Continuous-Time Optimization Approach for Medium-Range Production Scheduling...
Transcript of 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:
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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.
[Figure2 abouthere.]
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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
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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* +�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©5èÄ^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.
[Figure3 abouthere.]
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.
[Figure4 abouthere.]
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
[Figure5 abouthere.]
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.
[Figure6 abouthere.]
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.
[Figure7 abouthere.]
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.
[Figure8 abouthere.]
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.
[Figure9 abouthere.]
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).
[Figure11abouthere.]
[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.
[Figure12abouthere.]
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.
[Figure13abouthere.]
Theimportantdatainclude:
Ë Schedulingperiod;
Ë Orders,asshown in Figure14,theamount,duedateandpriority for eachcustomerorder;
[Figure14abouthere.]
Ë Inventoriesof bothfinal productsandintermediatematerials;
Ë Processingrecipes,asshown in Figure15, the processingstepsrequiredfor makingeach
productandotherrelatedinformationneedto bespecified;
29
[Figure15abouthere.]
Ë Informationof theunits,asshown in Figure16;
[Figure16abouthere.]
Ë Unit-productsuitabilities,for example,Figure17shows theType1 Unit SuitabilityForm;
[Figure17abouthere.]
Ë Processingtimes/rates,for example,Figure18showstheOperation3ProcessingTimeForm;
[Figure18abouthere.]
Ë Sequence-dependentclean-uprequirements,for example,Figure 19 shows the transition
tablefor theType1 units.
[Figure19abouthere.]
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.
[Figure20abouthere.]
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.
[Figure21abouthere.]
[Figure22abouthere.]
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.
[Figure23abouthere.]
7.5 ResultsOutput and DatabaseUpdate
The solutionto the short-termschedulingmodel is organizedin readableformats,including the
scheduletableandtheGanttChart,andthedatabaseis updatedaccordingly. Figure24 shows the
Ganttchartrepresentationof ascheduleobtainedin theuserinterface.
[Figure24abouthere.]
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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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
Operation1 Operation2
Operation3
Operation1 Operation2
Operation1 Operation3
Operation1 Operation2
Operation1
Operation1 Operation2
Operation2 Operation3
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
Figure6: Detailedschedulefor TimeHorizon2 (U1-U10:units;P3-P13:products.)41
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
Figure7: Detailedschedulefor TimeHorizon3 (U1-U10:units;P3-P29:products.)42
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
Figure8: Detailedschedulefor TimeHorizon4 (U1-U10:units;P2-P29:products.)43
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
Figure9: Detailedschedulefor TimeHorizon5 (U1-U10:units;P2-P29:products.)44
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
Figure10: Detailedschedulefor TimeHorizon6 (U1-U10:units;P4-P28:products.)45
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