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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).�jayanth@cmu.edu�grossmann@cmu.edu- 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 � ���:��¿*Ʀ� 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 »  ¹^À³Éwp Md ¸ ; ( 8©ª2�³;S´¦�³; � �«Ë (28)
·y]�¸³¹^À³ÉÊ8 ·«]�¸ º ¹ o �zº À º É�QÌÇ Â »  ¹^À³Éwp Md ¸ ; ( �³;S´l8©ª@ͳ; � �«Ë (29)
·«] �z� À^É � Ç Â »  � À³É�p M- � ; � ��Ë (30)
·y] � ¹³À^É � ·«] �zº ¹ o �zº À º É�QÌÇ Â »  ¹^À³Éwp Md � ;S´a8©ª2ͳ; � �«Ë (31)
·«]�¸ � À^É � ·«]�¸ o �zº��zº À º É�Q Ç Â »  � À³É�p Md ¸ ; ( 8©ª2ͳ; � �«Ë (32)
Ç ¸ Md ¸ � A ; � (33)
Ç Â Md ¸ � A ; ( (34)
·y]�¸³¹^À³ÉÊ8 0 ; ( �z´¦� � �«Ë (35)Md ¸ � 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�?®Øרp 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
Solutions Found Over RTS Progress
Current SolutionBest Solution so far
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 »  ¹³À^É , transitiontimes ��â<¹ andduedatesV  .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¨è ¹ areusedto
representassignmentof product�to unit ´ . Herewepresentthemodelfor makespanminimization.
Continuousvariables] %  áéÀ³É and ]³Ë  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 è ¹ Q ǹ�îðï ¡æñ^òäódñ^ôæ¤ ç èÂ î ¹ î Q=� Â� î áb� ª ; � � � è � �öõ�9� è �z´� $µ ÂS÷ µ è ÷ µÜá�� Ï ,à (39)
Ç î¥ïCø ô º  î¬ùú  �Â� î áb� A ; � ,±I� Ï !à (40)
Ç î ï'ø ô º  î ùú  �Â� î á)Q ǹyï ¡ðñ^òðódñ^ôæ¤ %  ¹ � A ; � ,±I� Ï !à (41)
ç è ¹ 8 %  ¹ ; � ,±I�z´l $µ  (42)
Ç Â ï'ø % Â ¹ � A ;S´a $µ Â (43)
ǹ«ï ¡æñ^ò¥ódñ^ô�¤ ç  ¹ � A ; � ,±I� Ï !à (44)
] %  á°À³É�Q ǹ«ï ¡æñ^ò¥ódñ^ôæ¤ ç è ¹ p2��»  ¹^À³É�QÈ�`â<¹T�� ] %  á î À^É ; Ï è � Ï ,àg� Ï è . Ï � � ,±I� � �«Ë (45)
] %  á°À³É�Q ǹ«ï ¡æñ^ò¥ódñ ô ¤ ç è ¹ p2��»  ¹^À³É�QÈ�`â<¹T�� �³Agq{� Â� î£áä� p-âÈQ>] %  î£á°À³É; � � � è "±K� �¨õ�9� è � Ï !àí� � �«Ë (46)
] %  á ò À³É Q ǹ«ï ¡æñ^ò¥ódñ^ôæ¤ ç è ¹ p2��»  ¹^À³É�QÈ�`â<¹T�� � % À³É ; � "±K� � ��Ë (47)
19
ç è ¹ 8 0 ; � �z´a {µ  (48)
ç è ¹ � A ; � �z´a {µ  (49)
� Â� î áû � 0S�'A*# ; � � � è "±K� �¨õ�9� è � Ï �à  (50)%  ¹ � 0S�'A*# ; � "±K�Ó´l $µ  (51)
] %  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 �ܵ 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
Solutions Found Over RTS Progress
Current SolutionBest Solution so far
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 �  p Ç � Ë ò� p  �
q5���:��¿*ÆÜ� p2��A7¿¦ª¦�+p Ç Ä ±2VPÄ=p Ç À ���/À/p2� & Ä�À H Q & Ä À J � (54)
] %  À³É�Q>»  À³É � ·«]�À º É ; � � � ��Ë (55)
¯¨Ä�Q & Ä�À³É 8 ·«]�À º É ; � � � �«Ë (56)
] %  À³É�Q>»  À³É � ] % ¹^À³É�Q�â  À^É�p)��Agq Md ¹�� ;w� � �z´2�7� �öõ� ´nͳ; � �«Ë (57)Md ¹ � A ;w� � �z´2�/ !U� s � (58)
Ç � �¨Â � p  � � ǹ � ¡ ¹Cùú  ¤ Ï�� �?U<¹T� p M ¹  ; � (59)
Ç �  � � A ; � (60)M- ¹�Q M ¹  � A ;w� � ´2�7� � �{´ (61)M- ¹wQ M ¹³á¦Q M á  � ª ;w� � �z´¦� Ï ���ð� � �{´l� Ï � (62)M ¹  Q Md á¦Q M á ¹ � ª ;w� � �z´¦� Ï ���ð� � �{´l� Ï � (63)
·«]�À³É 8 0 ; � ��Ë (64)& Ä�À³É 8 0 ; � � � �«Ë (65)
] %  À³É 8 0 ; � � � �«Ë (66)
0��  � �ÎA ; � � � (67)Md ¹ � 0S�CA*# ; � �Ó´ (68)
In formulation(PDP),theprecedencerelationshipbetweentwo tasks�and ´ is modeledwith
the help of the binary variableMd ¹ . Thus
M-Â ¹ � A if task�
precedestask ´ , elseMd 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.
ReferencesAarts,E.H.L. andKorst,J. (1989)SimulatedAnnealingandBoltzmannMachines:A Stochastic
Approachto CombinatorialOptimizationandNeuralComputing,JohnWiley, New York.
Balasubramanian,J.andGrossmann,I.E. (2002)A novel branchandboundalgorithmfor schedul-
ing flowshopplantswith uncertainprocessingtimes.Computers and Chemical Engineering, 26,
41.
Arnold, K. andGosling,J. (1998)The Java ProgrammingLanguage,SecondEdition, Addison-
Wesley, Reading,MA.
32
Battiti, R.andTecchiolli,G. (1994)Thereactivetabu search.ORSA Journal on Computing, 6, 126.
Birewar, D. B. and Grossmann,I. E. (1989) Efficient optimization algorithms for zero-wait
schedulingof multiproductbatchplants.Industrial and Engineering Chemistry Research, 28,
1333.
Campbell,H.G., Dudek, R.A. and Smith, M.L. (1970) A heuristicalgorithm for the n-job m-
machinesequencingproblem.Management Science, 16, 630.
Chanas,S. andKamburowski, J. (1981)Theuseof fuzzy variablesin PERT. Fuzzy Sets and Sys-
tems, 5, 11.
Chen,S.-J.andHwang,C.-L. (1992)FuzzyMultiple AttributeDecisionMaking,Springer, Berlin,
1992.
Dubois,D. andPrade,H. (1987)Themeanvalueof a fuzzy number. Fuzzy Sets and Systems, 24,
279.
Dubois,D. andPrade,H. (1988)PossibilityTheory:An Approachto ComputerizedProcessingof
Uncertainty, PlenumPress,New York, 1988.
Fortemps,P. (1997)Jobshopschedulingwith imprecisedurations:a fuzzyapproach.IEEE Trans-
actions on Fuzzy Systems, v 5 n 4, 557.
Fortemps,P. andRoubens,M. (1996)Rankinganddefuzzificationmethodsbasedon areacom-
pensation.Fuzzy Sets and Systems, 82, 319.
Galluzzo,M.; Ducato,R.; Bartolozzi, V. andPicciotto,A. (2001)Expert control of DO in the
aerobicreactorof anactivatedsludgeprocess.Computers and Chemical Engineering, 25, 619.
Glover, F. (1990)Tabu search:a tutorial. Interfaces, 20, 74.
Hagstrom,J.N. (1988)Computationalcomplexity of PERT problems.Networks, 18, 139.
Hapke, M. andSlowinski, R. (1996)Fuzzypriority heuristicsfor projectscheduling.Fuzzy Sets
and Systems, 83, 291.
Honkomp, S.J.;Mockus,L.; Reklaitis,G. V. (1999)A framework for scheduleevaluationwith
processinguncertaintyComputers and Chemical Engineering, 23, 595.
Hui, C.-W., Gupta,A. andvan der Meulen,H.A.J. (2000)A novel MILP formulationfor short-
termschedulingof multi-stagemulti-productbatchplantswith sequence-dependentconstraints.
Computers and Chemical Engineering, 24, 2705.
Ierapetritou,M. G. andPistikopoulos,E. N. (1996)Batchplantdesignandoperationsunderun-
certainty. Industrial and Engineering Chemistry Research, 35, 772.
33
Ishibuchi,H., Yamamoto,N., Murata,T. andTanaka,H. (1994)Geneticalgorithmsandneighbor-
hoodsearchalgorithmsfor fuzzy flowshopschedulingproblems.Fuzzy Sets and Systems, 67,
81.
Kubic,W. L. (Jr.) andStein,F. P. (1988)A theoryof designreliability usingprobabilityandfuzzy
sets.AIChE Journal, 34, 583.
Lootsma,F.A. (1989)StochasticandfuzzyPERT. European Journal of Operational Research, 43,
174.
Malcolm, D.G., Roseboom,J.H.,Clark, C.E.andFazar, W. (1959)Applicatin of a techniquefor
researchanddevelopmentprogramevaluation.Operations Research, 12, 16.
Majozi,T. andZhu,X. X. (2001)A combinedfuzzysettheoryandMILP approachin integrationof
planningandschedulingof batchplants.Presentedat AIChE AnnualConference,Reno,2001.
McCahon,C.S.andLee,E.S.(1992)Fuzzyjob sequencingfor a flow shop.European Journal of
Operational Research, 62, 294.
McDonald,C.M. andKarimi, I. (1997)Planningandschedulingof parallelsemi-continuouspro-
cess.2: short-termscheduling.Industrial and Engineering Chemistry Research, 36, 2701.
Nowicki, E. andSmutnicki,C. (1998)The flowshopwith parallelmachines:a tabu searchap-
proach.European Journal of Operational Research, 106, 226.
Ow, P.S. andMorton, T.E. (1988)Filteredbeamsearchin scheduling.International Journal of
Production Research, 26, 35.
Ozelkan,E.C.andDuckstein,L. (1999)Optimalfuzzycounterpartsof schedulingrules.European
Journal of Operational Research, 113, 593.
Pekny, J.F.; Miller, D.L. (1991)Exactsolutionof theno-wait flowshopschedulingproblemwith a
comparisonto heuristicmethodsComputers and Chemical Engineering, 15, 741..
Petkov, S. B. andMaranas,C. D. (1997)Multiperiod planningandschedulingof multiproduct
batchplantsunderdemanduncertainty. Industrial and Engineering Chemistry Research, 36,
4864.
Pinto,J.andGrossmann,I.E. (1995)A continuoustime mixedintegerlinearprogrammingmodel
for short-termschedulingof multistagebatchplants.Industrial and Engineering Chemistry Re-
search, 34, 3037.
Pinto,J.andGrossmann,I.E. (1996)An alternateMILP modelfor short-termschedulingof batch
plantswith pre-orderingconstraints.Industrial and Engineering Chemistry Research, 35, 338.
Postlethwaite,B. andEdgar, C.(2000)MIMO fuzzymodel-basedcontrollerChemical Engineering
Research and Design, Part A: Transactions of the Institute of Chemical Engineers, 78, 557.
34
Rajagopalan,D. andKarimi, I. A. (1989)Completiontimesin serialmixed-storagemultiproduct
processeswith transferandset-uptimes.Computers and Chemical Engineering, 13, 175.
Sand,G, Engell, A., Markert, A, Schultz,R. andSchulzC. (2000) Approximationof an ideal
onlineschedulerfor amultiproductbatchplant.Computers and Chemical Engineering, 24, 361.
Schmidt,C. (1998)OptimalSchedulingof theAgricultural ChemicalNew ProductDevelopment
Process.Ph.D. Thesis,Departmentof ChemicalEngineering,CarnegieMellonUniversity, Pitts-
burgh,PA.
Schmidt,C. andGrossmann,I.E. (1996)Optimizationmodelsfor theschedulingof testingtasks
in new productdevelopment.Industrial and Engineering Chemistry Research, 35, 3498.
Schmidt,C. and Grossmann,I.E. (2000) The exact overall time distribution of a project with
uncertaintaskdurations.European Journal of Operational Research, v 126n 3, 614.
Shah,N. (1998) Single-andmultisite planningandscheduling:currentstatusand future chal-
lenges.In Proceedingsof the Conferenceon Foundationsof ComputerAided ProcessOpera-
tionsFOCAPO98,Snowbird, USA, 75-90.
Slowinski, R. andHapke, M. (Eds.)(2000)SchedulingunderFuzziness,PhysicaVerlag,Heidel-
berg.
Tarifa, E. andChiotti, O. (1995)Flexibility vs costsin multiproductbatchplant design: a cal-
culationalgorithm.Chemical Engineering Research and Design, Part A: Transactions of the
Institute of Chemical Engineers, 73, 931.
van Laarhoven,P.J.M., Aarts,E.H.L. andLenstra,J.K. (1992)Jobshopschedulingby simulated
annealing.Operations Research, 40, 113.
Wang,J.R.(1999)A fuzzysetapproachto activity shedulingfor productdevelopment.Journal of
the Operational Research Society, 50, 1217.
Wets,R. J.-B. (1974)Stochasticprogrammingwith fixed recourse: the equivalentdeterministic
programSIAM Review, 16, 309.
Zadeh,L. (1965)Fuzzysets.Information and Control, 8, 338.
35
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