecontrolmechanismsshowninFig. 1arepresentedinthefollowing
subsections. To improve appearance and distinctionamong the
variables in the causal-loop diagrams, we changed theletter style
according to the variable style; stock (state) variablesare written
in capital letters,ow variables in small plain lettersand auxiliary
variables in small italic letters. Stocks integrate theirows,
characterizethestateofthesystem, givesystemsinertiaand provide it
with memory. The arrows (inuence lines) representthe relations
among variables. The direction of the inuence
linesdisplaysthedirectionoftheeffect. Thesign+ or
oneachinuencelineexhibits thesignof theeffect. A+ () signsignies
that the variables change in the same (opposite) direction.3.3.1.
Material release rate control mechanismFig. 2 depicts the material
release rate control mechanism
(forthenotationthereadermustrefertoTable1). Sincethecornerstoneof
DBRsystems is thesynchronizationof resources andmaterial
utilization, thematerial releasescheduleisthedrivingforce for the
production planning. Material release (MaterialReleaseRate,
MRR)isbasedonorderrelease(OrderReleaseRate,ORR). Morespecically,
theplanof orderrelease(PlannedOrderRelease Rate, PORR) is set by
means of a pipeline delay of Demand(D) withadurationequal
toMaterial ReleaseTime(MRT). PORRincreases ORDER RELEASE (OR)
backlog, which is depleted by
ORR.ORRislimitedbyMATERIALINVENTORY(MI). MRTisdenedbymeansof
DemandDueDate(DDD), MeanProductionTime(MPT),DandPTB. The denitionof
MRT is basedonthe schedulinglogic of backward innite loading (Park
& Bobrowski, 1989;Sabuncuoglu&Karapinar, 1999);
theMRTvalueispredictedbyback scheduling fromDDD by means ofMPT
andPTB, in orderDto be fullled on time. The related equations are
given in AppendixA (Eqs. (A.1)(A.6)).3.3.2. Material procurement
control mechanismInFig. 3,
theLoop1describestheDBRlogicforthematerialprocurement control
mechanism. More specically, Material OrderRope (MOR) is theropeof
theDBRlogic andit is denedbyassumingthat thematerial
inventoryismonitoredfororderattherateatwhichmaterial
isreleasedintheproductionprocessof theshop. Material For Order
Increase Rate(MFOIR) increasesMATERIALFORORDER(MFO),
whichisdepletedbyMaterial ForOrderDecreaseRate(MFODR). Material
Order(MO) isdenedbymeans of MFO. Material Procurement (MP) is the
pipeline delay ofMaterial Order(MO) withdurationequalstoMaterial
LeadTime(MLT). Material Procurement Rate (MPR) increases
MATERIALINVENTORY (MI), which is depleted by Material Usage Rate
(MUR).TheMaterialprocurementcontrolmechanismreferstoareviewsystem
applying the DBR logic. In particular, this system is a peri-odic
order quantity review system with probabilistic demand
andvariableorder quantitythat equals MFO. Theorder
quantityisbasedonalot-for-lot approach(Silver, Pyke, &Peterson,
1998;Steele et al., 2005); it is equal to the material consumption
of theTable 2Performance measures.Performance measures of
manufacturing processARM Average value of material inventory,
kgAWIP1 Average value of WIP1, itemsAWIP2 Average value of WIP2,
itemsAFP Average value of nished product inventory, itemsADB
Average value of demand backlog, itemsDBD Demand Backlog Delay;
i.e. total time duration of demand backlogoccurrence, daysALT
Average lead time, daysAPR Average value of CCR production rate,
items/dayPI Production Index measuring the efciency of DBR-based
PPCapproach; i.e. the average value of the ratio of actual CCR
productionrate values over the magnitude assuming innite inventory
WIP1,dimensionlessPerformance measures of PTB evaluation
processAPTB Average value of PTB, daysPTB PTB at the end of the
simulation process, daysFig. 1. Generic causal-loop diagram of the
SD model.692 P. Georgiadis, A. Politou/ Computers & Industrial
Engineering 65 (2013) 689703previousperiod.
TherelatedequationsaregiveninAppendixA(Eqs. (A.7)(A.15)).3.3.3. PTB
control mechanismThe PTB control mechanismis illustrated in Fig. 4.
Morespecically, bymonitoringPTBvaluesadecisioniscontinuouslymade
whether or not to increase or decrease its level and to whatextent.
The values of PTB increase (PTB Increase) and decrease
(PTBDecrease)dependonthediscrepancy(PTBDiscrepancy)betweentheDesiredPTBandtheactuallevelofPTB.
DesiredPTBisbasedon DEMAND BACKLOG (DB), DDD, MPT and Expected
Demand (ED),whichisaforecastedvaluebasedonthetimeseriesof D.
Themagnitude of each increase or decrease is proportional to the
PTBDiscrepancyat thespecictime. Specically,
PTBDiscrepancyismultipliedbyparameters K1for increaseandK2for
decrease,which characterize alternative PTB planning policies.
Values of K1or K2 equal to1, inparticular, refer to
PTBplanningpoliciescharacterizedbymatching-timeresponsiveness.
Insuchpolicies,practically, PTB reaches Desired PTB in one time
unit. Values of
K1orK2greaterthan1refertoPTBplanningpoliciescharacterizedby
high-time responsiveness. In such policies, PTB reaches DesiredPTB
in less than one time unit. Values of K1 or K2
smallerthan1refertoPTBplanningpoliciescharacterizedbylow-timeresponsiveness(i.e.
PTBreachesDesiredPTBinmoretimeunits).PTB control mechanismis based
on the stock managementstructure suggested by Sterman (1989).
Because of its central rolein the model, the related equations are
given in Section 3.4.3.3.4. Flow-shop production control
mechanismTheow-shopproductioncontrol mechanismis depictedinFig. 5.
ProductionRate1(PR1)iscontrolledbyCapacity1(Cap1),WIP0 and Material
Factor (MF), whereas Production Rate 2 (PR2)
islimitedbyCapacity2(Cap2)andWIP1.
ProductionRate3(PR3)islimitedbyCapacity3(Cap3) andWIP2.
PR3increases FINISHEDPRODUCTINVENTORY(FPI). Thecontrol
mechanismisbasedonlimitation functions considering the capacity and
inventoryFig. 2. Causal-loop diagram of the material release rate
control mechanism.Fig. 3. Causal-loop diagram of the material
procurement control mechanism.Fig. 4. Causal-loop diagram of the
PTB control mechanism.P. Georgiadis, A. Politou/ Computers &
Industrial Engineering 65 (2013) 689703 693constraints (Georgiadis
& Michaloudis, 2012). The relatedequations are given in
Appendix A (Eqs. (A.16)(A.22)).3.3.5. Shipments rate control
mechanismTheshipmentsratecontrol mechanismisdepictedinFig. 6.Loop 2
(DB Increase Rate (DBIR), DB, Shipments Rate (SR), DB
IncreaseRate(DBIR)) controlstheSRof thedemandfulllment
process.PlannedDemandFulllment(PDF)isthepipelinedelayofDwithduration
equals toDDD. DBIR increasesDB, which isdepleted
byDBDecreaseRate(DBDR). IncaseofproductSRislessthanPDF,DBIR gets a
positive value. However, in case of delayed fulllmentof D, SR is
greater than PDF and thus DBDR gets a positive value. SRdecreases
ORDERBACKLOG(OB), whichis increasedbyD. Thecontrol mechanism is
based on the assumption that all the demandissatised,
evenwithdelay. TherelatedequationsaregiveninAppendix A (Eqs.
(A.23)(A.29)).3.4. Mathematical formulationIn SD discipline the
development of the mathematical model isusually presented as a
stock-ow diagram that captures the modelstructure and the
interrelationships among the variables (Sterman,2000). The stock-ow
diagram is translated to a system of differen-tial equations,
whichis thensolvedvia simulation. High-levelgraphical simulation
programs support such an analysis. Theembeddedmathematical
equations are dividedinto two maincategories: the stock (state)
equations, relating the
accumulationswithinthesystemofthenetowrates,
andtherateequations,deningtheowsamongthestocksasfunctionsoftime.
InSDmodels, the stock and ow perspective represents time
asunfoldingcontinuously;eventscanhappenatanytime;changecan occur
continuously. The general mathematical representationof stocks and
ows is given by the following equations:Stockt Ztt0Inflowt
Outflowtdt Stockt0 3Inflowt f Stockt; Et; P; Outflowt gStockt; Et;
P 4where E(t) any exogenous variable and P system
parameters.Thegenericstock-ow diagramofthedevelopedSDmodelisgiven
in Fig. 7. The SD model consists of two modules: ow-shopmain
module; and performance evaluation module which providesthe
performance dynamics.Belowweprovidethemathematical
formulationforthePTBcontrol mechanism:PTBt Ztt0PTB Increaset PTB
Decreasetdt PTBt0;PTBt 0 DDD25EDt EDt dt 1c Dt EDt dt 6Desired PTBt
minDDD; MPT EDtdt DBt 7PTB Increaset maxK1 PTB Discrepancyt; 0=dt
8PTB Decreaset maxK2 PTB Discrepancyt; 0=dt 9PTB Discrepancyt
Desired PTBt PTBt 10By Eq. (6), ED is a rst-order exponential
smoothing of D at time t,with smoothing factor 1/c. Desired PTB is
dened in Eq. (7) by meansofDDD, MPT, EDandDB. PTBincrease rate
(PTBIncrease)andPTBdecrease rate (PTB Decrease) are the increase
and decrease decisionsper day. ByEq. (8) (byEq. (9)), PTBIncrease
(PTBDecrease) isFig. 5. Causal-loop diagram of the ow-shop
production control mechanism.Fig. 6. Causal-loop diagram of the
shipments rate control mechanism.694 P. Georgiadis, A. Politou/
Computers & Industrial Engineering 65 (2013) 689703proportional
to the positive (negative) part of PTB
Discrepancybetweenthedesiredandactual PTB, multipliedbyK1(K2).
Themagnitude of PTB Discrepancy is given by Eq.
(10).Theequationsfor therest of control mechanisms showninFig. 1
are given in Appendix A. The appendix provides also the
per-formancemeasures. PerformancemeasuresgiveninEqs. (A.30)(A.35)
are based on common measures suggested by the
literature(Betterton&Cox, 2009; Cook, 1994; Duclos
&Spencer, 1995;Guide, 1996). DBD(DemandBacklog Delay) measures
the timeduration of DB occurrence (Eq. (A.36)). ALT (Average Lead
Time) isdeneddividingOB(Orders Backlog) byASR(AverageShipmentsRate)
(Eq. (A.37)). PI (Production Index) is the average value of
theratioof theCCRproductionrateoveritsrespectiveupperlimit.The
latter is obtained by assuming the inventory WIP1 as innitive(Eq.
(A.38)). Therefore, PI is a measure of how efciently the WIP1is
managed in order to maximize the CCR production rate. Finally,APTB
(Average PTB) is the average value of PTB (Eq. (A.39)) and
PTBrefers to the value of PTB at the end of the simulation process
(Eq.(A.40)).The entire mathematical model is a non-linear model of
13 statevariables, 21 ow variables and a considerable number of
auxiliaryvariables and constants (2 array auxiliary variables, 44
scalarvariables and 13parameters). The model is developed
inthesimulation software Powersim2.5c.3.5. Model validationTo build
condence in themodel and to checkits quality, weusedtests
suggestedby the SDliterature (Sterman, 2000). Inparticular, we
tested that every equation of the model isdimensional consistent.
Besides, we conducted extreme-conditiontests checking whether the
model behaves realistically even underextremepolicies. For
instance, we checkedthat if there is nodemandforproducts(D = 0),
PR1, PR2, PR3, ED, DBandDesiredPTB equal zero; if there is no
available capacity for production inthe rst operation of the shop
(Cap1_M = 0), PR1, PR2, PR3equal zero, DBequalsthetotal amount of
demandbackloggedthroughout the planning horizon and Desired PTB
equals itsmaximum possible value (i.e. equals DDD). Integration
error testswere subsequently conducted. In our model we used the
Euler nu-mericmethodsincetheintegrationmethodRungeKuttashouldbe
avoided in models with random disturbances such as this one(demand
is not constant). We choose a simulation horizon of 300working days
(the rst 50 days are considered as transient period)to be able to
analyze short/medium-term decisions. Moreover, weset the
integrating time step (dt) initially at 0.25 days,
signicantlyshorter than the shortest value of the models time
constants andranthemodel. Thenwecut thedt inhalf
andranthemodelagain. The results did not signicantly change, so we
chosedt = 0.125 days (=1 working hour).4. Control parameters and
sets of valuesA complete numerical investigation of the SD models
responsetoPTBpolicieswouldrequirethesystematicstudyof
problemswithvarious levels of thesystemparameters. Sucha
detailedexperimental design is practically impossible because of
the largenumber of model parameters (in total 13). Consequently,
weconcentrate on two parameters controlling the demand,
oneparameter controlling the demand due date, two
operationalparameterscontrollingtheshopandtwoparameterscontrollingthe
PTB. These control parameters are presented in the
followingsubsections. The rest of system parameters are shown in
Table 3.Fig. 7. Generic stock-ow diagram of the SD model.P.
Georgiadis, A. Politou/ Computers & Industrial Engineering 65
(2013) 689703 6954.1. Demand control
parametersUsingtheparametersaandb,
weconnectthemeanvalueofdemand(Dm) tothemeanvalueof
CCRcapacity(CapCCR,m) andthe standard deviation of demand (DSD), as
follows:Dm a CapCCR;m11DSD b Dm12The parameter b stands for the
coefcient variance of demand. Theparameters a and b are examined in
two levels; 0.9, 0.98 for a and 0,0.2 for b.4.2. Demand due date
control parameterThe demand due date control parameter is the DDD.
It isexamined intwolevels;2and4. ThesevaluesofDDDarebasedonmodel
runningwithCap2_Mequals 10items/day,
ProtectiveCapacityequals0.1andCapacitySwitchequals1.
Forthiscase,bymeans of (2), themeanvalueof thetotal
productiontime(MPT) equals 0.28 days/item. Additionally, if a = 0.9
(a = 0.98)meaning that the mean value of daily demand is 9
(9.8)items/order, MPT equals 2.52 (2.72) days/order.4.3. Flow-shop
control parametersTheow-shopcontrol
parametersaretheprotectivecapacityand the capacity switch.
Protective Capacity (Atwater et al.,2004; Betterton & Cox,
2009) connects the mean values
ofproductiontimeinCCRoperationandrest shopoperations,
asfollows:Protective CapacityMean of production time in CCR Mean of
production time in no CCR Mean of production time in CCRThe
Protective Capacity is examined in two levels; 0.1 and 0.3.Capacity
Switch may take the values of 0 and 1; 0 denotes thatall
theproductiontimes of theshoparekept constant duringsimulationrun,
whereas1denotesthatproductiontimesfollowexponential
distributions.4.4. Production time-buffer control
parametersThedecisionparametersthatfullydescribethePTBplanningpoliciesarethecontrol
parametersK1andK2(seeEqs. (8)and(9)). Recall fromSection 3.3.3 that
low-time responsive PTBpoliciesrefertovaluesofK1 < 1.
Thesepoliciesleadtodelayingthe starting of production process (see
Eq. (A.4) of MRT denitioninAppendixA)
andconsequentlytolowerinventories. Forthisreason, the parameter K1
is examined in a range from 0 to 1. Withregard to K2, the parameter
is examined in a range from 0 to 0.25.The shorter range of K2
compared to that of K1 is explained by thefact that we wish to be
more conservative in reducing the values ofPTB in order to keep low
the possibility of DB occurrence.5. Dynamics of PTB control
mechanism and propertiesBy assigning specic values to the control
parameters andrunningthemodel,
weobtainthedynamicsofstocksandowsthroughouttheplanninghorizon.
Weconsiderasthebasecasethe following set of control parameters: a =
0.9, b = 0.2, protectivecapacity = 0.1, capacity switch = 1. For
the base case, the dynamicsof PTB control mechanism for PTB control
parameters K1 = 0.2 andK2 = 0.15andDDD = 2 days are showninFig. 8a.
The case ofDDD = 4 daysisshowninFig. 9a. Figs.
8aand9aillustratethedecisionstoincreaseordecreasePTBvaluesonadailybasis,
forthe given set of PTB control parameters. Figs. 8b and 9b
illustratethe dynamics of PTB under different K1 and K2 values.5.1.
Transient response and dynamic equilibriumFig. 10 (Fig. 11a)
illustrates the response of actual level of PTBfor three different
sets of PTB control parameters K1 and K2and DDD = 2 days (DDD = 4
days). We observe that in case ofDDD = 2 days, the actual level of
PTB does not reach anyequilibriumevenwhenthesimulationhorizonis
doubled(i.e.600 days). This is explained by the fact that for a =
0.9, MPT equals2.52 days/order (see Section 4.2). Given that DDD =
2 days, there isnot enough time for the production to be completed
on time.However, in case of DDD = 4 days, there is enough time for
theproductiontobecompletedontimeandtheactuallevel
ofPTBreachesadynamicequilibrium; i.e. thetotal increaseof
PTBisbalanced by its total decrease throughout the simulation
horizon.More specically, PTB reaches a dynamic equilibrium towards
thevalueof0.34 dayswithdifferenttransientperiods.
Thedifferenttransient periods are shown more clearly in Fig.
11b.Table 3System parameters remaining constant throughout the
simulation process.Parameter Value Unitc 3 daysCapCCR,m10
items/dayInitial value of MI 120 kgMF 2 kg/itemMLT 3 daysr timestep
daysFig. 8. Dynamic behavior of PTB (base case, DDD = 2 days).696
P. Georgiadis, A. Politou/ Computers & Industrial Engineering
65 (2013) 6897035.2. Adaptability to demand changesFig.
12illustratesthe response of the actual levelofPTB for
astepincreaseindemand. Inparticular, theFigureillustratestheresults
for a stepincrease withmagnitude 1item/day(about11%of
themeanvalueof thedailydemand)onthe100thday.Fig. 12a(Fig. 12b)
provesthat thestepincreaseindemandforDDD = 2 days(DDD = 4
days)resultsinagradualincreaseofPTB,whichisbalancedinahigher
valuecomparedtothat withoutstepincrease. This increaseof PTBis
explainedbyanincreaseofDesiredPTBduetotheincreaseofED,
DBandD(seeEqs. (6)and(7)).5.3. Robustness to sudden disturbancesThe
robustness of the PTB control mechanismto suddendisturbances in the
internal and external environment is given inFigs. 13and 14. Fig.
13illustrates theresponse ofactuallevelofPTB when a CCR breakdown
occurs from the 100th day up to the110th day. It is shown that
breakdown results in increase of PTBandDB, whichare
counterbalancedlater on, inbothcases ofDDD; counterbalance is
completed earlier in the case ofDDD = 4 days.Fig. 14 illustrates
the dynamics of PTB actual level in case of apulse increase in
demand with magnitude 90 items/day (equal toFig. 10. Dynamic
behavior of PTB under different K1 and K2 values (base case, DDD =
2 days).Fig. 9. Dynamic behavior of PTB (base case, DDD = 4
days).Fig. 11. Transient response and dynamic equilibrium of PTB
under different K1 and K2 values (base case, DDD = 4 days).P.
Georgiadis, A. Politou/ Computers & Industrial Engineering 65
(2013) 689703 697ten times the Dm) on the 100th day. The pulse
increase in demandresults in a sharp increase of PTB and DB, that
are counterbalancedlater on, in both cases of DDD; counterbalance
is completed earlierin the case of DDD = 4 days .6. The effect of
PTB policies on shops performance: Numericalinvestigation and
concluding
discussionInordertoevaluatethesystemsperformanceinadynamicequilibrium
condition, data is collected after the transient period(50
days)toavoidirregularitiesduringthatperiod. Thesystemsperformance,
in terms of performance measures given in Table
2,isexaminedunder32combinationsof demand, DDDandow-shop control
parameters, generating by the sets of levels given inSections 4.1,
4.2 and 4.3. In every experiment, two critical decisionsare made:
how much material to order (MO) on a daily basis andthe adjustment
of PTB on hourly basis.At rst, we examine each of the above 32
combinations undertwosetsof levelsof PTBcontrol
parameters/factorsK1andK2;K1inlevels0.5and1,
andK2inlevels0.125and0.25. Foreachcombination of K1 and K2, three
repeat simulation runs allow
theuseofANOVAtodeterminewhetherthePTBcontrolparametersaffect
signicantly the performance measures. Therefore, the totalnumber of
simulation runs is 32 4 3 (=384).The ANOVA results (P-values and
Partial Eta Squared) for thesesimulationruns(initial
ANOVA)arepresentedinTable4. SinceP-values are the lowest signicance
levels to reject the nullhypothesisthat theindependent parameter
doesnot affect theindicated performance measures, P-values less
than the 0.05 levelof signicanceshowstatistical signicance.
Besides, Partial EtaSquared (PES) reects the signicance of the
independentFig. 12. Adaptability of PTB in case of a step increase
in demand (base case, K1 = 0.2 and K2 = 0.15).Fig. 13. Robustness
of PTB in case of CCR breakdown (base case, K1 = 0.2 and K2 =
0.15).Fig. 14. Robustness of PTB in case of a pulse increase in
demand (base case, K1 = 0.2 and K2 = 0.15).698 P. Georgiadis, A.
Politou/ Computers & Industrial Engineering 65 (2013)
689703Table 4P-values of initial ANOVA for the effects of all
control variables on performance measures.Indicates minor signicant
effect (P-value 60.05 and PES 6 0.5).Indicates major signicant
effect (P-value 60.05 and PES > 0.5).P. Georgiadis, A. Politou/
Computers & Industrial Engineering 65 (2013) 689703
699parameter comparedtotheerrors signicance; thehigher thevalues of
PES, the higher is the effect of the independent parameterto the
dependent factor. Table 4 presents all the rst, second
andthird-orderresults(P-values)of theinitial
ANOVAandonlythesignicant higher-order results. For the signicant
effects (P-value60.05), the corresponding results are classied
regarding the PESvalueintotwocategories: (i) PESvaluesequal
orlessthan0.5denoting minor effect of the independent parameter to
thedependent factor and(ii) PESvalues higher than0.5denotingmajor
effect of the independent parameter to the dependent factor.The
results presented in Table 4 indicate that the demand, DDDand
ow-shop control parameters have signicant rst-ordereffects on the
majority of performance measures of manufacturingand PTB evaluation
processes. These rst-order effects (by meansof estimated marginal
means of performance measures) arepresented in Table 5. For
example, increase of the values of a andb results in increase of
AWIP1, AWIP2, ADB, DBD and ALT; increaseof protective capacity
value results in decrease of DBD; increase ofDDD value results in
decrease of ADB and DBD.In addition, the results shown in Table 5
indicate the effects of K1and K2 onthe performance measures of
manufacturing and PTB evalu-ation processes. It is noticeable that
there is no signicant rst-ordereffect of K2. For signicant effects,
Table 5 illustrates these resultsfor rst-order effects of K1 and
K2. Table 5 indicates that: Parameter K1 has signicant effect on
the majority ofperformancemeasuresof manufacturingprocess(i.e.
onthemeasures ARM, AWIP1, AWIP2, DBD, APR and PI) Parameters K1 and
K2 do not have signicant effect on the per-formance measures of PTB
evaluation process. However, thesemeasures are inuenced by the
demand, DDD andow-shop control parameters.The above observations
lead to the necessity of morethoroughly examination of PTB control
parameters/factors K1 andK2. Therefore, inasecondANOVA,
weexamineeachof the32combinations of demand, DDD and ow-shop
control parametersunder 100 sets of levels of PTB control
parameters/factors K1 andK2 generating by the combinations of their
levels: K1 from 0.1 to1 with step of 0.1;K2 from 0.025 to 0.25 with
step of 0.025. Foreachcombinationof K1 andK2, three repeat
simulationrunsallowtheuseof
ANOVAtodeterminewhetherthePTBcontrolparameters affect signicantly
the performance measuresand select their optimumvalues among the
considered ones.Therefore, the total number of simulationruns is 32
100 3(=9,200).The ANOVA results for each of 32 combinations of
demand, DDDandow-shopcontrolparameters(secondANOVA)indicatethatthe
effectK1 K2 onall performancemeasures ofmanufacturingprocessis
insignicant (for level of signicanceequal to0.05).Consequently,
there is no meaning to track a specic combinationof K1andK2inorder
tooptimizetheperformancemeasures.Althoughit is inprinciple
riskytogeneralize onthe basis ofnumerical examples, the embedded
PTB control mechanism leadsto the conjecture that the performance
measures are indeed robustto moderate changes of the control
parameters K1 and K2 for thestudied cases of demand, DDD and
ow-shop control parameters ofthe shop.It is very interesting that
the insensitivity of performancemeasures to changes in PTB control
parameters that was identiedthrough the above ANOVA analysis is not
coincidental, nor peculiartoaparticularcombinationof thedemand,
DDDandow-shopcontrol parameters. Extensivesimulationresults, not
shownforbrevity, revealthattheperformancemeasuresareindeedrobustto
changes in PTB control parameters for a wide range of demand,Table
5First-order Estimated Marginal Means (EMMs) of performance
measures connected to control variables (initial ANOVA).Indicates
no signicant effect.700 P. Georgiadis, A. Politou/ Computers &
Industrial Engineering 65 (2013) 689703DDDandow-shopcontrol
parameters. Therobustness of theperformance measures to PTB control
parameters and to
demandcharacteristicsisanextremelypositivepropertyoftheproposedmethodology,
sincefromonehandaccurateforecastsofdemandare in many real-world
applications difcult to obtain and on theother hand this property
provides production managers with ex-ibility in decision making
with regard to parameters K1 and K2.Besides, as it is shown in
Table 6, for all the 32 combinations,the effect K1 K2 is
insignicant on the performance measures ofPTBevaluationprocess(for
level of signicanceequal to0.05).The insignicance of K1 K2 effect
on APTB and PTB reinforce theevidence that the dynamic PTB control
mechanism is indeed robustto changes in PTB control parameters for
a wide range of demand,DDD and ow-shop control parameters.7.
Summary, limitations and directions for model extensionsThis paper
was aimedto introduce a dynamic PTB controlmechanism for DBR-based
PPC of ow-shop manufacturingsystems. We developed a SD model for a
three-operation, single-product, capacitatedow-shopsystemthat
purchasesonetypeof rawmaterial.
Weconsideredanormallydistributeddemandandexponentiallydistributedproductiontimes.
IntegratingthePTB control mechanism into the SD model, we examined
the shopperformance under different demand, DDDand shop settings.We
proved the insensitivity of performance measures
ofmanufacturingandPTBevaluationprocesses tochanges
inPTBcontrolparameters.
TheresultsobtainedbyextensivenumericalinvestigationandANOVAanalysisrevealedthatthePTBcontrolparameters
are indeed robust to changes in the values of demand,DDD and
operational parameters.TheproposedPTBcontrol
mechanismprovidesthefollowingpossibilities: (i) decision making on
the magnitude of PTB
withouttrackingaspeciccombinationofcontrolparametersK1andK2(byassuminginitial
values for PTB, K1andK2, managers maydecide on PTB values which are
based on the evaluation of DesiredPTB values), (ii) ability to
integrate real-time disturbances(machine failures, demandincrease)
intoPTB-relateddecisionsand(iii) dynamicadaptationof
PTBtochangesininternal andexternal shopenvironment. Inaddition, the
employedcontrolmechanism provides the ability to consider the
non-linear,non-stationaryanduncertainnatureof
productionprocessthatcharacterizesthemajorityof
ow-shopmanufacturingsystems.Withoutthiscontrol mechanism, it
wouldbeimpossibletogetthe adaptive behavior of the manufacturing
systemtowardsall the possible changes. However, the learning period
is aprecondition for successful real-world applications. This is
alignedwiththetransientperiodoccurrencethroughoutthesimulationprocess
presented in this paper. After all,the control mechanismis
self-correcting; its feedback structure ensures that
forecasterrors, changes in the structure of the shop environment
and evenself-generated overreactions can eventually be
corrected.The proposed real-time PTB control mechanism
faceslimitations. The continuous monitoringandadjustment of
PTBrequires the use of real-time controllers,
whichinautomatedproduction systems are integrated in their IT
infrastructure.However, ina more traditional owshop, the
controllers
arehuman-drivenandconsequentlythemonitoringandadjustmentof PTB
practically takes place in longer periods, resulting insuboptimal
shop performance. For example, for the base case, forDDD = 2 days
and forK1 = 0.2 andK2 = 0.15, the average value ofdemand backlog
(ADB) equals 0.51 items, when dt equals0.125 days. However, for dt
= 0.5 days, ADB equals 19.37 items.Therefore, in automated ow-shop
manufacturing systems, theoor manager in order to exploit the
advantages of the developedmodel inapplyingDBR-basedPPCmust
estimatethevaluesofDemandDueDateandPTBcontrol
parametersandascertaintheinitial values of stock variables. In case
of real-time
disturbances,themanagerhastoensuretheupdatingofthevaluesofmodelparameters.In
case of traditional ow-shop systems that apply
DBR-basedPPCwithhuman-drivencontrollers, the oor manager, at
thebeginning of each day, must decide upon the value of PTB. In
thisdecision, it is suggestedtofollowasimplerule.
Inparticular,basedonthePTBvalueofthepreviousdayandtheDesiredPTBvalue(seeEq.
(7)), themanager has tocomputethevalueofPTBDiscrepancy(seeEq.
(10)). Then, thecurrentvalueof PTBiscalculated (see Eqs. (5), (8),
and (9)) by using the assumed valuesof control parameters K1 and
K2. Finally, the current PTB value isusedtocalculatethevaluesof
Material ReleaseTime(MRT)(seeEq. (A.4)), Planned Order Release Rate
(PORR) (see Eq. (A.2)), OrderRelease Rate (ORR) (see Eq. (A.3))
andMaterial Release Rate (MRR)(see Eq. (A.6)), and to schedule the
rawmaterial release inproduction line.The results presented in this
paper certainly do not exhaust thepossibilitiesofinvestigating
allthefactorsaffecting thedynamicDBR-based PPC of ow-shop
manufacturing systems. For example,it is worthwhile to study the
proposed PPC system assuming multiproducts under different shop
settings and to integrate costelements into the proposed control
mechanism. Finally, thedevelopment of self-adaptive mechanisms for
PTB controlparameters (K1 and K2) may have added-value in the
developmentof more comprehensive DBR-based PPC systems.Table
6Resultsof secondANOVAfortheeffectsof PTBcontrol
variablesonperformancemeasures of PTB evaluation process.a b
Prot.cap.Cap.switchDDD APTB PTBP-value PES P-value PES0.9 0 0.1 0
2* * * *0.9 0 0.1 0 4* * * *0.9 0 0.1 1 2 0.91 0.24 0.75 0.260.9 0
0.1 1 4* * * *0.9 0 0.3 0 2* * * *0.9 0 0.3 0 4* * * *0.9 0 0.3 1 2
0.77 0.26 0.48 0.290.9 0 0.3 1 4* * * *0.9 0.2 0.1 0 2 0.52 0.29
0.37 0.300.9 0.2 0.1 0 4 0.25 0.31 0.42 0.300.9 0.2 0.1 1 2 0.27
0.31 0.34 0.300.9 0.2 0.1 1 4 0.66 0.27 0.80 0.260.9 0.2 0.3 0 2
0.95 0.23 0.46 0.290.9 0.2 0.3 0 4 0.79 0.26 0.82 0.250.9 0.2 0.3 1
2 0.49 0.29 0.33 0.300.9 0.2 0.3 1 4 0.88 0.24 0.85 0.250.98 0 0.1
0 2* * * *0.98 0 0.1 0 4* * * *0.98 0 0.1 1 2 0.94 0.23 0.37
0.300.98 0 0.1 1 4 0.49 0.29* *0.98 0 0.3 0 2* * * *0.98 0 0.3 0 4*
* * *0.98 0 0.3 1 2 0.44 0.29 0.64 0.270.98 0 0.3 1 4 0.46 0.29*
*0.98 0.2 0.1 0 2 0.55 0.28 0.96 0.220.98 0.2 0.1 0 4 0.71 0.27
0.82 0.250.98 0.2 0.1 1 2 0.51 0.29 0.66 0.270.98 0.2 0.1 1 4 0.39
0.30 0.16 0.330.98 0.2 0.3 0 2 0.98 0.21 0.40 0.300.98 0.2 0.3 0 4
0.76 0.26 0.72 0.270.98 0.2 0.3 1 2 0.79 0.26 0.92 0.240.98 0.2 0.3
1 4 0.73 0.26 0.38 0.30*Indicates that the value of performance
measure obtained by all simulation runsis the same.P. Georgiadis,
A. Politou/ Computers & Industrial Engineering 65 (2013) 689703
701Appendix A. Equations of control mechanisms and
performancemeasuresA.1. Material release rate control
mechanismBased on backward innite loading scheduling (Park
andBobrowski, 1989; Sabuncuoglu and Karapinar, 1999).ORt Ztt0PORRt
ORRtdt ORt0; ORt 0 0 A:1PORRt Dt MRT A:2ORRt minORt=dt; MIt=MF=dt
A:3MRTt max0; DDD MPT Dt dt PTBt A:4MPT X3i11Capi MA:5MRRt ORRt MF
A:6A.2. Material procurement control mechanismBased on a periodic
order quantity review systemwithprobabilistic demand and variable
order quantity that equalsMFO (Silver et al., 1998; Steele et al.,
2005).MORt ORRt r MF A:7MFOt Ztt0MFOIRt MFODRtdt MFOt0; MFOt 0 0
A:8MFOIRt MORt A:9MFODRt MFOtdt; ifan order is given at time t0;
otherwise(A:10MOt MFOt; ifan order is given at time t0;
otherwise
A:11MPt MOt MLT A:12MIt Ztt0MPRt MURtdt MIt0; MIt 0 0 A:13MPRt
MPt=dt A:14MURt minMRRt; MIt=dt A:15A.3. Flow-shop production
control mechanismBasedonlimitationfunctions considering the
capacity andinventory constraints (Georgiadis and Michaloudis,
2012).WIP0t Ztt0MRRt MCRtdt WIP0t0; WIP0t 0 0 A:16MCRt PR1t MF
A:17WIP1t Ztt0PR1t PR2tdt WIP1t0; WIP1t 0 0 A:18PR1t minWIP0tMF dt;
Cap1t ; PR2t minWIP1tdt; Cap2t A:19WIP2t Ztt0PR2t PR3tdt WIP2t0;
WIP2t 0 0 A:20PR3t minWIP2tdt; Cap3t A:21FPIt Ztt0PR3t SRtdt FPIt0;
FPIt 0 0 A:22A.4. Shipments rate control mechanismBased on the
assumption that all the demand is satised, evenwith delay.PDFt Dt
DDD A:23DBt Ztt0DBIRt DBDRtdt DBt0; DBt 0 0 A:24DBIRt maxPDFt=dt
SRt; 0 A:25DBDRt minmaxSRt PDFt=dt; 0; DBt=dt A:26SRt minPDFt DBt;
FPIt=dt A:27OBt Ztt0ORTt SRtdt OBt0; OBt 0 0 A:28ORTt Dt=dt
A:29A.5. Performance measures of manufacturing processARM
PTt1MItTA:30AWIP1 PTt1WIP1;tTA:31AWIP2 PTt1WIP2;tTA:32AFP
PTt1FPItTA:33ADB PTt1DBtTA:34APR PTt1PR2;tTA:35DBD XTt1tt ; where
tt t; ifDBt> 00; otherwise
A:36ALT OBTASRT; whereASRT PTt1SRtTaverage shipments rate A:37PI
PTt1PR2;tCap2;tTA:38A.6. Performance measures of PTB evaluation
processAPTB PTt1PTBtTA:39PTB PTBt T see Eq: 5; in Section 3:4
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