dynamic optimization of a copolymerization reactor using tabu search

Dynamic optimization of a copolymerization reactor using tabu search

P. Anand a, M. Bhagvanth Rao b, Ch. Venkateswarlu c,n

a Chemical Engineering Sciences Division Indian Institute of Chemical Technology Hyderabad 500 007, Indiab Anurag group of Institutions Hyderabad 501301, Indiac Chemical Engineering Department Padmasri Dr BV Raju Institute of Technology Narsapur 502313, India

a r t i c l e i n f o

Article history:Received 18 July 2013Received in revised form12 July 2014Accepted 18 July 2014This paper was recommended forpublication by Dr. A.B. Rad

Keywords:Dynamic optimizationPolymerizationSimulationTabu searchIterative dynamic programming

a b s t r a c t

A novel multistage dynamic optimization strategy based on meta-heuristic tabu search (TS) is proposedand evaluated through sequential and simultaneous implementation procedures by applying it to asemi-batch styrene–acrylonitrile (SAN) copolymerization reactor. The adaptive memory and responsiveexploration features of TS are exploited to design the dynamic optimization strategy and compute theoptimal control policies for temperature and monomer addition rate so as to achieve the desired productquality parameters expressed in terms of single and multiple objectives. The dynamic optimizationresults of TS sequential and TS simultaneous implementation strategies are analyzed and compared withthose of a conventional optimization technique based on iterative dynamic programming (IDP). Thesimulation results demonstrate the usefulness of TS for optimal control of transient dynamic systems.

& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Batch and semi-batch reactors are well suited for the production ofpolymers of varying grades whose quality is assessed in terms ofstrength, stiffness and processability. The properties of the polymersproduced in these reactors are closely related to the operatingconditions specified for the reactors. Ensuring product quality is amajor issue in a polymerization process since the molecular ormorphological properties of a polymer product strongly affects itsphysical, chemical, thermal, rheological and mechanical properties aswell as polymer applications [1]. In free radical copolymerization,different reactivities of comonomers can cause composition driftsunless a more reactive monomer is added to the reactor continuouslyto maintain a constant mole ratio. It is well known that two styrene–acrylonitrile (SAN) copolymers differing more than 4% in averageacrylonitrile content are incompatible and result in products with poorphysical and mechanical properties [2]. Significant variations in batchto batch copolymerization can also result in off-specification products.The copolymer composition influences the end use properties of thefinal product with respect to its flexibility, strength and glass transitiontemperature, where as the molecular weight (MW) and molecularweight distribution (MWD) affects the important end use propertiessuch as viscosity, elasticity, strength, toughness and solvent resistance.

The determination of the open-loop time varying control policiesthat maximizes or minimizes a given performance index is referred to

as optimal control/dynamic optimization. These optimal controlpolicies that ensure the satisfaction of the product property require-ments and the operational constraints can be calculated off-line, andare implemented on-line such that the system is operated in accor-dance with these control policies. There has been tremendous intereston the optimal control/dynamic optimization of polymerization reac-tors. Most of the studies on optimal control of polymerization reactorsare based on classical methods of solution. When the process model iscomplex and its dimension is large, it becomes tedious to compute theoptimal control policies by conventional optimal control techniques.For problems, where the Hamiltonian is a linear function in control,the optimal control problem becomes singular and the classicalanalytical methods are unable to provide a complete solution [3].Pontryagin's maximum principle is a classical control technique thathas been widely used to solve the optimal control problems ofpolymerization reactors [4–6]. However, this method requires goodinitial guess for the control inputs and the rate of convergence of thesolution is very sensitive to this guess. In case of complex systems, thesearch space quite often becomes very narrow and choosing the initialguess for the search region requires tedious effort. Chang et al. [7] useda two step method to compute the reactor temperature and timeprofiles to obtain a polymer with a prescribed molecular weightdistribution in a free radical polymerization reactor. The parametersearch of this method requires complicated numerical calculations tosolve the nonlinear algebraic equations. Salhi et al. [8] employedcontrol vector parameterization method to find the optimal jackettemperature profile to maximize the productivity in a batch emulsioncopolymerization reactor of styrene and α-methyl styrene. The control

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n Corresponding author. Tel.: þ91 8458 2222000; fax: þ91 8458 222002.E-mail address: [email protected] (Ch. Venkateswarlu).

Please cite this article as: Anand P, et al. Dynamic optimization of a copolymerization reactor using tabu search. ISA Transactions (2014),http://dx.doi.org/10.1016/j.isatra.2014.07.014i

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vector parameterization technique requires a set of trial functions withweights to pre-specify the functional form of the control and a deepphysical insight into the process behavior. Among the derivative freemethods, iterative dynamic programming (IDP) is a popular methodthat has been applied to solve different optimal control problems [9–12]. Vicente et al. [13] employed iterative dynamic programming tocompute the time optimal monomer and chain transfer agent feedprofiles to attain the desired copolymer composition and molecularweight distribution in a semi-batch methylmethacrylate reactor.Though this method is reliable in finding the global optimum andcan be applied to problems where the objective function is non-differentiable, it has low efficiency and computationally time consum-ing. Of recent is the class of stochastic optimization methods derivedfrom the principles of natural phenomena that are becoming increas-ingly popular to solve complex optimization problems due to theirflexibility and ease of operation. Various stochastic search andoptimization algorithms such as differential evolution (DE) [14–16],genetic algorithm(GA) [17–20], simulated annealing(SA) [21,22] andant colony optimization (CO) [23] have been successfully employed toachieve the global optimum of complex engineering problems.

Tabu search (TS) is a meta-heuristic approach and it is one of themost efficient optimization techniques that incorporates adaptivememory and responsive exploration to overcome the limitations oflocal optimality. The use of adaptive memory enables TS to “learn” andcreates a more flexible and effective search strategy than “memoryless” methods such as SA and GA. TS differs from other stochasticoptimization techniques by maintaining lists of previous solutions byusing a memory that helps to guide the search process. It exploits thememory to generate a sequence of progressively improving solutionsthrough repetitive modification of the current solution. TS uses aneighborhood search approach to explore the search space thatfacilitates escaping the solutions from local optima. The memory inTS allows to drive the method forward to discover regions that harborone or more solutions better than the current best. A set ofcoordinated strategies such as intensification and diversificationemployed in TS allow to explore the search space more thoroughly,thus helping to avoid becoming stuck in local optima. TS originallydeveloped by Glover and Laguna [24] has now become an establishedsearch procedure and has been successfully applied to solve a widespectrum of optimization problems [25–27]. The theoretical aspects ofTS has been investigated by various researches [28,29]. TS has beenrecently applied to solve design and synthesis problems in chemicalengineering [30,31]. However, the potentiality of TS has not beenexplored to solve dynamic optimization problems in chemical/poly-merization reactors.

This study explores the use of tabu search (TS) as a tool fordynamic optimization of a polymerization reactor. A multistagedynamic optimization methodology based on TS is presented andevaluated by applying it to a semi-batch reactor producing styrene–acrylonitrile (SAN) copolymer. In SAN copolymerization, styrene andacrylonitrile are the monomers, and xylene and azobisisobutyroni-trile (AIBN) are the solvent and initiator, respectively. Due to thedifferent reactivities of the comonomers, composition drift occursunless a more reactive monomer is added to the reactor in order tomaintain a constant mole ratio. It has been reported that the SANcopolymers differing by more than 4% in the average acrylonitrilecontent are incompatible resulting in poor physical and mechanicalproperties. The control objective is to maintain the copolymercomposition and the molecular weight as constants during thecourse of polymerization by manipulating the reactor variables suchas the monomer addition rate and reactor temperature. The SANcopolymerization is commercially important and optimal control isnecessary to maintain the copolymer composition and molecularweight distribution at their target values. Thus this system forms anideal test bed for design and implementation of the proposedmultivariable open loop optimal control strategy. In SAN polymer-ization, the copolymer composition, MW, MWD and PD are theproduct quality parameters. The objective is to find the optimalcontrol policies for monomer addition rate and reactor temperatureto produce a polymer with the desired copolymer composition andmolecular weight distribution. When a polymer quality parameter iscontrolled by one manipulated variable, the uncontrolled productquality parameter may deviate from its desired specification and thisrequires the need for simultaneous optimization of more than oneobjective. In this work, TS is designed and implemented to deter-mine the optimal control policies that satisfy the individual andmultiple objectives of SAN copolymerization reactor. Multistagedynamic optimization by TS is carried out by using sequential andsimultaneous implementation strategies. The results of TS strategiesare compared with those of another widely used discrete multistagedynamic optimization strategy, namely, iterative dynamic program-ming (IDP). The multistage dynamic optimization strategies basedon TS and IDP to compute the optimal control policies of our workare different than those reported in recent literature [32,33]. In thework of Pahija [32] batch cycle time has been discretized into stagesand optimal control policies are determined by using dynamicoptimization methods based on optimization package solvers. Pisti-kopolous [33] presented a parametric programming approach fordynamic optimization where present states and future decisionvariables are considered as parameters and the present decisions

Nomenclature

If initiator concentration in feedkd initiator decomposition rate constantkfij chain transfer rate constant of species i, jkpij propagation rate constant species i, jktcij combination termination rate constant of species i, jktdij disproportionation termination rate constant of

species i, jMif monomer concentration in reaction in feed of species iP total growing polymer concentration of type-1 [mol/l]Pi moment of total number MWD of radicals of type-1Q total growing polymer concentration of typeQi moment of total number MWD of radicals of species irij monomer reactivity ratioui manipulated variable of species iui manipulated variable of species iV reactor volume

wi molecular weight of monomer of speciesL length of time stepN number of grid pointsP number of time stagesq pass numberr region size used for allowable controlR number of allowable values for control used at eachtf residence time or batch timexi state variablex state vectorγ region contraction factorη region restoration factorϕ molar ratio of monomerϕf monomer mol ration in feed streamϕs desired value of molar ratio of monomer in reaction

mixtureϕt cross termination factor

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are used to represent the optimization variables. Before implement-ing the TS for dynamic optimization of copolymerization reactor, itsefficacy is tested by applying it to five test functions of differentcomplexities. In all these cases, the method has achieved globalminimum in respective objective functions while determining thefunction variables exactly. The implementation results of TS for thetest problems are given in Appendix B.

2. Multistage dynamic optimization

Dynamic optimization is increasingly used in batch and semi-batch process operations, in which the process variables undergosignificant changes. The major objective in these operations is todetermine the time varying open-loop control policies that max-imize or minimize the objective function specifying the systemperformance as a function of process variables and their changes.Optimizing the objective function means, for example, eitherachieving a desired product quality or maximizing the productyield of a batch process. The general open-loop optimal controlproblem with fixed terminal time, considering a lumped para-meter batch/semi-batch process can be stated as follows.

Find a control vector u(t) over tf [to, tf] to maximize (minimize)a performance index J(x,u):

J x;uð Þ ¼Z tf

t0ϕ xðtÞ;uðtÞ; t½ �dt ð1Þ

Subject to

_xðtÞ ¼ f ðxðtÞ;uðtÞ; tÞ; xðt0Þ ¼ x0 ð2Þ

h xðtÞ;uðtÞ½ � ¼ 0 ð3Þ

g xðtÞ;uðtÞ½ �r0 ð4Þ

xLrxðtÞrxU ð5Þ

uLruðtÞruU ð6Þ

where J is the performance index, x is the vector of state variables, u isthe vector of control variables. Eq. (2) is the system of ordinarydifferential equations with their initial conditions, Eqs. (3) and (4) arethe equality and inequality algebraic constraints and Eqs. (5) and (6)are upper and lower bounds in the state and control variablesrespectively. For multistage dynamic optimization, the optimal controlproblem is considered by dividing the entire batch duration into finitenumber of time instants referred to discrete stages. The controlvariables and the corresponding state variables that satisfy theobjective function are evaluated in stage wise manner by usingsequential or simultaneous solution methods. The sequential andsimultaneous implementation methodologies for multistage dynamicoptimization of a nonlinear process are presented as follows.

2.1. Sequential implementation methodology

Discretize the process into N stages. Define fi, ui and xi as theobjective function, control vector and state vector, respectively, forstage i. Here, the procedure adopted for solving the optimalcontrol problem using tabu search is similar to the dynamicprogramming based on the principle of optimality [34]. Thisprocedure is briefed in the following steps:

1. The optimum value of the objective function, f1[x1] for stage1 driven by the best control vector u1 along with the state

vector x1 is represented as

f 1 x1� �¼min

u1f 1o x1;u1� � ð7Þ

2. The value of the objective function, f2[x2] for stage 2 determinedbased on the best control vector u2 along with the state vectorx2 is given by

f 2 x2� �¼ f 1 x1

� �þminu2

f 2o x2;u2� � ð8Þ

3. Recursive generalization of the above procedure for the kthstage is represented by

f k xkh i

¼ f k�1 xk�1h i

þminuk

f ko xk;ukh i

ð9Þ

In this procedure, f ko represents the performance index of stage k.

2.2. Simultaneous implementation methodology

Discretize the process into N stages and define fi, ui, xi (i¼1, …,N) as the objective function, control vector and state vector,respectively, for all stages. The simultaneous implementation oftabu search procedure involves the following iterative steps.

1. Set iteration j¼0 and initialize ui (i¼1, …, N), for all stages.2. The optimum objective function values for all stages, f i [xi]

(i¼1, …, N) driven by the best control vectors, ui (i¼1, …, N) ofthe respective stages along with their state vectors, xi (i¼1, …,N) are evaluated as

f i xih i

¼ minui ;i ¼ 1;::::;N

f io xi;uih i

ð10Þ

where f ioði¼ 1; ::::;NÞ represents the performance index ofeach stage.

3. Set j¼ jþ1, go to step 2 and repeat the procedure based on thepreviously updated values of ui (i¼1, …, N).

4. Stop the iterations when the convergence in objectives isobtained.

3. Tabu search algorithm

Tabu search (TS) is a meta-heuristic problem solving approachthat incorporates adaptive memory and responsive exploration,which makes it to penetrate complexities that often confoundalternative approaches. The adaptive memory feature of TS allowsthe implementation of procedures that are capable of searchingthe solution space economically and effectively. Responsiveexploration integrates intelligent search mechanisms to exploitgood solution features while exploring new promising regions.

TS begins by choosing an initial solution xo. The neighborsolutions are generated by modifying the existing solution througha sequence of moves. The best new neighbor, xn is used as thestarting point for the next iteration unless it is in Tabu list. Thus,even if no neighbor solutions are better than the initial solution,the best solution is still chosen as the starting point for the nextiteration. A record of the best solutions ever found, xn is separatelymaintained. In addition, the adaptive memory in tabu lists guidesthe search process by taking the advantage of historical informa-tion. This memory enables TS to make strategic choices andachieve responsive exploration. The flow chart of the TS algorithmis shown in Fig. 1. The elements involved in tabu search procedureare discussed below.

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3.1. Neighbors generation and neighborhood search

The search procedure in TS requires to find the global optimumof a function f(x) over a solution space X which contains all of thepossible solutions x € X. This requires defining a neighborhoodstructure with respect to the solution space and an initial solution.The initial solution can be generated from preliminary heuristics orfrom the knowledge of the process to be studied. Moves are definedto modify the current solution to form a neighborhood of possiblenext solutions. Thus for each neighbor xi, a neighborhood N(xi) isdefined such that N(xi)CX. The number of solutions that TS visits inthe search process is the product of number of neighbor solutions ateach iteration, N(xi), and the number of iterations, k. In general, kshould be sufficient enough to thoroughly search the solution space.On each iteration of the search, the function is evaluated for all x € N(xi). The entire neighborhood is searched and an improving move ischosen by selecting the move xi which most improves the value of f(xi) and that move is labeled as xiþ1 (denoted by xn in Fig. 1). Thesearch moves to the next iteration, by looking at the neighborhoodof the accepted move, N(xiþ1). Thus TS algorithm starts searchingwith a present solution and constructs a set of feasible solutionsfrom the present one based on the neighborhood by using a historyrecord of the search.

3.2. Tabu list

A tabu list T is introduced to avoid cycling of the previously visitedsolution associated with the best new neighbor, xn. TS makes use ofthe tabu list to force the search away from solutions selected inrecent iteration. Thus tabu list keeps information of the last visitedsolutions. The most recent T moves are included into the tabu list andthe tabu list is changed dynamically as the search process goes on.Memory stored in tabu list helps to escape local optimal solutionsand to prevent cycling. The information in tabu list helps to guide themove from the current solution to the next solution. Tabu list isupdated in each iteration to keep track of the search process.

3.3. Short term memory and long term memory

Tabu list is built such that it stores information in the form ofa recency based short term memory (RSM) and a frequency basedlong term memory (FRM). In the search process, when the bestneighbor is not better than the current solution, it is classified astabu and added to the recency based tabu list. As new solutions areadded, older solutions will be released from the bottom. Thus, therecency based tabu list stores the most recently visited solutionsand prevents revisiting unpromising solutions for a specifiednumber of iterations. Long term memory is dependent on thefrequency that a solution has been visited. Once the maximumnumber of elements in the frequency based tabu list has beenattained, the solution with the smallest frequency index will bereplaced. Long-term memory complements short-term memory inselecting future moves. It enables TS to define a region that ishistorically attractive and deserves to be explored better.

3.4. Intensification and diversification

Intensification and diversification strategies are used to gen-erate neighbors that provide a high probability of finding optimalsolution based on the information in tabu list. Intensificationstrategies are employed to search promising areas more thor-oughly around areas that have been historically found good. Thesestrategies are commonly implemented based on long term mem-ory whose components are used to generate neighbors for searchintensification [24]. The intensification of TS is enhanced bymodifying the distribution of the neighbour solutions for a bettersearch around the current solution and by gradually reducing theoverall search space to improve the precision of the final solution.In generating neighbor solutions, a coefficient, α is used to controlthe difference between the current and new neighbor solution.The change from the current point is multiplied by α in theprocess of constructing new neighbors. The coefficient α takes the

Fig. 1. Flow chart of the TS algorithm.






format of a sine function:

α¼ 12

1þ siniθπNneigh

� �� ð11Þ

Here i is index of the neighbor, Nneigh is the total number ofneighbor solutions generated at each iteration and θ is a para-meter that controls the oscillation period of α. Inclusion of αcauses the neighbors to be distributed closer to the currentsolution. Since the closer area to the current solution is promisingcompared to other areas, the sine function with α helps TS tobetter explore attractive areas. Thus the use of α more neighbourswithin areas closer to the current solution.

Diversification strategies are employed to broadly search the entirefeasible region thus avoiding the search to stick in local optima. Thesestrategies encourage examining unvisited regions by generatingsolutions significantly different from those searched before [24].A frequency based tabu list is used to keep track of the search area.When the maximum of the frequency index is larger than apredefined threshold, the search process will be restarted by repla-cing the current solution with a new randomly generated solution.

3.5. Aspiration criterion

Tabu search conditions may sometimes forbid moves leading tounvisited solutions. Aspiration criterion is a condition under whichthe tabu status of a certain move can be overridden. In order toavoid feasible missing solutions during the search process, aspira-tion criterion is used to invalidate the tabu property in certaincases and maintains an appropriate balance between diversifica-tion and intensification. An aspiration criterion that is designedbased on a sigmoid function [24] is given by

S kð Þ ¼ 11þe�σ k�kcenter�Mð Þ ð12Þ

where kcenter is a tuning parameter, k is the current iterationnumber and σ is another tuning parameter. A random number Pthat lies between 0 and 1 is generated from a uniform distributionat each iteration. If P is greater than S(k), the tabu property isactive and the best non-tabu neighbor is used as a new startingpoint. If P is less than or equal to S(k), the aspiration criterionoverrides the tabu property.

The two parameters σ and kcenter involved in the sigmoidfunction, Eq. (10), play an important role in controlling the balancebetween intensification and diversification. A smaller σ valueresults in a flatter sigmoid function thus favoring search diversifica-tion. A smaller kcenter value results in a higher probability of theaspiration criterion thus intensifying the search. Generally, a flatsigmoid function with large kcenter can be used to favor a diversifiedsearch. The recommended ranges for kcenter and σ are 0.3–0.7 and 5/M–10/M, respectively, where M refers to maximum number ofiterations. M should provide sufficient opportunity to thoroughlysearch the solution space. A larger M results in a higher probabilityof finding good solutions at the cost of a longer computation time.The value of NNeigh in Eq. (9) also affects the choice for the value ofM since the number of functional evaluations is the product ofthem. If NNeigh is larger, M can be smaller. For a specific problem,additional tuning can be performed if computation time is suffi-ciently important. However, depending on the complexity of theoptimization problem, these ranges can be relaxed while exploringtheir suitability for such problems.

3.6. Stopping criteria

A stopping criterion is required to stop the search process whenthe optimum is found. The simplest form of stopping criterion is afixed number of iterations or a given computational effort. The

termination criteria is considered effective and practical due toease of implementation [30,35–38]. Other stopping criteria usedfor the search process include the maximum time termination andtermination-on convergence [24]. The equation representing thetermination on convergence criteria is given by

f kðxÞ� f k�ΓðxÞf k�Γ ðxÞ

�� oδ ð13Þ

where δ is the ratio of the change in the value of the objectivefunction, Γ¼ηM, and η is the fraction of the maximum iterations(M) possible over which the change in the objective function iscompared. According to this criterion, if the improvement over Γgenerations is no longer than a threshold (δ), continuing furtheriterations are considered to be ineffective, and the search shouldbe stopped. In the above equation, η results in larger savings incomputation time, where as a larger value of η maintains a highprobability of obtaining good solutions at the expense of moretime. A value of η between 0.3 and 0.5 is found to provide betterresults. A larger value of η will lead to premature termination witha less computational time, where as a smaller δ requires morenumber of iterations.

4. Process description

Styrene acrylonitrile (SAN) copolymer is commercially impor-tant and optimal control of the reactor is needed to achieve thedesired polymer properties. In this work, the process of solutioncopolymerization of styrene and acrylonitrile takes place in a semibatch reactor is considered for implementation of the proposeddynamic optimization strategies. Xylene and AIBN are used assolvent and initiator. The feed is a mixture of monomers, solventand initiator which enters the reactor in semi-batch mode. Initialvolume of the reactor is 1.01 l. The solvent mole fraction fs¼0.25and initiator concentration I0¼0.05 mol/l are taken as the initialdesign parameters. The mol ratio of monomers,M1/M2 in the feed is1.5. The mathematical model of the process is given in Appendix A.

This model is in the general form of Eq. (1), where the set ofstate variables, X and the control vector U are given by

X tð Þ ¼ ½M1 tð Þ;M2 tð Þ; I tð Þ;V tð Þ; λ0 tð Þ; λ1 tð Þ; λ2 tð Þ�T ð14Þand

U tð Þ ¼ T tð Þ;u tð Þ½ �T ð15ÞIn the above equations, M1 and M2 are the molar concentrations

of the unreacted monomers (styrene and acrylonitrile) at time t, I isthe concentration of the unreacted initiator, V is the volume of thereaction mixture, λk (k¼0, 1, …) is the kth moment of molecularweight distribution of the dead copolymer, T is the temperature ofthe reaction mixture, and u is the volumetric flow rate of the feedmixture to the reactor. More details of reaction kinetics andnumerical data pertaining to this system can be referred in [39].

5. Design and implementation of TS

The design and implementation of TS for multistage dynamicoptimization of polymerization reactor is presented as follows.

5.1. Control policies and objectives

Optimal control of SAN copolymerization involves the determina-tion of control policies to maintain the specified objectives whilesatisfying the constraints. The objectives are specified as minimizingthe deviations of copolymer composition, F1 and molecular weight,MW from their respective desired values during the entire span ofthe reaction. For satisfying these objectives, monomer addition rate,






u and reactor temperature, T are selected as control variables. If onlyone polymer quality parameter is controlled by manipulating onecontrol variable, uncontrolled property parameters may deviate fromtheir desired values as the reaction proceeds. Therefore simultaneousoptimization of desired polymer quality characteristics is desirable bymanipulating the decision variables.

The optimal control problem is considered as optimizing thesingle objectives as well as both the objectives simultaneously. Theobjectives of SAN copolymerization are formulated as

J1 ¼ 1�MW tð Þ=MWd� �2 ð16Þ

J2 ¼ 1�F1 tð Þ=F1d� �2 ð17Þ

J3 ¼ ½1�F1 tð Þ=F1d�2þ 1�MW tð Þ=MWd� �2 ð18Þ

Here MW and F1 are the molecular weight and copolymer composi-tion, andMWd and F1d are their respective desired values. The notationt here refers discrete time. The copolymer composition refers to thecomposition of the polymer which is expressed as a function of thecurrent ratio of monomers and the relative reactivities of monomersreacting with themselves or with the comonomer. The equationrepresenting the copolymer composition is given in Appendix A. Thedesired copolymer composition (F1d) and number average molecularweight (MWd) are specified as 0.58 and 30,000, respectively. Thepurpose of single objective optimization is to determine the tempera-ture (T) policy to achieve the desired MW based on the objectivefunction defined in Eq. (14) and the monomer feed rate (u) policy tosatisfy the desired F1 based on the objective function in Eq. (15).

The hard constraints are set as

0ru tð Þr0:07 l= min� �

320rT tð Þr368 Kð ÞV tð Þr4:0 ð19Þ

These constraints on operating variables are chosen based onthe requirements for reaction rate, heat transfer limitation andreactor safety. Determination of temperature policy, T to satisfy J1(Eq. (14)) monomer feed policy, u to satisfy J2 (Eq. (15)) and areconsidered as single objective optimization problems. Determina-tion of u and T policies to satisfy J3 (Eq. (16)) is considered as aproblem of simultaneous optimization.

5.2. Implementation of TS

The optimal control problem of SAN copolymerization reactoris considered as a problem of dynamic optimization by dividingthe entire span of reaction time into finite number of time instantswhich can also be referred as discrete stages. The total duration ofreaction is fixed as 300 min. Thus u and T profiles within theranges of their constraints for the entire duration of reaction areequally discretized into 19 stages having 20 time points. Thesediscrete control sequences for feed flow and temperature are givenby

u¼ ½u1;u2; :::…::;u20�T ð20Þ

T ¼ ½T1; T2; :::…::; T20�T ð21Þ

In single objective optimization problems, the temperature(T)policy is determined to maximize MW based on the objectivefunction defined in Eq. (14) and monomer feed rate (u) policy isdetermined to maximize F1 based on the objective functiondefined in Eq. (15). In simultaneous optimization problem, thedetermination of the dual policies is based on the objectivefunction defined in Eq. (16). Sequential and simultaneous imple-mentation of Tabu search is carried out as follows

Fig. 2. Sequential implementation scheme of TS for dynamic optimization.






5.2.1. Sequential implementationTS implementation is carried to compute the optimal control

policies in polymerization reactor using the procedure described inSection 2.1 and the flow scheme shown in Fig. 2. This method isreferred here as TS sequential. Initially, the control vector is specifiedto be a constant value at each of the 20 points representing 19 stageswithin the constraints specified by Eq. (17). Since the processoperation is considered for a time duration of 300 min, the durationof each stage corresponds to15 min. TS explores the search space offeasible solutions by a sequence of moves. The control input at thebeginning of the first stage is chosen at the lower bound of the inputspace. For MW optimization, ten neighbors are generated at the endof the first stage by introducing random changes in the search spaceof T with an incremental variation of �0.4 to 0.4. The process model(Appendix A) is integrated for a duration of 15 min with a time stepof 1 min from the beginning to the end of the first stage and theobjective function values are evaluated for all the generated neigh-bors at the end of this stage. The iterative convergence of TSestablishes the best control input (T) along with its objectivefunction. This becomes the optimal control point for the first stage,which is then used as a starting point for the second stage solution.For second stage solution, random neighbors are generated at theend of the second stage around the optimal T of the first stage. Themodel is integrated from the beginning to the end of the first stagebased on the initial control point (T) and from the beginning to theend of the second stage for each of the neighbors generated at theend of second stage. The optimal control inputs for successive stagesuntil the end of last stage are established in a similar manner. Thecontrol input values thus determined at the end of each stagerepresents the optimal control policy for T. For F1 optimization, tenneighbors are generated at the end of first stage with an incrementalvariation of �1.0�10–6 to 1.0�10–6. In analogous manner, optimal

control policy for u is determined adapting the TS procedure of as inT policy. For establishing dual control policies for T and u, multistagedynamic optimization by TS is carried out by considering incrementalvariations in T and uwith in the limits of �0.4 to 0.4, and �1.0�10–6 to 1.0�10–6, respectively. This case requires the evaluation of theobjective function values for 100 neighbor combinations at each ofthe control point representing T and u.

5.2.2. Simultaneous implementationIn this approach, computation of optimal control policies by TS is

carried out by using the procedure illustrated in Section 2.2 and theflow scheme shown in Fig. 3. This method is referred here as TSsimultaneous. For determining the temperature policy for MWmaximization, the control input T is initially set as a constant valuefor all the 19 stages, the duration of each stage corresponds to15 min. Initially, neighbors are generated for each of these stages byintroducing random changes in the search space using the incre-mental variations for control input as in sequential implementationof TS. Thus 10 neighbors are generated for control input T at eachstage end. The control point at the beginning of the first stage ischosen at its lower constraint value. The process model is integratedfor the first stage duration for each of these neighbor points using atime step of 1 min. The objective function values are evaluated forall the generated neighbors at the end of this stage and the bestneighbor is selected. The objectives for the neighbors at the end ofthe second stage are evaluated for each of the neighbor points ofthis stage by integrating the model equations from the beginning ofthe first stage to the end of the second stage and the best neighboris selected. This procedure is continued until the last stage and thiscompletes the first iteration. The multistage solution using TS isrepeated for the next iteration by generating neighbors of each

Fig. 3. Simultaneous implementation scheme of TS for dynamic optimization.






successive stage based on the best neighbors selected in each stageof previous iteration. This procedure is continued until the conver-gence in the desired objective function at the end of last stage isachieved. The control input values thus determined at the end ofeach stage represents the optimal control policy for T. Similarprocedure is followed to determine the optimal feed rate policyrepresenting F1 maximization. Multistage dynamic optimization fordetermining the dual control policies of T and u that maximizesMWand F1 is carried out by generating random neighbors of each stagefor both the control inputs and implementing the TS procedure. Inthis case, the evaluation of objective function at each stage requires100 input neighbor combinations.

6. Design and implementation of IDP

In order to compare the sequential and simultaneous imple-mentation methodologies of TS, a multistage dynamic optimizationstrategy based on iterative dynamic programming (IDP) is alsopresented. The optimal control problem is to find the control u(t) inthe time interval tk�1rtrtk, so as to satisfy the desired objectives.The optimal control policy is approximated by a piecewise constantcontrol policy over P time stages, each of length L, where L¼tf/P, sothat the time interval tk�1rtrtk is considered with the constantcontrol u(t)¼u(k�1). The problem then is to find u(0), u(1),… u(P�1) that satisfy the performance index.

The algorithm of iterative dynamic programming (IDP) consistsof the following steps [40]:

1. Divide the time interval tf into P stages, each of length L.2. Choose the number of x-grid points N, the number of allow-

able values M for the control u, the region r for the control

values, the region contraction factor γ and the region restora-tion factorη.

3. Choose the number of iterations used in every pass and thenumber of passes. Set the pass number index q¼1 and theiteration number index j¼1.

4. Set r j¼ηq�1 rin, where rin is initial region size.5. By choosing N values of control inside allowable region

integrate Eq. (1). N times to generate the x-grid at eachtime stage.

6. Starting at the last time stage P, corresponding to time tf–L, foreach x-grid point, integrate Eq. (1). from tf–L to tf for all the Mallowable values of control. Choose the control that satisfiesthe performance index and store the value of control asunk(P�1) for use in the next step of the algorithm.

7. Step back to stage P�1, corresponding to time tf –2L, andintegrate Eq. (1). from tf –2L to tf –L for each x-grid point with

Fig. 4. Temperature policy and molecular weight. Fig. 5. Process responses for temperature policy.






theM allowable values of control. To continue integration fromtf –L to tf choose the control from step 5 that corresponds tothe grid point closest to the resulting x at tf–L. Compare the Mvalues of the performance index and store the values ofcontrol, unk(P�2) that gives the desired optimum value.

8. Continue the procedure until stage 1, corresponding to the initialtime t¼0 is reached. Store the control policy that satisfies theperformance index and store the corresponding x-trajectory.

9. Reduce the search region for allowable control values by afactor γ: r jþ1¼ γ r j

The updated control trajectory is

ujkþ1ðPÞ ¼ unj

k ðPÞþDrj

where unjk ðPÞ is the optimum value obtained in the previous

iteration and D is a diagonal matrix of random numbersvarying from �1 to 1.

10. Increment the iteration index j by 1 and go to step 3. Continuethe procedure for a specified number of iterations for each pass.

11. Increment the pass number index q by 1 and go to step 4.Continue the procedure for a specified number passes.

The IDP procedure is implemented to determine the optimalcontrol policies for temperature T, feed rate u and for both T and u.The control sequences and the objectives are specified in Section 5.1.The calculations are initialized by choosing the control inputs atthe lower bound of the input space. The control values, M for eachgrid point, the region restoration factor, η and the region contrac-tion factor, γ are selected appropriately. The diagonal matrix, D isgenerated randomly with in the range of �1 to 1. The initialtemperature policy for all stages is set as 333 1K. The parametersfor this policy are set as P¼100, rin¼15, γ¼0.85 and η¼0.1. Theinitial feed rate policy for all stages is set as 0.00025 (l/min). Theparameters of this policy are chosen as P¼50, rin¼5, γ¼0.75 andη¼0.5. The parameters for simultaneous optimization are set asP¼100, rin¼15, γ¼0.85 and η¼0.1. The optimal solution for single

control policies as well as dual control policies is achieved with in5 passes, each pass with 50 iterations.

7. Results and discussion

Sequential and simultaneous implementation of TS is carriedout to determine the optimal control policies that satisfy theindividual and multiple objectives of SAN copolymerization reac-tor. Considering the wt% of acrylonitrile in the polymer to be in therange of 25–35%, the desired values of copolymer composition(F1D) and number average molecular weight (MWD) are chosen as0.58 and 30,000, respectively. The convergence criteria for theobjectives are set as 0.0001. This convergence is defined based on

Fig. 6. Monomer feed policy and copolymer composition. Fig. 7. Process responses for monomer feed policy.






the absolute error between the current and previous iterations ofTS. The elements of tabu search for computing the optimal controlpolicies in SAN polymerization reactor is set as follows. The sizesof both recency and frequency based tabu lists are fixed as 50. Thesizes of these lists are chosen such that they forbid revisiting ofunpromising solutions in the search process. An intensificationstrategy with the coefficient α taking the format of a sine functionin Eq. (9) is employed, in which the value of θ is assigned to be4.0001. An aspiration criterion based on a sigmoid function givenin Eq. (10) is employed with kcenter as 0.3 and σ as 7/M, where Mrefers specified number of iterations. The number of neighbors foreach control input of each stage is set as 10.

7.1. Temperature policy to achieve the desired molecular weight

The computation of temperature (T) policy for multistagedynamic optimization of polymerization reactor using the TS sequen-tial and TS simultaneous methodologies is carried out according tothe design and implementation procedure given in Section 5. Theresults of these methodologies are further compared with thedynamic optimization results of IDP. The IDP is implemented tocompute the optimal temperature policy iteratively and recursivelyfrom the last stage to first stage. This method involves the heuristicselection of grid points and the penalty parameters such as regioncontraction factor and region restoration factor. The temperaturepolicies computed by TS and IDP to achieve the desired molecularweight (MW) along with the process responses are shown inFigs. 4 and 5. Fig. 4 depicts the results of the temperature policiesonly when the molecular weight (MW) is controlled. The responsesof polydispersity, conversion and copolymer composition due to thetemperature policies are shown in Fig. 5. These results show that thetemperature policies generated by TS sequential and TS simultaneousalgorithms are with in the range of the constraints specified for

temperature. The MW values obtained at the end of the reactionbased on the optimal temperature policies of TS sequential and TSsimultaneous are 29,983 and 29,976, respectively, which are almostnear to the desired value of 30,000. The optimal temperature policiesand the values of MW evaluated by both the TS algorithms areobserved to be in close agreement. The temperature policy and MWof IDP in Fig. 4 shows considerable deviation than those of TSprofiles. The MW obtained at the end of the reaction due to optimaltemperature policy of IDP is found to be19,640, which is far awayfrom the desired value. The results in Fig. 5 also show a significantdeviation of IDP responses. From the results in Fig. 5, it is observedthat better control of MW by TS leads to improved polydispersity.However, the uncontrolled copolymer composition (F1) is found todeviate from its desired value when only MW is controlled.

7.2. Monomer feed rate policy to obtain the desired copolymercomposition

Feed rate (u) policy for multistage dynamic optimization ofpolymerization reactor is determined using TS sequential and TSsimultaneous, and the results are further compared with thedynamic optimization results of IDP. Fig. 6 shows the results ofmonomer feed policy along with the copolymer composition whenthe reactor temperature is kept at 333 K. The process responses ofMW, polydispersity and conversion corresponding to the feed ratepolicies of these methods are shown in Fig. 7. It is observed thatthe copolymer composition obtained due to all these methods isalmost nearer to the desired value, and the final monomerconversion is found to be about 48%. However, the uncontrolledMW is found to deviate considerably from its desired value whenonly copolymer composition (F1) is controlled. The copolymercomposition control results show that the TS and IDP exhibitsalmost similar trend in response behaviors.

Fig. 8. Dual control policies and objectives.






7.3. Dual policies for optimizing molecular weight and copolymercomposition

Dual control policies of T and u for multistage dynamicoptimization of polymerization reactor are determined using TSsequential and TS simultaneous and the results are furthercompared with the dynamic optimization results of IDP. Fig. 8depicts the results of the dual control policies of these methodsalong with the objective function values. The process responses ofpolydispersity and conversion corresponding to the dual controlpolicies of these methods are shown in Fig. 9. These results showthat both the temperature and feed rate policies of TS maintainMW and F1 almost near to their desired values. However, theresults of IDP exhibit considerable deviation than those of TS. TheMW values obtained at the end of the reaction by TS sequential, TSsimultaneous and IDP are 29,986, 29,973 and 28,981, respectively,

and the corresponding F1 values obtained at the end of thereaction are 0.5759, 0.5752, and 0.5564, respectively. These resultsshow the better performance of TS over IDP.

7.4. Effect of design parameters and computational efficiencies

The performance of TS is further studied with respect to theeffect of the parameters σ and kcenter involved in the sigmoidfunction, Eq. (10) of the search intensification scheme. TS sequentialstrategy is considered for this study as it exhibits better performancefor dynamic optimization of the polymerization reactor. The case ofdual policies involved in the control of both molecular weight andcopolymer composition is considered to study the effect of theseparameters. The effect of kcenter is studied by varying it from 0.3 to0.7 with an increment of 0.1 by keeping the values of σ andneighbors as 5/M and 10, respectively. When kcenter is changed from0.3 to 0.7, the end value of MW is found to change from 29,968 to29,962 and the F1 is found to change from 0.5680 to 0.5635. Thecorresponding normalized absolute error is found to vary from0.0153 to 0.00751. The same trend is observed when the value ofσ is changed from 5/M to 7/M. The effect of the neighbors on theperformance of TS is studied by varying them from 10 to 22 with anincrement of 2 while keeping the values of kcenter and σ as 0.3 and 7/M, respectively. When the neighbors are changed from 10 to 22, therespective changes observed in the end values of MW and F1 are29,980–29,986, and 0.5748 to 0.5759, respectively. The correspond-ing normalized absolute error and the time of execution are found tochange from 0.009631 to 0.0075346 and from 5 s to 8 s, respectively.This shows that the effect of kcenter and σ with in their specifiedranges is marginal on the results, whereas the execution timeincreases with the increase of neighbors. Similar trend is observedtowards the effect of kcenter, σ and neighbors on the results of thesingle control policies of temperature and monomer feed rate.

The computational efficiencies and the normalized absolute errorvalues of TS sequential, TS simultaneous and IDP are shown inTable 1. The computational efficiencies are calculated by executingthese methods on Intel Core Duo CPU with 988 MB Ram, using Cprogramming. The execution and implementation times givenTable 1 are in seconds. The execution time refers to the time untilthe solution is converged to result the optimal control policy. Theimplementation time is the time taken to generate the modelsolution in accordance with the optimal control policy. These resultsshow that the execution times required for the iterative convergenceof the optimal control policies by TS sequential and TS simultaneousare much lower to that of IDP. From the results of TS strategies, it isobserved that the execution time required for TS simultaneous isconsiderably higher than that of TS sequential. In TS sequential, thesolution requires iterative convergence at each of the 20 stages,where as in TS simultaneous, the solution moves from the beginningof the first stage to the end of the last stage and iterated from thebeginning to the end to obtain convergence in solution. The iterative

Table 1Comparison of the methods of dynamic optimization.

Strategy Control policy Computational efficiency Normalized absolute error Memory storage, k

Convergence time (s) Implementation time (s)

TS Sequential T 1.89 16.83 0.000566 2224u 2.19 13.81 0.01379 2076T & u 4.25 19.21 0.007535 3084

TS Simultaneous T 8.26 17.12 0.000066 3827u 7.37 15.86 0.01210 2843T & u 13.05 19.78 0.009175 3982

IDP T 38.59 226.09 0.3453 7108u 26.87 165.00 0.03448 5272T & u 73.90 179.06 0.0764 18768

Fig. 9. Process responses for dual control policies.






convergence procedure of TS simultaneous thus required highercomputational effort than the sequential implementation procedure.However, the execution times required for implementing the optimalpolicies of these methods is found to have marginal difference. Theperformance of the three methods is also evaluated based on thenormalized absolute error which is defined as the difference betweenthe values of the desired and end objectives. These TS results showthat the absolute error due to T policy is found to be much lower tothat of u policy, however, a marginal difference is observed in theerrors of single and dual policies of TS sequential and TS simulta-neous strategies. The performance of the three methods is evaluatedbased on the normalized absolute error which is defined as thedifference between the values of the desired and end objectives. Theperformance of TS sequential and IDP is also evaluated based oncomputational cost and memory storage. The memory storagerequirement of Tabu search and IDP is given in Table 1 along withthe computational efficiencies of these methods. Simultaneously, theproposed TS technique is also compared with another evolutionaryalgorithm, differential evolution (DE) [41]. These results show that TSalso has significant advantage over DE in solution quality. The resultsthus show the better performance of tabu search over IDP. In TSstrategies, TS sequential is found to converge faster than TS simulta-neous with a marginal difference in the normalized absolute error.

8. Conclusions

A multistage dynamic optimization methodology based on tabusearch (TS) is presented for computing the optimal control policiesto achieve the desired product quality parameters of the polymerproduced in a semi-batch styrene–acrylonitrile (SAN) copolymer-ization reactor. The performance of the TS methodology is com-pared to a conventional dynamic optimization technique calledIterative dynamic programming (IDP). Iterative solution of multi-stage dynamic optimization problem for TS is formulated such thatit evaluates the optimal control policies successively from the firststage to last stage, and the optimal policy computation at eachcurrent stage is based on the optimal policies determined forprevious stages, where as the iterative solution of multistagedynamic optimization by IDP is performed recursively from thelast stage to the first stage. The results have shown the efficacy ofTS for determining the optimal control policies satisfying thedesired polymer product characteristics. This TS based dynamicoptimization strategy can be applied to solve the optimal controlproblems of other complex dynamic systems.

Acknowledgment

Financial assistance from DST through the grant SR/FTP/ETA-011/2009 is gratefully acknowledged.

Appendix A. Mathematical model of SAN system

The following kinetic model is used to describe the homo-geneous solution free-radical copolymerization of styrene withacrylonitrile [39].

Initiation:

I-2R

RþM1⟹ki1

P10

RþM2⟹ki2

Q01

Propagation:

Pn;mþM1⟹kp11Pnþ1;m

Pn;mþM2⟹kp12

Qn;mþ1

Qn;mþM1⟹kp21

Pnþ1;m

Qn;mþM2⟹kp22

Qn;mþ1

Combination termination:

Pn;mþPr;q⟹ktc11Mnþ r;mþq

Pn;mþQr;q⟹ktc12

Mnþ r;mþq

Qn;mþQr;q⟹ktc22

Mnþ r;mþq

Disproportionation termination:

Pn;mþPr;q⟹ktd11Mn;mþMr;q

Pn;mþQr;q⟹ktd12

Mn;mþMr;q

Qn;mþQr;q⟹ktd22

Mn;mþMr;q

Chain transfer:

Pn;mþM1⟹kf 11Mn;mþP10

Pn;mþM2⟹kf 12

Mn;mþQ01

Qn;mþM1⟹kf 21

Mn;mþP10

Qn;mþM2⟹kf 22

Mn;mþQ01 ðA:1Þ

where Pn,m represents a growing polymer chain with n units ofmonomer 1 (styrene) and m units of monomer 2 (acrylonitrile)with monomer 1 on the chain end. Similarly, Qn,m representsgrowing copolymer chain with monomer 2 on the end. Mn,m

denotes inactive or dead polymer.The copolymer molecular weight (MW) and molecular weight

distribution (MWD) are computed using three leading moments ofthe total number average copolymers. The instantaneous kthmoment is given by:

λdk ¼ ∑1

n ¼ 1∑1

m ¼ 1ðnw1þmw2ÞkMn;m; k¼ 0;1;2; ::::::: ðA:2Þ

where w1 and w2 are the molecular weights of styrene andacrylonitrile, respectively.

The total number average chain length (Xn), the total weightaverage chain length (Xw) and the polydispersity index (PD) areexpressed as:

Xn ¼ λd1=λd0

Xw ¼ λd2=λd1

PD¼ Xw=Xn ðA:3Þ

The modeling equations of the semi-batch copolymerizationreactor are given as follows:

dM1=dt ¼ uðM1f �M1Þ=V�½ðkp11þkf11ÞPþðkp21þkf21ÞQ �M1

dM2=dt ¼ uðM2f �M2Þ=V�½ðkp22þkf22ÞQþðkp12þkf12ÞP�M2

dI=dt ¼ uðIf � IÞ=V�kdI

dV=dt ¼ u ðA:4Þ

The live polymer moments are given by:

P ¼ ½2f kdI=ððktc11þktd11Þþ2βðktc12þktd12Þþβ2ðktc22þktd22ÞÞ�1=2

where

β¼ ðkp12þkf12Þ=ðkp21þkf21ÞΦ ; Φ¼M1=M2






Pseudo state approximation to live polymers leads to the followinglive polymer moment equations:

P1 ¼ ½w1C1α1þα1γQ1=r1þw1ðα1Pþα1γQ=r1Þ�=ð1�α1ÞQ1 ¼ ½w2C2α2þα2P1=γr2

þw2ðα2Qþα2P=γr2Þ�=ð1�α2ÞP2 ¼ ½w1

2C1α1þα1γQ2=r1þ2w1α1P1

þ2w1α1γQ1=r1þw12ðα1Pþα1γQ=r1Þ�=ð1�α1Þ

Q2 ¼ ½w22C2α2þα2P2=γr2þ2w2α2P1=γr2

þ2w2α2Q1þw22ðα2P=γr2þα2Q Þ�=ð1�α2Þ ðA:5Þ

C1 ¼ kf 11Pþkf 21Q� �

M1� �

=kp 11M1 C2 ¼ ½ðkf22Qþkf12PÞM2�=kp22M2

where

r1 ¼ kp11=kp12; r2 ¼ kp22=kp21; γ ¼ kp21=kp12α1 ¼ kp11M1=½ðkp11þkf11ÞM1þðkp12þkf12ÞM2

þðktc11þktd11ÞPþðktc12þktd12ÞQ �α2 ¼ kp22M2=½ðkp22þkf22ÞM2þðkp21þkf21ÞM1

þðktc22þktd22ÞQþðktc12þktd12ÞP�The moment equations for dead polymers are given by:

dλ0d=dt ¼ ðktc11=2þktd11ÞP2þðktc22=2þktd22ÞQ2þðktc12þ2ktd12ÞÞPQ

þðkf11M1þkf12M2ÞPþðkf22M2þkf21M1ÞQ�ðλd0=VÞudλ1

d=dt ¼ ðktc11Pþktd11Pþktc12Qþktd12Qþkf11M1

þkf12M2ÞP1þðktc22Qþktd22Qþktc12Pþktd12P

þkf22M2þKf12M1ÞQ1�ðλ1duÞ=Vdλ2

d=dt ¼ ðktc11Pþktd11Pþktc12Qþktd12Qþkf11M1

þkf12M2ÞP2þðktc22Qþktd22Qþktc12P

þktd12Pþkf22M2þkf21M1ÞQ2þktc11P12þktc22Q1

2

þ2ktc12P1Q1�λ2du=V ðA:6Þ

The rate constants,kd, kfij,kpij, ktcijand ktdij are Arrhenius functions oftemperature, the pre-exponential factors and activation energiesof which are reported elsewhere [39].

The instantaneous copolymer composition F1 is determined bythe relative reactivities of the monomers (r1 and r2) and the bulkphase monomer mol fractions (f1 and f2) as given by:

F1 ¼ ðr1f 12þ f 1f 2Þ=ðr1f 12þ2f 1f 2þr2f 22Þ

or in terms of monomer mol ratio (Φ)

F1 ¼ ðr1Φ2þΦÞ=ðr1Φ2þ2Φþr2Þ ðA:7ÞThe conversion of monomer 1 (styrene) is defined as:

x1 ¼ ½V0M10þZ t

0uðtÞM1f dt�VM1ðtÞ�=½V0M10þ

Z t

0uðtÞM1f dt�

ðA:8Þ

Appendix B

B.1. Application of TS for test cases [42]:

The optimization ability of TS is tested by applying it to fiveoptimization functions which differ in complexity and the numberof variables (parameters). The adopted benchmark functions arelisted as follows.

B2. Rosenbrocks function

Definition: 100n(x1–x22)2þ(x1–1)2

Global minima obtained: (1, 1), Value at global minima: 0.0

B.3. Beale function

Definition: (1.5–x1þx1� 2)2þ(2.25–x1þx1� 22)2þ(2.625–x1þ

x1x23)2

Global minima obtained: (3, 0.5), Value at global minima: 0.0

B.4. Freudenstein and Roth function

Definition: (–13þx1þ ((5–x22)–2) x2)2þ(–29þx1þ((x2þ1)x2–14)x2)2

Global minima obtained: (4, 5), Value at global minima: 0.0Local minima obtained: (11.4125; –0.89685), Value at local

minima: 48.98425

B.5. quadratic test function

Definition: (x1þ2x2–7)2þ(2x1þx2–5)2

Global minima obtained: (1, 3), Value at global minima: 0.0

B.6. Hooks and Jeeves test function

Definition: x1–x2þ2� 12þx22þ2x1x2

Global minima obtained: (–1, 1.5), Value at global minima: –1.25

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dynamic optimization of a copolymerization reactor using tabu search

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