A Fuzzy Set Aproach to Activity Scheduling

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A Fuzzy Set Approach to Activity Scheduling for Product Development

Author(s): J. R. WangSource: The Journal of the Operational Research Society, Vol. 50, No. 12 (Dec., 1999), pp. 1217-1228Published by: Palgrave Macmillan Journalson behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/3010631.

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Journal of the Operational Research Society (1999) 50, 1217-1228 ( 1999 Oper-ationalResear-ch ociety Ltd. All rights reser-ved.0160-5682/99 $15.00http://www.stockton-press.co.u/jors

A fuzzy set approach o activity schedulingforproductdevelopmentJR WangFeng Chia Univer-sity,Tcaiwatn,OCIndustriesneed to effectively managetheir productdevelopmentprocessesto reducethe productdevelopment ime andcost. Due to incompletedesign informationat the early stage of productdevelopment, he durationof each activity isdifficult to estimate accurately.The objective of this research is to develop a methodology to schedule productdevelopmentprojectshavingimprecise emporal nformation.The researchproblem s formulatedas a frizzyconstraintsatisfactionproblemand a new methodbasedon possibilitytheoiy is proposedto determninehe satisfactiondegreesoffuzzy temporalconstraints.Based on the proposedmethod, a fuzzy schedulingprocedure s developedto constructaschedulewith the leastpossibilityof being lateandto maximizethe satisfactiondegreesof all fuizzy emporalconstraints.Moreover, he computationalefficiency of the proposedapproach s also discussed. The proposed methodology canproducemore satisfactory chedulesin an uncertainproductdevelopmentenvironment.Keywords: projectscheduling;fuzzy set theory;productdevelopment

IntroductionIn recent years, new products and changes in existingproductsare occurringat an increasing rate, causing thelife cycle of a product to decrease. To maintain marketshares,industriesneed to effectively managetheirproductdevelopment processes in order to reduce the productdevelopment time and bring their products to market asearlyas possible. A productdevelopmentprocess is consid-ered as a series of several phases: product specification,conceptual design, detail design, prototype, and testing.'Each phase of a productdevelopmentprocess consists of aset of interrelatedactivities. A modem-day project forproductdevelopment,for exampleto develop an electronicproduct,usually consists of thousandsof activities. It isdifficultfor an individualprojectmanager o keep trackofall activities. Unlike the manufacturingprocess, it is alsodifficultto predictaccuratelythe durationof an activity atthe project initialization stage, due to the high level ofdesign imprecision and engineering changes frequentlyoccurringduringthe productdevelopment.2' The impre-cise temporal informationfurthermakes project manage-ment tasks more difficult. Poor managementof productdevelopment projects may result in a critical delay ofproductsgoing to marketand cause great sales losses.The concept of concurrent engineering (CE)4 hasemerged to reduce the lead time of productdevelopmentCorresponldenice: r. Juite (Ray,)Wang, Associ(te Professor, Dept. ofIndiustrialEngineering,Feng Chia University,100 WVenhwaoad, Seat-wen, Taichiuing, ailvan 407, ROC.E-mail: rdwang Wfcu.edu.tw

by coordinatingvarious life cycle considerationsconcur-rently. A product development project based on the CEconcept is perfonmedby teams with membersfrom differ-ent disciplines. An R&D organizationusually has severalprojects under development simulataneously. Therefore,activities for variousprojectshave to compete for limitedavailable resources in order not to delay projects. Thecomplex resourcedependenciesamong activities within aproductdevelopmentprojectand betweendifferentprojectsfurthercomplicatethe management asks.The objective of this research s to develop a methodol-ogy to schedule product development projects havingimprecise temporal infoirnation. To effectively allocatescarce resources to activities in an uncertain productdevelopment project, it is necessaiy to incorporate theimprecise temporal informnationnto the problem solvingprocess.Fuzzy set theory5 s used to represent he imprecisetemporalinformationand our researchproblemis formu-lated as a fuzzy constraintsatisfactionproblem.6A newmethodbased on possibility theory7 s proposedto handlethe fuzzy temporalconstraints.Since the durationof eachactivity is imprecise, the actual project finish time isdifficult to predict. A fuzzy scheduling procedurebasedon beam search8 s developedto determinea schedulewiththe least possibility of being late. The start time of eachactivity is assignedwith the maximumsatisfactiondegreesof all fuzzy temporalconstraints.In the next section, the literature elatedto this researchis presented.The activity schedulingproblemfor productdevelopmentis then modelled with the fuzzy set theory.Anew method is introduced to measure the satisfaction

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1218 Journalf heOperationalesearchocietyol. 0,No, 2degreeof a fuzzy temporalconstraint,anda fuzzy schedul-ing procedure s developedto produce he most satisfactoryschedules.An illustrativeexampleandsome computationalresults are also presented.Finally, we provide a summaryof this researcheffort.Literature reviewThe project evaluationand review technique (PERT) andcriticalpathmethod (CPM) have long been establishedinindustryas tools forplanningandmanagingprojects.9Oneof their shortfalls s theirrigid analysisand specificationofa project structure.Many studies have been conductedonextendingthe classical conceptof PERTto deal with morecomplex problems.GeneralActivity Network (GAN)9 is astochastic project network with a more flexible networkstructure in the form of stochastic event nodes. Eachactivity (on arc) is assigned with an execution probabilityin addition to its stochastic duration.Several researchershave developed scheduling approacheson GAN with alimitednumberof resources. 0 11A GAN modelrepresentsthe stochastictemporal nformationwith probabilitydistri-butions.Therefore,the computationrequiredfor temporalanalysisor resourceallocationdecisions is based on prob-ability theory. The objective is to minimize the expectedprojectmakespan,expectedprojectlateness, etc. However,schedulingof activities on GAN with limited resourcesiscomputationally too expensive and theoretically toocomplex. Moreover, a product development project isusually unique in nature and is often described as being'open-ended' or 'ill-defined.' Therefore, it is difficult tocollect enoughdatato obtainthe distribution f the randomvariablefor each activity duration.'2The activitynetworkwith GeneralizedPrecedenceRela-tions (GPR)13generalizes the classical PERT with strictprecedence relations between activities. Four types ofprecedence relations that are present in many practicalsituationsare introduced o providemore generalrepresen-tation. Several researchers have extended the resource-constrainedprojectscheduling problemwith GPRto mini-mize the projectmakespan.4 However, all temporalpara-meters in an activity network with GPR are precisenumbers. Therefore, the satisfaction of a temporalconstraint s eithertrue or false.

In contrast o the schedulingmodels with GAN or GPR,this researchrepresents he imprecisetemporal nformationwith fuzzy set theory. The computation required fortemporal analysis or resource allocation decisions arebased on fuzzy arithmetic and fuzzy ranking methods5that are less complex than probabilitytheory. In addition,as the temporalparametersarerepresentedwith fuzzy sets,the satisfactionof a fuzzy temporalconstraint s not abouttruthor falsehood,but is a degree within [0, 1].6The schedulingof a productdevelopmentproj ct may beviewed as a resource-constrained project scheduling

problem, which has been investigated extensively in theliterature.'5'16 Since the resource constrainedprojectsche-dulingproblemis NP-complete,optimizationapproaches'6are not suitablefor solving practical-sizedproblems.There-fore, heuristic approaches'7-'9 have been developed toconstructa schedule effective for practicaluse. With theadvent of concurrent engineering, Belhe and Kusiak20consideredthe flow of informationin the design processas a pull system like a just-in-timemanufacturing ystemanddevelopeda dynamicschedulingheuristicto minimizeweighted lateness.Little research has been performed on extending theresource-constrained project scheduling problem withfuzzy set theory. The CPM and PERT networks havebeen extended with fuzzy durationfor each activity.12 21However, the resource capacity constraints are ignored.Hapke and Slowinski22applied twelve dispatchingruleswith fuzzy sets to generatea set of schedules and the bestschedule with the minimum fuzzy makespanis selected.However, it is known that the dispatching rules onlyconsider local aspects of the solution space. The computa-tional efficiency of their approachis not investigated. Inaddition, their approachonly determines the fuzzy starttime for each activity.The assignmentof a crisp starttimefor each activity to execute a project is not discussed.

Modeling the scheduling problem with fuzzy set theoryRepresentation of imprecise temporal information withfuzzynumbersIn many practicalsituations,the temporalparametersof aproduct development project, such as the duration ofparticularactivities or the project deadline, often cannotbe given precisely at the projectinitializationstage due toincomplete design information.Unless a productdevelop-ment project being scheduled is quite similar to previousprojects, previous experience is of limited relevance.Therefore, statistical approaches9-11may not be suitablein this situation.However, one may estimate the activityduration romthe activity thatperformsa similarfunctionin previous projects. Informationabout what durationismore plausible than anotheris often available and can bepredicted by experienced project managers. Therefore,fuzzy set theorymay providean alternativeand convenientframework or handlingthis issue.12,23Let t be an activitydurationand [tL(x) e the membershipfunction of t to estimate the possibility that the activityduration is preciselyx, for all x belongingto a time scale.If the activity durationis precisely known with xo, then,tt(xo)= 1, and for all x :Axo, [lt(x) = 0 (Figure la). If theduration is known and the value of t is between twoextreme time points xo and xl, then pft(x)= 1 for allx E [xo,xl]; otherwiseMt-(x)= 0 (Figure Ib). If the durationis fuzzily known, the more or less possible values of t are



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Wang-Fuzzyetapproachoactivitycheduling 219restrictedby fut(x)that takes the values between [0, 1]. Thevalues belonging to the core of t (i.e., {xlt-(x) = 1} areconsideredas the most possible values for t andthe valuesoutside the support of t (i.e., {xl,ut(x)= O} are the leastpossible ones. The values outside the core and inside thesupportof t are in between (Figure Ic). For computationalefficiency, the trapezoidalfuzzy numbert = (a, b, c, d) isused to representthe imprecise activity duration.A dura-tion between b and c is the most possible durationand aduration ess thana and largerthan d is the least possibleone (Figure Id).The ready time and deadline of a project are oftendetermined by the preference of project managers. Forexample, a project manager may expect that a projectshould be completedbetween d, and d2, but no later thand3, because it may delay the productenteringthe market.Fuzzy sets can also be used to represent he preferencesofproject managers regarding different values of projectdeadline. Let d be a project deadline and [1a(x) be themembership unctionto represent he preferenceof assign-ing x to the projectdeadline, for all x belonging to a timescale. The values belongingto the core of d are consideredas the most preferredvalues and the values outside thesupport of d are the least preferable ones. The valuesoutsidethe core andinside the supportof d are in between.For example, the preferredprojectdeadline can be repre-sented as a trapezoidalfuzzy numberd = (dl, dl, d2,d3)(Figure2). Similarly,the preferredreadytime of a projectcan also be representedas a trapezoidal fuzzy numberb = (bl, b2,b3, b3) (Figure2).

/k~~~~~~~~p8ti ~ ~~A t '20 Time 0 - Timexo Xo XI

(a) (b)O. '.................... -.

a bc d Time ? a b c d > Tinie(c) (d)

Figure 1 Interpretationf fuzzy activityduration.

Readytime Deadline

b1 b2 b3 d, d2 d3 TimeFigure 2 Representationf the project eady imeanddeadlinewithfuzzynumbers.

Modelingthe activityschedulingproblem or productdevelopmentThe activity scheduling problem with imprecise temporalparameterss describedas follows. A productdevelopmentproject P has a ready time b and a deadline d betweenwhich all its activities ai(1 < i


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1220 Journalf heOperationalesearchociety ol. 0,No. 2The latest finish time of activity ai is defined:

Min{ifti, Ifij E t1}, if ai precededsa1id, if ai has no successors

(4)* Precedenceconstraints

If activityai precedes a1in the partialorder, hen:stj? tj < stj. (5)

It means that the finish time of activity ai should be lessthan or equal to the start time of activity aj.* Resource capacity constraintsForany time t, let J = {ailsti< t < sti Dti}. Then for alltime t, and all resourcesrk E {r,, r2, . rq}, such that:Eaic-Jnik,


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JWang-Fuzzyetapproachoactivitycheduling 221Given two fuzzy numbersM and N, the four indices aredefined and used to assess the possible relationshipsbetween fuzzy numbersM andN:

PG(M, N) =fl([N, +oo))sup minC'(,uM),MM(v)) (13)u;( > ,,

PSG(M, N) =f1lM(]N, +oo))= sup inf min(y,u(u),1 ,u(v)) (14)

NG(M, N) = N&([N,+oo))= inf sup max(1 - [L,(u), ,uk(v)) (15)It v;vI, It

NSG(M, N) = Ni(]N, +oo))= 1 - supmin(,Y(u), Pk(V)) (16)it '< v

The four indices take values in [0, 1]. PG(M, N) (respec-tively, PSG(M, N)) implies that the gradeof possibility ofproposition M is greater han or equalto N' (respectively,'M is strictly greater than N'). NG(M, N) (respectively,NSG(M, N)) implies that the gradeof necessity of proposi-tion 'M is greaterthanor equal to N' (respectively, 'M isstrictly greaterthanN').The four rankingindices can be interpretedgeometri-cally forbetterunderstanding.Consider wo fuzzy numbersm=(2,4,4,7) and h=(3,6,6,9) as shown in Figures 5and 6. PG(mi,h) is used to comparethe left leg (the worstpart)of n with the right leg (the best part)of mi.The indexwill be high, when the left leg of h is smaller than or equalto the right leg of uin.NG(mi,h) comparesthe left legs ofbothmiand h. The index will be larger, f the left leg of h issmaller than the left leg of mi. One can interpretthatPG(mi,h) is the upperbound of the constraint'M > N' tobe satisfiedandNG(mi,h) is its lower bound.PSG(mi,h) andNSG(m,h) can be interpreted imilarly.Accordingto Eqs.(13)(16), four indices between miand h are calculated asfollows:

PG(mi,h) = 0.67 PSG(i, h) = 0.17NG(m,h) = 0.20 NSG(mii, ) = 0.00

'UAL U'j (U) sup-r (v,.1.0 .......

0.67 ,- PG(m,n)/ ~~NGQMi,i)

0.20 1 2 3 4 5 6 7 8 9 10 u,v

Figure5 Geometricalnterpretationf two ndicesPG(mi, )andNG(inm,)

A pi (u) inf I:-,jk(v)I.0i. )NSG(m,n)

PSG(in,W)0.0J10 1 2 3~ 4 5 6 7 8 9 10 u,vFigure 6 Geometrical nterpretation f two indices PSG(im-,)andNSG(im,).

Satisfactiondegree of a fuzzy constraintAccordingto the four indices defined above, the satisfac-tion degree of a fuzzy temporal constraint can be deter-mined.DefinitionGiven two fuzzy temporalparameters andb.The satisfac-tion degree of the fuzzy temporal constraint 'a > b' isdefined as the weightedsum of PG(a, b) andNG(a, b):

g(a > b) = , x PG(a,b) + (1-,f) x NG(a, b), (17)where ,B s the preferenceratio,0 < ,B< 1.

PG(a, b) andNG(a, b) use the right leg (optimistic)andleft leg (pessimistic) of a to compare with the left leg of b.We may interpret PG(a, b) as the possibility of 'a > b' andNG(a, b) as the necessity of 'a > b.'Therefore he satisfac-tion degreethat 'a is greater han or equalto b' is betweenNG(a, b) and PG(a, b). A parameter B,called preferenceratio, is defined to incorporatethe attitude of a projectmanagertowardspessimism versus optimism. If the atti-tude of a project manageris optimistic, then ,Bshould begreaterthan 0.5. On the otherhand, /Bshould be less than0.5, if his/her attitude s towardpessimism.According to Eq. (17), the following rule is used toidentifythe relationshipbetween two fuzzy temporalpara-metersa and b:

If g(a b) > g(b )a) then a belse if g(a b) < g(b a) then b >, a

else a = bSimilarly, the satisfaction degree of the fuzzy temporalconstraint a > b' can also be defined.DefinitionGiventwo fuzzy temporalparameters and b. The satisfac-tion degree of the fuzzy temporal constraint a > b isdefined as the weightedsum of PSG(a, b) andNSG(a,b):

g(a > b) x PSG(a, b) +-r x NSG(a,b),18)where ,Bis the preference ratio, 0 < ,B< 1.



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1222 Journalf heOperationalesearchocietyol. 0,No. 2For example, consider the same example of two fuzzynumbers mandh shown in Figures 5 and 6. The satisfactiondegrees of fuzzy temporalconstraints im> n' and 'im> n'are calculatedas follows (assume /B= 0.5):

g(m ) n) = 0.5 x 0.67 + 0.5 x 0.2 = 0.435g(m > h) = 0.5 x 0.17 + 0.5 x 0.0 = 0.085

ScheduleriskDue to the imprecise activity duration, the actual projectfinish time for a productdevelopmentproject is difficult topredict. Therefore, this paper intends to find a schedulewith the leastpossibilityof being late, corresponding o thefuzzy project deadline defined by project managers. Aschedule risk is defined to determine the possibility of aproject schedule to be late in an uncertainproductdevel-opmentenvironment.Definition (Schedule risk)Givena scheduleS, the riskof S to be late is definedas thesatisfaction degree of the fuzzy constraint'project finishtimeft is greater han project deadline d:

Risk(S)= gt > d)= /Bx PSG(ft, d) + (1 -,) x NSG(ft, d), (19)

where /3 s the preferenceratio,0 < ,B< 1.A schedule is late if its project finish time passes theproject deadline. When both project finish time and dead-line are fuzzy, the schedule being late is not either true orfalse, but is a degree within [0, 1]. Therefore,a schedulerisk is definedin terms of the satisfactiondegree of fuzzy

temporal constraint 'ft > d.'

Fuzzy scheduling procedureThe resource-constrainedproject scheduling problem isNP-hard and it seems unpromisingto use any exact algo-rithms to solve practice-size problems.17In this paper, afuzzy schedulingprocedurewithbeam search8 s developedto producea schedule(fuzzy start ime assignmentfor eachactivity) with the least possibility of being late in reason-able computing time. Then, the crisp start time of eachactivity is determinedbased on possibility theoryto maxi-mize the satisfaction degrees of all fuzzy temporalconstraints.The proposed fuzzy scheduling procedure generates aschedule by proceeding forwardchronologically. At anytime point, the eligible set of activities that can be sched-uled is determined according to their precedenceconstraints. Due to the limited resource capacities, theseactivities may contend for the same resources.A minimaltotal tardiness heuristic is used to choose the conflicting

activities for delay to ensure the feasibility with respect totemporal and resource capacity constraints at any timepoint during the schedule construction.The fuzzy scheduling procedure s described as follows.Fuzzy Scheduling Procedure:Denote:E: set of eligible activities with all their predecessorscompleted.S: set of activities that have been scheduled.V: set of events to characterize the completion ofactivities.P: set of activities in process.R: a vector to indicate resourceavailability.D: the minimum numberof delayingactivities to resolveresource conflicts.DS: delaying set that contains alternative groups D ofdelaying activities.ino,, the current ime.AR(E): aggregated resource requirementsfor activitiesin E.Step 1: (Initialization)Set E


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JWang-Fuzzyetapproachoactivitycheduling 223(2) Updatethe earlieststart imes of affected succeedingactivities;(3) V


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1224 Journalf heOperationalesearchocietyol. 0,No. 2

J (e , e', e, e) (a,bcd) (e ,e ,e,,e3)

0 est' /2 st, 1s \

0 s bc TimeFigure7 Assignmentf crisp tartome i foractivity i.lstl and the left leg of st1. Accordingly,s1 is determinedasfollows:

c' = PG(lst, sti) = o (26)S1=(-o,S*]n[S*,d] =s (27)

In summary, his paperdevelopsa unifiedframeworkbasedon possibility theoryto determinethe satisfactiondegreesof temporalconstraintsand to assign a crisp start time toeach activitywith the maximumsatisfactiondegrees of allfuzzy temporalconstraints.Illustrated exampleInthis section, an exampleis used to illustrate he approachdeveloped. A product development project consists oftwelve activities, which is representedwith the precedencegraph shown in Figure 8. Three types of resourcesrl, r2,and r3 are requiredto performthe project and the avail-ability of resources is R = (3, 3, 3). Assume that the readytime and deadline of the projectare (0, 3, 3, 3) and (110,120, 120, 130) respectively. The correspondingactivityinformation s listed in Table 1. The beam width is set toone and the preferenceratio ,B s set to 0.5 for the averagecase in this example.The iterationsperformedby the developed fuzzy sche-duling procedure are demonstratedin Table 2. Duringiterations 1 and 2, no resource conflicts are observed. Initeration 3, activity a4 needs three units of resource r3which have been used by activities a2 and a3. Therefore,activitya4has to be delayed.Aftera2 anda3 arecompleted,the identifiedeligible set E in iteration4 includes a4, a5,and a6. A resource conflict is identified, because theresource availability (R = (3, 3, 3)) does not meet the

6 9eFigure8 Example f precedenceraphwithtwelveactivities.

Table1 Activitynformationorrespondingo theprecedencegraph n FigureActivityNo. Duration Resources Requireda, (10, 12, 12, 14) (1, 1, 0)a2 (3, 4, 4, 7) (1, 2, 2)a3 (9, 10, 10, 14) (1, 0, 1)a4 (9, 10, 10, 14) (2, 1, 3)a5 (13, 15, 15, 20) (3, 1, 0)a6 (23, 25, 25, 30) (1, 2, 3)a7 (6, 7, 7, 9) (3, 2, 3)a8 (5, 5, 5, 9) (1, 1, 1)a9 (4, 6, 6, 11) (2, 3, 1)a1o (18, 20, 20, 25) (1, 0, 1)a,, (12, 15, 16, 20) (0, 2, 1)a12 (25, 31, 42, 47) (2, 3, 1)

aggregated resource requirements (AR(E) = (6, 4, 6)).Thus, step 5 is used to determine the delaying activities.Table 3 shows the delaying set DS, which contains thealternative groups of delaying activities to resolve theresourceconflicts. We can identify thatDI listed in Table3 has the minimal total tardiness (0, 0, 0, 28). Therefore,activities a4 and a5, are selected for delay and then a6 canbe processed.The similarprocessiteratesuntil all activitiesare completed.A frizzyscheduleis generatedandlisted in Table 4. Theactivities are scheduled in the following order:(a,,a2, a3,a6, a5, a4, a9, a8, a7, alo, a,I,a12). The obtainedfuzzy project makespan is ft = (122, 144, 155, 200) which

Table2 The terationserformedy thefuzzyschedulingprocedure,as R = (3, 3, 3)Iteration Eligible Conflicting Delayed Activities in Activitiesno. set resources activities process completed

1 a1 None None a1 a12 a2, a3 None None a2, a3 a23 a4 r3 a4 a2 a34 a4, a5, a6 rl, r2, r3 a4, a5 a6 a65 a4, a5, a9 rl, r2, r3 a4,a9 a5 a56 a4,a8,a9 rI,r2,r3 a8,a9 a4 a47 a7,a8,a9 rl, r2, r3 a7,a8 a9 a98 a7,a8 ri a7 a8 a89 a7, a10 r, a1o a7 a710 a1o,all None None a1o,all a,111 None None None a1o a1o12 a12 None None a12 a12

Table3 The identified elayingetin iterationDelaying set Group of delaying Total(DS) activities tardinessDI {a4,a5} (0, 0, 0, 28)D2 {a4,a6} (0, 0, 0, 49)D3- a a6) (0, 0, 0, 55)



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JWang-Fuzzyetapproachoactivitycheduling 225Table 4 The schedulegenerated or the precedencegraphin Figure8, as R = (3, 3, 3)

Activity Fuzzy start time Fuzzy latest start Fuzzyfinish time Start Sat.(ai, i = 1 - 12) (sti) time (Isti) (fti) time (Si) degreea1 (0,3,3,3) (-1, 29, 41, 65) (10,15,15,17) 3 1.00a2 (10,15,15,17) (13,41,53,75) (13,19,19,24) 15 1.00a3 (10,15,15,17) (0,30,42,65) (19,25,25,31) 15 1.00a4 (55,65,65,81) (20,45,57,78) (64,75,75,95) 63 0.74a5 (42,50,50,61) (14,40,52,74) (55,65,65,81) 50 1.00a6 (19,25,25,31) (-7, 27, 38, 60) (42,50,50,61) 25 1.00a7 (73,86,86,114) (34,55,67,87) (79,93,93,123) 79 0.42a8 (68,81,81,105) (24,53,64,82) (73,86,86,114) 74 0.45a9 (64,75,75,95) (23,52,63,83) (68,81,81,105) 71 0.61alo (79,93,93,123) (33,58,69,87) (97,113,113,153) 83 0.25all (79,93,93,123) (43,62,74,93) (91,108,109,143) 85 0.42a12 (97,113,113,153) (63,78,89,105) (122,144,155,200) 101 0.25Fuzzy Makespan (122,144,155,200) Projectrisk 0.875

overlaps' with the planned project due dated= (110, 120, 120, 130). According to Eq. (19), theproject risk is calculated:

Risk(P)= 0.5 x PSG(ft,d) + (1 - 0.5) x NSG(ft,d)= 0.5 x 1.00 + 0.5 x 0.75=0.875

The last two columns in Table4 display the crisp start imeandthe corresponding atisfactiondegree for each activity,according to Eq. (23). For example, the crisp starttime foractivity a4 is calculated(see Figure 9):a = PG(lst4,st4) = 0.74.S4 = 074(-oo, 1st4]n0?74st4= 62.42 63.

In order to reduce the risk of the projectbeing late, projectmanagersmay consider including more resources in theproject. It is observed from Table 2 that resourcer1 is inconflict from iterations 4 to 9. Therefore assigning anadditional one unit of r1 may obtain the most benefit.Table 5 shows iterations performed when the resourceavailability s changedto R = (4, 3, 3). Table6 summarizesthe obtained schedule and the fuzzy project makespanisreduced to (109, 129, 140, 180) with the risk level 0.65.One can observe that resource r3 becomes the bottleneckresource,becauseit is in conflictfromiterations3 to 9. If r3is also increasedone unit, then the shortermakespan(92,113, 125, 155) with risk level 0.438 can be obtained(Table7). If one accepts this risk level, then (s)he may assign

1st4 (20,45,57,78) t4 =(55,65,65,81)1.0 A....0.74

62.42 TimeFigure9 Assignmentf crisp tart imeS4foractivity 4.

R = (4, 3, 4) to the project.Otherwise, (s)he may increasethe resourceavailabilityagain. In summary, he developedmethodology allows project managers o evaluate decisionsforresourceallocation n anuncertainproductdevelopmentenvironment.Computational experimentsPrimarycomputationalexperiments are conductedto testthe proposedfuzzy schedulingprocedureon a set of bench-mark problems to evaluate its efficiency. The set ofproblemscontains threeproblemtypes including30 activ-ities, 60 activities and 90 activities. Each problem typecontains 86 problems. The benchmark problems arerandomly selected from the project scheduling problemlibrary(PSLIB).8Due to the fact that all the benchmarkproblems createdfrom PSLIB are deterministic, we randomly fuzzify itsdata. For each crisp duration ti, four numbers that areused to define the fuzzified activity durationti = (ai, bi, ci, di) are randomly generated from the interval(T x ti, ti + (1 - y) x ti) and ranked in ascending order. InTable 5 The iterationsperformedby the fuzzy schedulingalgo-rithm, as R = (4, 3, 3)Iteration Eligible Conflicting Delayed Activities in Activitiesno. set resources activities process completed

1 a, None None a, a,2 a2, a3 None None a., a3 a23 a4 r3 a4 a3 a34 a4, a5, a6 r1, r2, 3 a4 a5, a6 a55 a4, a8 r2, r3 a4, a8 a6 a66 a4,a8,a9 r1,r2,r3 a8,a9 a4 a47 a7, a8,a9 I12,-2, 3 a7,a8 a9 a98 a7,a8 r3 a7 a8 a89 a7, al0 r3 alo a7 a710 a1o,a, I None None alo, a, I all11 None None None a1o alo12 a12 None None a12 a12



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1226 Journalf heOperationalesearchociety ol. 0,No. 2Table 6 The generatedschedule, as R = (4, 3, 3)

Activity Fuzzystart Fuzzy latest start Fuzzy inish time Start Sat.(ai, i 1 - 12) time(sti) time(lsti) (fti) time (si) degreea, (0,3,3,3) (-1,29,41,65) (10,15,15,17) 3 1.00a2 (10,15,15,17) (13,41,53,75) (13,19,19,24) 15 1.00a3 (10,15,15,17) (0,30,42,65) (19,25,25,31) 15 1.00a4 (42,50,50,61) (20,45,57,78) (51,60,60,75) 50 1.00a5 (19,25,25,31) (14,40,52,74) (32,40,40,51) 25 1.00a6 (19,25,25,31) (-7,27,38,60) (42,50,50,61) 25 1.00a7 (60,71,71,94) (34,55,67,87) (66,78,78,103) 70 0.87a8 (55,66,66,85) (24,53,64,82) (60,71,71,94) 66 0.93a9 (51,60,60,75) (23,52,63,83) (55,66,66,85) 60 1.00alo (66,78,78,103) (33,58,69,87) (84,98,98,133) 75 0.70all (66,78,78,103) (43,62,74,93) (78,93,94,123) 77 0.87a12 (84,98,98,133) (63,78,89,105) (109,129,140,180) 94 0.70Fuzzy Makespan (109,129,140,180) Projectrisk 0.650

Table 7 The generatedschedule,as R = (4, 3, 4)Activity Fuzzystart Fuzzy latest start Fuzzy inish Start Sat.(ai, i 1 '12) time (sti) time (lsti) time (fti) time(si) degreea, (0,3,3,3) (-1,29,41,65) (10,15,15,17) 3.0 1.00a2 (10,15,15,17) (13,41,53,75) (13,19,19,24) 15.0 1.00a3 (10,15,15,17) (0,30,42,65) (19,25,25,31) 15.0 1.00a4 (13,19,19,24) (20,45,57,78) (22,29,29,38) 19.0 1.00a5 (22,29,29,38) (14,40,52,74) (35,44,44,58) 29.0 1.00a6 (22,29,29,38) (-7,27,38,60) (45,54,54,68) 29.0 1.00a7 (49,60,60,78) (34,55,67,87) (55,67,67,87) 60.0 1.00a8 (35,44,44,58) (24,53,64,82) (40,49,49,67) 44.0 1.00a9 (45,54,54,68) (23,52,63,83) (49,60,60,78) 54.0 1.00alo (49,60,60,78) (33,58,69,87) (67,80,80,108) 60.0 1.00all (55,67,67,87) (43,62,74,93) (67,82,83,107) 67.0 1.00a12 (67,82,83,108) (63,78,89,105) (92,113,125,155) 87.0 1.00Fuzzy Makespan (92,113,125,155) Projectrisk 0.438

the currentexperiment,y is set to 0.8. The projectreadytime and deadlineare fuzzified in the similarway.The optimal solutions to these problems were foundusing the fuzzy version of the A* algorithm29see Appen-dix). Table 8 shows the computational esults for the threeproblem sets regardingdifferentvalues of beamwidth. Theaverage deviations from the optimumsolutionsare almostless than 0.1, except for the problem set III for beamwidth = 1. The solution quality can be improved byincreasing the beam width (Figure 10). Moreover, theTable 8 Computational esultsregardingdifferentbeam widths

Beam width1 3 5Problem No. ofset Activities Avg dev Std dev Avg dev Std dev Avg dev Std dev

I 30 0.084 0.111 0.052 0.095 0.043 0.083II 60 0.065 0.106 0.031 0.052 0.028 0.066III 90 0.116 0.125 0.088 0.116 0.076 0.100

average CPU time is increased linearly, as the beamwidth increases (Figure 11).ConclusionThe contributionof this research was to develop a fuzzyactivity scheduling model to handle uncertain temporalinformation in product development. A new method wasdeveloped to determine the satisfaction degrees of fuzzytemporal constraints.Based on the proposed method, a



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JWang-Fuzzyetapproachoactivitycheduling 2270.140 30.120 160.100 Ag

. 0.080 0,076> 0.060

o0.040 * 0.028X 0.020 30.000 1 3 5 Beam width

Figure10 Plotof average eviationsromoptimalolutionsorthreeproblemets.1.00.90:50.8080.7a- 0.6' 0.5 V.4#0' 0.4 /g 0.3 325? 0.20.10o.0

1 2 3 4 5 6 7 8 9 BearnwidtlhFigure11 AverageCPU imeregardingifferenteamwidths orproblemetI.fuzzy scheduling procedure was developed not only toobtain a schedule with the least possibility of being late,but also to maximize the satisfactiondegrees of all fuzzytemporalconstraints.Finally, computationalstudies wereprovidedto examine the solution quality. It is concludedthat the proposedmethodologycan producemore satisfac-tory schedules in an uncertainproductdevelopmentenvir-onment.

Appendix: The fuzzy version of A* algorithm foractivity schedulingA* algorithm, which is a kind of the best-first searchalgorithm, s described n most introductoryartificial ntel-ligence texts.30This paperadaptsthe A* schedulingalgo-rithm developed by Bell and Park29 o obtain optimumsolutionsforexamining he solutionqualityof the proposedfuzzy schedulingprocedure.At each step of the A* searchprocess, the most promising node is chosen for furtherexpansion. If the selected node has no resource conflict,then an optimal schedule is found and the searchprocessterminates uccessfully. Otherwise,we expandthe selectednode by delaying some activities to generate the node'ssuccessors for resolving resource conflicts. Newly gener-atednodes arethen addedto the set OPEN which have notbeen expanded.Again the most promisingnode is selectedand the processcontinues.Note that the evaluationfunctionof a node is determinedby the schedule risk accordingtoEq. (19). In addition, in order to obtain better computa-

tionalperformance, he following pruningrules29areused:a node n weakly dominatesn', if(1) n and n' have the same scheduledset of activities.(2) Let E be the set of eligible activities. Both n and n'have the same eligible set of activities. If the earlieststarttimes of all activities, n E for node n are lessthan or equal to the correspondingactivities in n'.Node n' can be pruned, f n weakly dominatesn'.The fuzzy version of A* algorithm is presented asfollows.A* algorithm

Step 1. Set OPEN *- OPENU {no}, where OPEN is theset storingunexpandednodes and no is the initialnode for a given problem.Step 2. If OPEN 0 0, then go to step 3; else exit withfailureand no solutionexists.Step 3. Remove from OPEN a node n for whichf(n) is

minimum. Set CLOSE


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1228 Journalf heOperationalesearchocietyol. 0,No. 2(2) If all activities have been completed, then stop

and return the status 'complete', else go toStep 3.

Step 3. (Resource allocation)For activity ai E E, do(1) sti *- t7,1, fti t10ov,EDi, S

A Fuzzy Set Aproach to Activity Scheduling

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