Design,Synthesisand Schedulingof Multipur poseBatch Plants

via an EffectiveContinuous-Time Formulation

X. Lin andC. A. Floudas1

Departmentof ChemicalEngineering,PrincetonUniversity, Princeton,N.J.08544-5263,USA

Abstract – Design,synthesisandschedulingissuesareconsideredsimultaneouslyfor multipur-

posebatchplants.A previouslyproposedcontinuous-timeformulationfor schedulingis extended

to incorporatedesignandsynthesis.Processingrecipesarerepresentedby theState-TaskNetwork.

Thesuperstructureof all possibleplantdesignsis constructedaccordingto thepotentialavailability

of all processing/storageunits.Theproposedmodeltakesinto accountthetrade-offs betweencap-

ital costs,revenuesandoperationalflexibility . Computationalstudiesarepresentedto illustratethe

effectivenessof theproposedformulation.Bothlinearandnonlinearmodelsareincluded,resulting

in MILP andMINLP problems,respectively. TheMILP problemsaresolvedusinga branchand

boundmethod.Globallyoptimalsolutionsareobtainedfor thenonconvex MINLP problemsbased

onakey propertythatarisesdueto thespecialstructureof theresultingmodels.Comparisonswith

previousapproachesarealsopresented.

Keywords – DesignandScheduling;MultipurposeBatchProcess;Continuous-TimeFormulation;

MILP; MINLP

1 Intr oduction

Multipurposebatchplantshave beenemployedextensively for themanufactureof many typesof

chemicals,particularlythosewhichareproducedin smallquantitiesandfor which theproduction

processesor thedemandpatternarelikely to change.In theseplants,a wide varietyof products

1To whom all correspondenceshould be addressed. Tel: (609) 258-4595; Fax:(609) 258-0211; E-mail:floudas@titan.princeton.edu

1

canbe producedvia differentprocessingrecipesby sharingavailablepiecesof equipment,raw

materialsand intermediates,utilities andproductiontime resources.The specialinherentoper-

ationalflexibility representsconsiderablecomplexity in the designandsynthesisof suchplants.

Therehave beenseveral publicationsin the areaof designandoperationof multipurposebatch

plants(e.g.,GrossmannandSargent (1979);SuhamiandMah (1982);Birewar andGrossmann

(1989);Papageorgaki andReklaitis(1990a,b,1993);Barbosa-PovoaandMacchietto(1994);Xia

andMacchietto(1997)). In many cases,schedulingstrategiesarenot incorporatedor integrated

very well, which may leadto over-designor under-design. In orderto ensurethat any resource

incorporatedin the designcanbeusedasefficiently aspossible,detailedconsiderationsof plant

schedulingmustbe taken into accountat the designstage.Therefore,it is importantto consider

design,synthesisandschedulingsimultaneously.

All formulationsfor designandschedulingof batchprocessescanbeclassifiedinto two groups

basedonthetimerepresentations.Examplesof discretetimeformulationsarefoundin Grossmann

and Sargent (1979); Suhamiand Mah (1982); Papageorgaki and Reklaitis (1990a,b);Barbosa-

PovoaandMacchietto(1994). GrossmannandSargent(1979)solvedtheproblemof optimalde-

signof sequentialmultiproductbatchprocessesasamixed-integernonlinearprogramming(MINLP)

problem.On thisbasis,SuhamiandMah(1982)studiedtheoptimaldesignof multipurposebatch

plant focusingon a restrictedform of the problemasthe “uniqueunit-to-taskassignment”case.

PapageorgakiandReklaitis(1990a)criticizedmany previousformulationsin omittingkey aspects

of thegeneralmultipurposeplant,suchasalternativeassignmentsof differentequipmentitemsto

eachproducttaskandsharingof the units of the sameequipmenttype amongmultiple tasksof

the sameor differentproducts.They proposeda formulationwhereflexible unit-to-taskalloca-

tions andnon-identicalparallelunits areconsidered.A decompositionstrategy is alsoproposed

to solve the resultingMINLP problemsby Papageorgaki andReklaitis(1990b). Barbosa-Povoa

andMacchietto(1994)presenteda detailedformulationof multipurposebatchplant designand

retrofit basedon the State-TaskNetwork (STN) descriptionandequally-spacedfixed event time

representationproposedby Kondili et al. (1993).

2

In recentyears,attemptshavebeenmadeto createcontinuous-timeformulations.Xia andMac-

chietto(1997)presentedaformulationbasedonthevariableeventtimeschedulingmodelof Zhang

andSargent(1996,1998). A stochasticmethodis usedto solve theresultingnonconvex MINLP

problemsdirectly, insteadof introducinga largenumberof auxiliary variablesandconstraintsto

reducetheMINLP into aMILP.

Ierapetritouand Floudas(1998a,b);Ierapetritouet al. (1999) proposeda novel continuous-

time mathematicalmodelfor thegeneralshort-termschedulingproblemof batch,continuousand

semicontinuousprocesses.It featuresthekey conceptof eventpointsandsetsof specialsequence

constraints.Market demandscanbe specifiedat the endof the time horizonor within the time

horizonwith intermediateduedates.

In this paper, weextendtheformulationto addresstheproblemof integrateddesign,synthesis

andschedulingof multipurposebatchplants.First,adefinitionof theproblemunderconsideration

is stated,and the representationof problemdatais discussed.This is followed by a detailed

descriptionof the proposedmathematicalformulation. Computationalresultsandcomparisons

with apreviouslyproposedformulationarealsopresented.

2 ProblemDefinition

Theintegrateddesign,synthesisandschedulingproblemfor multipurposebatchplantsconsidered

in thispaperis statedasfollows:

Given

� Productionrecipes(i.e., the processingtimes for eachtask at the suitableunits, and the

amountof thematerialsrequiredfor theproductionof eachproduct);

� Potentiallyavailableprocessing/storageequipmentandtheir rangesof capacities;

� Materialstoragepolicy;

� Productionrequirement;

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� Thetimehorizonunderconsideration;

Determine

� Thenumber, typeandsizeof equipmentitems;

� A feasibleoperationalschedule;

- Theoptimalsequenceof taskstakingplacein eachunit;

- Theamountof materialbeingprocessedateachtime in eachunit;

- Theprocessingtimeof eachtaskin eachunit;

soasto optimizea performancecriterion,for example,to minimizethecapitalcostor to maximize

theoverallprofit.

3 ProcessRecipeand Plant SuperstructureRepresentations

Thefirst stageof solvingthedesign,synthesisandschedulingproblemis to developageneralrep-

resentationof theprocessrecipe.In thiswork,weemploy theconceptof State-TaskNetwork(STN)

proposedby Kondili etal. (1993).TheSTNis adirectedgraphwith two typesof distinctivenodes:

thestatenodesdenotedby a circle andthe tasknodesdenotedby a rectanglebox. Figure1 gives

an illustrationof the STN descriptionof a batchprecessnamedBM process,in which two final

products,S5andS6,areproducedfrom two raw materials,S1andS2,throughfour tasks,T1–T4,

involving two intermediatematerials,S3andS4.

In additionto theprocessrecipe,the informationof potentiallyavailablepiecesof equipment

and their suitability for different tasksis usedto constructa superstructureof the plant under

considerationthat includesall possibledesigns.For example,basedon the BM processrecipe

in Figure1 andequipmentdatain Table1, we areableto establisha superstructureof the BM

plant,asshown in Figure2. Decisionsontheplantstructureinvolvethreeprocessingunitsandone

storagevessel.Full connectivity of theprocessingunits/storagevesselnetwork is assumed.

4

4 Mathematical Formulation

To formulatethemathematicalmodelfor integrateddesign,synthesisandschedulingof multipur-

posebatchplants,werequirethefollowing indices,sets,parameters,andvariables:

Indices:

�tasks;� units; � states;

� eventpointsrepresentingthebeginningof a taskor utilizationof aunit.

Sets:

�tasks;

���tasksthatcanbeperformedin unit (j);

��tasksthateitherproduceor consumestate(s);

��processingtasks;

� �storagetasks;

�units;

���unitsthatcanperformtask(i);

���storageunits;

�all involvedstates;

� �statesthatcanonly bestoredin dedicatedstorageunits;

�eventpointswithin thetimehorizon.

Parameters:

� ����� ������ proportionsof state(s)produced,consumedby task(i), respectively;

� ��� ��� ��� ��� ��� constantterm, coefficient andexponentof variableterm of processingtime of

task(i) in unit (j), respectively;

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�� � � �� � � �� � constantterm,coefficientandexponentof variabletermof capitalcostof unit (j),

respectively;

�time horizon;

� priceof state(s);

! � market requirementfor state(s)at theendof timehorizon;

"$# �&%� � "'#)(+*�minimumandmaximumpossiblesizesof unit (j), respectively .

Variables:

,.- �0/ binaryvariablesto determineif unit (j) exists;

� - �0/ positivevariablesthatdeterminethesizeof unit (j);

1324- � � � / binaryvariablesthatassignthebeginningof task(i) ateventpoint (n);

5024- � � � / binaryvariablesthatassigntheutilizationof unit (j) ateventpoint (n);

6 - � � � � � / positive variablesthatdeterminethe amountof materialundertakingtask(i) in

unit (j) ateventpoint (n);

7 - � � � / positive variablesthat determinethe amountof state(s) beingdeliveredto the

marketateventpoint (n);

�98 - � � � / positivevariablesthatdeterminetheamountof state(s)ateventpoint (n);

8 � - � � � � � / time thattask(i) startsin unit (j) ateventpoint (n);

8;: - � � � � � / time thattask(i) finishesin unit (j) while it startsateventpoint (n).

Basedon thisnotationthemathematicalmodelinvolvesthefollowing constraints:

ExistenceConstraints

5<24- � � � />= ,.- �</ �@? �BA � � � A � (1)

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Theseconstraintsexpresstherequirementthataunit canbeutilizedonly if it exists.

Unit SizeConstraints

" # �&%� ,.- �0/C=D� - �0/C= " #E(;*� ,.- �0/ �@? �BA �(2)

Theseconstraintsdeterminethe rangeof the sizeof eachunit. If e(j) equalsone,that is, a unit

exists,thenConstraint(2) correspondsto thelower andupperboundson thesizeof theunit, s(j).

If e(j) equalszero,thens(j) becomeszero.

AllocationConstraints

F�HG9IKJ 1324- � � � /ML 5<24- � � � / �N? �BA � � � A � (3)

Theseconstraintsexpressthat in eachunit (j) andat any eventpoint (n) at mostoneof the tasks

that canbe performedin this unit (i.e.,� A ���

) shouldtake place. If unit (j) is utilized at event

point (n), that is, yv(j,n) equals1, thenoneof thewv(i,n) variablesshouldbeactivated.If unit (j)

is not utilized at eventpoint (n), thenall correspondingwv(i,n) variablestake zerovalues,that is,

noassignmentsof tasksaremade.

CapacityConstraints

6 - � � � � � /O= " #)(+*� 1324- � � � / �N? � A � � �BA ��� � � A � (4)

6 - � � � � � /C=P� - �</ �N? � A � � �QA ��� � � A � (5)

Theseconstraintsexpressthe requirementthat thebatch-sizeshouldbewithin themaximumca-

pacityof a unit (j). If wv(i,n) equalszero,that is, task(i) doesnot take placeat eventpoint (n),

thenthefirst constraintenforcesb(i,j,n) to bezero.If wv(i,n) equalsone,thenthesecondconstraint

7

restrictsb(i,j,n) to bewithin theavailablecapacityof unit (j), s(j).

MaterialBalances

�8 - � � � /MLR�98 - � � �TSVU / S 7 - � � � /XW F�HG9I�Y �

�Z� F�[G]\;^ 6 - � � � � �_S`U /XW F�HG9IZY � � �Z�

F�aGb\c^ 6 - � � � � � /

? �dA � � � A � (6)

where ������ =fe � � ���hg e representthe proportionof state(s) consumedor producedby task(i),

respectively. Accordingto theseconstraintstheamountof materialof state(s) at eventpoint (n)

is equalto thatat eventpoint (n-1) adjustedby any amountsproducedor consumedbetweenthe

eventpoints(n-1)and(n) andtheamountrequiredby themarketateventpoint (n) within thetime

horizon.

StorageConstraints

�98 - � � � /MLPe �@? �dA � � � � A � (7)

Theseconstraintsenforcethat thosestatesthatcanonly bestoredin dedicatedstorageunitshave

to beconsumedby someprocessingtaskor storagetaskimmediatelyafterthey areproduced.

DemandConstraints

F%iGbj 7 - � � � / g ! � �@? �dA � (8)

Theseconstraintsrepresenttherequirementto produceat leastasmuchasrequiredby themarket.

8

DurationConstraints:Processingtask

8 : - � � � � � /kLl8 � - � � � � � /mW � ��� 1324- � � � /XW � ��� 6 - � � � � � /�n ^ J �@? � A �� � �QA ��� � � A � (9)

In Constraints(9), theprocessingtime takesa generallynonlinearform consistingof a fixedterm

anda variableterm dependingon the batch-size.When� ���

equalone,theseconstraintsbecome

linearasaspecialcase.

DurationConstraints:Storagetask

8 : - � � � � � / g 8 � - � � � � � / �o? � A � � � �BA ��� � � A � (10)

8 : - � � � � �qp ( ��� /kL � �N? � A �r� � �BA ���(11)

Theseconstraintsexpressthat the durationof storagetaskscantake any positive valueas long

asthey endat the endof the time horizon. The conceptof storagetaskis introducedto make it

possibleto treatdedicatedstorageconstraintsandprocessingconstraintsin a unifiedandgeneral

way.

SequenceConstraints:

Sametaskin thesameunit

8� - � � � � � W U / g 8 : - � � � � � / �@? � A � � �QA ��� � � A � � �tsL �qp ( ��� (12)

Theseconstraintsstatethat task(i) startingat event point (n+1) shouldstartafter the endof the

sametaskperformedin thesameunit (j) whichhasalreadystartedateventpoint (n).

9

Differenttasksin thesameunit

8 � - � � � � � W U / g 8 : - �vu � � � � / S � -+UOSw1324- �Zu � � /�/? �BA � � � A ��� � � u A ��� � � sL � u � � A � � �tsL �qp ( ��� (13)

Theseconstraintsarewrittenfor tasks(� � � u

) thatareperformedin thesameunit (j). If bothtasksare

performedin thesameunit they shouldbeat mostconsecutive. This is expressedby Constraints

(13) becauseif 1324- � u � � /'L U which meansthat task(� u) takesplaceat unit (j) at eventpoint (n),

thenthesecondtermof theright handsideof (13) becomeszeroforcing thestartingtime of task

(i) ateventpoint (n+1) to begreaterthantheendtimeof task(� u) ateventpoint (n); otherwise,the

right handsideof (13)becomesnegativeandtheconstraintis trivially satisfied.

Differenttasksin differentunits

8 � - � � � � � W U / g 8 : - �vu � � u � � / S � -;UxSy1324- �vu � � /�/? � � � u A � � �BA ��� � � u A ���&z � � sL{� u � � A � � �|sL �qp ( �Z� (14)

Constraints(14)arewritten for differenttasks(� � � u

) thatareperformedin differentunits(� � � u ) but

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

unit (� u ) ateventpoint (n) (i.e., 1324- � u � � /ML U ), thenwehave 8 � - � � � � � W U / g 8;: - � u � � u � � / andhence

task(i) in unit (j) hasto startaftertheendof task(� u) in unit (� u ).

“Zero-wait” condition

8 � - � � � � � W U /C=`8 : - � u � � u � � / S � -~}3Sw1�24- � � � W U / Sw1�24- � u � � /�/? � � � u A � � �QA ��� � � u A ���&z � � A � � �|sL �4p ( �Z� (15)

10

Constraints(15)arewrittenfor differenttasks(� � � u

) thattakeplaceconsecutively with “zero-wait”

conditiondueto storagerestrictionsontheintermediatematerial.Combinedwith Constraints(13)

and(14), theseconstraintsenforcethat task(i) in unit (j) at eventpoint (n+1) startsimmediately

aftertheendof task(i’) in unit (j’) ateventpoint (n) if bothof themareactivated.

TimeHorizonConstraints

8 : - � � � � � /C= � �@? � A � � �BA ��� � � A � (16)

8 � - � � � � � /C= � �@? � A � � �BA ��� � � A � (17)

Thetime horizonconstraintsrepresenttherequirementthatevery taskshouldstartandendwithin

thetimehorizon(H).

Objective: Minimize

F� - �� � ,.- �0/XW �� � � - �0/r�n J / S F

�F% � 7 - � � � / (18)

Theobjectiveis to minimizethecapitalcostsof units,whichconsistof afixedtermandagenerally

nonlinearterm dependingon the sizesof the units, minus profits due to productsales. Other

performancecriteriacanalsobeincorporated.

It shouldbepointedout thatthecasewheredifferentunitssharethesametaskscanbeaccom-

modatedin theaboveformulationby consideringeachtaskin eachunit asadifferenttaskwith the

samefeatures.Weshouldalsonotethatdueto thenonlinearmodelsof processingtimeandcapital

cost,theresultingmathematicalprogrammingmodelis a nonconvex MINLP problem.Therefore,

deterministicglobaloptimizationmethodsareneededto determinetheglobaloptimalsolution.

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5 Computational Studies

Theabove mathematicalformulationis appliedto two examplestaken from Xia andMacchietto

(1997). For eachexample,both linear andnonlinearcasesarestudied. MINOPT, an advanced

modelinglanguageandalgorithmicframework proposedby SchweigerandFloudas(1997)(URL:

http://titan.princeton.edu/MINOPT/),is usedto establishand solve the resultingMILP/MINLP

mathematicalprogrammingproblems.TheMILP problemsaresolvedusingCPLEX,abranchand

boundmethod.For MINLP problems,MINOPT implementsseveralalgorithmsincludingGener-

alizedBendersDecomposition(GBD), OuterApproximationwith EqualityRelaxation(OA/ER),

andOuterApproximationwith EqualityRelaxationandAugmentedPenalty(OA/ER/AP).A de-

tailed accountof the theoreticaland algorithmic issuesfor MINLP problemscan be found in

Floudas(1995).All thecomputationsaredoneonaHP-C160workstation.

BM Plant

Theprocessrecipe,equipmentdataandplantsuperstructurearethosewe discussedin a previous

sessionasBM processandBM plant (seeFigure1, Table1 andFigure2, respectively). Further

informationon theproductionrequirementandrelevantmaterialdataareshown in Table2. The

timehorizonunderconsiderationis 12hours.

In the caseof BMFIX, the processingtimesof processingtasksandcapitalcostmodelsare

providedaslinearforms. Five eventpointsareusedandtheresultingMILP probleminvolves59

binary variables,175 continuousvariablesand332 constraints.It is solved in 0.4 secondsCPU

time andtheobjective functionvalueis Ub���<��� . Theoptimalplantdesignis shown in Figure3, in

whichall thethreeprocessingunitsareselectedwhile thestoragevesselis not. Thecorresponding

optimaloperationalscheduleis representedin Figure4.

In thecaseof BMNON, theprocessingtime andcapitalcostmodelsarebothnonlinear. Four

eventpointsareneededandtheresultingMINLP probleminvolves48 binaryvariables,142con-

tinuousvariablesand262constraints.It is solvedin 0.75secondsCPUtimeandthecorresponding

objective functionvalueis � �����<� � � . Theoptimalplantstructureis thesameasthatin BMFIX, as

12

shown in Figure3. Theoperationalscheduleobtainedis representedin Figure5.

KPS Plant

Theprocessrecipeis shown in Figure6. Two final productsareproducedfrom threeraw mate-

rials throughheating,threereactionsandseparation,involving four intermediatematerials.The

equipmentdataaregiven in Table3. Four processingunits and four storagevesselsareunder

consideration.Full connectivity of the equipmentnetwork is assumed.Theplant superstructure

is thenconstructedasshown in Figure7. Productionrequirementsandrelevantmaterialdataare

givenin Table4. Thetimehorizonunderconsiderationis also12hours.

In the linear caseof KPSLIN, six event pointsareusedandthe resultingMILP problemin-

volves128 binary variables,341 continuousvariablesand877 constraints.It is solved in 22.49

secondsCPUtimeandtheobjective functionvalueis ����}<������� . Theoptimalplantdesignis shown

in Figure8, in whichall four processingunitsarechosenandonly onestoragevesselis selectedto

hold oneof the intermediatematerials.Thecorrespondingoptimaloperationalscheduleis repre-

sentedin Figure10.

In thenonlinearcaseof KPSNON,fiveeventpointsareneededandtheresultingMINLP prob-

lem involves108 binary variables,287 continuousvariablesand722 constraints.It is solved in

7.31secondsCPUtime andthecorrespondingobjective functionvalueis � � e � ����� . Comparedto

thesolutionof KPSLIN, only onereactoris includedin theoptimalplantstructurein thiscase,as

shown in Figure9, which is dueto theincreasedcapitalcostof theotherreactor. Theoperational

scheduleobtainedis representedin Figure11.

Comparisonwith Other Approaches

Table5 shows the resultsof the proposedformulationcomparedwith the resultsfound in litera-

ture. Xia andMacchietto(1997)transformedthe formulationthey presentedinto an alternative

onewithoutgiving thenecessarydetailsof thetransformationandbasedthedatathey providedon

thetransformedone.Therefore,in additionto thereporteddata,thecorrespondingdatawe obtain

accordingto their original formulationarealsopresentedhere,which in our point of view, reflect

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theactualsizeof their resultingmathematicalmodels.It is shown that the formulationproposed

in this paperhasthe following advantages:(i) It givesrise to a simplermixed-integeroptimiza-

tion problemmainly in termsof a smallernumberof binaryvariables.Theproposedformulation

introduces59, 48, 128 and108 binary variablesin BMFIX, BMNON, KPSLIN andKPSNON,

respectively, comparedto 124 in the first exampleand288 in the secondexamplerequiredfor

theformulationof Xia andMacchietto(1997);(ii) Theoptimalsolutionobtainedcorrespondsto a

betterobjectivefunctionvalueandconsequentlyabetterintegrateddesignandschedulingstrategy.

In BMFIX andBMNON, theproposedformulationleadsto anobjective functionvalueof 195.6

and3557.35respectively, which arelower thanthevalueof 197.23and3576.49from theformu-

lation of Xia andMacchietto(1997). In KPSLIN andKPSNON,we alsoachieve bettervalues

of 572.808and490.433comparedto 585.62and495.11;(iii) The computationaleffort required

is significantlyreduced,which makesit very promisingto solve large-scaleindustrialproblems.

It takesonly 0.4 secondsin BMFIX, 0.75 secondsin BMNON, 22.49secondsin KPSLIN and

7.31secondsin KPSNON,respectively, on a HP-C160workstation,while theformulationof Xia

andMacchietto(1997)required1821.19seconds,2998.46seconds,2407.62secondsand7849.23

seconds,respectively, onaSunUltra station-1.

Local versusGlobal Solution

It is interestingtonotethatin thenonlinearnonconvex cases,BMNON andKPSNON,thesolutions

obtainedthroughMINOPT (SchweigerandFloudas(1997)),which is a local MINLP solver, are

alsotheglobaloptimalsolutions.This is provenby solving the lower-boundingproblem,which

givesexactly thesamesolution,that is, thesameobjective functionvalue,plantstructuredesign

andoperationalschedule.Thenonlinearitycomesfrom thevariabletermsin theprocessingtime

model and capital cost model (seeConstraints(9) and the objective function (18)). When the

exponentialconstantsof theseterms,thatis,� ���

and�� �

, aregreaterthanone,thesenonlinearterms

of continuousvariablesareconvex. We cantransformthe original nonconvex MINLP problem

to a lower-boundingconvex problemby splitting the nonlinearequalityconstraint(9) into two

14

inequalityconstraintsandthenrelaxingtheconcaveonewith underestimatingtechniques(Floudas

(2000);Adjimanetal. (1998a,b))to a linearconstraintasfollows:

8 : - � � � � � /C=`8 � - � � � � � /mW � ��� 1324- � � � /XW � ��� - " #E(+*� /�n ^ Ja��� 6 - � � � � � /? � A �� � �BA ��� � � A � (19)

It is foundthatonly theconvex inequalityconstraintis active at theoptimalsolutionpoint of the

resultinglower-boundingconvex problem. In otherwords,theequalityconstraintin theoriginal

problemcanbechangedto the correspondingconvex inequalityconstraintwithout changingthe

optimalsolution,whichtransformsthemathematicalmodelfrom anonconvex problemto aconvex

oneandguaranteesautomaticallytheglobaloptimalityof thesolutionobtained.Suchfastconver-

genceto theglobaloptimalsolutionin only oneiterationis observedinfrequentlyandit represents

anexcellentelementof theproposedapproach.Thisspecialpropertyis relatedto thespecialforms

of theobjective functionandtheconstraints,andtheboundsof therelevantvariables.

6 Conclusions

In this paper, a continuous-timeformulation is proposedfor integrateddesign,synthesisand

schedulingof multipurposebatchplantsonthebasisof apreviouslypresentedshort-timeschedul-

ing model. Two computationalstudiesarepresentedto demonstratetheeffectivenessof thepro-

posedformulation.Thecomputationalresultsarecomparedwith thosein literatureandshow that

theproposedformulationresultsin smallersizeMILP/MINLP mathematicalmodelsprimarily in

termsof binaryvariablesandbetterobjective valuescanbeaccomplishedwith significantlyless

computationalefforts. An interestingpropertyconcerningthe global optimality of the solutions

obtainedin nonconvex nonlinearcasesis alsodiscussed.

15

Acknowledgments

The authorsgratefully acknowledgesupportfrom the National ScienceFoundation,the Mobil

TechnologyCompany, andElf-AtochemCompany.

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17

List of Figures

Figure1 State-TaskNetwork Representationfor BM Process

Figure2 PlantSuperstructurefor BM Plant

Figure3 OptimalPlantStructurefor BMFIX andBMNON

Figure4 GanttChartfor BMFIX

Figure5 GanttChartfor BMNON

Figure6 State-TaskNetwork Representationfor KPSProcess

Figure7 PlantSuperstructurefor KPSPlant

Figure8 OptimalPlantStructurefor KPSLIN

Figure9 OptimalPlantStructurefor KPSNON

Figure10 GanttChartfor KPSLIN (thevaluein parenthesisfor eachunit denotesits sizeif in-

stalled)

Figure11 Gantt Chart for KPSNON(the value in parenthesisfor eachunit denotesits size if

installed)

List of Tables

Table1 EquipmentDatafor BM Plant

Table2 MaterialDatafor BM Plant

Table3 EquipmentDatafor KPSPlant

Table4 MaterialDatafor KPSPlant

Table5 ResultsandComparisons(t: reportedbasedon transformedformulation; o: recounted

basedonoriginal formulation;*: SunUltra station-1; **: HP-C160workstation)

T1

0.6

T2

T3

0.6

S5

1.0

0.4

S2

S1

S3

S4

T4

S6

1.0

1.0

1.0

1.0

0.4

1.0

Figure1: State-TaskNetwork Representationfor BM Process

V5

V1

V2

U1a

U1b

V4

U2

V6

Figure2: PlantSuperstructurefor BM Plant

V5

V1

V2

U1a

U1b

U2

V6

Figure3: OptimalPlantStructurefor BMFIX andBMNON

01

23

45

67

89

1011

12

U1

a (

50

)T

1

19.

200

T1

48.

000

T1

48.

000

U1

b (

50

)T

2

12.

800

T2

32.

000

U2

(8

0)

T3

32.

000

T4

80.

000

T3

80.

000

V4

(N

O)

Figure4: GanttChartfor BMFIX

01

23

45

67

89

1011

12

U1

a (

50

)T

2

22.

400

U1

b (

50

)T

1

33.

600

T1

24.

000

U2

(5

6)

T3

56.

000

T4

40.

000

V4

(N

O)

Figure5: GanttChartfor BMNON

Hea

ting

Pro

duct

1

0.4

Hot

A

0.6 In

tBC

0.5

Fee

d B

0.

5

Rea

ctio

n 2

0.6

Rea

ctio

n 1

IntA

B 0.

8

Pro

duct

2

0.9

0.1

Fee

d C

Impu

re E

0.2

Sep

arat

ion

Rea

ctio

n 3

0.4

Fee

d A

Figure6: State-TaskNetwork Representationfor KPSProcess

V3

H

R2

R1

V4

V5 V

6

V7

St

V8

V9

V1

V2

Figure7: PlantSuperstructurefor KPSPlant

V9

St

V8

V1

V2

V3

H

R1

R2

V6

Figure8: OptimalPlantStructurefor KPSLIN

V9

St

V8

HV

1

V2

V3

R2

V6

Figure9: OptimalPlantStructurefor KPSNON

01

23

45

67

89

1011

12

H

(2

0)

Hea

ting

20.0

0H

eatin

g15

.56

Hea

ting

4.44

R1

(5

0)

Rea

ctio

n 1

30.0

0R

eact

ion

123

.33

Rea

ctio

n 2

38.8

9R

eact

ion

329

.17

R2

(7

0)

Rea

ctio

n 2

50.0

0R

eact

ion

337

.50

Rea

ctio

n 1

6.67

Rea

ctio

n 2

11.1

1

St

(5

0)

Sep

arat

ion

37.5

0S

epar

atio

n29

.17

V4

(n

o)

V5

(n

o)

V6

(13

.3)

IntA

B3.

75

V7

(n

o) Figure10: GanttChartfor KPSLIN

01

23

45

67

89

1011

12

H

(20

)H

eatin

g

20.0

0

R1

(no

)

R2

(70

)R

eact

ion

1

30.0

0

Rea

ctio

n 2

50.0

0

Rea

ctio

n 3

33.3

3

St

(50

)S

epar

atio

n

33.3

3

V4

(no

)

V5

(n

o)

V6

(10

)In

tAB

3.33

IntA

B

3.33

IntA

B

6.67

V7

(no

) Figure11: GanttChartfor KPSNON

Unit Capacity Suitability TaskTimeModel CostModelBMFIX BMNON BMFIX BMNON

Units1a,1b 50-150 T1, T2 }<� e }<� e�W�e � e�e � 6 ��� ��� } exW|e ��� � } e � e�W|e �&� � ��� �Unit 2 50-200 T3 � � e � � e�W|e � e U 6 ��� � ��exW U�� e�� ��e�W U�� e�� ��� �

T4 }<� e }<� e�W|e � e�e � 6 ��� �Vessel4 10-100 S4 – – U�� e�W�e ��U � U�� e�W�e ��U � ��� ���

Table1: EquipmentDatafor BM Plant

State StorageCapacity Price RequirementBMFIX BMNON

S1 Unlimited 0 0 0S2 Unlimited 0 0 0S3 0 0 0 0S4 Vessel4 0 0 0S5 Unlimited 0.04 80.0 40.0S6 Unlimited 0.015 80.0 40.0

Table2: MaterialDatafor BM Plant

Unit Capacity Suitability TaskTimeModel CostModelKPSLIN KPSNON KPSLIN KPSNON

Heater 20-50 Heating U�� exW�e � e�e ��� 6 U�� e�W�e � e�e ��� 6 ��� ��� U e�e � exW|e ��} � U e�e � exW�e ��} �Reaction1 }<� exW�e � e }���� 6 }<� e�W�e � e }���� 6 ��� ���

Reactor1 50-70 Reaction2 }<� exW�e � e }���� 6 }<� e�W�e � e }���� 6 ��� ��� U]� e � exW|e ��� � U]� e � e�W|e �&� � ��� �Reaction3 U�� exW�e � e U ��� 6 U�� e�W�e � e U ��� 6 ��� ���Reaction1 }<� exW�e � e U��� 6 }<� e�W�e � e Ub��� 6 �����Z�

Reactor2 70 Reaction2 }<� exW�e � e U��� 6 }<� e�W�e � e Ub��� 6 �����Z� U]} e � e U]} e � eReaction3 U�� exW�e � e�e � � 6 U�� e�W�e � e�e � � 6 �����Z�

Still 50-80 Separation }<� exW�e � e�e���� 6 }<� e�W�e � e�e���� 6 ��� ��� U]� e � exW|e � ��� U]� e � e�W|e � ��� ��� �Vessel4 10-30 I1(Hot A) – – ��e � exW|e ��U � ��e � e�W�e ��U �Vessel5 10-60 I2(IntBC) – – Ub�<� exW|e ��U � U]�0� exW|e �KU � ��� �Vessel6 10-70 I3(IntAB) – – U e � exW|e ��U � U e � exW|e �KU � ��� �Vessel7 50-100 I4(ImpureE) – – } e � exW|e ��} � } e � exW|e �&} � ��� �

Table3: EquipmentDatafor KPSPlant

State StorageCapacity Price RequirementKPSLIN KPSNON

FeedA Unlimited -0.001 0 0FeedB Unlimited -0.002 0 0FeedC Unlimited -0.0015 0 0Hot A Vessel4 0 0 0IntBC Vessel5 0 0 0IntAB Vessel6 0 0 0

ImpureE Vessel7 0 0 0Product1 Unlimited 0.02 40.0 20.0Product2 Unlimited 0.03 60.0 30.0

Table4: MaterialDatafor KPSPlant

Cost Integer Continuous CPUCase Formulation ($ U e�� ) Variables Variable Constraints (sec)

Xia and 197.23 ��e � }�} � ��� � Ub��}<U���Ub���BMFIX Macchietto(1997) U]} ��� U e � � }�} e��

ThisWork 195.6 59 175 332 e � � ���Xia and 3576.49 ��e � }�} � ��� � }�������� � ���

BMNON Macchietto(1997) U]} � � U e � � }�} e �ThisWork 3557.35 48 142 262 e ����� ���Xia and 585.62 ��} � ��� � U]}�} � } ��e �<����} �

KPSLIN Macchietto(1997) }���� � } e U � � }�� �ThisWork 572.898 128 341 877 }�}<� � �����Xia and 495.11 ��} � ��� � U]}�} � ��� � ����} � �

KPSNON Macchietto(1997) }���� � } e U � � }�� �ThisWork 490.433 108 287 722 �<� � U ���

Table5: ResultsandComparisons(t: reportedbasedon transformedformulation;o: recountedbasedonoriginal formulation;*: SunUltra station-1; **: HP-C160workstation)

Abstract

Design,synthesisand schedulingissuesare consideredsimultaneouslyfor multipurposebatch

plants. A previously proposedcontinuous-timeformulationfor schedulingis extendedto incor-

poratedesignandsynthesis.Processingrecipesarerepresentedby theState-TaskNetwork. The

superstructureof all possibleplantdesignsis constructedaccordingto thepotentialavailability of

all processing/storageunits.Theproposedmodeltakesinto accountthetrade-offs betweencapital

costs,revenuesandoperationalflexibility . Computationalstudiesarepresentedto illustratetheef-

fectivenessof theproposedformulation.Both linearandnonlinearmodelsareincluded,resulting

in MILP andMINLP problems,respectively. TheMILP problemsaresolvedusinga branchand

boundmethod.Globallyoptimalsolutionsareobtainedfor thenonconvex MINLP problemsbased

on a key propertythatarisesdueto thespecialstructureof the involvedproblems.Comparisons

with anotherapproacharealsopresented.