Multiobjective optimization of a free radical bulk polymerizationreactor using genetic algorithm

Sanjeev Garg, Santosh K. Gupta*a

Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, Indiaskgupta@iitk.ernet.in

(Received: June 3, 1998; revised: August 17, 1998)

SUMMARY: A multiobjective optimization technique has been developed for free radical bulk polymeriza-tion reactors using genetic algorithm. The polymerization of methyl methacrylate in a batch reactor has beenstudied as an example. The two objective functions which are minimized are the total reaction time and thepolydispersity index of the polymer product. Simultaneously, end-point constraints are incorporated to attaindesired values of the monomer conversion (xm) and the number average chain length (ln). A nondominatedsorting genetic algorithm (NSGA) has been adapted to obtain the optimal control variable (temperature)history. It has been shown that the optimal solution converges to a unique point and no Pareto set is obtained.It has been observed that the optimal solution obtained using the NSGA for multiobjective function optimiza-tion compares very well with the solution obtained using the simple genetic algorithm (SGA) for a singleobjective function optimization problem, in which only the total reaction time is minimized and the two end-point constraints onxm andln are satisfied.

IntroductionThe physical properties of any polymer depend largely onits molecular weight distribution (MWD). From the pointof view of applications, it is desirable to have a highmolecular weight product, often with a narrow MWD (attimes, though, broad MWD products may be more desir-able). Martin et al.1) and Nunes et al.2) showed that nar-rowing the MWD improves the thermal properties, stress-strain relationships, impact resistance, hardness andstrength of the polymer. In order to produce such materi-als in industrial polymerization reactors, we must haveappropriate (optimal) operating conditions. Several stu-dies have been reported on the optimization of polymeri-zation reactors, and these have been reviewed recently byFarber3), Chakravarthy et al.4) and Mitra et al.5) A detaileddiscussion is, therefore, not being provided here. Thethrust in the recent past has been on the optimization ofreactors using multiple objective functions, as well asend-point (product) constraints. It is found that the oper-ating variables for polymerization reactors influence theproperties of the product in conflicting ways, i.e., thesecomplex systems are such that any desirable change inone objective function often leads to a detrimental changein another objective function. A multi-criteria analysis foroptimization of the operating conditions is, therefore,quite important for industrial reactors.

In the present study, we discuss the multiobjective opti-mization of a free-radical bulk polymerization batch reac-tor which produces a desired end-product. The example

system chosen is the polymerization of methyl methacryl-ate (MMA). An important objective function for this sys-tem is the minimization of the final reaction time,tf . Thisleads to higher productivity. The other objective functionis the minimization of the polydispersity index (PDI orQ) of the polymer product. This ensures good physicalproperties of the polymer manufactured. Two end-pointconstraints are also used. The first is to attain a desired(high) value of the monomer conversion,xm. This ensureslow post-reactor processing costs. The other importantend-point constraint is to produce polymer having adesired value of the number average chain length,ln.The temperature history,T(t), alone [rather thanT(t) aswell as an initiator addition history] is used as the ‘con-trol’ or operating variable. Sacks et al.6) studied a similaroptimization problem (but with a single objective func-tion only involving a weighted average of these indivi-dual objectives) over twenty five years ago, using theearly models for the Trommsdorff7) effect and using rela-tively elementary optimization techniques. With morepowerful and robust optimization techniques now avail-able4, 8–12) which can solve problems involving multipleobjectives and end-point constraints correctly, it is appro-priate that these be applied to study such problems, andsee if similar solutions are obtained or not.

This study accounts for the gel (or Trommsdorff)effect7) which is exhibited during the polymerization ofMMA. The free-volume theory of Vrentas and Duda13) isused to model this manifestation of diffusional resistance.

Macromol. Theory Simul.8, No. 1 i WILEY-VCH Verlag GmbH, D-69451 Weinheim 1999 1022-1344/99/0101–0046$17.50+.50/0

a On leave to: The Department of Chemical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore119260. Email: cheskg@nus.edu.sg.

46 Macromol. Theory Simul.8, 46–53 (1999)

Multiobjectiveoptimizationof a freeradical bulk polymerization reactorusinggenetic algorithm 47

Thedetailedtheory andtherateconstantsandparametersrequiredfor describing theTrommsdorff7) effect aregivenin our previouspapers4,14,15), and are not being repeatedhere. An adapted version5) (known as NondominatedSorting Genetic Algorithm, NSGA) of genetic algo-rithm8,9), anartificial intelligencebasedtechnique,is usedfor obtaining optimal temperature histories.Mit ra et al.5)

haveusedthis techniquefor theoptimization of anindus-trial nylon 6 reactor. By formulating a multi-objectiveoptimization problem for the PMMA batch reactor weintend to investigate quantitatively the tradeoffs amongindividual objectivesandtry to generatea wider rangeofalternative operatingpolicies.

FormulationThemathematicalmodel for thepresentstudyis basedonthe general free radical polymerization kinetic schemeshownin Tab.1. It consists mainly of four major steps:initiation, propagation, termination and chain transfer.The mass balance and moment equations for methylmethacrylate (MMA) polymerization in a semi-batchreactorcaneasilybewritten,basedon thekinetic scheme,and are reported elsewhere4,12,14). Theseare not repro-ducedherefor thesakeof brevity. Similarly, thevaluesofthe differentparameters to integratethe model equationsare also extensively reported4,12,14). In general, the statevariableequationscanbewritten in theform:

dx/dt = f(x, u); x (t = 0) = x0 (1)

wherex is thestatevariablevector givenby

x = [I, M, R, k0, k1, k2, l0, l1, l2, fm1]T (2)

andu is thecontrolvariablevector(hereit is a scalar)

u (t) = u(t) = T(t) (3)

The initial valueproblem (IVP) given by the ordinarydifferential equations(ODEs) in Eq. (1) canbeintegratedusing the D02EJFsubroutine of the NAG library for anygiven T(t) and initial values of the statevariables.ThissubroutineusesGear’s technique16) for integrating a setofstiff ODEs.Provision wasmade in thealgorithm for self-adjustmentof theerror-tolerance(TOL in thecode).

The monomerconversion in a semi-batch reactor atanytime, t, is definedas

xm = (1 – M/fm1) (4)

wherefm1 is thetotal monomeraddedtill time t. Themul-tiobjective function optimization problem studiedin thiswork is

Min I [T(t)] 3 [I1, I2]T = [tf , Qf]T (5a)

subjectto

xmf = xmd l TOL1 (5b)

lnf = lnd l TOL2 (5c)

dx/dt = f(x,u) x (t = 0) = x0 (5d)

Tmin f Tf Tmax (5e)

wherexmd andlnd arethe desiredvaluesof the monomerconversionandof thenumberaveragechainlength at thefinal (total reaction) time, tf . TOL1 andTOL2 aretakenas0.005 and 20.00, respectively. The constraints on thevalues,xmf andlnf, are incorporatedin the first objectivefunction, I1, in the form of penalty functions with the(large)weightage factors,w1 andw2

I1 � tf � w1 1ÿ xmf

xmd

� �2

� w2 1ÿ lnf

lnd

� �2

�6�

Minimization of I1 leadsto an increasein the produc-tion capacity through a reduction of tf , while simulta-neously giving preferenceto solutions satisfying the endrequirements.The secondobjectivefunction, I2, also in-corporatesthe violationsof xmf andlnf from their desiredvalueswith thesameweightage factors,w1 andw2

I2 � Qf � w1 1ÿ xmf

xmd

� �2

� w2 1ÿ lnf

lnd

� �2

�7�

Minimization of Qf ensuresa narrow breadth of themolecularweight distribution of the product, and, hence,productwith desiredphysicalproperties,while simultane-ously satisfying theend-pointrequirementsonxm andln.

The two objective functions are usually conflicting innature and, therefore, provide an excellent opportunityfor carrying out multiobjective function optimization.The solution of the multiobjective optimization problem

Tab.1. Kinetic schemefor additionpolymerization

Initiation I gggskd 2R

R + M gggski P1

Propagation Pn + M gggskp Pn+1

Terminationby combination Pn + Pm gggsktc Dn+m

Terminationby disproportionation Pn + Pm gggsktd Dn + Dm

Chaintransferto monomer Pn + M gggskf P1 + Dn

Chaintransferto monomer Pn + S gggsks S+ Dn

via solvent S+ M ggggsfast S+ P1

or

S+ M gggsks Dn + P1

ktc , kf andks aretakenaszeroin thepresent study(bulk polymer-izationof MMA)

48 S.Garg, S.K. Gupta

describedin Eq. (5) is obtainedusing the nondominatedsorting genetic algorithm (NSGA8–10)) adaptedby ourgroupso asto apply for control variableswhich arecon-tinuousfunctionsof time. Details of this adapted NSGAare given elsewhere5). The Appendix provides a shortintroduction. Theoptimization techniqueusually(but notalways) leadsto severalfeasible solutions which satisfythe end-point constraintson xmf andlnf (within specifiedvaluesof the tolerances).Thesesolutions form what isreferredto as a Paretoset. The latter comprises severalnon-inferior or nondominated (optimal) points whichhave the characteristic that if we go from one point toanotheron this set,one objective function improvesbuttheother worsens.

Resultsand discussionThe adapted NSGA codefor multiobjective optimizationwasrun for a PMMA batchreactor for thefollowing case(referredto asthereferencecase)

xmd = 0.94(l0.005) (8a)

lnd = 1850(l20) (8b)

608Cf T(t) f 908C (8c)

Thesevaluesarequite closeto thoseusedby Vaid andGupta17) and Chakravarthy et al.4), and are identical tothoseusedby Garg et al.12) for single objective functionstudies.The samevalues of xmd and lnd are being usedheresothatwe cancomparetheresultsfor thesingle andthe double objective function optimizations.The compu-ter code developed herein was tested using standardchecks5,12). The CPU time requiredfor running the pro-gramfor about30 generationson a DECa 1000 was69 s.Tab.2 givesthe valuesof the severalparametersusedinthis work.

Fig. 1 shows all the feasible solutions (i. e., thosesatis-fying the end-point constraints in Eqs. (8a) and (8b) atdifferent values of Ng, the generation number. In theinitial generations,the feasiblepointsmovearoundin thetf – Qf space,but as the generationsevolve, the feasible pointsmove towardsa uniquepoint. It maybeadded that

there are 61 feasiblepoints in Fig. 1 (Ng = 10 and 30),severalbeingduplicatesor beingalmostindistinguishablefrom the pointsshown.The pointsareshownfor genera-tions 15 to 30 in a highly expanded scale in Fig. 2. Thesolution of our multiobjective function optimizationpro-blem with end-point constraints (Eq. (5)), thus, is auniquepoint (takenaspoint A in Fig. 2) with

tf = 1696.96s xmf = 0.9396

Qf = 11.6275 lnf = 1850.58(9)

Point A has been selectedfrom among the severalpoints in Fig. 2 suchthat the end-point values of xm and

Tab.2. Computational parametersusedin this studya)

Nchr = 70 q = 15Nga = 10 TOL1 = 0.005Ng,max = 30 TOL2 = 20.00NP = 100 Tk

max = + 15.08C; k = 1, 2, ....,Nga

Nsim = 100 Tkmin = –15.08C; k = 1, 2, ....,Nga

Nstr = 7 tf0 = 4000spc = 0.99 w1 = w2 = 0.256106

pm = 0.0 a = 2

a) Parametersfor NSGA describedin the Appendixaswell asin ref.5)

Fig. 1. Qf vs. tf space showing convergenceof the feasiblesolutionsover the generations. For Ng = 10 and 30, 61 finalpointswereobtained (Np = 100),xmd = 0.94,lnd = 1850

Fig. 2. Feasible solutionsover the generationswith expandedscales.PointA is takenastheoptimalsolution

Multiobjectiveoptimizationof a freeradical bulk polymerization reactorusinggenetic algorithm 49

ln lie closestto the desiredvalues.No Paretoset of non-inferior solutions is obtainedin this study. A similar con-clusion (unique solution) is obtained for other combina-tionsof xmd andlnd, asobserved from Tab.3.

Theoptimal temperaturehistory, Topt(t), for theoptimalsolutiondescribed in Eq. (9) (referencecase)is shown inFig. 3. It maybeemphasizedthat in this study, a heat-bal-anceequationis not used.Hence,theoptimal temperaturehistories,Topt(t), in the well-stirredbatchreactormust beimplementedusing appropriate computer-control. Thismay be relatively easyto achieve (using heating from ajacket fluid), at leastduring the early stagesof reaction,whenthe temperatureof the reaction massgoesup rela-tively slowly. However, trying to maintain the tempera-ture of the reactionmassat Topt(t) during the later stagesof reactionwhentheTrommsdorff effect manifestsitself,maynot beastrivial, becauseof theexcessive amountsofthe exothermicheat of reaction released in a relativelyshortperiod of time.Thefocusof thisstudyis to computeTopt(t), and not the constraints associatedwith its indus-trial implementation.

The optimal temperature history correspondingto ourprevious12) optimization study using a single objectivefunctiononly, anddescribedby

Min I [T(t)] 3 tf (10a)

subjectto

xmf = xmd = 0.94 (10b)

lnf = lnd = 1850 (10c)

dx/dt = f(x,T) x (t = 0) = x0 (10d)

608C f Tf 908C (10e)

is alsoshown in Fig. 3. The(unique)optimal solution forEq. (10) wasfoundusingthesimpleGA (SGA)8, 9), as

tf = 1656.56s xmf = 0.9354

Qf = 11.2323 lnf = 1854.99(11)

Theend-point valuesof xmf andlnf in Eq. (11) arequitesimilar to thosein Eq. (9). It is observedthatthetwo tem-perature histories in Fig. 3 are more or less the same,except for some deviation in the early stagesof reaction.Fig. 4–6 show the variations of xm, ln and Q with timefor the optimal solutions for both thesingleandmultiob-jective cases(Eqs.(5) and(7)). Again, the variationsforthe two optimization problemsarenot too significant. Itmay be notedthat the value of Q shootsup from about2.0 to about11.6 in anextremelyshortperiod of time dur-ing which the Trommsdorff effect manifests itself. Thehigh value of Q is becauseof the production of materialhaving very high molecular weights (instantaneousvalues) during this short interval of time (essentially, abimodal molecular weightdistribution is expectedfor thefinal product).

Thetwo interesting results from thepresentstudy, thus,are(i) theuniquenessof theoptimal solutionof themultiob-

jectiveoptimization problem(Eq. (5)), and(ii) the fact that the single optimal solution obtainedfor

the two-objective-function problemis similar to thatobtained in the single objective function problem(Eq. (10)).

In the latter problem, the polydispersity index of theproductis not considered.A considerableamount of con-troversy hasexisted in theopenliteratureon theeffect ofincorporating the PDI in the optimizationproblem usingsomewhat indirect approaches18) in which an objectivefunction is used which incorporatesseveral objectiveswith questionable values of weightage factors. Theseapproachessuffer from a drawback19,20) that certainopti-mal solutions canbemissed, irrespective of thevaluesofthe weightage factorsassociatedwith the severalindivi-dual objectives.This happensif the non-convexity of theobjectivefunction givesrise to a duality gap.To the bestof our knowledge,this is thefirst time thata formally cor-rectprocedurehasbeenusedto solve a problemincorpor-ating objectivefunctionsinvolving the minimization of tf

Tab.3. Effect of different [I ]0, xmd andlnd valueson the opti-mal solution

�I �0mol Nmÿ3

xmd lnd xmf lnf tf /s Qf

15.48a) 0.94 1850 0.9397 1850.6 1697 11.6315.48 0.94 1900 0.8951 1900.8 1697 10.4015.48 0.92 1850 0.9186 1860.3 1616 11.1125.80 0.94 1800 0.9409 1798.6 2182 9.6425.80 0.92 1800 0.9208 1801.4 2020 9.9425.80 0.94 1850 0.9401 1848.0 2626 10.48

a) Referencecase.

Fig. 3. Temperaturehistory for the optimal solution (point Ain Fig. 2). TheSGAhistoryis alsoshownfor comparision

50 S.Garg, S.K. Gupta

and Qf , while ensuring desiredvaluesof xmf and lnf. Itmay be pointed out that Chakravarthyet al.4) had foundsome indications in their study that minimization of tf

alonepossibly ensuresthe minimization of Qf in the pre-senceof end-point constraints on xmf and lnf. They hadnot, however, actually solvedthe problem involving theminimizationof both tf andQf .

ConclusionsA unique solution hasbeenobtainedfor the multiobjec-tive optimizationproblemdescribedin Eq. (5). No Paretosethasbeenobtained.Temperature history hasbeenusedasthe control variable while minimizing Qf aswell as tf ,while constraining xmf andlnf to lie at desiredvalues.Anadapted NSGA technique has been used to obtain theoptimal solution. This is the first time in the openlitera-

ture that a rigorously correctprocedure hasbeenusedtostudy this interesting multiobjective optimization pro-blemin polymerreactionengineering.

Nomenclature

Dn deadpolymermoleculehavingn repeatunitsI vectorof objectivefunctionsI molesof initiator at anytime t (in mol)I1, I2 objectivefunctionskd, kp, kt rateconstantsfor initiation, propagationand

termination in presenceof the gel and glasseffects(in s–1, or m3 N mol–1 N s–1)

M molesof monomerin theliquid phase(in mol)Nchr total numberof binarydigits in chromosomeNg generationnumberNga numberof u valueswhich GA generatesNp numberof chromosomesNsim numberof u valuesafter interpolationNstr numberof binarydigits representingeachof

thecontrolvariablevaluesPn growingpolymerradicalhavingn repeatunitspc crossoverprobabilitypm mutationprobability

Q polydispersityindex 3�k2 � l2��k0 � l0��k1 � l1�2

" #q desirednumber(approx.)of Paretopoints

requiredto begeneratedR molesof primaryradical;universalgasconstant

(in atm N m3 N mol–1 N K–1)T temperatureof thereactionmixtureat time t

(in K)TOL1, TOL2 allowedtoleranceson xmf andlnf, respectivelyt time (in s)tf final reactiontime (in s)u controlvector(scalar, u, in this work)w1, w2 weightagefactors

Fig. 4. Variation of monomer conversion with time for theoptimal temperaturehistoriesshownin Fig. 3

Fig. 5. Variationof thenumberaverage chainlengthwith timefor thetemperaturehistoriesshownin Fig. 3

Fig. 6. Variation of the polydispersity index with time for thetemperaturehistoriesshownin Fig. 3

Multiobjectiveoptimizationof a freeradical bulk polymerization reactorusinggenetic algorithm 51

x vectorrepresentingstatevariablesxm monomerconversion(molar)at time t

Greekletters

a exponentcontrollingthesharingeffectkk kth (k = 0, 1, 2, ...) momentof live polymer

radicals(Pn)

3Xvn�1

nkPn

" #(in mol)

lk kth (k = 0, 1, 2, ...) momentof dead(Dn) poly-merchains

3Xvn�1

nkDn

" #(in mol)

ln numberaveragechainlengthat time tfm1 netmonomeraddedtill time t

Subscripts/Superscripts

d desiredvaluesf final values(at t = tf )max maximumvaluemin minimumvalue0 initial value(at time t = 0)

Appendix: Detailsof the nondominatedsortinggeneticalgorithm 5)

Fig. 7 showsa flowchart of the algorithm (NSGA) usedin this work. More details areprovidedin ref.5)

1. At generationnumber, Ng = 0, a population havingNp members(calledchromosomes) is generated(initializepopulationin Fig. 7). Eachchromosomein thepopulationcarriesthe information of one digitized control variablehistory [digitized setof valuesof the temperature,u(t) 3T(t)]. We discretizeour control variablehistory, T(t), intermsof Nga equispacedpointsin 0 f t f tf 0 (tf 0, aninitialestimate of tf , is to be supplied).Thus, eachof the Np

chromosomes(calledstrings)comprisesof a sequenceofNga numbers(calledsubstrings).Eachof thesesubstrings,in turn, comprises a set of Nstr binary numbers(0 or 1).Eachchromosome,therefore,hasNchr = Nga6Nstr binarydigits. TheNchr individualbinaries in eachof theNp chro-mosomesare generated using a randomnumber genera-tion subroutine.

The complete binary string (sequence of Nchr binaries)of the i th chromosome, when decoded into real numbers,u�i�P;k , and interpolated(mapped)betweenthe upper (u fumax) andlower (u F umin) boundsof thecontrol variables,u, at that location, gives a digitized u�i�P -history (a set ofNga real values), �u�i�P;1; u

�i�P;2; :::; u

�i�P;Nga�, representinga T(t)

history. Thus, there is a set of Np chromosomesin theinitial population, each representing a digitized uP(t)[3T(t)] history, andeachappropriatelycodedin theformof a stringof Nchr binaries.

The decoded and adaptively mapped discretizedvalues,u�i�P;k , arecurve-fittedpiece-wise(splines)to obtaina continuous function, U�i�P �t�. A piece-wise cubic Her-mite subroutine (E01BFF from the NAG library) is usedto do this. This continuousfunction is againdigitized togive Nsim (FNga) valuesof the control variable, [U�i�P;1 ; l =1, 2,...,Nsim].

Thesemorecloselyspaced,discretizedvaluesof U�i�P �t�are fed to the simulation package, D02EJF (of NAGlibrary) which integratesthestatevariableequations (Eq.(1)) startingwith the given initial conditions and termi-nating at thestopping condition, xmf = xmd. The valuesofthe two objectivefunctions,I �i�1 and I �i�2 [at the final reac-tion time t = tf , see Eqs.(6) and(7)], arecomputedfor alltheNp chromosomes.

Oneadditional point needsto beemphasized. Thecom-puter codes involving GA usually maximize a fitnessfunction, F�i�m , rather than minimize objective functions,I �i�m , m = 1, 2. Hence, we define fitnessfunctionsto con-vert the minimization problem to an equivalent maximi-zation problemasfollows:

F�i�1 3 1=�1� I �i�1 �F�i�2 3 1=�1� I �i�2 � �12�

All the feasible points from among the Np chromo-somesareidentified (for plotting) at this stage.Thefeasi-ble pointsor chromosomesarethosesatisfying Eqs. (5b)and(5c).

Fig. 7. A flowchart of the adapted NSGA. The numbersinsomeof theboxescorrespondto thesectionsin theAppendix

52 S.Garg, S.K. Gupta

2. A chromosome, i1, is said to be dominated byanotherchromosome,i2, (for thepresentproblemof mini-mizationof I or maximizationof F), if

F�i1�1 a F�i2�1 �a�

aswell as

F�i1�2 a F�i2�2 �b�

then

i1 is dominatedby i2 (c) (13)

We testeachof the NP chromosomesin the populationagainstall othersto sort out all dominatedchromosomes.As soonasa chromosomeis found to be dominated,it isnot checkedfor dominancewith any other chromosomein the population. When all chromosomeshave beencheckedfor dominance,andall dominatedchromosomeshave been identified, the rest of the chromosomesaregiven a front number, FRONT= 1. Thesechromosomeshaving FRONT = 1 are called nondominated chromo-somes.

3. All nondominatedchromosomesarethenassignedadummyfitnessvalue,F�1, equalto NP.

4. Thereafterthesedummyfitnessvaluesaremodifiedaccording to the sharing proceduredescribed in item 6below, to assigna sharedfitness value.Sharing is done tomaintaindiversityin thenondominatedchromosomes.

5. In order to identify chromosomes for other fronts,we temporarily discardall nondominatedchromosomes.Theremaining chromosomesareagaincheckedfor domi-nanceusing Eq. (13) and new nondominated chromo-somesaresortedandgiven a front number, FRONT= 2.Again, the new nondominatedchromosomes(in FRONT2) aregiven a dummy fitnessvalue, F�2, which is slightlysmaller thanthe lowestof theshared fitnessvaluesof thepreviousfront. The sharingof the dummy fitnessvaluesis performedagain, anda sharedfitnessvalue is assignedto each nondominated chromosome.This procedureiscontinued until all NP chromosomes have beengiven afront number.

6. Sharing: Sharing is performed amongthe membersof the i th front (having ni members) using the followingprocedure:

(a) For eachchromosome,j, in front i, the dimension-less distance,djk , of this chromosomefrom any otherchromosome, k, (including j) in the (same)front is calcu-latedusing

djk �XNga

i�1

��u�j�P;i ÿ u�k�P;i �=�umaxP;i ÿ umin

P;i ��2( )1=2

�14�

(b) We calculatethenichecount, mj , using

mj �Xni

k�1

Sh�djk� �15�

where

Sh�djk� � 1ÿ djk

rshare

� �a

; if djk a rshare

0; otherwise

8<: �16�

rshare is givenby

rshare� 12q1=�Nga�1� �17�

In Eq. (17), q is the number of Paretooptimal pointsdesired(we have usedq = 15 in our study). The para-meter, a, is anexponent which controlsthesharingeffect(we haveuseda = 2 in thepresentstudy).

(c) The dummyfitness,F�i , of eachchromosome,j, infront i, is modified by dividing F�i by the chromosome’sniche count,mj , to calculate thesharedfitnessvalue, F�ij 9,asfollows:

F�ij 9 �F�imj

�18�

7. The stochastic remainder roulette wheel selectionprocedureis usedon thesharedfitnessvalues,anda mat-ing pool of NP chromosomesis generated. This procedureinvolves proportionate selection,wherefirst the numberof copiesmade of eachchromosomeis equalto the inte-ger part of the valueof F�ij 9=F

�ij 9. Here,F

�ij 9 is the average

of theshared fitnessvalues of all theNP chromosomesinthe population. Additional copies of the j th chromosomein the i th front (to makea total of NP in the mating pool)aremadethereafter, usinga roulettewheelwith probabil-ity proportional to thefractionalpartof F�ij 9=F

�ij 9.

8. After the mating pool is created, crossover takesplace to producethe new population(next generation).This operation takesplace at the chromosome(binary)level. Two chromosomesarerandomly selectedfrom themating pool, a crossing site is selected(randomly again),and portions of the chromosomesbefore and after thecrossingsite are exchanged.For example,for seven-bitchromosomeswith crossing siteafter the third binary, thecrossoveris describedby thefollowing:

1 0 0 | 1 1 1 1 1 0 0 0 1 0 0ggggs

1 1 0 | 0 1 0 0 1 1 0 1 1 1 1 (19)(old generation) (newgeneration)

While performing crossovers,only pcNP chromosomesare crossed, the remaining being left untouched (pc isreferredto asthecrossover probability).

9. Another operation,called mutation, is also usedtoimprove the next generation. The mutation operator

Multiobjectiveoptimizationof a freeradical bulk polymerization reactorusinggenetic algorithm 53

changesa binarynumber from 1 to 0 or vice versa, with aprobability pm. This operationis carriedout for eachofthebits in thepopulation,againusing appropriaterandomnumbers.The needfor mutation leadsto a local searcharoundthe currentsolution andhelpsmaintain the diver-sity of thepopulation.

This completesonegeneration of NSGA.Thesesets ofoperationsarecarriedout from onegeneration to thenextuntil the number of generationsequals the maximumnumber, Maxgen, specifiedat thestarting of the programasaninput parameter.

Acknowledgement:Financial supportfrom the DepartmentofScience and Technology, Government of India, New Delhi[through Grant III-5(59)/95-ET] is gratefully acknowledged.The correspondence author also gratefully acknowledges thehospitalityandkindnesses(andthe excellentinfrastructure)thathe is enjoyingasa Visiting Professorat the National Universityof Singapore, wherethis paperwasrevised.

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