continuous-time optimization approach for medium-range production scheduling...

Continuous-Time Optimization Approachfor Medium-Range

Production Schedulingof a Multi-Pr oduct Batch Plant

XiaoxiaLin andChristodoulosA. Floudas�

Departmentof ChemicalEngineering

PrincetonUniversity

Princeton,NJ08544-5263

SwetaModi andNikolaM. Juhasz

ATOFINA Chemicals,Inc.

900FirstAvenue

King of Prussia,PA 19406-0936

Abstract: The medium-rangeproductionschedulingproblemof a multi-productbatchplant is

studied.Themethodologyconsistsof a decompositionof thewholeschedulingperiodto succes-

sive shorthorizons.A mathematicalmodelis proposedto determineeachshorthorizonandthe

productsto be included. Thena novel continuous-timeformulationfor short-termschedulingof

batchprocesseswith multipleintermediateduedatesis appliedto eachtimehorizonselected,lead-

ing to a large-scalemixed-integerlinearprogramming(MILP) problem.Specialstructuresof the

problemarefurtherexploitedto improve thecomputationalperformance.An integratedgraphical

userinterfaceimplementingtheproposedoptimizationframework is presented.Theeffectiveness

of the proposedapproachis illustratedwith a large-scaleindustrialcasestudy that featuresthe

productionof thirty fivedifferentproductsaccordingto abasic3-stagerecipeandits variationsby

sharingtenpiecesof equipment.

Keywords: medium-rangescheduling,multi-productbatchprocess,decomposition,continuous-

time formulation,MILP.

1 Intr oduction

In multi-productbatchplants,differentproductsaremanufacturedvia thesameor similarsequence

of operationsbysharingavailablepiecesof equipment,intermediatematerialsandotherproduction�Author to whom all correspondenceshouldbe addressed;Tel: (609) 258-4595;Fax: (609) 258-0211;E-mail:

[email protected].

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resources.They have long beenacceptedfor the manufactureof chemicalsthat areproducedin

small quantitiesandfor which theproductionprocessor the demandpatternis likely to change.

The inherentoperationalflexibility of this type of plant providesthe platform for greatsavings

reflectedin agoodproductionschedule.

Theresearchareaof productionschedulingandplanningof multi-productandmulti-purpose

chemicalprocesseshasreceivedgreatattentionin thelastdecade.Oneof themostrecentreviews

of therelatedworksis thatof Shah1, whichfirst examineddifferenttechniquesfor optimizingpro-

ductionschedulesat individualsites,with anemphasison formalmathematicalmethods,andthen

focusedon progressin the overall planningof productionanddistribution in multi-site flexible

manufacturingsystems.In anotherreview, Pekny andReklaitis2 discussedthe natureandchar-

acteristicsof thescheduling/planningproblemsin chemicalprocessingindustriesandpointedout

the key implicationsfor the solutionmethodologyfor theseproblems.Most of the work in this

areahasdealtwith eitherthe long-termplanningproblemor the short-termschedulingproblem.

Long-termplanningor capacityexpansionproblemsinvolve identifying the timing, locationof

additionalfacilities over a relatively long time horizon3. Short-termschedulingmodelsaddress

detailedsequencingof variousoperationaltasksover shorttime periods.All of themathematical

modelsin the literaturecanbeclassifiedinto two maingroupsbasedon thetime representations.

Early attemptsrely on the discretizationof the time horizoninto a numberof intervals of equal

duration4 � 5. Thisapproachis adiscreteapproximationof thetimehorizonandresultsin anunnec-

essaryincreaseof theoverall sizeof themathematicalmodel.Recentwork aimsatdevelopingef-

ficientcontinuous-timemodels6 � 7 � 8 � 9 � 10 � 11 � 12. However, it shouldbepointedout thatall slot-based

formulations6 � 7 � 8 restrictthetime representationandresultby definition in suboptimalsolutions.

Floudasandcoworkers13 � 14 � 15 proposeda novel truecontinuous-timemathematicalmodelfor the

generalshort-termschedulingproblemof batch,continuousandsemicontinuousprocesses,which

is thebasisof thework presentedin this paper. Lin andFloudas16 furtherextendedthis modelto

incorporateschedulingissuesin thedesignandsynthesisof multipurposebatchprocesses.

Therestof thispaperis organizedasfollows. We will first presenttheprobleminvestigatedin

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this work. Thentheoverall framework is proposedanddetailedformulationsof a decomposition

modelanda short-termschedulingmodelarediscussed.Computationalresultsfrom anindustrial

casestudy arealso given. At the end, an integratedgraphicaluserinterfaceimplementingthe

proposedoptimizationframework is presented.

2 ProblemDescription

In this work, we investigatethemedium-rangeproductionschedulingproblemof a multi-product

batchplant,which is definedasfollows: Given(i) theproductionrecipe(i.e., theprocessingtimes

for eachtaskat thesuitableunits,andtheamountof thematerialsrequiredfor theproductionof

eachproduct),(ii) theavailableunitsandtheir capacitylimits, (iii) theavailablestoragecapacity

for eachof the materials,and(iv) the medium-rangetime horizonunderconsideration,thenthe

objectiveis to determine(i) theoptimalsequenceof taskstakingplacein eachunit, (ii) theamount

of materialbeingprocessedat eachtime in eachunit, and(iii) the processingtime of eachtask

in eachunit, soasto satisfythemarket requirementsexpressedasspecificamountsof productsat

giventime instanceswithin thetimehorizon.

In thebatchplant investigated,therearethreetypesof operations:Operations1, 2 and3. Up

to sixty differentproductscanbeproduced.For eachof them,oneof theprocessingrecipesshown

in Figure1 is applied.Therecipesarerepresentedin theform of State-TaskNetwork (STN)4, in

which thestatenodeis denotedby a circle andthe tasknodeby a rectanglebox. Someproducts

sharethesameOperation1 step.

Theplanthasthreetypesof units: four Type1 units(Units 1� 4) for Operation1, threeType

2 units (Units 5� 7) for Operation2, andthreeType 3 units (Units 8� 10) for Operation3. The

informationon which unitsaresuitablefor eachproductis given. Type1 unitsandType3 units

areutilized in a batchmode,while Type2 unitsoperatein a continuousmode.Thecapacitylimit

of eachType 1 unit variesfrom oneproductto another, while the capacitylimit of eachType 3

unit is thesamefor all suitableproducts.Theprocessingtime or processingrateof eachtaskin

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thesuitableunits is alsospecified.Whenswitchedfrom onetypeof productto another, theunits

needcleaningup between.Thedataabove arerelatively fixed,which seldomchangeover a long

period.

[Figure1 abouthere.]

Thetimehorizonconsideredfor productionschedulingis aslongasawholemonth.Customer

ordersaredistributedthroughoutthetimehorizonwith specifiedamounts,duedatesandpriorities.

We assume(i) no limitation on raw materials,and(ii) unlimitedstoragecapacityfor all materials

basedonanalysisof thespecificsituationin theplant.

3 Overall Framework

Theoverall methodologyfor solving themedium-rangeproductionschedulingproblemis to de-

composethelargeandcomplex problemto smallershort-termschedulingsub-problemsin succes-

sive time horizons. The flowchart is shown in Figure2. The first stepis to input relevant data.

Then,a mathematicalmodelfor thedeterminationof thecurrenttime horizonandcorresponding

productsthatshouldbeincludedis formulatedandsolved.Accordingto thesolutionof thedecom-

positionmodel,a short-termschedulingmodelis formulatedusingthe informationon customer

orders,inventorylevelsandprocessingrecipes.TheresultingMILP problemis a large-scalecom-

plex problemwhich requiresa largecomputationaleffort for its solutionandexhibits difficulties

in obtainingglobal optimality. It is solved iteratively by usingcut-off valuesuntil a satisfactory

feasiblesolutionis obtained.Thenthesolutionis outputandthenext timehorizonis to besolved.

Theabove procedureis appliediteratively until thewholeschedulingperiodunderconsideration

is finished.


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4 DecompositionModel

A key issuethat arisesin the rolling horizonapproachdescribedabove is the determinationof

the time horizon and thoseproductsthat shouldbe consideredfor eachshort-termscheduling

sub-problem. We develop a two-level mathematicalformulation that effectively addressesthis

issuetaking into accountthe trade-off betweendemandssatisfaction,unit utilization andmodel

complexity. In thefirst level, thetime horizonis determinedandthemainproductsthatshouldbe

consideredfor all processingstepsareidentified,while in thesecondlevel, additionalproductsare

identifiedto go throughtheOperation1 stepif needed.

4.1 Level 1 Formulation

Themathematicalmodelinvolvesthefollowing notations:

Sets:�days;

�products;

��Operation1 groups;

�productsin Operation1 group( � ) whichsharethesameOperation1 step;

�� Type3 units.

�� Type3 unitssuitablefor product(� ).

Parameters:�� numberof eventpointsperday;

�� amountof demandfor product(� ) dueatday( );

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�� total amountof demandfor product(� ) underconsideration;

�� total numberof Type1 unitssuitablefor productionof Operation1 group( � );�� total numberof otherunitssuitablefor productionof product(� );

"! � $# capacityof Type3 unit ( � );

% � �� # fixedprocessingtime in Type3 unit ( � ) for productionof product(� );

�&� �(' � � � � customerpriority of demandfor product(� ) dueatday( );

!*)&��! � � � minimum numberof daysin advanceto startprocessingto satisfydemandfor

product(� ) dueonday( ), whichcanbederivedfrom theamountof thedemand,

unit capacitiesandtaskprocessingtimes;

+,� � overall weight of product(� ) basedon the priority, amountandduedateof its

first demand;

-� � complexity index for processingof Operation1 group( � ), .0/ for someproducts

with specialrestrictions,= 1 for others;

��1 � maximumnumberof +,2435�7698$67�;: variablesallowedasthecomplexity limit of the

resultingschedulingmodel;

< � upperboundon theratio of thetime for which theType3 unitscanbeusedfor

selectedproductsover thetimehorizon;

= coefficient in objective functionaccountingfor relative importanceof products

inclusioncomparedto horizonmaximization.

Variables:

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!?>�3 : binary, whetheror not to includeday( );

�&� ' 3 � : binary, whetheror not to includeproduct(� );

� � 3 � : binary, whetheror notOperation1 group( � ) is included.

Basedontheabovedefinitionsof parametersandvariables,thefollowing constraintsandobjective

functionareformulated:

HorizonContinuity

!?>�3 :A@ !?>43 CB / : D FEG� 6 IHJ ?KML7N5OQP(1)

Theseconstraintsensurethatsuccessivedaysstartingfrom thefirst oneareselectedto form acon-

tinuoustimehorizon.

Inclusionof productswith demandsin currenthorizon

�&� ' 3 � :R@ !?>43 : D � EG� 6 SEG� 6 �� .UT P (2)

Theseconstraintsstatethat if thereis a demandfor product(� ) dueon day( ) andday(

) is in-

cludedin thecurrenttimehorizon,thenproduct(� ) shouldbeconsideredfor thishorizon.

Inclusionof productswith demandsin thefuture

�&� ' 3 � :R@ !V>43 XW !*)&��! � � � : D � EG� 6 SEG� 6Y!*)&��! � � � .ZT P (3)

Theseconstraintsexpressthe requirementthat if a demandfor product(� ) dueon day( ) needs!*)&��! � � � day(s)in advanceto startprocessingin orderto satisfytheduedateandday(

[W !\)&��! � � � )7

is includedin thehorizon,thenproduct(� ) hasto betakeninto accountfor thishorizon.

�&� ' 3 � :R@ !?>�3 [W / : D � EI� 6 ]EG� 6 �&� �^' � � � � is highP

(4)

Theseconstraintsstatethatif ademandis of highpriority, thenit is pushedonedayforward.These

constraintsarenot necessary, however, this explicit considerationof demandprioritiescanleadto

solutionsthatsatisfydemandswith highprioritiesto a largerextent.

Definition for variables� � 3 � :� � 3 � :`_ a��b�c�d �&� ' 3 � : D � EI�� (5)

� � 3 � :`@ �&� ' 3 � : D � EG�� 6 � EG��P (6)

Theseconstraintsrelate � � 3 � : variableswith corresponding�&� ' 3 � : variables. An Operation1

group( � ) is included,that is, � � 3 � : J / , if andonly if oneor moreof theproductsthatbelongto

thisOperation1 groupareincluded,thatis, �&� ' 3 � : J / .Modelcomplexity limit

e afb�gVh � � 3 � :jik�� il m� � nB a��boc �&� ' 3 � :pil��q�� r ika� b�s !V>43 :i�� _t ��1 � P (7)

The left handsideof this constraintgivesan estimateof the numberof +�2u3��f698$6v�;: binary vari-

ables(seeschedulingmodelin next section)in theresultingschedulingproblem.Becauseit can

be usedto representthe scaleandcomplexity of the schedulingproblem,this constraintkeeps

the schedulingproblemundertractablesizeby imposinganupperboundon the total numberof3��f698$6v��: combinations.

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Type3 unit utilization limit

a��boc �&� ' 3 � : �� a#�b�w �(x m! � $#�y�^�#�blw � x"z % � �� #${&|}�� ~| _�< � a� b�s !?>�3 :nil��Xi |}�� |�P (8)

The left handsiderepresentsa lower boundon the total numberof hoursfor which the Type 3

unitsareutilized to satisfydemandsof selectedproductsunderconsideration,for example,within

two weeks.Thus,thisconstraintlimits theselectedproductsby consideringtheutilizationof Type

3 units.

Objective: Maximizationof durationof horizonandproductsincluded

a� b�s !?>�3 : B =�a��boc +,� � i �&� ' 3 � : P (9)

Thefirst termin theobjective maximizesthedurationof thetime horizon,while thesecondterm

aimsat including asmany productsaspossible. = is usedto balancethe relative importanceof

thesetwo targets.

The formulationdescribedabove is a mixed-integernonlinearprogramming(MINLP) prob-

lem dueto thebilineartermsof binaryvariablesin constraint(7). We introduceadditionalbinary

variables� 3 � 6 : and � 3 � 6 : and the following constraintsto replacethe bilinear productsof�&� ' 3 � :ni !?>�3 : and � � 3 � :ni !?>�3 : .Definition for variables� 3 � 6 :

� 3 � 6 :`_ �&� ' 3 � : D � EI� 6 ]EG� (10)

� 3 � 6 :`_ !?>43 : D � EG� 6 �EG� (11)

� 3 � 6 :`@ �&� ' 3 � : B� !V>43 : W / D � EI� 6 ]EG��P (12)

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This setof linear constraintsareequivalent to � 3 � 6 : = �&� ' 3 � :�i !V>43 : because� 3 � 6 : are

binaryvariables.

Definition for variables� 3 � 6 :� 3 � 6 :�_ � � 3 � : D � EI�� 6 SEG� (13)

� 3 � 6 :�_ !?>43 : D � EI�� 6 SE�� (14)

� 3 � 6 :�@ � � 3 � : B� !?>43 : W / D � EI�� 6 ]EG��P (15)

Similarly, theseconstraintsareequivalentto � 3 � 6 : = � � 3 � :ni !V>43 : .Now, constraint(7) canbereformulatedasfollows:

Reformulationof constraintonmodelcomplexity limit

z a7b�gVh e �� il -� � a� b�s � 3 � 6 : rVB a��b�c e �� a� b�s � 3 � 6 : r�{ i�� _t ��1 � P (16)

By includingconstraints(10)–(15)andreplacingconstraint(7) with constraint(16), theorig-

inal MINLP problemis transformedto anMILP problemandcanbesolvedto globaloptimality

effectively.

4.2 Level 2 Formulation

After the time horizon and the main productsare determinedin the first level, a secondlevel

mathematicalmodelis formulatedto investigatetheutilization of the Type1 units, in which the

first stepin theprocessingsequence,theOperation1 step,is performed,andto includeadditional

products,if necessary, to gothroughtheOperation1 stepto ensurethattheType1 unitsareutilized

efficiently.

Thesecond-level mathematicalmodelinvolvesthefollowing notations:

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Sets:��Operation1 groups;

�productsin Operation1 group( � );

� � Type1 units;

� � Type1 unitssuitablefor Operation1 group( � ).Parameters:� 1 � 0-1parameterto indicatewhetherproduct(� ) is selectedin thefirst level;

�� total amountof demandfor post-Operation1intermediatematerialof Operation

1 group( � ); m! �&� � # capacityof Type1 unit ( � ) for processingof Operation1 group( � );% � � � # fixedprocessingtime in Type1 unit ( � ) for processingof Operation1 group( � );

< lower boundon the fractionof the time horizonfor which theType1 unitsare

utilized;

�durationof thetimehorizondeterminedin thefirst level.

Variables:

� � 3 � : binary, whetheror not to includeOperation1 group( � ).

Thefollowing constraintsandobjective functionareformulated:

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Inclusionof productsselectedin first level

� � 3 � :A@�� 1 � D � EG�� 6 � EG�joP (17)

Theseconstraintsensurethat if a productis selectedin the first level, that is, � 1 � J / , thenthe

Operation1 groupto which thisproductbelongsis included,thatis, � � 3 � : J / .Type1 unit utilization

a7b�g*h � � 3 � : �� a#�b�w\^d "! �&� � # �y��#�b�w\^d z % � � � #${&|M� � �| @Z< i � i |}� � |�P (18)

Theleft handsiderepresentsalowerboundonthetotalnumberof hoursfor whichtheType1 units

areutilized to satisfydemandsof selectedOperation1 groups.Thus,by imposinga lower bound

suchastheoneontheaboveright handside,thisconstraintexpressestherequirementthatenough

productsshouldbeincludedto utilize theType1 unitsefficiently.

Objective: Minimizationof Operation1 groupsincluded

a7b�g*h � � 3 � : P (19)

Theobjective in thesecondlevel is to minimizethetotal numberof productsincludedto limit the

sizeandcomplexity of theresultingschedulingproblemaslongasefficientutilizationof theType

1 unitsis ensured.

Thesecondlevel mathematicalmodelleadsto anMILP problemandcanbesolvedeasily.

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5 Short-Term SchedulingFormulation

5.1 BasicFormulation

After eachtime sub-horizonandcorrespondingproductsto be includedaredeterminedwith the

decompositionmodel,a continuous-timeformulationfor short-termschedulingwith multiple in-

termediateduedatesis applied. This formulationis basedon Floudas,IerapetritouandHene’s

works13 � 14 � 15, featuringthe novel conceptof event points and formulation of specialsequence

constraints.

Theformulationis presentedin detailsasfollows:

Sets:�tasks;�

� taskswhichcanbeperformedin unit (8 );� Ntaskswhicheitherproduceor consumeprocessstate( � );

�units;

��unitswhicharesuitablefor performingtask( � );

�eventpointswithin thetimehorizon;

�materialstates( � ).

Parameters:�� denotesthe minimal capacityallowed of the specificunit (8 ) whenperforming

task( � );�[� LQ�� denotesthemaximalcapacityallowedof the specificunit (8 ) whenperforming

task( � );13

��!?2 � time whenunit (8 ) startsbeingavailable;

�� *N market requirementfor state( � ) at theendof timehorizon;

� � N5� 6v�?�N5� proportionof state( � ) produced,consumedfrom task( � ), respectively;

= � � constanttermof processingtimeof task( � ) in unit (8 );� � � variabletermof processingtimeof task( � ) in unit (8 ) expressingthetimerequired

by theunit to processoneunit of materialperformingtask( � );H time horizon;

2*! 1 � N relativevalueof state( � ) in thesequenceof materialsfor thecorrespondingprod-

uct;

2*! 1�� N relativevalueof thecorrespondingproductindicatingits priority;

2*! 1 *N relative value of the correspondingproductindicating its importanceto fulfill

futuredemands;

�� 1 �� clean-uptimesof unitswhenswitchedfrom task( �� ) to task( � ) ��u� N�� demandfor state( � ) at eventpoint ( � ), which is specifiedbasedon relative time

atwhichthedemandhasto befulfilled, thenumberof stagesrequiredto produce

thefinal product,andthenumberof othertasksthatmaytake placein thesame

unit;

�q� N�� duetime for demandfor state( � ) ateventpoint ( � );

�&� � N�� priority of demandfor state( � ) ateventpoint ( � );

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< constantcoefficient in objective functionbalancingmeetingdemandswith inter-

mediateduedatesandoverallproduction.

Variables:+�2u3��f698$6v��: binaryvariablesthatassignthebeginningof task( � ) in unit (8 ) ateventpoint ( � );

>*2u3¡8$6v��: binaryvariablesthatassigntheutilizationof unit (8 ) ateventpoint ( � );

¢ 35�7698$6v��: amountof materialundertakingtask( � ) in unit (8 ) ateventpoint ( � );

�p£ � 3��k: initial amountof state( � ) ;

�p£ 3^�$6v��: amountof state( � ) ateventpoint ( � );

�p£,¤ 3��k: amountof state( � ) at theendof thehorizon;

� 3^�$6v��: amountof state( � ) deliveredateventpoint ( � ) ;

�¦¥ 3^�$6v��: slackvariablefor theamountof state( � ) not meetingthedemandat eventpoint

( � ) ;

£ N 35�7698$6v��: time thattask( � ) startsin unit (8 ) ateventpoint ( � );

£,§ 35�7698$6v��: time thattask( � ) finishesin unit (8 ) while it startsateventpoint ( � );

AllocationConstraints

a �¨b�©«ª +�2u3��f698$6v��: J >\2u3«8$6v�;: D~8 E¬� 6� EG�®P (20)

Theseconstraintsexpressthat in eachunit (8 ) andat aneventpoint ( � ) only oneof thetasksthat

canbeperformedin this unit (i.e., � E�� ) shouldtake place. If unit (8 ) is utilized at eventpoint

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( � ), thatis, >\2u3¡8$67�;: equal1, thenoneof the +�2u35�7698$6v��: variablesshouldbeactivated.If unit (8 ) is

notutilizedateventpoint ( � ), thenall +�2u3��f698$6v�;: variablestakezerovalues,thatis noassignments

of tasksaremade.

MaterialBalances

�p£ 3^�$6v�¯ N�O : J �°£ � 3^�l: B a �¨bo©�± � � N�� a� bl²f³ ¢ 35�7698$6v�¯ N�O : D�� E � (21)

�°£ 3��?67�;: J �p£ 3^�$6v� W / : W´� 3^�$6v��: B a �¡b�© ± � � N5� a� b�²f³ ¢ 3��76(8$6v� W / : B a �¡b�© ± � � N5� a� b�²f³ ¢ 3��f698$6v�;:D�� E � 6� EG� (22)�p£,¤ 3��l: J �p£ 3^�$6v� KMLfN�O : B a �¨bo©�± � � N5� a� b�²f³ ¢ 3��f698$6v� K}LfN�O : D�� E � (23)

where � � N5� _ T 6v� � N�� @ T representtheproportionof state( � ) consumedby or producedfrom task

( � ), respectively. Accordingto theseconstraintstheamountof materialof state( � ) at eventpoint

( � ) is equalto thatateventpoint ( � W / ) adjustedby any amountsdeliveredateventpoint ( � ) and

producedor consumedbetweentheeventpoints( � W / ) and( � ).

CapacityConstraints

¢ 35�7698$6v��:µ@ � � �� +,2435�7698$67�;: Du� E � 6G8 E¬�$� 6´� E�� (24)¢ 3��76(8$6v�;:µ_ � � L�� +,2u3��76(8$6v�;: Du� E � 6I8 E¬�$� 6� EG�®P (25)

Theseconstraintsexpressthe minimal andmaximalallowed capacityof a unit (8 ), respectively,

whenperformingtask( � ). If +,2435�7698$67�;: equalsone,thenconstraints(24) and(25) correspondto

lower andupperboundson the batch-size,¢ 35�7698$6v��: . If +�2u3��f698$6v�;: equalszero, then

¢ 3��f698$6v��:becomeszero.

In themultiproductplantthatwestudy, thereis nophysicalrestrictionontheminimalcapacity

of units and parameters�X� �� are set to zero. However, if it is not allowed or not suitableto

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operatewith smallbatch-sizesdueto otherconsiderationsnot includedin this model,appropriate

artificial valuescanbe incorporated.Therearealsocasesin which someunits, for example,the

Type1 units,arealwaysoperatedin full capacitiesto produceasmuchaspossible.Thenthetwo

inequalityconstraints(24)and(25)arecombinedandgivethefollowing equalityconstraint:

¢ 35�7698$67�;: J � � L�� +�2u3��f698$6v�;: Du� E � 6I8 E¬�$� 6� EG� (26)

where

� is thesetof Operation1 tasks.

DurationConstraints

£ § 3��f698$6v��: J £ N 35�7698$6v��: B = � � +,2u3��76(8$6v�;: B � � � ¢ 3��f698$6v�;: Du� E � 6I8 E¬�$� 6� EG� (27)

where= � � arethefixedprocessingtimesfor batchtasks(Operation1 andOperation3) andzerofor

continuoustasks(Operation2),� � � aretheinverseof processingratesfor continuoustasksandzero

for batchtasksrespectively. Thedurationconstraintsexpressthedependenceof thetime duration

of task( � ) in unit (8 ) at eventpoint ( � ) on theamountof materialbeingprocessed.If +�2u3��f698$6v�;:equalsone,thenthelasttwo termsin constraints(27)areaddedto

£ N 3��f698$6v�;: . If +,2u3��76(8$6v�;: equals

zero, then the last two termsbecomezerodue to the capacityconstraints(24), (25) andhence£,§ 3��f698$6v�;: J £ N 3��f698$6v��: .SequenceConstraints:

Sametaskin thesameunit

£ N 3��f698$6v� B / :µ@ £ § 35�7698$67�;: Du� E � 6I8 E¶�$� 6� EG� 6v� HJ � KMLfN�O9P (28)

Thesequenceconstraints(28) statethat task( � ) startingin unit (8 ) at eventpoint ( � B / ) should

17

startafter theendof thesametaskperformedin thesameunit which hasalreadystartedat event

point (n).

Differenttasksin thesameunit

Thefollowing setof constraints(29)establishestherelationshipbetweenthestartingtimeof a

task( � ) at eventpoint ( � B / ) andtheendingtime of task( �� ) at eventpoint ( � ) whenthesetasks

takeplacein thesameunit (8 ).£ N 3��f698$6v� B / :�@ £ § 35� 698$67�;: B �� 1 �� +,2435�7698$67�;: W·� 3 / W +�2u3�� 698$67�;:7:D~8 E¬� 67� E � � 6� E

�� 6¬� HJ � 6¬� EI� 6v� HJ � K}LfN�OQP (29)

Constraints(29)arewrittenfor tasks( �f6v�� ) thatareperformedin thesameunit (8 ). If bothtasksare

performedin thesameunit they shouldbeat mostconsecutive. This is expressedby constraints

(29)becauseif +,2435� 698$6v�;: J / whichmeansthattask( � ) takesplacein unit (8 ) ateventpoint ( � ),

thenthelasttermof constraint(29)becomeszeroforcing thestartingtimeof task( � ) in unit (8 ) at

eventpoint ( � B / ) to begreaterthantheendingtimeof task( � ) in unit (8 ) ateventpoint ( � ) plus

therequiredclean-uptime; otherwisetheright handsideof constraint(29) becomesnegativeand

theconstraintis trivially satisfied.

Differenttasksin differentunits

£ N 35�7698$67� B / :R@ £ § 3�� 698 6v��: W´� 3 / W +�2u3�� 698 67�;:7:D~8$698 E¬� 6v� E � � 6v� E�� 6v� HJ � 6´� E�� 6v� HJ � K}LfN�OQP (30)

Constraints(30)arewritten for differenttasks( �f6v� ) thatareperformedin differentunits(8$6(8 ) but

take placeconsecutively accordingto the productionrecipe. Note that if task( � ) takesplacein

unit (8 ) at eventpoint ( � ) (i.e., +�2u3�� 698 6v�;: J / ), thenwe have£ N 35�7698$6v� B / :¸@ £ § 35� 698 6v��: and

hencetask( � ) in unit (8 ) hasto startaftertheendof task( �� ) in unit (8$ ). Otherwisetheright hand

sidebecomesnegativeandtheconstraintis trivially satisfied.

18

Constraintsfor Demandswith IntermediateDueDates

� 3��$6v�;: B �¹¥ 3��?67�;: J �^�q� N�� D�� E � 6� EG�ºP (31)

Theseconstraintsrepresentthatproductsaredeliveredat eventpointswhendemandsexist. The

slackvariables,�¦¥ 3^�$6v��: , areintroducedto give moreflexibility to themodelin handlingpartial

fulfillment of demands.Underfeasibleconditions,someor all of thesevariablescanbefixed to

zeroto ensurethatsomeor all of thedemandswithin thetimehorizonaremet.

DueDatesConstraints

£ N 35�7698$6v��:µ_ �q� N�� D�� E � 6v� E � N 698 E¬��(P (32)

Theseconstraintsensurethesatisfactionof productdemandby thecorrespondingduedate.

Constraintsfor Demandsat theEndof theTimeHorizon

�p£�¤ 3^�l:»@ �� *N D�� E � 6� EG�ºP (33)

Theseconstraintsensurethesatisfactionof demandswhich shouldbemetat theendof the time

horizon.

Unit AvailableTimeConstraints

£ N 35�7698$6v��:µ@Z��!V2 � W´� 3 / W +�2u3��f698$6v��:7: Du� E � 698 E¬�$� 67� EI�®P (34)

Theseconstraintsrepresenttherequirementof notstartingany taskuntil theunit isavailable.When

19

+�2u35�7698$6v��: equalszero,whichmeansthetaskis notactivated,theconstraintis relaxedandbecomes

trivial.

TimeHorizonConstraints

£ § 35�7698$6v��:µ_ � Du� E � 6(8 E¬�$� 6v� EG� (35)£ N 35�7698$67�;:µ_ � Du� E � 698 E¬�� 6v� E��ºP (36)

Thetimehorizonconstraintsrepresenttherequirementthateverytaskstartandendwithin thetime

horizon(H).

Objective: Maximizationof production

W a N a � �¼� � N�� i �¦¥ 3^�$6v��: B <�a N 2*! 1 VN i�2*! 1�� N i�2*! 1 � N i �p£,¤ 3��l: P (37)

The objective shown in (37) is the maximizationof productionin termsof relative valueof all

statesminusthepenaltytermfor notmeetingdemandsat intermediateduedates.

5.2 Additional Constraints

Themathematicalformulationdescribedabove resultsin anMILP problem,which canbesolved

by a commercialMILP solver suchasCPLEX. Mainly becauseof its original conceptof event

points,theproposedformulationoutperformssomeotherexistingdiscrete-timeandpseudo-continuous-

time formulationsin termsof reducingsignificantlythe sizeof resultingmathematicalprogram-

mingproblemandthustherequiredcomputationalresources.However, solvingtheresultingMILP

problemsis still very challenging(e.g.,thesolutionsrequireconsiderableCPUtime for proof of

globaloptimality), which reflectsthe inherentcomplexity of thespecificphysicalconditions.To

improvethemodelingandsolution,thefollowing constraintsareincorporated:

20

TighterSequenceConstraintsfor Operation1

BecauseOperation1 is the first stepin the task sequencefor all productsand we want to

maximizetheoverall productionin principle,thetiming of theOperation1 taskscanbeenforced

to beastight aspossible.

Sametaskin thesameunit

£ N 3��f698$6v� B / :R_ £ § 35�7698$67�;: B�� 3�� W +�2u3��f698$6v��: W +�2u3��f698$6v� B / :7:Du� E � 6G8 E¶�$� 6� EG� 6v� HJ � K}LfN�OQP (38)

Theseadditionalsequenceconstraintsfor the sameOperation1 task in the sameType 1 unit,

combinedwith constraints(28), enforce“zero-wait” condition on the task taking placeat two

consecutiveeventpoints.Namely, if +,2435�7698$67�;: J +�2u3��f698$6v� B / : J / , thatis, task( � ) takesplace

in unit (8 ) at botheventpoint ( � ) and( � B / ), then£ N 3��f698$6v� B / : J £ § 35�7698$6v��: , which statethat

in unit (8 ), task( � ) startingateventpoint ( � B / ) startsimmediatelyaftertheendof thesametask

whichhasalreadystartedateventpoint (n); otherwisetheseconstraintsarerelaxed.

Differenttasksin thesameunit

£ N 3��f698$6v� B / :R_ £ § 3�� 698$67�;: B �� 1 ��*B�� 3�� W +�2u3�� 698$67�;: W +�2u35�7698$6v� B / :f:D~8 E¬�� 6v� E � � 6¬� E�� 6� HJ � 6� EG� 6v� HJ � KMLfN�O (39)

where��

is thesetof Type1 units.

Accordingto thesenew constraintsandconstraints(29), if +,2u3�� 698$6v��: J +,2435�7698$67� B / : J /(that is, task( � ) takesplacein Type 1 unit (8 ) at event point ( � ) andtask( � ) takesplacein the

sameType1 unit at eventpoint ( � B / )), then£ N 3��f698$6v� B / : J £ § 3�� 698$6v��: B �� 1 �� requiringthat

task( � ) in Type1 unit (8 ) at eventpoint ( � B / ) startimmediatelyaftertask( �� ) in thesameType

1 unit ateventpoint ( � ) endsandnecessarycleaningis donefor theType1 unit. Otherwise,these

21

constraintsaretrivial.

RestrictionsonBinaryVariables

For eachproduct,basedon theinformationof overallamountof demandsandmaximalbatch-

sizesof relatedtasksperformedin suitableunits, lower boundson the total numberof activated

taskscanbespecifiedfor Operation1, Operation2 andOperation3, respectively.

a� b�²f³ a��b�½ +�2u3��f698$6v�;:`@ � � Du� E � (40)

where� � areparameterscalculatedbasedonrelevantdata.

Theseconstraintsreducethecombinatorialcomplexity of theMILP problemsandimprovethe

computationalperformance.

SpecialRestrictions

Restriction1

It is given that Operation1 for a specificproductshouldrun in oneof the Type 1 units in

campaignmode,namely, a prespecifiedminimum number, , of batchesneedsto be performed

consecutively oncestarted.This arrangementis dueto the comparatively long clean-uptime for

theType1 unit requiredto switchfrom this productto others.Thecorrespondingconstraintscan

beformulatedasfollows:

+�2u3��f698$6v��:R@�+�2u3��f698$6v� W / : Du� E � 9¾ 6(8 E¬�$9¾ 6v� EG� 6-�¿_��_t (41)+�2u3��f698$6v�;:R@�+,2u3��76(8$6v� W / : W / W / a�kÀ �(Á � � Á ��À*Â +�2u3��f698$6v� :Du� E � (¾ 698 E¬��9¾ 6v� EG� 6v� . (42)+�2u3��f698$6v�;:R_�+,2u3��76(8$6v� W / : Du� E � 9¾ 698 E¬�$9¾ 67� EI� 6v��@�� KMLfN�OuW B � (43)

where

� 9¾is thesetof Operation1 tasksand

�$9¾is thesetof Type1 unitsdedicatedto them.

22

Constraints(41)and(42)statethatif +�2u3��f698$6v� W / : equalsoneand+,2435�7698$67� W �$: � +�2u3��f698$6v� W ": , whichdonot all exist for � lessthan , arenot all equalto one,whichmeanstask( � ) hastaken

placein unit (8 ) at event point ( � W / ) andtherehave beenlessthan batchesperformed,then+�2u35�7698$6v��: equalsone,that is, task( � ) shouldtake placein unit (8 ) at eventpoint ( � ); otherwise,

theseconstraintsbecometrivial. Constraint(43) ensuresthat the Operation1 taskdoesnot get

startedaftereventpoint ( � KML7N5OqW B / ) becausetheType1 unit won’t beableto perform batches

in theremainingperiod.

Restriction2

A subsetof theproductsaredescribedasblack. They arealsorequiredto beput togetherfor

Operation2 andOperation3 soasto avoid clean-upasmuchaspossible.

+�2u35�7698$6v��:R_�� W a��«bo©5Ã¨Ä^Å~Æl©¡ª e +�2u3�� 698$6v� : B +,2u3�� 698$6v� : rDu� E � Ç � ½nÈ 698 E¬�$� 6v�p67� 67� EG� 6v� qÉ � É � Ê (44)

where

�Ç � È and

�Ç � ½nÈ are the setof Operation2 andOperation3 tasksfor black productsand

non-blackproducts,respectively. If Type2 unit or Type3 unit (8 ) is utilized for blackproductsat

botheventpoint ( �q ) and( �q ), thentheright handsideof constraints(44) is zero,whichstatesthat

no taskfor non-blackproductsis allowedin thesameunit at any eventpoint ( � ) between( �u ) and

( �q Ê ). Otherwise,theseconstraintsbecometrivial.

Restriction3

Productionof a specialcategoryof productsonly needsto go throughOperation1 andOpera-

tion 3. It is requiredthattheseOperation1 andOperation3 tasksrun for 1-2weeksin acampaign

mode.Therefore,demandsfor this category of productswithin a periodof time (e.g.,onemonth)

areput together. A Type1 unit anda Type3 unit arethendedicatedto the“combined”Operation

1 taskandOperation3 task,respectively. Therelativeorderof original tasksfor differentproducts

canbesimplybasedontheduedatesof demands.Productionof thiscategoryof productsis treated

separately. ThededicatedType1 unit andType3 unit areexcludedfrom availableresourcesfor

23

otherproductsduring the dedicatedtime intervals. This approachwill be illustratedthroughthe

computationalstudyin next section.

6 Computational Study

6.1 ProblemOverview

Theproposedrolling horizonapproachis appliedto anindustrialcasestudyin whichdetailedpro-

ductionschedulesareto bedeterminedto satisfycustomerordersfor variousproductsdistributed

within awholemonth.

The distribution of demandsthroughoutthe whole monthunderconsiderationis plot in Fig-

ure 3. Therearefive main categoriesof productsandthirty five differentproductsarerequired

to beproducedin this month. It is assumedthatno final productis availableat thebeginningof

themonth.However, lowerboundson theamountsof initially availableintermediatematerialsare

provided.


The processingrecipesto make theseproductsareshown in Figure1. The Operation1 and

Operation3 stepsareperformedin abatchmode,while theOperation2 stepin acontinuousmode.

Theprocessingtime or processingrateof eachstepis dependentonboththeproductandtheunit,

with Operation1, Operation2, andOperation3 in therangesof 6-11hours,0.15-0.25units/hour,

and12-16hours,respectively. Capacitiesof thefourType1unitsvaryfrom1.125units/batchto3.5

units/batch,while capacitiesof thethreeType3 unitsareeither4.5units/batchor 3.5units/batch.

Theclean-uptimerequiredrangesfrom 2 hoursto 36hours,dependingontheunit andtheproduct

sequenceinvolved.

6.2 DetailedSchedules

“Campaign mode” production

24

We first considerschedulingof the specialcategory of productsthat requireoperationin the

“campaignmode”(Category4 in Figure3). Demandsfor all productsin thiscategory in thewhole

montharegroupedtogether. OneType1 unit andoneType3 unit arededicatedto theproduction

of theseproductsand the detailedscheduleof the Operation1 tasksandOperation3 tasksfor

differentproductsaredeterminedbasedontheir relativeduedates.Thestartingtimeof production

of all theseproductsis determinedsothatall theduedatesof thedemandsfor theseproductscanbe

satisfied.Then,theunitsandtime intervalswhich areusedareexcludedfrom availableresources

for theproductionof otherproducts.Therelativescheduleobtainedfor productionof thiscategory

of productsis shown in Figure4.


The rolling horizon approachis then appliedfor the productionof the remainingproducts

to breakdown the large schedulingprobleminto several short-termschedulingsub-problemsin

successive time horizons. Therearemainly two typesof connectionsbetweenconsecutive time

horizons:initial availabletime of unitsandintermediatematerials.Thedecompositionandshort-

termschedulingmodelsareimplementedwith MINOPT17 andsolvedwith CPLEX,acommercial

MILP solver. TheCPUtime requiredto obtaineachsolutionrangesfrom 15min to about7 hours

onanHP J-2240workstation.

Horizon 1

With parameter ��1 � of 1500, the first horizon is determinedto be the first 5 daysof the

monthand8 main productsareidentifiedto be includedin this horizonaccordingto Level 1 of

thedecompositionmodel.No additionalproductis identifiedto undergotheOperation1 stepfrom

Level 2 of thedecompositionmodel.

Twentyeventpointsareusedin theshort-termschedulingmodelfor this horizon,which leads

to 1320binaryvariables,3968continuousvariablesand21912constraints.Two feasiblesolutions

aregeneratedandthedetailedscheduleacceptedis shown in Figure5.

25


Horizon 2

The secondhorizon is determinedto be the next 5 daysof the month, that is, from the 6th

dayto the10thday, and6 mainproductsareidentifiedto beincludedaccordingto Level 1 of the

decompositionmodelwith ��1 � of 1000. No additionalproductis identifiedfrom Level 2 of the

decompositionmodel.

Twentythreeeventpointsareusedin theshort-termschedulingmodelfor this horizon,which

leadsto 897 binary variables,2576 continuousvariablesand 14260constraints. One feasible

solution is obtainedand accepted.The detailedscheduleis shown in Figure 6. Note that the

startingtimesof theType1 unitscorrespondto thefinishingtimesof thesameunitsin theprevious

horizon.


Horizon 3

Thethird horizonis determinedto be from the11thdayto the14thdayof themonthand10

mainproductsareidentifiedto beincludedaccordingto Level 1 of thedecompositionmodelwith ��1 � of 1500.No additionalproductis identifiedfrom Level 2 of thedecompositionmodel.

Nineteenevent points are usedin the short-termschedulingmodel for this horizon, which

leadsto 1216binaryvariables,3918continuousvariablesand22086constraints.Threefeasible

solutionsareobtainedbeforethelastoneis accepted.Thedetailedscheduleis shown in Figure7.


Horizon 4

Thefourth horizonis determinedto befrom the15thdayto the19thdayof themonthand8

mainproductsareidentifiedto beincludedaccordingto Level 1 of thedecompositionmodelwith ��1 � of 2000.Sevenadditionalproductareidentifiedto undergo theOperation1 stepfrom Level

2 of thedecompositionmodelwith < of 80%.

26

Twentyoneeventpointsareusedin theshort-termschedulingmodelfor this horizon,which

leadsto 1827binaryvariables,5831continuousvariablesand37298constraints.Threefeasible

solutionsareobtainedbeforethelastoneis accepted.Thedetailedscheduleis shown in Figure8.


Horizon 5

Thefifth horizonis determinedto befrom the20thdayto the24thdayof themonthand9 main

productsareidentifiedto beincludedaccordingto Level 1 of thedecompositionmodelwith ��1 �of 2000. Five additionalproductareidentifiedfrom Level 2 of thedecompositionmodelwith < of 30%.

Twentyoneeventpointsareusedin theshort-termschedulingmodelfor this horizon,which

leadsto 2016binary variables,6471continuousvariablesand41866constraints.One feasible

solutionis obtainedandaccepted.Thedetailedscheduleis shown in Figure9.


Horizon 6

Thereareonly six daysremainingin themonth.It is foundthatthedemandsin this remaining

periodcanbefulfilled veryeasilyandonly 4.5daysis actuallyneeded.Therefore,thelasthorizon

is chosento be4.5daysstartingfrom the25thdayof themonthandall of the11 productsfor the

remainingdemandsareincluded.

Teneventpointsareusedin theshort-termschedulingmodelfor this horizon,which leadsto

580binaryvariables,1820continuousvariablesand6723constraints.Five feasiblesolutionsare

obtainedbeforewe acceptthelastoneasa satisfactoryschedule.Thedetailedscheduleis shown

in Figure10.

[Figure10abouthere.]

27

6.3 Summary of Results

As describedin detail in theprevioussection,thedecompositionmodelandtheschedulingmodel

areusediteratively, moving forward the schedulinghorizon. In summary, the wholescheduling

periodis decomposedinto six time horizons,eachvaryingfrom 4 to 5 dayslong andincluding6

to 15products,asshown in Table1.

[Table1 abouthere.]

A sketchof theschedulesobtainedfor thewholemonthisgivenin Figure11. It shouldbenoted

thattheType1 unitsaremostlyidle towardstheendof thewholeperiodbecauseno demandsare

specifiedfor thecomingperiod.Theproductionschedulesobtainedfor 28.5daysnot only satisfy

all demandsin all categories,thoughsomeof the duedatesarerelaxed, but alsoproduce9.1%

morethanthedemandsin overall (seeTable2).


[Table2 abouthere.]

Anotherimportantcriterionfor judgingtheproductionscheduleis theefficiency of unit utiliza-

tion. As shown in Table3, theschedulesobtainedin thiswork ensurethattheunits,especiallythe

Type3 units,areutilizedefficiently. TheType3 unitsareutilizedintensively throughoutthewhole

period,which indicatesthat they arebottlenecksof the overall productionasfar asthe demand

structurein thiscasestudyis concerned.

[Table3 abouthere.]

7 Integrated Graphical User Interface

A graphicaluserinterfacehasbeendevelopedto integratevariouscomponentsrequiredto apply

the proposedoptimizationframework to the medium-rangeproductionschedulingproblemsys-

tematicallyandeffectively. Theflowchartis shown in Figure12. After theuserinputsall relevant

28

data,whichis storedin adatabase,theschedulingfor productsthatrequireproductionin the“cam-

paignmode”overarelatively longperiodis performedfirst. Then,basedontheinformationin the

database,the two-level decompositionmodeis generatedandthensolved with an MILP solver,

CPLEX. Next, the short-termschedulingmodel is formulated. The resultingMILP problemis

solvediteratively by usingcut-off valuesuntil asatisfactoryfeasiblesolutionis obtained.Thenthe

solutionis outputin readableformats,for example,theGanttChart,andthedatabaseis updated

accordingto thesolutionto move towardsthenext time horizon. Theabove procedureis applied

iteratively until the wholeschedulingperiodis finished. The softwareis developedin extensive

Visual Basicwith M.S. Accesssupportingthe database.It consistsof the following five main

functionalmodules.


7.1 Data Manipulation

All of theinformationis storedin a backgrounddatabase,andcanbeentered,changedor deleted

throughthe userinterface. Figure13 shows the main window of the userinterfacewith forms

openedto accessvariousdata.


Theimportantdatainclude:

Ë Schedulingperiod;

Ë Orders,asshown in Figure14,theamount,duedateandpriority for eachcustomerorder;


Ë Inventoriesof bothfinal productsandintermediatematerials;

Ë Processingrecipes,asshown in Figure15, the processingstepsrequiredfor makingeach

productandotherrelatedinformationneedto bespecified;

29


Ë Informationof theunits,asshown in Figure16;


Ë Unit-productsuitabilities,for example,Figure17shows theType1 Unit SuitabilityForm;


Ë Processingtimes/rates,for example,Figure18showstheOperation3ProcessingTimeForm;


Ë Sequence-dependentclean-uprequirements,for example,Figure 19 shows the transition

tablefor theType1 units.


7.2 Campaign-ModeProduction Scheduling

Productsthat arerequiredto go throughthe campaignmodearehandledseparately. Theappro-

priateproductsandsuitableunits are identifiedanda certainperiodof time is dedicatedto the

campaign-modeproductionof theseproducts.Figure20 showsscheduletablesfor thecampaign-

modeproductiongeneratedin theuserinterface.


30

7.3 Decomposition

Basedon the informationin thedatabase,thedecompositionmodelis generatedandthensolved

with anMILP solver, to determinethecurrenttimehorizonandto identify productsto beincluded

for the schedulingproblem. Figure21 shows CPLEX solving the decompositionmodelandthe

resultsarepresentedin Figure22.



7.4 Short-term Scheduling

Accordingto thesolutionof thedecompositionmodel,theshort-termschedulingmodelis formu-

latedto determinethedetailedschedulefor thecurrenthorizon.Theresultinglarge-scalecomplex

MILP problemis solvediteratively by usingcut-off valuesuntil a satisfactoryfeasiblesolutionis

obtained.Figure23showsCPLEXsolvingtheschedulingmodel.


7.5 ResultsOutput and DatabaseUpdate

The solutionto the short-termschedulingmodel is organizedin readableformats,including the

scheduletableandtheGanttChart,andthedatabaseis updatedaccordingly. Figure24 shows the

Ganttchartrepresentationof ascheduleobtainedin theuserinterface.


Furthermore,a wide varietyof additionalfeaturesareincorporatedto make thegraphicaluser

interfaceasuser-friendlyaspossible.For example,theusercanrequestsearchfunctionsonvarious

dataformsto accessspecificinformationefficiently.

31

8 Conclusions

In this paper, themedium-rangeproductionschedulingproblemof a multi-productbatchplant is

investigated.Threebasictypesof operationareinvolvedandtenpiecesof equipmentareshared

to produceup to sixty differentproducts.Theschedulinghorizonconsideredis onemonth,even

thoughlongerhorizonscanbeaddressedwith the proposedframework. Theoverall approachis

to decomposethelargeandcomplex problemfor thewholeschedulingperiodinto smallershort-

term schedulingsub-problemsin successive time horizons.A two-level decompositionmodelis

proposedto determinethe currenthorizon and identify thoseproductsto be included. Then a

continuous-timeformulationfor short-termschedulingof batchprocesseswith multiple interme-

diate due datesis introduced. This procedureis appliediteratively until the whole scheduling

periodis completed.Theeffectivenessof thisproposedrolling horizonapproachis illustratedwith

computationalresultsfrom an industrialcasestudy. A graphicaluserinterfacedevelopedto in-

tegratevariouscomponentsto apply the proposedoptimizationframework systematicallyis also

presented.

Acknowledgments:Theauthorsgratefullyacknowledgesupportfrom theNationalScienceFoun-

dationandfrom ATOFINA Chemicals,Inc.

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34

List of Figures

1 State-TaskNetwork of productionrecipes . . . . . . . . . . . . . . . . . . . . . . 362 Flowchartof therolling horizonapproach . . . . . . . . . . . . . . . . . . . . . . 373 Distributionof demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Productionschedulefor a specialcategory of productsin campaignmode(U4,

U10: units;Pc1-Pc6:products.) . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Detailedschedulefor TimeHorizon1 (U1-U10:units;P1-P8:products.). . . . . . 406 Detailedschedulefor TimeHorizon2 (U1-U10:units;P3-P13:products.) . . . . . 417 Detailedschedulefor TimeHorizon3 (U1-U10:units;P3-P29:products.) . . . . . 428 Detailedschedulefor TimeHorizon4 (U1-U10:units;P2-P29:products.) . . . . . 439 Detailedschedulefor TimeHorizon5 (U1-U10:units;P2-P29:products.) . . . . . 4410 Detailedschedulefor TimeHorizon6 (U1-U10:units;P4-P28:products.) . . . . . 4511 Sketchof productionschedulefor thewholemonth(U1-U10:units) . . . . . . . . 4612 Flowchartof theintegrateduserinterface . . . . . . . . . . . . . . . . . . . . . . 4713 Dataformsin theintegratedgraphicaluserinterface. . . . . . . . . . . . . . . . . 4814 OrdersFormin theintegratedgraphicaluserinterface . . . . . . . . . . . . . . . . 4915 ProductsFormin theintegratedgraphicaluserinterface . . . . . . . . . . . . . . . 5016 UnitsForm in theintegratedgraphicaluserinterface. . . . . . . . . . . . . . . . . 5117 Type1 Unit SuitabilityFormin theintegratedgraphicaluserinterface . . . . . . . 5218 Operation3 ProcessingTimeFormin theintegratedgraphicaluserinterface . . . . 5319 Type1 Unit TransitionTablein theintegratedgraphicaluserinterface . . . . . . . 5420 Scheduletablesfor campaign-modeproductionobtainedin the integratedgraphi-

caluserinterface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521 CPLEXsolvingthedecompositionmodelin theintegratedgraphicaluserinterface 5622 Decompositionresultsshown in theintegratedgraphicaluserinterface . . . . . . . 5723 CPLEX solving the short-termschedulingmodelin the integratedgraphicaluser

interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5824 A scheduleobtainedin theintegratedgraphicaluserinterface . . . . . . . . . . . . 59

35

Operation3

F I1 I2 P

I1 I2 I3 P

P I

F

F

( a )

( b )

( c )

P

1-x

I1F1

I1 I2

P

F

1-xx

xI3F2

I1F1

I2F2

I2

Px

1-x

I P

P

F

IF

F P

F P

( d )

( e )

( f )

( g )

( h )

( i )

( j )

Operation1 Operation2 Operation3

Operation1 Operation3 Operation2 Operation3

Operation1 Operation3


Operation3




Operation1



Operation1

Figure1: State-TaskNetwork of productionrecipes

36

Decomposition ModelGeneration & Solution

Scheduling ModelSolution

Results Output

Solutionsatisfactory ?

Whole periodfinished

?

End

Scheduling Model

No

Generation

Data Input

Start

Yes

Yes

No

Figure2: Flowchartof therolling horizonapproach

37

Figure3: Distributionof demands

38

0

298.

8

Pc1

Pc1

Pc2

Pc2

Pc3

Pc4

Pc4

Pc5

Pc2

Pc1

Pc1

Pc1

Pc1

Pc6

Pc4

Pc4

Pc2

Pc2

Pc2

Pc2

P

c2

Pc2

Pc2

Pc2

2.37

52.

375

2.37

52.

375

2.37

52.

375

2.37

52.

375

3.25

3.25

3.19

72.

375

2.37

52.

375

2.37

53.

253.

25 2

.72.

375

2.37

52.

375

2.37

52.

125

2.37

5

Pc2

Pc2

Pc2

Pc2

Pc2

Pc2

Pc4

Pc4

Pc6

Pc1

Pc1

Pc1

Pc2

Pc5

Pc4

Pc4

Pc3

Pc2

Pc2

Pc1

3.5

3.5

3.5

3.5

1.25

2.7

32.

125

2.37

53.

53.

53.

53.

53.

53.

53.

53.

197

31.

5

211.

6

(hr)

(hr)

U4

U10

3.5

Figure4: Productionschedulefor a specialcategory of productsin campaignmode(U4, U10:units;Pc1-Pc6:products.) 39

01

23

45

(day

)

U1

N2

P5

1.21

4N

3

P5

1.21

4N

6

P5

1.21

4N

7

P5

1.21

4N

8

P5

1.21

4N

9

P5

1.21

4N

10

P5

1.21

4N

11

P5

1.21

4N

12

P5

1.21

4N

13

P5

1.21

4N

14

P5

1.21

4N

15

P5

1.21

4N

16

P5

1.21

4N

17

P5

1.21

4N

18

P5

1.21

4N

19

P5

1.21

4N

20

P5

1.21

4

U2

N1

P5

1.21

4N

5

P5

1.21

4N

6

P5

1.21

4N

7

P5

1.21

4N

8

P5

1.21

4N

9

P5

1.21

4N

10

P5

1.21

4N

11

P5

1.21

4N

12

P5

1.21

4N

13

P5

1.21

4N

14

P5

1.21

4N

15

P5

1.21

4N

16

P5

1.21

4N

17

P5

1.21

4N

18

P5

1.21

4N

19

P5

1.21

4N

20

P5

1.21

4

U3

N2

P5

1.21

4N

4

P5

1.21

4N

5

P4

1.12

5N

6

P4

1.12

5N

7

P5

1.21

4N

8

P5

1.21

4N

9

P4

1.12

5N

10

P8

1.12

5N

11

P5

1.21

4N

12

P8

1.12

5N

14

P5

1.21

4N

15

P5

1.21

4N

16

P5

1.21

4N

17

P5

1.21

4N

18

P5

1.21

4N

19

P5

1.21

4N

20

P5

1.21

4

U4

N4

P1

3.37

5N

5

P1

3.37

5N

6

P5

3.37

5N

9

P1

3.37

5N

10

P1

3.37

5N

14

P5

3.37

5N

15

P4

3.37

5N

18

P5

3.37

5N

19

P1

3.37

5N

20

P1

3.37

5

U5

N1

P5

0.91

4N

2

P5

0.40

2N

9

P5

0.70

0N

10

P5

3.95

9N

14

P5

1.49

6N

15

P5

4.29

5N

18 P5

0.13

5N

19

P5

1.93

5

U6

N1

P5

1.87

5N

6

P4

1.12

5N

12 P4

0.22

5N

15 P4

0.20

0N

16

P5

3.38

1N

17 P5

0.13

5N20

P5

1.79

7

U7

N5

P1

3.37

5N

6

P1

1.87

4N

9

P2

1.08

7N

10

P1

4.87

6N

15

P1

3.37

5N

16

P5

0.98

3N

20

P1

3.37

5

U8

N1

P5

4.50

0N

5

P5

4.50

0N

6

P3

4.50

0N

9

P5

4.50

0N

10

P2

4.50

0N

14

P3

4.50

0N

15

P5

4.50

0N

16

P3

4.50

0N

17

P6

1.99

5

U9

N1

P3

4.50

0N

4

P5

4.50

0N

7

P1

4.50

0N

8

P5

4.50

0N

9

P3

4.50

0N

15

P5

4.50

0N

16

P4

4.50

0N

17

P5

4.50

0

U1

0N

1

P3

3.50

0N

6

P5

3.50

0N

7

P5

3.50

0N

8

P5

3.50

0N

10

P5

3.50

0N

13

P3

3.50

0N

16

P1

3.50

0N

17

P1

3.50

0N

19

P5

3.50

0

Figure5: Detailedschedulefor TimeHorizon1 (U1-U10:units;P1-P8:products.)40

45

67

89

10 (

day)

U1

N3

P6

1.21

4N

4

P6

1.21

4N

5

P6

1.21

4N

6

P6

1.21

4N

7

P6

1.21

4N

8

P6

1.21

4N

9

P6

1.21

4N

11

P6

1.21

4N

12

P6

1.21

4N

13

P6

1.21

4N

14

P6

1.21

4N

15

P6

1.21

4N

16

P6

1.21

4N

17

P6

1.21

4N

18

P6

1.21

4N

19

P6

1.21

4N

20

P6

1.21

4N

21

P6

1.21

4N

22

P6

1.21

4N

23

P6

1.21

4

U2

N4

P6

1.21

4N

5

P6

1.21

4N

6

P6

1.21

4N

7

P6

1.21

4N

8

P6

1.21

4N

9

P6

1.21

4N

10

P6

1.21

4N

11

P6

1.21

4N

12

P6

1.21

4N

13

P6

1.21

4N

14

P6

1.21

4N

15

P6

1.21

4N

16

P6

1.21

4N

17

P6

1.21

4N

18

P6

1.21

4N

19

P6

1.21

4N

20

P6

1.21

4N

21

P6

1.21

4N

22

P6

1.21

4N

23

P6

1.21

4

U3

N3

P6

1.21

4N

4

P6

1.21

4N

5

P6

1.21

4N

7

P6

1.21

4N

8

P6

1.21

4N

11

P6

1.21

4N

12

P6

1.21

4N

13

P6

1.21

4N

14

P6

1.21

4N

15

P6

1.21

4N

16

P6

1.21

4N

17

P6

1.21

4N

18

P6

1.21

4N

19

P6

1.21

4N

20

P6

1.21

4N

21

P6

1.21

4N

22

P6

1.21

4N

23

P6

1.21

4

U4

U5

N3

P3

0.90

0N

5

P3

0.90

0N

18

P3

4.50

0N

23

P4

1.87

5

U6

N9

P5

4.50

0N

23

P4

3.11

4

U7

N1

P13

3.37

5N

21

P5

3.82

8N

22

P4

0.99

7N23

P5

0.67

2

U8

N3

P6

4.50

0N4

P6

4.50

0N5

P3

4.50

0N

6

P3

4.50

0N

10

P11

4.50

0N

12

P13

4.50

0N

14

P4

4.50

0N

15

P4

4.50

0N

16

P11

4.50

0N

18

P4

4.50

0N

19

P3

4.50

0

U9

N1

P4

4.50

0N

2

P6

4.50

0N

11

P13

4.50

0N

12

P4

4.50

0N

13

P4

4.50

0N

15

P13

4.50

0N

19

P11

4.50

0N

20

P4

4.50

0N

21

P6

4.50

0N22

P5

4.50

0

U1

0


910

1112

1314

(da

y)

U1

N1

P3

1.21

4N

2

P10

1.12

5N

3

P3

1.21

4N

5

P7

1.12

5N

6

P3

1.21

4N

7

P11

1.12

5N

8

P3

1.21

4N

9

P7

1.12

5N

10

P3

1.21

4N

11

P3

1.21

4N

12

P3

1.21

4N

13

P3

1.21

4N

14

P3

1.21

4N

15

P11

1.12

5N

16

P3

1.21

4N

17

P3

1.21

4N

18

P3

1.21

4N

19

P3

1.21

4

U2

N1

P10

1.12

5N

2

P7

1.12

5N

3

P7

1.12

5N

4

P3

1.21

4N

6

P3

1.21

4N

7

P10

1.12

5N

8

P3

1.21

4N

9

P3

1.21

4N

10

P3

1.21

4N

11

P3

1.21

4N

12

P3

1.21

4N

13

P3

1.21

4N

14

P11

1.12

5N

15

P11

1.12

5N

16

P3

1.21

4N

17

P3

1.21

4N

18

P3

1.21

4N

19

P3

1.21

4

U3

N1

P7

1.12

5N

3

P11

1.12

5N

4

P11

1.12

5N

5

P3

1.21

4N

6

P3

1.21

4N

7

P3

1.21

4N

8

P3

1.21

4N

9

P7

1.12

5N

11

P3

1.21

4N

12

P29

1.12

5N

13

P3

1.21

4N

14

P29

1.12

5N

16

P3

1.21

4N

17

P29

1.12

5N

18

P3

1.21

4

U4

U5

N2

P10

2.29

6

U6

N1

P11

1.25

0N

2

P17

1.35

4N

9

P10

2.25

0

U7

N2

P17

1.98

4N

19

P7

3.50

0

U8

N3

P3

4.50

0N

7

P23

4.50

0N

8

P4

4.50

0N

9

P4

4.50

0N

10

P7

4.50

0N

11

P11

4.50

0N

12

P4

4.50

0N

13

P4

4.50

0

U9

N1

P26

4.50

0N

2

P4

4.50

0N

3

P17

4.50

0N

10

P13

2.23

3N

12

P10

4.50

0N

15

P11

4.50

0

U1

0


13.5

1414

.515

15.5

1616

.517

17.5

1818

.519

(da

y)

U1

N1

P8

1.12

5N

2

P12

1.12

5N

3

P9

1.12

5N

4

P9

1.12

5N

5

P12

1.12

5N

6

P8

1.12

5N

7

P12

1.12

5N

8

P2

1.21

4N

9

P12

1.12

5N

12

P2

1.21

4N

13

P2

1.21

4N

14

P2

1.21

4N

15

P2

1.21

4N

16

P2

1.21

4N

17

P2

1.21

4N

19

P2

1.21

4N

20

P2

1.21

4N

21

P2

1.21

4

U2

N2

P24

1.12

5N

3

P12

1.12

5N

4

P17

1.12

5N

5

P17

1.12

5N

6

P17

1.12

5N

7

P12

1.12

5N

8

P2

1.21

4N

11

P17

1.12

5N

12

P17

1.12

5N

13

P8

1.12

5N

14

P12

1.12

5N

15

P8

1.12

5N

16

P9

1.12

5N

18

P12

1.12

5N

19

P12

1.12

5N

21

P12

1.12

5

U3

N1

P24

1.12

5N

3

P29

1.12

5N

4

P29

1.12

5N

5

P8

1.12

5N

6

P8

1.12

5N

7

P8

1.12

5N

8

P8

1.12

5N

9

P29

1.12

5N

10

P15

1.12

5N

11

P15

1.12

5N

12

P29

1.12

5N

14

P15

1.12

5N

15

P8

1.12

5N

16

P8

1.12

5N

21

P2

1.21

4

U4

N1

P24

3.37

5N

2

P13

3.37

5N

3

P13

3.37

5N

5

P13

3.37

5N

6

P13

3.37

5N

12

P13

3.37

5N

13

P13

3.37

5N

14

P24

3.37

5N

15

P2

3.37

5N

20

P13

3.37

5N

21

P13

3.37

5

U5

N9

P16

4.50

0

U6

U7

N2

P7

1.00

0N

3

P13

2.25

6N

5

P13

0.62

0N

6

P13

2.25

9N

10

P13

7.48

1N

14

P13

0.61

8N

19

P13

2.96

9N

20

P13

2.96

9N

21

P13

3.06

4

U8

N2

P7

3.25

0N

4

P24

4.50

0N5

P11

4.50

0N

6

P4

4.50

0N

8

P4

4.50

0N

9

P2

3.32

9N

13

P4

4.50

0N

14

P13

4.50

0N

15

P24

4.50

0N16

P4

4.50

0N

18

P13

2.35

8

U9

N1

P24

3.66

4N2

P11

4.50

0N

4

P4

4.50

0N

6

P13

2.87

6N

8

P4

4.50

0N

9

P7

1.25

0N

14

P11

4.50

0N

16

P4

4.50

0N

17

P11

4.50

0

U1

0N

19

P13

3.50

0N

20

P13

2.96

9N

21

P4

3.50

0


18.5

1919

.520

20.5

2121

.522

22.5

2323

.524

(da

y)

U1

U2

N2

P27

1.12

5N

3

P22

1.13

8N

4

P22

1.13

8N

5

P22

1.13

8N

6

P22

1.13

8N

7

P22

1.13

8N

8

P22

1.13

8N

15

P2

1.21

4

U3

N8

P27

1.12

5N

12

P27

1.12

5N

21

P29

1.12

5

U4

N2

P13

3.37

5N

4

P13

3.37

5N

5

P13

3.37

5N

9

P2

3.37

5N

14

P27

3.37

5N

15

P27

3.37

5N

17

P23

3.37

5N

19

P29

3.37

5N

21

P12

3.37

5

U5

N9

P17

3.85

3N

10 P11

0.1N12

P12

0.37

1N13

P17

0.70

0N

14

P2

4.50

0N

16

P2

4.29

5

U6

N9

P17

0.75

9N

10

P27

2.25

0N

14

P12

4.29

8N

20

P27

2.27

9N

21

P27

2.21

6

U7

N1

P13

2.85

0N

3 P13

0.29

8N

4

P13

4.50

0N

10

P12

1.86

7N

13

P17

1.43

8N

14

P13

4.50

0N

16 P17

0.21

4N

19

P13

2.74

2N

21 P2

0.20

6

U8

N2

P16

4.50

0N

3

P13

4.20

5N

6

P8

4.50

0N

8

P11

3.50

0N

14

P11

1.12

5N

15

P13

4.50

0N

16

P2

4.50

0N

17

P17

3.46

4N

19

P23

3.37

5N20

P13

2.74

2

U9

N1

P22

4.49

5N

4

P13

1.50

0N

6

P8

4.50

0N8

P13

4.50

0N

11

P25

4.50

0N

14

P23

1.80

8N

17

P12

2.03

6N

19

P12

4.50

0N

20

P22

4.50

0

U1

0N

1

P13

3.50

0N

8

P28

3.50

0N

9

P28

3.50

0N

14

P27

2.25

0N

17

P17

3.50

0N

21

P27

2.27

9


2424

.525

25.5

2626

.527

27.5

2828

.5 (

day)

U1

U2

U3

U4

N2

P4

3.37

5N

8

P4

3.37

5

U5

N2

P4

1.61

7N

3

P20

4.48

8

U6

N3

P27

3.38

1N

5

P4

0.69

9N

9

P12

6.14

1

U7

N1

P15

3.37

5N

2

P18

0.76

9N

3

P20

2.90

5N

4

P19

4.94

8N

5

P9

3.37

5N

6

P12

1.94

8

U8

N1

P8

2.25

0N

2

P4

4.50

0N

3

P4

4.50

0N

4

P4

4.50

0N

5

P20

1.51

8N

6

P20

4.50

0N

7

P19

0.44

8N

9

P21

4.50

0N

10

P12

4.50

0

U9

N2

P28

4.50

0N

4

P18

0.76

9N

5

P20

4.50

0N

6

P21

4.50

0N

7

P19

4.50

0N

8

P9

3.37

5N

10

P12

3.59

0

U1

0N

2

P4

3.50

0N

3

P15

3.37

5N

4

P4

3.50

0N

6

P27

3.50

0N

8

P27

2.09

6N

9

P4

2.07

9N

10

P4

3.50

0


01

2 3

45

67

89

1011

1213

1415

1617

1819

2021

2223

2425

2627

28

U1

U2

U3

U4

U5

U6

U7

U8

U9

U1

0

Figure11: Sketchof productionschedulefor thewholemonth(U1-U10:units)46

Decomposition ModelGeneration & Solution

Scheduling ModelSolution

Results Output

Solutionsatisfactory ?

Whole periodfinished

?

End

Data Input

Start

Scheduling ModelGeneration

"Campaign-Mode"

Database

Production Scheduling

Yes

Yes

No

No

Figure12: Flowchartof theintegrateduserinterface

47

Figure13: Dataformsin theintegratedgraphicaluserinterface

48

Figure14: OrdersFormin theintegratedgraphicaluserinterface

49

Figure15: ProductsForm in theintegratedgraphicaluserinterface

50

Figure16: UnitsFormin theintegratedgraphicaluserinterface

51

Figure17: Type1 Unit SuitabilityFormin theintegratedgraphicaluserinterface

52

Figure18: Operation3 ProcessingTimeFormin theintegratedgraphicaluserinterface

53

Figure19: Type1 Unit TransitionTablein theintegratedgraphicaluserinterface

54

Figure20: Scheduletablesfor campaign-modeproductionobtainedin the integratedgraphicaluserinterface

55

Figure21: CPLEXsolvingthedecompositionmodelin theintegratedgraphicaluserinterface

56

Figure22: Decompositionresultsshown in theintegratedgraphicaluserinterface

57

Figure23: CPLEXsolvingtheshort-termschedulingmodelin theintegratedgraphicaluserinter-face

58

Figure24: A scheduleobtainedin theintegratedgraphicaluserinterface

59

List of Tables

1 Decompositionof thewholeschedulingperiodinto successivehorizons . . . . . . 612 Comparisonsof demandsandproductionthroughproposedschedules . . . . . . . 623 Unit utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

60

TimeHorizon 1 2 3 4 5 6numberof days 5 5 4 5 5 4.5

numberof mainproducts 8 6 10 8 9 11numberof additionalproducts 0 0 0 7 5 0

Table1: Decompositionof thewholeschedulingperiodinto successivehorizons

61

Product Demand ProductionCategory1 325.8 350.6Category2 105 113.5Category3 16.5 17.0Category4 54.5 61.1Category5 36.9 45.3

Overall 538.7 587.5(+9.1%)

Table2: Comparisonsof demandsandproductionthroughproposedschedules

62

Unit U1 U2 U3 U4 U5 U6 U7 U8 U9 U10TimeUsed(hrs) 444.6 507.3 457.1 534.8 357.2 346.9 433.8 650.0 652.4 663.8Ì � � Ç w¼N Ç �Â9Í7Î Ï � LQÐvN (%) 65.0 74.2 66.8 78.2 52.2 50.7 63.4 95.0 95.4 97.1

Table3: Unit utilization

63

continuous-time optimization approach for medium-range production scheduling...

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