Scheduling Optimization under Uncertainty - An Alternative...

SchedulingOptimizationunderUncertainty-

An Alternative Approach

J.Balasubramanian�

andI. E. Grossmann�

Departmentof ChemicalEngineering,Carnegie Mellon University

Pittsburgh,PA 15213,USA

March2002

Abstract

The prevalent approachto the treatmentof processingtime uncertaintiesin production

schedulingproblemsis throughtheuseof probabilisticmodels.Apart from requiringdetailed

informationaboutprobabilitydistribution functions,this approachalsohasthedrawbackthat

the computationalexpenseof solving thesemodelsis very high. In this work, we presenta

non-probabilistictreatmentof schedulingoptimizationunderuncertainty, basedon concepts

from fuzzy settheoryandinterval arithmetic,to describethe imprecisionanduncertaintyin

the taskdurations. We first provide a brief review on the fuzzy setapproach,comparingit

with theprobabilisticapproach.We thenpresentMILP modelsderivedfrom applyingthisap-

proachto two differentproblems- flowshopschedulingandnew productdevelopmentprocess

scheduling.ResultsindicatethattheseMILP modelsarecomputationallytractablefor reason-

ably sizedproblems.We alsodescribetabu searchimplementationsin orderto handlelarger

problems.

1 Intr oduction

With increasedinterestin productionschedulingin chemicalengineeringin recentyears,therehas

beenmuchwork addressingthe optimal schedulingof a wide rangeof sytems- from individual

productionunitsto geographicallydistributedmultisitesupplychains(seeShah,1998,for review).�[email protected]�[email protected] Correspondingauthor

1

A significantamountof the work in this areahasfocussedon the developmentof deterministic

models,wheretheproblemdatais assumedto beknown in advance.In reality, though,therecan

be uncertaintyin a numberof factorssuchasprocessingtimesandcosts.As a result,a number

of papersin the recentyearshave addressedschedulingin the faceof uncertaintiesin different

parameters- for e.g.,demands(IerapetritouandPistikopoulos,1996;Petkov andMaranas,1997;

Sandet al., 2000)andprocessingtimes(Honkomp et al., 1999;SchmidtandGrossmann,2000;

BalasubramanianandGrossmann,2001).

Theprevalentapproachto thetreatmentof theseuncertaintiesis throughtheuseof probabilis-

tic modelsthatdescribetheuncertainparametersin termsof probabilitydistributions. However,

theevaluationandoptimizationof thesemodelsis computationallyexpensive,eitherbecauseof the

largenumberof scenariosresultingfrom adiscreterepresentationof theuncertainty(Wets,1974),

or theneedto usecomplicatedmultiple integrationtechniqueswhentheuncertaintyis represented

by continuousdistributions(SchmidtandGrossmann,2000).Furthermore,theuseof probabilistic

modelsis realisticonly whenthesedescriptionsof theuncertainparametersis available,sayfrom

historicdata. Whensuchdatais not available(for examplein New ProductDevelopment,when

thetaskshaveneverbeenor only rarelyperformedbefore),wedonothaveenoughinformationfor

inferringor deriving theprobabilisticmodels.In suchsituations,wehaveto resortto analternative

treatmentof uncertainty. For example,themodelermaybeableto approximatethedurationof the

tasksandspecifythelongestandshortestdurationsor theinterval in which thedurationbelongsat

differentlevelsof confidence.

In thiswork, wedraw uponconceptsfrom fuzzysettheoryandinterval arithmeticto describe

theimprecisionanduncertaintiesin thedurationsof batchprocessingtasks.Indeed,this approach

hasbeenreceiving increasingattentionrecently. McCahonandLee(1992)werethefirst to illus-

tratetheapplicationof fuzzy settheoryasa meansof analyzingperformancecharacteristicsfor a

flowshopsystem.They modifiedtheCampbell,DudekandSmithsequencingheuristic(Campbell

et al., 1970)for the caseof fuzzy processingtimes. However, their procedurewascumbersome

for fairly largeproblemsizes.Most of thework in applyingfuzzy settheoryto schedulingopti-

mizationhasprimarily focussedon usingheuristicsearchtechniquessuchassimulatedannealing

andgeneticalgorithmsto obtainnear-optimalsolutions.For example,Ishibuchi et al. (1994)ex-

aminedflowshopschedulingproblemswith fuzzy duedates,wherethe membershipfunction of

the fuzzy duedateindicatesthe gradeof satisfactionof the decisionmaker with the completion

time of a job. Theseauthorspresentedresultsfrom applyinggeneticalgorithmsto two typesof

problems- i) maximizingtheminimumgradeof satisfactionover all jobs,andii) maximizingthe

totalgradeof satisfaction.Fortemps(1997)addressedtheproblemof minimizingthemakespanof

jobshopswith fuzzy durationsusingsimulatedannealing.Theauthoralsointroduceda new com-

2

parisonmethodfor fuzzy numbers.A recentpaper(OzelkanandDuckstein,1999) investigates

the necessaryconditionsfor optimality of fuzzy counterpartsof classical(deterministic)optimal

schedulingrules and emphasizesthe importanceof using ranking functionsthat satisfy certain

properties.In theareaof projectscheduling,ChanasandKamburowski (1981)laid thetheoretical

foundationsfor approachingtheproblemthroughthefuzzy setframework. Lootsma(1989)con-

trastedthe fuzzy setapproachwith the well-known stochasticPERT (Malcolm et al., 1959)and

discussedsomeof thepitfalls associatedwith stochasticPERT, includingthecomputationalcom-

plexity of PERT (Hagstrom,1988). Hapke andSlowinski (1996)andWang(1999)addressedthe

resource-constrainedprojectschedulingproblem.While theformerappliedheuristicdispatching

rulesto generatetheschedules,thelatterformulatedtheproblemasa fuzzyconstraintsatisfaction

problembasedon possibility theory(DuboisandPrade,1988)anduseda beamsearch(Ow and

Morton, 1988)to obtaina schedulewith theleastpossibilityof beinglate. We refer thereaderto

Slowinski andHapke(2000)asanexcellentoverview of schedulingunderfuzziness.In thechem-

ical engineeringliterature,thetheoryof fuzzy setshasprimarily beenappliedfor processcontrol

(seefor examplesof recentworks,PostlethwaitheandEdgar, 2000;Galluzoet al., 2001)although

therehave alsobeenapplicationsin theareaof design(Kubic andStein,1988;Tarifa andChiotti,

1995).Majozi andZhu (2001)recentlypresentedanapplicationof fuzzysettheoryto theoptimal

allocationof operatorsto batchplants.

Unlike the previously mentionedpapers,which have relied on heuristicsto obtain near-

optimal solutions,we show how it is possibleto develop Mixed Integer Linear Programming

(MILP) modelsfor differentschedulingproblemsthat usea fuzzy representationof uncertainty.

We alsocomparetheMILP approachwith heuristictechniquesfor flowshopproblems.As a brief

introductionto theconceptsof this non-probabilisticapproach,we presentresultsfrom flowshop

optimizationunderuncertainty. The durationsof the tasksaredescribedby fuzzy numbers(as

opposedto deterministicvaluesor probabilisticdescriptions).We presentMILP modelsfor se-

lecting theoptimalschedulefor two typesof flowshopsystems- (i) multistageflowshopswith a

singleprocessingunit in eachstage(Birewar andGrossmann,1989;Pekny andMiller, 1991),and

(ii) multistageflowshopswith multiple parallelunits in eachstage(PintoandGrossmann,1995).

We thenaddresstheproblemof schedulingtheNew ProductDevelopment(NPD) processfor an

agriculturalchemicalunderuncertainty(SchmidtandGrossmann,1996).In theNPD process,the

potentialproductmustpassa seriesof teststhatassessits safety, efficacy andenvironmentalim-

pact.Thecosts,durationsandprobabilitiesof successof varioustestingtasksareoftennotknown

with certaintywhenthe schedulehasto be constructed.We considerthe problemof optimizing

thescheduleof testingtaskswhile taking into accounttheuncertaintiesin thedurationsandsuc-

cessof the tasks,a problemwhich hasnot beenreportedbefore. Theoptimizationof the testing

3

scheduleconsistsin introducingprecedenceconstraintsbetweenthetests,in additionto thegiven

setof technologicalprecedenceconstraints.We presentanMILP formulationfor theobjective of

maximizingthe differenceof the expectedvalueof the fuzzy incomeandthe expectedcostof a

testingschedule.

The paperis organizedas follows. Section2 definesthe problemsof interest. Section3

presentsa brief overview of fuzzy settheoryandcomparesmathematicaloperationsin thefuzzy

framework with theirprobabilisticcounterparts.After anoverview of ourapproachto theproblem

in Section4, we presentresultsfrom optimizationof flowshopsin Sections5 and6 andtheNPD

processin Section7. Finally, theconclusionson this work aredrawn.

2 ProblemStatement

We considerin this paperthefollowing generalproblem.Givenarea setof tasks(ordersor tests)

to beperformedusingcertainresources(processingequipment).Thesetaskshave their durations

specifiedasfuzzy numbers.The objective is to determinean allocationof the resourcesto the

tasksthat is optimal with respectto a certainmeasuresuchasthe makespanor the profit. Our

principalobjectiveis todemonstratetheapplicationof conceptsfromfuzzysettheoryto scheduling

optimizationwhentheprocessingtimesareuncertain.Wepresentapplicationsfrom flowshopand

NPDtestingoptimization.

3 Comparison of Fuzzy Setsand Numbers with Probabilistic

Approach

In this section,we review key conceptsfrom the theory of fuzzy setsthat will be usedfor the

schedulingmodels.Wealsocomparethemwith theirprobabilisticcounterparts.

3.1 Definitions

A classicalset � of a universe� is a collectionof elementsor objectsthatarewell definedand

possesssomecommonproperties.An element� of theuniverse� mayeitherbelongto (true or 1)� or not (false or 0). Zadeh(1965)introducedtheconceptof a fuzzysetin which themembership

of an elementto a setneednot be just binary-valued(i.e., 0-1), but could be any valueover the

interval [0,1] dependingon the degreeto which the elementbelongsto the set. The higher the

degreeof belonging,higherthemembershipvalue.

4

A fuzzy set �� of the universe� is specifiedby a membershipfunction � �� which takes

its valuein theinterval [0,1]. For eachelement� of � , thequantity �� specifiesthedegreeto

which � belongsin�

. Thus, �� is completelycharacterizedby thesetof orderedpairs:

�� ! "�$# (1)

Thesupportof a fuzzyset �� is a (crisp/classical)subsetof X givenby

%'&)(*( �+�� , "� �-�� /.102# (2)

The 3 -level set( 3 -cut) of a fuzzyset �� is acrispsubsetof X givenby

��45�� , 6� �7�� /893:# ;<3= >�?0@�'ACB (3)

A fuzzy numberis definedasa boundedsupportfuzzy setof the real line D whose 3 -cutsare

closedintervals.

3.2 Arithmetic Operations- a Comparisonwith Probabilistic Computations

Arithmetic operationson fuzzy numbersthatwill beusedfor theobjective functionandthecon-

straints,suchas addition, subtraction,taking the maximumof two fuzzy numbersare defined

throughthe extensionprinciple (Zadeh,1965),which allows the extensionof operationson real

numbersto fuzzy numbers.In theschedulingproblemsof interestto us, theprincipalarithmetic

operationsthatareinvolvedareaddition(computingthefuzzy end-timeof a taskgiventhefuzzy

starttime andduration)andmaximization(computingthestarttime of a taskasthemaximumof

theendtimesof precedingtasks).Hence,we restrictour descriptionof arithmeticoperationson

fuzzynumbersto thesetwo operations.

3.2.1 Addition

1. FuzzyNumbers

If �� and �E aretwo fuzzynumbers,theiradditioncanbeaccomplishedby using 3 -level cuts.

If �� 4��GF �IH4 ��KJ4 B and �E 4��LFNM H4 � M J4 B , thentheresult �O which is theadditionof �� and �E can

bedefinedby the 3 -level setsasbelow:

�O 4P� �� 4 �?QR�K�E 45�LF � H4 Q M H4 �� J4 Q M J4 B ;<3> =�?0S�'ATB (4)

This clearly shows that the lower boundof �O is the sumof the lower boundsof �� and �Eandsimilarly, theupperboundof �O is thesumof theupperboundsof �� and �E . Figure1

providesanexample.

5

10 35t

mX(t)

20

1.0

05 25

t30

1.0

0

1.0

15 65t

450

YX Z

Figure1: Addition of TFNs

2. RandomVariables

If � andE

aretwo independentrandomvariableswith probabilitydensityfunctions( UWVYX s)

givenby Z7[\��]^� and Z7_`��]^� , andO

is a randomvariablethatis thesumof � andE

, the UWVaXofO

is obtainedby convolving thedistribution of � with thatofE

. As is well known, the

expectedvalueofO

is givenby theadditionof theexpectedvaluesof � andE

.

O � �bQ EZdc��]^� � Z�[e��]^��fgZ7_h��]^� �jilknmo m Z�[e� & � p-Z7_h��] q & � p7r & (5)s F O B � s F �"B2Q s F E B (6)

3.2.2 Maximum Operator

1. FuzzyNumbers

If �� and �E aretwo fuzzy numbers,their maximumcanalsobe obtainedby using 3 -level

cuts.Thus,

�O 4P��tlu-v �w�� 4 ��E 4 � �xFNtlu-v �� H4 � M H4 �y� tlu-v �� J4 � M J4 �zB ;<3{ >�|0S�'ACB (7)

In general,this operationrequiresinfinite computations,i.e., evaluatingmaximafor every3} ~�|0S�'ACB . However, goodapproximationscanbe obtainedby performingthesecompu-

tationsat specificvaluesof 3 , ratherthan at all values. As canbe seenin Figure2, the

maximumof two TFNsneednotbeanotherTFN.

2. RandomVariables

If � andE

aretwo independentrandomvariableswith UWVYX s givenby Z�[R��]^� and Z7_��]^� ,and

Ois a randomvariablethat is the maximumof � and

E, the cumulative distribution

function ( ��VYX ) ofO

is computedby multiplying the �WVaX of � with the �WVaX ofE

.

Unlikethecaseof theadditionoperation,theexpectedvalueof themaximumof two random

6

Figure2: Maximum of TFNs

variablescannotbeobtainedby takingthemaximumof theexpectedvalues.In fact,taking

themaximumof theexpectedvaluesresultsin a lowerboundto theactualvalue.

O � tlu-v ��"� E �X c+��]^� � X�[R��]^� p7X+_��]^� (8)s F�� "� E ��B�8 �� F s �� s � E �zB (9)

3.3 ObjectiveFunction Comparisons

3.3.1 FuzzyNumbers

Sincewe areinterestedin selectinga schedulewith theoptimum(minimum)makespan,we have

to comparethe fuzzy makespansof potentialschedules.In a probabilistictreatment,we maybe

interestedin minimizingtheexpectedvalueof themakespan.However, with ourapproach,matters

arenotsostraightforward.

Thereexistsa largebodyof literaturethatdealswith thecomparisonof fuzzy numbers(see

ChenandHwang,1992, for an overview) anda numberof indiceshave beenproposed.More

recently, FortempsandRoubens(1996)proposeda methodfor comparingfuzzy numbersbased

on the compensationof the areasdeterminedby the membershipfunctions,which hasa very

desirablepropertythatallows usto modelthedistancebetweentwo fuzzy numbers.This method

is alsorelatedto themeanvalueof a fuzzy numberasdefinedin DuboisandPrade(1987)in the

framework of Dempster-Shafertheory. In the framework of Dempster-Shafer’s theory, a fuzzy

number �� is consideredto defineasetof possibleprobabilitydistributions;hencethemeanvalue

of the fuzzy number �� is the set of meanvaluescalculatedaccordingto all theseprobability

distributions.

Thelowermeanvalue �l� anduppermeanvalue � � arecalculatedfrom thefollowing distri-

bution functionsX��7�� and X � �� asfollows.

X � �� %'&)( � ��[ ��]^�7� ]/��# (10)

7

X �'�� Z � Agq{l�[ ��]^�7� ]/8��# (11)

�l� � i knmo m �lp7r2X�� (12)

� � � ilknmo m �lp7r2X � �� (13)

Thus,themeanvalueof thefuzzynumber �� is theintervalF �l�'�� B . Now, if weassumethat

all the valuesin the intervalF �l�'�� B areequallylikely, thenthe naturalchoicefor a singlereal

numberwouldbethemid-pointof this interval. Theareacompensationmethodbasicallyinvolves

computingthefollowing integral:� �l�w�� 0@��\p iY�� H4 Q>� J4 ��r)3 (14)

FortempsandRoubens(1996)prove that the areacompensationprocedureprovidesa real

numberthatis themid-pointof theinterval correspondingto themeanvalueof a fuzzynumber, in

thesenseof theDempster-Shafertheory, wherebothsidesof thepossibilitydistribution inducea

probabilitymeasure.Thus,wehave:� �l�w�� 0@��\p i �� H4 Q>� J4 ��r)3� 0@��\p)��a��Q>� � � (15)

Thearea-compensationintegral in (14)canbeusedto comparethefuzzymakespansof poten-

tial schedules.If we areinterestedin minimizing themakespan,a schedulewith fuzzy makespan�� is preferredoveraschedulewith fuzzymakespan �� if� �l�w�� /� � �l�� .

3.3.2 Probabilistic Case

Supposewe wish to computethe expectedvalueof a function of several randomvariablessuch

asO �� '�C�'�'�C�^�a�� . The first stepis to obtainthe joint U�VYX of the randomvariables,

i.e., Zw�� '�'�'�T��K�� , which in thecaseof independentrandomvariablesis merelytheproductof the

individual U�VYX s. Then,to obtainthe �WVYX ofO

, X c��]^� , we have to computea multiple integral

overapolytope U thatis parametricin t.

X c��]^� � i1i p�p�p i' <¡£¢¥¤ Z �� '�C�'�C��K�`� p7r¦� � p�p�p�r¦�K�Ul��]^� � �-� � ��]^�`�� 9§ � ��]^��C�'�'�C� � �¨��]^�/��K��©§y��]^�� A¦��ª@�'�C�'�T�^�K�/�`��]«# (16)

OnceX�c ��]^� hasbeenobtained,the UWVYX canbeobtainedthroughdifferentiationandtheexpected

valueofO

maybecomputed.However, computingtheparametricmultiple integral in (16) is an

expensiveoperationandis not tractable,evenwhenthenumberof randomvariablesis moderately

large.

8

3.3.3 Comparison

It is instructiveto comparethestepsinvolvedin computingthemakespanin thefuzzysetcasewith

theprobabilisticcase.

Whenthe processingtimesaregivenby fuzzy numbers,the makespan,which is a function

of thesefuzzy numbers,canbecomputedveryefficiently (althoughapproximately)throughinter-

val arithmeticat various 3 -levels,asin (4) and(7) andthe objective computedthroughthe one-

dimensionalintegral in (14). However, in theprobabilisticcase,thedistribution of themakespan

is very difficult to obtain,sinceit hasto beobtainedthroughseveralparametricmultiple integrals

asin (16).

Furthermore,we notethefollowing interestingpropertiesof theareacompensationoperator,

which have their counterpartsin the probabilisticapproach.The resemblanceof� �l�� to the

expectedvalueoperators �� is indeedstriking (see(6) and(9)).� �l� ��bQ �E � � � �a� ��Q � �a� �E � (17)� �l� �� F ��>��E B¬�8 �� F�� l�:��y� � �l�2�E ��B (18)

4 Overview of Approach

The key ideasthat we will useto apply the conceptspresentedin the previous sectionare as

follows:

1. Evaluation of agivenscheduleunderuncertainty

Thisstepinvolvesthecalculationof thestart-timesandend-timesatvarious3 -levels,through

a setof recurrencerelations.Thesecomputationstypically involve theuseof interval arith-

metic.

2. Generalizationof Expressions

Oncetheexpressionsfor calculatingthestart-andend-timesfor thetasksfor agivensched-

ule (sequenceof tasks)have beenderived,thenext stepis to generalizetheseto accountfor

all possibleschedules.This is donefirst by introducingbinaryvariablesthatmodeleitherthe

precedencerelationshipsbetweenthetasks,or theallocationof tasksto specificprocessing

units.

3. Optimization

Theareacompensationintegral will beusedto comparetheobjective functionsof potential

schedules.Theoptimizationmodelsusea discretizationapproximationto thecalculationof

9

theone-dimensionalintegral in (14).1 Furthermore,thebenefitsof usinga largenumberof

discretizationpointsarealsolesspronounced- with verysmallimprovementsin theestima-

tion of the integral andanorderof magnitudelarger computationtime. In somecases,we

alsoconsiderthegenerationof movesfor local searchtechniques.

5 The FlowshopProblem

Stage 2Stage 1 Stage 3Reactor Centri fuge Tray Dryer

Product

Figure 3: FlowshopPlant

Flowshopplants(seeFig.3)aremultiproductbatchplantswherethejobsassociatedwith the

manufacturingof productordersusethesamesetof processingunits in thesameorder(Birewar

andGrossmann,1989).A solutionto theflowshopschedulingproblemspecifiestheorderin which

jobsareprocessedin eachunit.

Givenare ® products� � �«¯l�'�'�'�°� , � "± , thatareto bemanufacturedin a flowshopplantwith²processingstages�³A¦�yª2�'�'�'�� ² � , ´l ¶µ . Eachproductrequiresprocessingin all of the

²stages

andfollows the samesequenceof operations,i.e., a permutationflowshopplant. In the caseof

Unlimited IntermediateStorage(UIS) modeof operation,anunlimitednumberof batchesmaybe

heldin storagebetweenstages.Thecomputationof themakespan(of agivenprocessingsequence)

for theUIS flowshopplant involvesa seriesof recurrencerelationsfor thecompletiontimes ·«]�¸³¹of productin position ( in thesequencein stage . Therelationsfor thecompletiontimesfor the

deterministiccasearegivenbelow (RajagopalanandKarimi, 1989):

·«]�¸³¹ � tlu-v �?·«]�¸ o �zº ¹��«·«]�¸ º ¹ o � ��Q>»@¸^¹ ; ( .�A¦�³;S´¼.jA (19)

·«] � ¹ � ·«] �zº ¹ o � Q>» � ¹ ;S´½.jA (20)

·y]�¸ � � ·«]�¸ o �zº�� Q=»S¸ � ; ( .jA (21)� % � ·«]³� ¾ (22)1A rectangularapproximationof this integral leadsto large errors,when the numberof discretizationpoints is

low; while a piecewisequadraticapproximation(Simpson’s rule) givesmuchbetterresults.In theresultspresented,

Simpson’s rule wasusedto approximatetheintegral in (14).

10

where »S¸³¹ is theprocessingtime of theproductin position ( in thesequencein stage and� % is

themakespanof thecorrespondingschedule.

For thecasewheretheprocessingtimesarerepresentedby fuzzynumbers,theaboverelations

have to be modified using the extensionprinciple. The integral in (14) is approximatedby a

summationof thefunctionvaluesat specificvaluesof 3 . Thus,theinterval [0,1] is discretizedto

points�-� � � � �'�C�'�'�C� � �# , with stepsize�h� � A7¿�� q�A7� .·«]�¸³¹^À � tlu-v �|·y]�¸ o �zº ¹ º À��«·«]�¸ º ¹ o �zº À��Q=»S¸³¹^À ; ( .ÁA¦�³;S´¼.ÁA¦��; � (23)

·«] � ¹^À � ·«] �zº ¹ o �zº ÀwQ>» � ¹^À ;S´½.ÁA¦�³; � (24)

·y]�¸ � À � ·«]�¸ o �zº��zº À�Q=»S¸ � À ; ( .�A¦�³; � (25)� % À � ·«]³� ¾\À ; � (26)

With the above relationsfor the completiontimes of the variousoperationsfor a specific

processingsequence,we canformulatean MILP modelfor selectingthe sequencewith optimal

makespan.In this MILP model,we introducebinary variablesM-Â ¸ which denotethe assignment

of product�

to position ( in the processingsequence.A sequencewith fuzzy makespan� � is

preferredover a sequencewith makespan�� ifO �� "� O ��y� , with

O �� calculatedusing

(14). TheMILP model(FSP)is asbelow:

(FSP) Min ÃCÄwÅ � ��:��¿*Æ¦� p2�³A7¿¦ª*��p-Ç À �:�/À`p)�?·«]³� ¾\À H QÈ·«]³� ¾\À J � (27)

·y]�¸³¹^À³ÉÊ8 ·«]�¸ o �zº ¹ º À º É�Q Ç Â » Â ¹^À³Éwp MdÂ ¸ ; ( 8©ª2�³;S´¦�³; � �«Ë (28)

·y]�¸³¹^À³ÉÊ8 ·«]�¸ º ¹ o �zº À º É�QÌÇ Â » Â ¹^À³Éwp MdÂ ¸ ; ( �³;S´l8©ª@Í³; � �«Ë (29)

·«] �z� À^É � Ç Â » Â � À³É�p M-Â � ; � ��Ë (30)

·y] � ¹³À^É � ·«] �zº ¹ o �zº À º É�QÌÇ Â » Â ¹^À³Éwp MdÂ � ;S´a8©ª2Í³; � �«Ë (31)

·«]�¸ � À^É � ·«]�¸ o �zº��zº À º É�Q Ç Â » Â � À³É�p MdÂ ¸ ; ( 8©ª2Í³; � �«Ë (32)

Ç ¸ MdÂ ¸ � A ; � (33)

Ç Â MdÂ ¸ � A ; ( (34)

·y]�¸³¹^À³ÉÊ8 0 ; ( �z´¦� � �«Ë (35)MdÂ ¸ � 0S�'Ad# ; � � ( (36)

wherethe subscriptsË indicatethe left andright end-pointsusedin the interval-arithmetic;

thus,ËW s �Î�-Ï ��Ð2# . In (FSP),(27)representsthediscreteapproximationof thearea-compensation

integral appliedto themakespan,with the �� s denotingtheSimpsoncoefficientsusedin theap-

proximationof the integral. Constraints(28)-(32)arethe generalizationof the completiontime

11

constraintsin (23)-(25) for any sequence.Constraints(33) and(34) arethe well-known assign-

mentconstraintsthat specifythat eachproduct�

shouldbe assignedto only 1 position,andthat

eachposition ( shouldbeassignedto only oneproduct.Model (FSP)canalsobeextendedto the

casewherethereareset-uptimesbetweentheprocessingof differentproducts.Theseset-uptimes

caneitherbesequencedependentor unit dependent.

5.1 Illustrati veExample

Table1: TFN Descriptionsfor 5-Product, 4-StageExample

Product Stage1 Stage2 Stage3 Stage4

1 (16.083,17,18.405) (41.680,44,45.036) (31.369,32,34.011) (21.505,22,22.205)

2 (20.056,22,23.490) (16.632,19,19.692) (23.169,24,27.099) (39.606,44,48.935)

3 (11.487,13,15.002) (28.634,30,31,787) (49.171,50,56,611) (30,513,33,37.988)

4 (48.799,50,56,274) (34.765,40,42,271) (14.403,15,15,259) (34.457,36,36.738)

5 (14.575,16,18.052) (17.832,20,22.181) (33.513,37,37.233) (25.122,27,31.279)

220 230 240 250 260 270 280 2900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Membership Function of Makespans of Different Sequences

Time

Mem

bers

hip

Opt.Seq. 5−2−3−1−4Seq. 5−2−3−4−1Seq. 2−1−3−4−5

Figure4: Membership Functions of Makespans

12

Figure4 presentsthemembershipfunctionsof themakespansof threeprocessingsequences

for a 5-product,4-stageexample. Thesemembershipfunctionsareobtainedby solving the FSP

modelandlookingupthe ·y] valuesat thevarious3 -levels.Thedatafor thisexamplecanbefound

in Table 1. The processingtimes are specifiedas triplets correspondingto a TriangularFuzzy

Number(TFN) description.Thecomputationswereperformedwith a21-pointdiscretization.The

CPUtime for solving theFSPmodelwasunder5 secs.Table2 summarizesthekey resultsfrom

this example.

Table2: Resultsfor the ExampleProblem

Makespan(time units)Sequence

Objective value (AC) Optimistic Most Lik ely Pessimistic

5-2-3-1-4 239.809 225.59 238 258.108

5-2-3-4-1 239.967 224.734 239 258.108

2-1-3-4-5 264.961 249 263 284.845

Theoptimalsequence(� � valueof 239.809units)wasdeterminedto be’5-2-3-1-4’ with a most

likely makespanvalue of 238 units, an optimistic makespanof 225.59units and a pessimistic

makespanof 258.108units. This informationis againobtainedby simply looking up the values

of thevariables·«]�ÑÓÒ� H , ·«]�ÑÓÒ � H and ·«]�ÑÓÒ � J . Also shown in Figure4 is themembershipfunctionof

the makespanof sequence’5-2-3-4-1’: note that althoughthis sequencehasa betteroptimistic

makespanthansequence’5-2-3-1-4’, it is not optimal with respectto the� � of the makespan,

with a marginally higher� � value. Indeed,if we changetheobjective functionso thatwe want

a sequencefor optimaloptimisticmakespan,we getsequence’5-2-3-4-1’ asoptimal. If we wish

to find thesequencewith minimumpessimisticmakespan,we obtaineitherof theabove two se-

quencessinceboth of themhave the minimum pessimisticmakespanof 258.108units. Finally,

sequence’2-1-3-4-5’ is sub-optimalwith respectto all measures-� � value, optimistic, most

likely andpessimisticmakespans.Clearly, sequences’5-2-3-1-4’ and’5-2-3-4-1’ arepreferrable

over sequence’2-1-3-4-5’ on two counts: 1) in termsof the� � values,and2) in termsof the

spreadof themakespan.While themakespanof sequence’5-2-3-1-4’ hasaspreadof 32.518units

(i.e.,258.108-225.59),sequence’2-1-3-4-5’ hasa spreadof 35.845units.

5.2 Computational Results

In the following examples,all MILP modelswere formulatedusing GAMS and solved using

CPLEX 7.0 on a PentiumIII/930 MHz machinerunning Linux. Table 3 displaysthe solution

13

timesrequiredfor model(FSP)whenappliedto problemsof differing sizes. In theseexamples,

thedurationswereassumedto be givenby asymmetricTriangularFuzzyNumbers(TFNs). The

resultsarefrom severalexamples,with theTFNsrandomlygenerated.In thesemodels,a21-point

discretizationwasusedfor evaluatingtheintegral in (14). As canbeseen,thecomputationaltimes

requiredarequite small. In the worst caseabouthalf an hour wasrequiredfor oneof the 12-

product,4-stageproblems.Note that the 12-productexamplescontainasmany as48 taskswith

uncertaindurations,which would posea formidabledimensionalityproblemto a probabilistic

model.

Table 3: Computational Timesfor FlowshopProblems

CPU Time (secs)Products Stages

Min. Mean Max.

5 4 4 4.5 6

4 27 101 2008

5 160 178 237

3 8 31 12212

4 296 788 1598

5.3 Remarks - Local Search Techniques

For largerproblemswherethenumberof productsto beprocessedis quite large,say20-50prod-

ucts,solvingmodel(FSP)to optimalitycanbeprohibitive. In thesecases,theuseof a localsearch

algorithmsuchasSimulatedAnnealing(Kirkpatrick et al., 1983;AartsandKorst,1989)or Tabu

Search(Glover,1990)maybepreferrable,sincethesealgorithmscanobtaingoodqualitysolutions

within reasonabletime. However, thesealgorithmsalsohave significantdrawbacks- they do not

provideany guaranteeonthequalityof thesolutionobtained,andit is oftenimpossibleto tell how

far thecurrentsolutionis from optimality.

In this section,weoutlineresultsfrom aTabu Searchimplementationfor flowshopoptimiza-

tion. WechoseTabu Searchfor theadvantagesthatit offersoverSimulatedAnnealingandGenetic

Algorithms. Tabu Searchis themostdeterministicof thethreetechniquesandalsohasfewer tun-

ableparameters.We implementeda variantof Tabu SearchcalledReactive Tabu Search(RTS)

(Battiti andTecchiolli, 1994;Schmidt,1998)which hasthe desirablepropertythat the lengthof

the tabu list is dynamicallyadjustedduring the progressof the algorithm. The dynamicadjust-

mentof the tabu list helpsto automaticallyintensify thesearchin promisingregionsor diversify

14

thesearchin unexploredregions. RTS storesall solutions(throughtheuseof hashtables,digital

searchtreesetc.) asthey arevisitedduringits progress,sothatit is possibleto recognizewhether

a particularsolutionhasbeenencounteredbefore. The frequency with which solutionsreappear

indicateswhetherthesearchshouldbeintensifiedor diversified.RTS alsoassumesthat if a num-

ber of solutionshave beenencounteredseveral times, the searchhasbeentrappedin a basinof

attraction.Consequently, a numberof randommovesaremadeto escapefrom thecurrentregion.

Therearea few tunableparametersusedby the RTS algorithmthat relateto the thresholdfor a

”frequentlyseen”solution,thenumberof frequentlyseensolutionsbeforeperformingtheescape

sequence,thefactorsby which thetabu list sizeis alteredetc. Thereaderis referredto Battiti and

Tecchiolli (1994)for a moredetaileddescriptionof theRTSalgorithm.

For theflowshopproblem,a permutationstringof theproductsis a naturalcharacterization

of thesolution.A movewasdefinedto beany pair-wiseinterchangeof theproductsin thepermu-

tation.Thustheneighborhoodof allowablemovesfrom asolutionis of size Ôa�?® � � . Furthermore,

the computationof the makespanof a given solutioncanbe donein ÔY�?®Õp ² p � � , where�

is

thenumberof 3 -levelsat which thecalculationis performed.Theworstcasecomplexity of one

RTS iterationis determinedby the evaluationof themakespans(an ÔY�?®Öp ² p � � operation)of

all the ÔY�?® � � neighbors.Thus,the complexity of oneRTS iterationis ÔY�?®Ø×¨p ² p � � . Figure

5 shows theCPUtime requiredfor 1 RTS iterationasa functionof theproblemsize(numberof

products)andthenumberof 3 -levelsat which thecomputationis performed.TheCPUtime for

onemove whenthemakespanevaluationsarecarriedout at 101 3 -levels for 3-stageproblemsis

givenby (37),where® denotesthenumberof products.This relationwasobtainedby regression.

TheCPUtime reportedis for a Java (Arnold andGosling,1998)implementationon a PentiumIII

machinerunningLinux.

] � � � � Æ@�£Ù*ÚÜÛepÜA'0 o Ò ® × q>ÙS�£�*Ù¦Ù\pÜA'0 o × ® � QÈ0@�£0¦Ý*ÆSA7�*®~q=0@�£Æ¦0@A�Ù¦Þ (37)

Figure6 displaystheinitial progressof theRTSfor a20-product,4-stageproblemwith sequence-

dependentset-uptimeson all stages.Theevaluationof a solutionwasperformedwith a 21-point

discretization.In the initial stages,the algorithmbehavesmuchlike a steepestdescentheuristic

wherethereis a rapiddecreasein theobjective function. Surprisinglyenough,evenin this early

stageof the algorithm,the escapesequences(sequencesof randommoveswhich seekto escape

theregionof acomplex attractorandtakethesearchinto anew, unexploredregion)areperformed,

as indicatedby the spikes in Figure6. The searchthen proceedsto locatethe local minimum

in the new searcharea. Figure7 displaysthe long-termprogressof the RTS over the courseof

5000iterations.Onecanclearlyobserve thetwo differenttime-scalesin thesearch.Thesolution

variesrapidlyoverashorttimehorizon,andtheescapesequencesmovethesearchinto unexplored

15

0

1

2

3

4

5

6

5 10 15 20 25 30

CP

U T

ime

(sec

s)ß

Number of Tasks

CPU Time for 1 RTS Iteration as a Function of Problem Size

101 Alpha-levels51 Alpha-levels21 Alpha-levels

Figure5: RTS Iteration Time

regionsoverthelongertimehorizons.Thesearchwasterminatedat theendof 5000iterationsand

the bestsolutionwasfound at iteration863,with an objective valueof 303.005.The total CPU

time requiredfor this problemwas around1400 secs,thus averagingabout0.28 secsper RTS

iteration.

An indication of the quality of the solution can be obtainedby solving the corresponding

FSPmodelto optimality. However, whenwe attemptedto solve the21-pointFSPmodelfor this

particularproblemwith GAMS/CPLEX, we could not even find an integer feasiblesolution in

1400seconds.TheLP relaxationof theobjective functionfor this problemwas280.3036,which

indicatesthat theRTS hasdonequitewell, comingwithin 8% of thebestpossiblesolutionin the

sametime (1400secs).Thearbitraryterminationcriteria for theRTS implies that theCPUtime

canbe adjustedasdesired. Furthermore,even after 50,000CPU secs,CPLEX could not solve

the21-pointFSPmodelto optimality - thebestsolutionobtainedwas295.3863andthegapwas

4.5%(thebestnodehada valueof 282.0548).Thus,wecannotethatsolvingtheseMILP models

to optimality canbecomputationallyexpensive. Consequently, theuseof local searchalgorithms

suchasRTSthatyieldnear-optimalsolutionsin significantlysmallercomputationaltimesis clearly

advantageous.

16

300

310

320

330

340

350

360

370

380

390

50 100 150 200 250

Obj

ectiv

e F

unct

ion

Iteration

Solutions Found Over RTS Progress

Current SolutionBest Solution so far

Figure 6: Initial Progressof RTS

300

310

320

330

340

350

360

370

380

390

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Obj

ectiv

e F

unct

ion

Iteration



Figure7: Long-term Progressof RTS

6 Multistage Flowshopwith Parallel Units

6.1 MILP Model

In thissection,weaddresstheproblemof optimizingtheschedulingof multistageflowshopplants

which have parallelunits in severalstages(seeFig. 8) - anextensionof two classicalscheduling

17

Stage 1 Stage 2 Stage 3

O rders

Figure 8: Multistage Flowshopwith Parallel Units

problems(flowshopandparallelmachinesproblems).Givena productionhorizonwith duedates

for thedemandsof severalproducts,thekey decisionsthatareto bemadearetheassignmentof

theproductsto theunitsaswell asthesequencingof productsthatareassignedto thesameunit.

Eachproductis to beprocessedonly onceby exactly oneunit of every stagethat it goesthrough.

Theobjective is usuallyminimizing themakespanor thetotal latenessof theorders.Wepresenta

MILP modelfor obtainingtheschedulewith minimumtotal lateness,whentheprocessingtimes

of theordersaregivenby fuzzynumbers.Theassumptionsinvolvedin themodelare:

1. Theprocessingtimesof thevariousproductsin theunitsaregivenby fuzzynumbers.

2. Duedatesaredeterministic.

3. Transitiontimesaredeterministicandonly equipmentdependent.

4. Thereis unlimitedintermediatestoragebetweenstages.

Assumption1 seemsreasonable,especiallywhenonly estimatesof theprocessingtimescan

be obtained,for instance,when new productsare being processedfor the first time. Relaxing

assumption2 will changethe natureof the problem. Transitiontimescanalsobe madefuzzy,

without muchproblem- however, introducingsequencedependenttransitionswill considerably

complicatethemodel.

Much work hasbeendoneon deterministicversionsof this problemof schedulingflowshop

plantswith parallelunits.PintoandGrossmann(1995)presentedacontinuous-timeMILP formu-

lation for minimizing the total weightedearlinessandtardinessof the orders. This formulation

was a slot-basedmodel which allocatedunits and orderson two parallel time coordinatesand

matchedthemwith tetra-indexed(stage-order-slot-unit) variables.A later paperby thesameau-

thors(PintoandGrossmann,1996)incorporatedpre-orderingconstraintsin themodelto replace

the tetra-indexed variablesby tri-indexed variables(order-stage-unit)and thus reducethe com-

putationaleffort requiredto solve themodel. McDonaldandKarimi (1997)developedan MILP

formulationfor theshort-termschedulingof single-stagemulti-productbatchplantswith parallel

semi-continuousprocessingunits.Morerecently, Hui et al. (2000)presentedanMILP formulation

whichusestri-indexedvariables(order-order-stage)to minimizethetotalearlinessandtardinessof

theorders.With theeliminationof theunit index, theformulationrequiredfewer binaryvariables

18

andreducedcomputationaltime.

For ourproblemwheretheprocessingtimesaregivenby fuzzynumbers,wepresentanMILP

modelwhich is a generalizationsof thetri-indexedmodel(Hui et al., 2000). We alsoformulated

an MILP model(M1) usingthe slot-basedapproachof Pinto andGrossmann(1995),but dueto

spacerestrictions,werestrictourpresentationto thetri-indexedmodel(M2). As before,thetiming

computationsareperformedfor different 3 -levelsandtheobjective is a discreteapproximationof

theareacompensationintegralappliedto thetotal lateness.

6.2 Tri-indexed model - (M2)

Givenareproducts(orders)� ,± andprocessingstages

Ï !à whichhaveprocessingunits ´a $µ¦á .Eachproduct

�hasto be processedon stagesà Â , and can be processedonly on units µ Â . The

parametersof interestaretheprocessingtimes » Â ¹³À^É , transitiontimes ��â<¹ andduedatesV Â .Themodelpresentedin Hui et al. (2000)usestri-indexedbinaryvariables� Âäã º Âæå º á to indicateif

product� � is processedbeforeproduct

� � in stageÏ. Binaries% Â ¹ areusedto representtheassignment

of product�

to the first processingpositionin unit ´ , while continuousvariablesç¨èÂ ¹ areusedto

representassignmentof product�to unit ´ . Herewepresentthemodelfor makespanminimization.

Continuousvariables] % Â áéÀ³É and ]³Ë Â áéÀ³É respectively denotethe startandendtimesof product�

in

stageÏat 3 -level

�.

(M2)tlêìë O Ä Å � ��:��¿*ÆÜ�+f/0S�£��fíÇ À ��/Àhfe� � % À H Q � % À J � (38)

s.t. ç èÂ ¹ Q Ç¹�îðï ¡æñ^òäódñ^ôæ¤ ç èÂ î ¹ î Q=� Â�Â î áb� ª ; � � � è � �öõ�9� è �z´� $µ ÂS÷ µ èÂ ÷ µÜá�� Ï ,à (39)

ÇÂ î¥ïCø ô º Â î¬ùú Â �Â�Â î áb� A ; � ,±I� Ï !à (40)

ÇÂ î ï'ø ô º Â î ùú Â �Â�Â î á)Q Ç¹yï ¡ðñ^òðódñ^ôæ¤ % Â ¹ � A ; � ,±I� Ï !à (41)

ç èÂ ¹ 8 % Â ¹ ; � ,±I�z´l $µ Â (42)

Ç Â ï'ø % Â ¹ � A ;S´a $µ Â (43)

Ç¹«ï ¡æñ^ò¥ódñ^ô�¤ ç Â ¹ � A ; � ,±I� Ï !à (44)

] % Â á°À³É�Q Ç¹«ï ¡æñ^ò¥ódñ^ôæ¤ ç èÂ ¹ p2��» Â ¹^À³É�QÈ�`â<¹T�� ] % Â á î À^É ; Ï è � Ï ,àg� Ï è . Ï � � ,±I� � �«Ë (45)

] % Â á°À³É�Q Ç¹«ï ¡æñ^ò¥ódñ ô ¤ ç èÂ ¹ p2��» Â ¹^À³É�QÈ�`â<¹T�� ³Agq{� Â�Â î£áä� p-âÈQ>] % Â î£á°À³É; � � � è "±K� �¨õ�9� è � Ï !àí� � �«Ë (46)

] % Â á ò À³É Q Ç¹«ï ¡æñ^ò¥ódñ^ôæ¤ ç èÂ ¹ p2��» Â ¹^À³É�QÈ�`â<¹T�� % À³É ; � "±K� � ��Ë (47)

19

ç èÂ ¹ 8 0 ; � �z´a {µ Â (48)

ç èÂ ¹ � A ; � �z´a {µ Â (49)

� Â�Â î áû � 0S�'A*# ; � � � è "±K� �¨õ�9� è � Ï �à Â (50)% Â ¹ � 0S�'A*# ; � "±K�Ó´l $µ Â (51)

] % Â á°À³É 8 0 ; � � Ï � � �«Ë (52)

(53)

Constraints(39) specifythat if orders�

and� è areconsecutive ordersandorder

�is assigned

to unit ´ , thenorder� è is alsoassignedto unit ´ . Constraints(40) and(41) ensurethateachorder�

hasat mostonesuccessorandonepredecessorrespectively in eachstageof processing,while

(42) specifiestherelationbetweentheassignmentand % variable.Note thatwith constraints(39)

and(42), the assignmentvariablescanbe relaxed ascontinuousvariableswith 0 and1 aslower

andupperbounds(for proof, pleaserefer to Hui et al., 2000). Constraints(43) statethat each

unit processesat mostonestartingorderand(44) statethat every orderhasto be processedby

a uniqueunit in eachstage.Constraints(45), (46) and(47) govern therelationshipsbetweenthe

startingtimesof anorderin successive stages,thestartingtimesof successive ordersin thesame

stageandthecomputationof themakespanrespectively. Theobjective(38) is to minimizethearea

compensationof themakespan- this ensuresthat themakespanof thescheduleis the largestend

timeof all orderson thefinal stage.

6.3 Computational Results

Unlike thecaseof theflowshopproblemdiscussedin Section5, theMILP modelsfor themulti-

stageflowshopwith parallelunitsdo not exhibit goodcomputationalperformanceastheproblem

sizeincreases(seeTable4). This is primarily dueto thelargenumberof binaryvariablesthatare

requiredfor modelingthis problem. Although thenumberof binaryvariablescanbereducedby

takingin to accounttheforbiddenassignmentsandpostulatinganappropriatenumberof slotsfor

the variousunits, it increasesdramaticallywith problemsizeascanbe seenfrom Table4. The

solutiontimesreportedarefor makespanminimization.

Model (M2) wasa betterformulationthan(M1), ascanbeseenfrom thereducedcomputa-

tional timesrequiredfor thesolution(exceptfor the10-product,15-unitsproblem).Both models

hadthe sameLP relaxation. Solutionof model (M1) for the 10- and12-productproblemswas

terminatedafter50,000CPUsecs.Model (M2) for the10-product(25 units)problemwassolved

in about4000CPUsecs,whereasfor model(M1), theoptimalitygapwas11.7%evenafter50,000

CPUsecs.For the12-productproblem,bothmodelscouldnotbesolvedto optimality, althoughfor

20

(M2) theoptimality gapwassmallerthanfor (M1) after50,000CPUsecs.Theoptimality gapfor

the10-and12-productproblemswith 25unitsis smallerthanthatfor the10-productproblemwith

15unitsmainlybecausethe25-unitproblemshadmany assignmentsthatwereforbidden,resulting

in morestructurefor thesolution.Eventhen,theseMILP modelsrequireexcessivecomputational

resources,ascanbeseenfrom theCPUtimes.

Table 4: Model Characteristics for Parallel FlowshopProblems

Binaries CPU secs Optimality Gap (%)N-L-(Units in Stgs)

M1 M2 M1 M2 M1 M2

5-5-(6,3,10,3,3) 241 225 15 18 0 0

10-5-(6,3,10,3,3) 721 700 . 50,000 4086 11.7 0

10-5-(3,2,3,4,3) 730 700 . 50,000 . 50,000 53.5 56.95

12-5-(6,3,10,3,3) 1000 960 . 50,000 . 50,000 14.5 6.5

The latenessminimizationMILP modelsalso exhibit similar computationalperformances.

Recall that, for the earlinessminimization problem,Pinto and Grossmann(1995) developeda

decompositionstrategy which determinesfeasibleassignmentsthatminimizein-processtime and

thenfurtherdeterminesa minimum-earlinessschedulethateliminatesunneccesarryset-ups.They

also usedpre-orderingconstraintsto obtain near-optimal solutions. However, an extensionof

theseideasto thecaseof uncertainprocessingtimesis not straightforward. Thus,a local search

techniquesuchastheRTS is particularlyrelevantfor solvinglargerinstancesof theseproblems.

6.4 RTS Implementation

In thissection,wediscusssometheissuesthatarecritical to thesuccessof theRTSimplementation

for this problem.

6.4.1 Solution Characterization and Evaluation

A solutionfor themultistageflowshopwith parallelunits is characterizedby a feasibleallocation

andsequencingof all the productsto differentprocessingunits in all the stages.Onepossible

representationof asolutionis asetof ü áðï H �Üµ H � strings,with eachstringdenotingthesequencing

of theproductsallocatedto theunit. Of course,theallocationof theproductsto thevariousunits

mustsatisfytheconstraintsthatforbid certainallocationsaswell as(39)-(44).Figure9 illustrates

onepossiblesolutionfor a problemwhere4 productsareto be processedin 2 stages,wherethe

21

first stagehas2 unitsin parallelandthesecondstagehasonly 1 unit. Thus,productA is processed

first on unit 1 in stage1 andis followedby B, while C andD areprocessedon unit 2 in stage1.

Theprocessingsequenceon theunit in stage2 is C-D-B-A.

A B

C D

Stage 1

C D

Stage 2

B A

Un i t 1

Un i t 2

Un i t 1

Figure9: Solution Representation- Example

Oncewe have a feasibleallocationandsequencingof the ordersin variousunits, the start

andendtimesof theproductsin thestagescanbeevaluatedwith expressionssimilar to (45) and

(46). Themakespanmaybecalculatedasin (47) andtheareacompensationappliedto thefuzzy

makespanto representtheobjectivevalue.Themakespanof a givensolutionmaybecomputedinÔY�|®~p ² p � � time.

6.4.2 Neighborhoodof a Solution

We considerinsertionmoves,wherewe selecta product ( � on a unit in stageÏ � andinsert it in

anotherpositionon the sameunit or any otherunit on the samestage. Clearly, thenthe neigh-

borhooddefinedby the insertionmove is of size Ôa�?® � p ² � , sinceÏ � canbechosenin M ways,( � canbe chosenin ® ways,andthe positionof insertioncanbe chosenin ÔY�|®,� ways. Since

thecomputationof themakespanof a givensolutiontakes ÔY�|®ûp ² p � � time, theevaluationof

thecompleteinsertionneighborhoodtakes ÔY�|®6×íp ² � p � � time. Figures10 and11 displaytwo

possibleinsertionmovesfrom thesolutionin Figure9 for the4-product2-stageexamplediscussed

earlier. Figure10 shows productA beingremovedfrom Unit 1 on Stage1 andinsertedaheadof

productsC andD onUnit 2 in thesamestage,while Figure11showstheinsertionof productD at

theendof theprocessingsequenceon Unit 1 in Stage2.

The computationalcomplexity of evaluatingthe completeinsertionneighborhoodwascon-

firmedby testson anumberof problems,with theresultthatevaluatingtheentireinsertionneigh-

borhoodwasdeemedtoo time-consuming.As a result,we useda reducedneighborhoodbasedon

theinsertionmove. Thegainsin computationalefficiency for thereducedneighborhoodin compar-

isonwith theentireneighborhoodarequitesignificant.For instance,theCPUtimefor oneiteration

of a25-product,5-stage,15-machineproblemwith a full evaluationof theinsertionneighborhood

22

Stage 1

A B

C D

Un i t 1

Un i t 2

Figure10: Insertion Move - Different Units

C D

Stage 2

B A

Un i t 1

Figure 11: Insertion Move - SameUnit

was4.9secs,while for thereducedneighborhoodimplementation,theCPUtime for oneiteration

was0.50secs.Pleasereferto AppendixA for adiscussionof thereducedneighborhood.

6.4.3 Computational Results

To seehow well theRTSalgorithmperformsonthemultistageparallelflowshopproblems,the10-

product,5-stage,15-unitsproblemsolvedin Section6.3wassolvedwith alimit of 50000iterations

(about2800CPUsecs).

Recallthat for model(M1), evenafter50,000CPUsecsof solutiontime, therewasanopti-

mality gapof over 50%(i.e., thebestsolutionpossiblewas113.38andthebestsolutionobtained

was250.086),while for (M2), the bestsolutionobtainedhadan objective valueof 280.932. It

is likely that the large numberof binaryvariablesandtheBig-M constraintsin theMILP model

resultin therebeingsuchpoorLP relaxations:thus,thebestsolutionpossiblemayhave a signifi-

cantlylargerobjectivevaluethanthatindicatedwhenthebranchandboundalgorithmterminated.

Indeed,fixing theorderingof a subsetof productson evenoneunit leadsto muchbetterbounds

on theobjective.

Using theRTS algorithm,thebestsolution(obtainedin 2800CPUsecs)hada makespanof

182.878. Note that the RTS algorithmobtaineda muchbettersolutionin a fraction of the time

requiredby the mathematicalprogrammingapproach.The key observation that we can derive

from this exampleis that local-searchtechniquessuchasthe RTS algorithmarewell suitedfor

solving theparallelflowshopproblemssincethey areableto provide goodquality solutionswith

reducedcomputationalrequirements.However, MILP modelsarenotcompletelyuselessfor these

23

problems,sincethey canbe usedto derive boundinginformationthat indicatethe quality of the

solutionsobtainedusingthelocal searchtechniques.

300

400

500

600

700

800

900

1000

0 5000 10000 15000 20000 25000

Obj

ectiv

e F

unct

ion

Iteration



Figure12: RTS Progress

Another advantagewith techniquessuchas RTS is that they usually scalequite well with

problemsize.Often,theiterationcomplexity grows only polynomiallywith problemsize.Figure

12displaystheprogressof anRTSimplementationfor a25-product,5-stage,15-machineproblem.

TheRTSalgorithmusedthereducedneighborhoodof theinsertionmove,asdescribedearlier;the

CPUtimefor 25,000iterationwas11,600secs,thusaveragingunder0.5secsperiteration.Clearly

then,our RTS implementationshows promisefor tacklingevenbiggerproblems.Thestartingso-

lution which wasobtainedby a randombalancedallocationof the jobsto thevariousunits in the

stageshadan objective valueof 776.197units. The bestsolution(shown below) obtainedhada

valueof 326.236unitsandwasobtainedat iteration2408.Thus,onunit 1 in stage1, productU�ý is

processedfirst, andis followedby productsU × , U �|þ etc. It is instructiveto notefrom Figure12that

thissolutionwasobtainedsoonafteroneof therandomescapeswasperformed,therebyvalidating

theutility of theescapesequencesin theRTS algorithm.

BestSolution obtained fr om the RTS algorithm

24

Stage1

Unit 1: U�ý`ÿ U × ÿ U �|þ ÿ U � × ÿ U�Ñ`ÿ U � ýíÿ U �� ÿ U � ÿ U��ÓÒ/ÿ U � ÒUnit 2: U��ÿ U �z� ÿ U � ÿ U � �/ÿ U��`ÿ U�� Unit 3: U � ÿ U � �`ÿ U � � ÿ U+�z�/ÿ U þ ÿ U+� × ÿ U+Ò/ÿ U+� � ÿ U � ÑStage2

Unit 1: U��ÿ U × ÿ U �z� ÿ U � � ÿ U þ ÿ U � × ÿ U��z�`ÿ U+� × ÿ U�� ÿ U�Òíÿ U � Ñ`ÿ U � ÒUnit 2: U�ý`ÿ U � ÿ U � ��ÿ U �|þ ÿ U � �`ÿ U � ÿ U�Ñ`ÿ U � ýíÿ U � ÿ U��ÓÒ/ÿ U�� ÿ U �� ÿ U��Stage3

Unit 1: U��ÿ U × ÿ U �z� ÿ U � ÿ U � � ÿ U � ýíÿ U � ÿ U+Ò/ÿ U � ÒUnit 2: U �|þ ÿ U þ ÿ U�Ñ`ÿ U+�ÓÒíÿ U ��Unit 3: U�ý`ÿ U � ÿ U � ��ÿ U � �/ÿ U � × ÿ U+� × ÿ U�� ÿ U+�z�/ÿ U � Ñ/ÿ U+� � ÿ U��Stage4

Unit 1: U � ÿ U+Ñ/ÿ U�� ÿ U � ÿ U+Ò/ÿ U � ÒUnit 2: U � �`ÿ U+�ÓÒUnit 3: U × ÿ U �|þ ÿ U � �/ÿ U � × ÿ U�� × ÿ U+� � ÿ U��Unit 4: U�ý`ÿ U � ÿ U��ÿ U þ ÿ U �z� ÿ U � � ÿ U � ý`ÿ U��z�íÿ U � Ñ`ÿ U ��Stage5

Unit 1: U��ÿ U � ÿ U � �`ÿ U+� � ÿ U��z�`ÿ U+� � ÿ U��Unit 2: U �z� ÿ U � × ÿ U � � ÿ U � ý/ÿ U � ÿ U �� ÿ U � ÑUnit 3: U�ý`ÿ U � ÿ U � ��ÿ U �|þ ÿ U × ÿ U þ ÿ U�Ñ`ÿ U�� × ÿ U��ÓÒ/ÿ U+Ò/ÿ U � Ò

Figure13 displaysthe variationof the sizeof the tabu list over the courseof the RTS. The

tabu list sizetypically oscillatesbetween5 and60asthesearchvariesbetweenintensificationand

diversification.

Althoughwehaveonly illustratedtheRTSimplentationfor themakespanminimizationprob-

lem,extendingit to latenessminimizationis a straightforwardtask,requiringonly minor changes

in thecomputationsrequiredat eachiteration.

7 The New Product DevelopmentProblem

In theNPD process,thepotentialproductmustpassa seriesof teststhatassessits safety, efficacy

andenvironmentalimpact.Thecosts,durationsandprobabilitiesof successof varioustestingtasks

areoftennot known with certaintywhentheschedulehasto beconstructed.Theoptimizationof

thetestingschedulecanbeconceivedasintroducingprecedenceconstraintsbetweenthetests,in

additionto thegivensetof technologicalprecedenceconstraints.Thereis aninherentcost-income

trade-off associatedwith theschedulingof theNPD process.This trade-off is betweenthegreater

25

0

10

20

30

40

50

60

70

0 5000 10000 15000 20000 25000

Tab

u Li

st L

engt

h

�

Iteration

Length of Tabu List Over RTS Progress

Figure13: Tabu List Size

incomeresultingfrom a shorterschedule(wheremany of thetestsareperformedin parallel)and

thelowerexpectedvalueof thetotal costfrom a longersequentialschedule.

Weconsidertheproblemof optimizingthescheduleof testingtaskswhile takinginto account

theuncertaintiesin thedurationsandsuccessof thetasks,a problemwhich hasnot beenreported

before. In previous work, SchmidtandGrossmann(1996)developedan MILP modelassuming

knowndurations.Theseauthorsalsoconsideredtheevaluationof thedistributionof thecompletion

time for a fixedschedule,givenuncertaintiesin durations(SchmidtandGrossmann,2000). The

extensionof their methodto optimizationis non-trivial, given the complexity of their procedure

for evaluatingthecompletiontimewith uncertaindurations.

To effectively addresstheproblem,we usea two-level descriptionof theuncertaintiesinher-

ent in theNPD process.We retaintheprobabilitydescriptionof thesuccessof the tests,i.e., for

every testthereis a givenprobability that theproductwill passthe test. However, insteadof as-

sumingthediscrete/continuousprobabilitydistributionsthatSchmidtandGrossmann(2000)used

to modeltheuncertaintyin testdurations,we draw uponconceptsof fuzzy settheoryto represent

the imprecisionanduncertaintyof information in the time parametersof the variousactivities.

Thus,theprocessingtimesarerepresentedby intervalsor fuzzy numbersinsteadof deterministic

26

valuesor probabilitydistributions.For agivenschedule,thecompletiontime is computedthrough

interval arithmeticby the extensionprinciple. This computationcanbe performedefficiently, in

contrastto thecasewhenweuseaprobabilisticdescriptionof thetestingdurations.Theexpected

valueof thecostof ascheduleis evaluatedusingtheprobabilisticframework, andassociatedwith

this successfulscheduleis a fuzzy descriptionof theproject’s completiontime. This completion

time is thentranslatedinto anequivalentincomedescription,therebycompletingtheevaluationof

agivenschedulein termsof theexpectedcostandthecompletiontime (income).

The optimizationof the scheduleconsidersas the objective the differenceof the expected

valueof thefuzzy incomeandtheexpectedcost.Formulation(PDP)representsaMILP modelfor

selectingtheoptimalscheduleundertaskuncertainty. It is assumedthatthereaderis familiarwith

thedeterministicversionof theproblem(SchmidtandGrossmann,1996).

(PDP) Max ( Ð��*Z � ] � ² ±5q Ç Â � Â p Ç � Ë ò� p Â �

q5��:��¿*ÆÜ� p2��A7¿¦ª¦�+p Ç Ä ±2VPÄ=p Ç À ��/À/p2� & Ä�À H Q & Ä À J � (54)

] % Â À³É�Q>» Â À³É � ·«]�À º É ; � � � ��Ë (55)

¯¨Ä�Q & Ä�À³É 8 ·«]�À º É ; � � � �«Ë (56)

] % Â À³É�Q>» Â À³É � ] % ¹^À³É�Q�â Â À^É�p)��Agq MdÂ ¹�� ;w� � �z´2�7� �öõ� ´nÍ³; � �«Ë (57)MdÂ ¹ � A ;w� � �z´2�/ !U� s � (58)

Ç � �¨Â � p Â � � Ç¹ � ¡ ¹Cùú Â ¤ Ï�� ?U<¹T� p M ¹ Â ; � (59)

Ç � Â � � A ; � (60)M-Â ¹�Q M ¹ Â � A ;w� � ´2�7� � �{´ (61)M-Â ¹wQ M ¹³á¦Q M á Â � ª ;w� � �z´¦� Ï ��ð� � �{´l� Ï � (62)M ¹ Â Q MdÂ á¦Q M á ¹ � ª ;w� � �z´¦� Ï ��ð� � �{´l� Ï � (63)

·«]�À³É 8 0 ; � ��Ë (64)& Ä�À³É 8 0 ; � � � �«Ë (65)

] % Â À³É 8 0 ; � � � �«Ë (66)

0�� Â � �ÎA ; � � � (67)MdÂ ¹ � 0S�CA*# ; � �Ó´ (68)

In formulation(PDP),theprecedencerelationshipbetweentwo tasks�and ´ is modeledwith

the help of the binary variableMdÂ ¹ . Thus

M-Â ¹ � A if task�

precedestask ´ , elseMdÂ ¹ � 0 . In

(PDP),(54)representstheobjectivefunction,whichis calculatedasthedifferenceof themaximum

possibleincome,² ± , andthecostof thescheduleandthe total decreasein incomedueto delay

27

in productrelease.Sincethe durationsof the varioustestingtasksaregiven by fuzzy numbers,

theschedulecompletiontimeandthedecreasein incomearealsofuzzynumbers.Themeanvalue

of thetotal decreasein incomeascalculatedby theareacompensationintegral is includedaspart

of theobjective function. The �� s denotetheSimpsoncoefficientsusedin theapproximationof

the integral; ¯öÄ denotethe points in time whenthe incomedecreaserate ±2V�Ä changesand U�¹indicatestheprobabilityof successof task ´ . Constraints(55) forcesthecompletiontime, ·y] , (at

every 3 -level)of thescheduleto begreaterthanthecompletiontimeof eachtask(atevery 3 -level),

while (56)aid in thecalculationof theexcesstimeover ¯¨Ä . Thetiming relationshipbetweentwo

tasks�

and ´ is modeledusingthe Big-M formulationin (57), which specifiesthat either task ´startsafter

�is complete,or task ´ startsbefore

�, or the relationshipbetweenthe two tasksis

unspecified.Constraints(58) fix thetechnologicalprecedencerelationships(givenby set U� s � )

betweencertaintasks�

and ´ , while (59) and(60) approximatetheexponentialin calculatingthe

cost term. Finally, (61)-(63)are logic cutsandvalid inequalitiesthat eliminatedirectedcycles

of length2 or 3, that strengthenthe relaxationof the MILP. We now presenttwo examplesthat

illustratetheperformanceof model(PDP).

7.1 The 9-TaskExample

Thedatafor this examplearetakenfrom SchmidtandGrossmann(1996)andaremodifiedto re-

flect theuncertaintyin testingdurations.In particular, the testingsuccessprobabilitiesandmost

likely valuesof thedurationsarethesameasthosein theabovepaper. However, thedurationsare

assumedto be given by asymmetricTFNs, with a left spreadof 5% anda right spreadof 20%.

Figure14 displaysthe technologicalprecedencesfor this problem,while Figure15 displaysthe

optimal schedulewhenthe computationsareperformedwith 20 intervals (21 points). Computa-

tionalresultsfrom two discretizationsareshown in Table5. Bothmodelsprovidethesameoptimal

solution,althoughthe21-pt discretizationmodelprovidesa betterestimateof themeanvalueof

theprofit. Thenumberof equationsandcontinuousvariablesin the21-ptmodelis 160%and85%

more than the 7-pt modeland the solution time is an orderof magnitudelarger. However, the

improvementin the estimationof the incomeis lessthan1%, which doesnot really warrantthe

orderof magnitudeincreasein thecomputationaltime.

7.2 The 65-TaskExample

The task datafor this exampleare taken from Schmidt (1998) and are modified to reflect the

uncertaintyin testingdurations. The durationsare assumedto be given by asymmetricTFNs,

with a left spreadof 5% anda right-spreadof 20%andwith thecentralvalueat thedata-pointin

28

H

G

E

Figure 14: Technological Prece-

dences

D

IG F

C

EB

H

A

Figure15: Optimal Schedule

Table5: Computational Resultsfor the 9-Task Example

Model Characteristics 7-pt Model 21-pt Model

Equations 1402 3754

Variables 492 856

Binaries 70 70

CPUsecs 601 7611

ScheduleDuration(asTFN) (19,20,24) (19,20,24)

ScheduleCost($1000s) 1033.6445 1033.6445

IncomeLoss(Mean)($1000s) 1156.9444 1120

Profit ($1000s) 3809.4110 3846.355

Schmidt(1998). Themaximumincomeis $25million andthe incomedecreaseis over a period

of 72 months.Figure16 displaysthetechnologicalprecedencesbetweenthevarioustasks,while

Figure17 shows the optimal scheduleobtainedby solving the MILP with a 7-pt discretization.

Theothermodelswith finerdiscretizationprovidedthesameoptimalsolution,althoughwith better

estimatesof theobjective function. Thesemodelssolve comparatively fasterthanthesmaller9-

taskexamplebecausetherearea numberof technologicalprecedenceconstraints(seeFigure16),

which leadsto morestructurein theschedule.Thenumberof equationsandcontinuousvariables

in the21-ptmodelis 80%and70%morethanthe7-pt modelandthesolutiontime is almostan

orderof magnitudemore.Theonly benefitof usingthe21-ptmodelis a moreaccurateestimation

of theduration,income(2%higherthanthe7-ptmodel).Similarly, the51-ptdiscretizationmodel

provideslessthana1%improvementin theestimationof theincomelosswhencomparedwith the

21-ptmodel,althoughthesolutiontimehasincreasedby a factorof 3.

29

29

32

28

31 30

27

59

62

58

26

57

61

25 24

56

65

23

55

22

54

53

63

21 20

52

60

5150

9

13

8

12

19

18

7

49

17

6

16

5

4847

4

15 14

3

46

2

45

1

11

10

44

4342

414039 38

64

37 36

35

34

33

Figure 16: Technological Prece-

dences

29

32

28

31

27

59

58

56 55

38 35

62

26

30

57

61

25

24

65

23

54

5343

22

63

21

20

52

60

51

50

47

40

9

13

19

8

12 7

18 17

6

49

16

5

48

15

4

11

10

14

3

46

2

45

1

44

42

41 39

64

3736

34

33

Figure17: Optimal Schedule

Table6: Computational Resultsfor the 65-Task Example

Model Characteristics 7-pt Model 21-pt Model 51-pt Model

Equations 148780 267192 520932

Variables 6834 8794 12994

Binaries 4061 4061 4061

CPUsecs 324 2786 8909

ScheduleDurn (TFN) (58.11,61.17,73.4) (58.11,61.17,73.4) (58.11,61.17,73.4)

ScheduleCost($1000s) 3147.2161 3147.2161 3147.2161

IncomeLoss(Mean)($1000s) 8487.9514 8194.8833 8119

Profit ($1000s) 13364.8326 13657.9006 13732.86

30

8 Conclusions

In this paperwe have applieda non-probabilisticapproachto the treatmentof processingtime

uncertaintyin two problems- theschedulingof (i) flowshopplantsand(ii) thenew productdevel-

opmentprocess.Thebenefitsof usingthenon-probabilisticapproachareasfollows:

1. Conceptssuchas fuzzy sets,interval arithmeticallow us to modeluncertaintyin cases

wherehistorical(probabilistic)dataarenot readilyavailable. Thesemodelsarefairly flexible in

their descriptionof uncertainty- for example,differenttypesof functionscanbe usedto model

theuncertaintyin theparameters.Herewe have usedonly TriangularFuzzyNumbers(TFNs) to

capturethe uncertaintyin the taskdurations.An advantageof this representationis the easeof

interpretationof the results,for instance,in termsof the most likely, optimistic andpessimistic

estimatesof the makespan,althoughan interpretationof the objective function value is not as

straightforwardasin theprobabilisticcase(for e.g.,in termsof theexpectedvalue).

2. Themostsignificantgainis thecomputationtime requiredto solve theoptimizationmod-

elsunderuncertainty. Thesemodelsneithersuffer from thecombinatorialexplosionof scenarios

thatdiscreteprobabilisticuncertaintyrepresentationsexhibit, nor do they requirecomplicatedin-

tegrationschemeswhichcontinuousprobabilisticmodelsrequire.TheMILP modelsdevelopedin

this work could be solved with reasonablecomputationaleffort for problemswherethe solution

wasmorestructured.Thesemodelsdid not performaswell whenwe attemptedto solve general

problemsto optimality.

3. Anothersignificantadvantageof thefuzzy-setapproach(over theprobabilisticapproach)

is thatheuristicsearchalgorithmsfor combinatorialoptimizationsuchastabu searchcanbeeasily

appliedto obtaingoodqualitysolutionsin reasonablecomputingtime. This is primarily dueto the

easein computingtheobjective functionof asolution.

4. The examplesconsideredshow that very goodestimatesof the uncertainmakespanand

incomecanbeobtainedby usingfairly coarsediscretizationsandthatthesemodelscanbesolved

with little computationaleffort. In theseexamplestheimprovementin theestimationof thecom-

pletion time by usinga denserdiscretizationwasnot significantenoughto warrantthe orderof

magnitudeincreasein computationtime required.

Appendix A - ReducedNeighborhood

The ideaof usinga reducedneighborhoodto improve the computationalcharacteristicsof local

searchalgorithmsis notnew. For instance,vanLaarhovenet al. (1992)usedtheconceptof critical

arcs in jobshopschedulingproblemsfor eliminatingmovesthat will definitely not improve the

31

solution. This idea was then extendedto the fuzzy jobshopschedulingproblemby Fortemps

(1997). A recentpaperby Nowicki andSmutnicki(1998)alsoemploys a specificneighborhood

definitionbasedon critical pathsfor flow shopswith parallelmachinesin a deterministicsetting.

With this reducedneighborhoodandfurtheradvancedimplementationsof tabu list queryingand

searching,theauthorswereableto dramaticallyimprovethespeedof their tabu searchalgorithm.

In our implementation,we build uponthe ideasin both Fortemps(1997)andNowicki and

Smutnicki(1998)to defineour reducedneighborhood.Thus,for a givensolution,we identify the

critical path(s)andconsideronly theinsertionof jobsthat lie on thecritical path(s).A move that

seeksto inserta job that doesnot lie on any of the critical pathsis considereduseless,in that it

will not improve the makespan.The main differencebetweenthe deterministicversionandour

caseis thattheremaybemany morecritical jobs,sincedifferentpathsmaybecritical at different3 -levels. Theidentificationof thecritical path(s)for a solutionis a straightforwardoperationand

doesnotcontributesignificantlyto thecomputationalcomplexity of aniteration.

However, we have to accountfor thememoryrequiredfor storingtheadditionalinformation

aboutthe critical jobs with every solution. If we storeinformationaboutall critical pathsat all3 -levels,thememoryrequiredbecomesexcessiveveryquickly. Therefore,in our implementation

wehavestoredthepredecessorinformationto beusedfor computingthecritical pathfrom only the3 � A level, which correspondsto themostpossiblerealizationof processingtimes. Checkingif

a move is uselessis donein constanttime usinga hashtablecontainingthecritical jobsassociated

with thesolution.Fortunately, with thesealgorithmicimprovements,thesizeof thereducedneigh-

borhoodis only ÔY�?®�p�� , where � is thenumberof jobson thecritical path.Now, � is often

muchsmallerthan ®¼p ² , thuswhencomparedwith the ÔY�|® � p ² � sizeof thecompleteinsertion

neighborhoodwe have a considerableimprovement.Also, the solutionsobtainedby performing

theuselessmovesdonotneedto beevaluatedin thereducedneighborhoodimplementationunlike

thecompleteimplementation,therebyleadingto a furtherreductionin thecomputationaleffort.

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