optimization of process synthesis and design problems: a modified differential evolution approach

Chemical Engineering Science 61 (2006) 47074721www.elsevier.com/locate/ces

Optimization of process synthesis and design problems:Amodied differential evolution approach

RakeshAngira, B.V. Babu

Department of Chemical Engineering, Birla Institute of Technology & Science, Pilani-333 031, IndiaReceived 21 November 2005; received in revised form 15 January 2006; accepted 4 March 2006

Available online 10 March 2006

Abstract

A large number of process synthesis and design problems in chemical engineering can be modeled as mixed integer nonlinear programming(MINLP) problems. They involve continuous (oating point) and integer variables. A common feature of this class of mathematical problemsis the potential existence of non-convexities due to the particular form of the objective function and/or the set of constraints. Due to theircombinatorial nature, these problems are considered to be difcult. In recent years, evolutionary algorithms (EAs) are gaining popularity fornding the optimal solution of nonlinear multimodal problems encountered in many engineering disciplines. In the present study, a novelmodied differential evolution [Angira, R., Babu, B.V., 2005a. Optimization of non-linear chemical processes using modied differentialevolution (MDE). Proceedings of the Second Indian International Conference on Articial Intelligence (IICAI-05), Pune, India, December2022, pp. 911923. Also available at http://discovery.bits-pilani.ac.in/discipline/chemical/bvb/publications.html], one of the evolutionaryalgorithms, is used for solving process synthesis and design problems. To illustrate the applicability and efciency of modied differentialevolution (MDE), seven test problems on process synthesis and design have been solved. These problems arise from the area of chemicalengineering, and represent difcult nonconvex optimization problems, with continuous and discrete variables. The performance of MDE iscompared with that of Genetic Algorithm, Evolution Strategy, and MINLP-Simplex Simulated Annealing (M-SIMPSA). 2006 Elsevier Ltd. All rights reserved.

Keywords: Optimization; Processes synthesis; Systems engineering; Design; Evolutionary algorithm; Mixed integer non-linear programming (MINLP)problems; Differential evolution (DE); Modied differential evolution (MDE)

1. Introduction

Process synthesis can be dened as the selection, arrange-ment, and operation of processing units so as to create anoptimal scheme. In other words, it is an act of determining theoptimal interconnection of processing units as well as the op-timal type and design of units within a process system. Theinterconnection of processing units is called the structure of theprocess system. When the performance of the system is spec-ied, the structure of the system and the performance of theprocessing units are not determined uniquely. This task is com-binatorial and open-ended in nature and has received a great

Corresponding author. Tel.: +91 1596 245073x205/224, +91 1596 244977;fax: +91 1596 244183.

E-mail address: [email protected] (B.V. Babu)URL: http://discovery.bits-pilani.ac.in/discipline/chemical/bvb/

(B.V. Babu).

0009-2509/$ - see front matter 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2006.03.004

deal of attention over the past twenty-ve years (Nishidaet al., 1981). Since the synthesis problem is open ended, ithas lead to the development of quite different approachessuch as thermodynamic targets (Linnhoff, 1981), heuristic(Rudd et al., 1973; Douglas, 1985), evolutionary methods(Stephanopoulos and Westerberg, 1976), and optimizationtechniques (Grossmann, 1985). The present study deals withthe structural ow sheet optimization problem that arises inthe latter approach.

The use of mathematical programming techniques for pro-cess synthesis has received considerable attention over the lasttwo d ecades. For example, nonlinear programming (NLP)technique for heat exchanger networks (Floudas et al., 1986),and mixed integer nonlinear programming (MINLP) models forstructural owsheet optimization (Kocis and Grossmann, 1987,1988, 1989; Floudas et al., 1989) to name a few. The majorreason for this increased interest lies in the fact that mathemat-ical programming techniques provide a systematic frameworkfor process synthesis. Also, there has been substantial progress

4708 R. Angira, B.V. Babu / Chemical Engineering Science 61 (2006) 47074721

in methods and software for solving optimization problems,development of powerful modeling languages (General Alge-braic Modeling System, GAMS), and technological advancesin computing.

In order to formulate the synthesis problem as a mathe-matical programming problem, a superstructure is postulatedwhich includes many alternate designs from which the optimalprocess will be selected. Once the superstructure is specied,the next task is to determine the optimal process ow sheetthrough structural and parameter optimization of the superstruc-ture (which requires the solution of a mixed integer optimiza-tion problem). In early 1980s, most of the process synthesis anddesign problems have been formulated as mixed-integer linearprogramming (MILP) problems. Although these formulations(e.g. Papoulias and Grossmann, 1983) have proved to be quitepowerful, they have the limitation that nonlinearities in the pro-cess equations cannot be treated explicitly and approximatedthrough the discretization. The need for the explicit handling ofthe nonlinearities in the synthesis problem motivates the use ofmixed-integer nonlinear programming (MINLP). MINLP prob-lems, however, are much more difcult to solve than MILPproblems for which Branch and Bound methods perform rea-sonably well.

A large number of process synthesis, design and control prob-lems in chemical engineering can be modeled as mixed inte-ger nonlinear programming problems (Grossmann and Sargent,1979; Kocis and Grossmann, 1987, 1988, 1989; Floudas et al.,1989; Salcedo, 1992; Ciric and Gu, 1994 etc.). They involvecontinuous (oating point) and integer variables. A commonfeature of this class of mathematical problem is the potentialexistence of nonconvexities due to the particular form of theobjective function and/or the set of constraints. Due to theircombinatorial nature, these problems are considered to be dif-cult.

The optimization of mixed integer nonlinear programmingproblems constitutes an active area of research. So far variousmethods such as branch and bound technique (Grossmann andSargent, 1979), outer-approximation (OA)/equality-relaxationalgorithm (Kocis and Grossmann, 1987, 1988, 1989), variantof OA method (Diwekar et al., 1992), adaptive random-searchmethod (MSGA by Salcedo, 1992), branch-and-reduce algo-rithm (Ryoo and Sahinidis, 1995), MINLP Simplex SimulatedAnnealingAlgorithm (M-SIMPSA by Cardoso et al., 1997), andgenetic algorithm & evolution strategies (Costa and Oliviera,2001) have been used for solving nonconvex MINLP problems.

Gradient optimization techniques have only been able totackle special formulations, where continuity or convexity hadto be imposed, or by exploiting special mathematical struc-tures. Stochastic algorithms, also known as adaptive randomsearch methods, have tackled MINLP problems, mostly in thearea of chemical engineering (Salcedo, 1992). These requireneither the prior step of identication or elimination of thesources of nonconvexities nor decomposition of the probleminto sub problems which have to be iteratively solved. How-ever, various problem-independent heuristics related to searchinterval compression and expansion and to shifting strategiesare required for their effectiveness (Salcedo, 1992). Also, for

large scale or very ill conditioned and highly constrainedfunctions, these methods require the application of successiverelaxations which may substantially increase the effort in iden-tifying feasible regions and attaining the global optimum andhence suited for small to medium scale problems. Cardosoet al. (1997) compared the performance of the M-SIMPSAwith MSGA (Salcedo, 1992). They concluded that for small-scale problems and with penalizing scheme its performanceis comparable to MSGA algorithm, however, for large scaleand/or ill conditioned problems, the M-SIMPSA algorithmperformed better. Costa and Oliviera (2001) studied seven testproblems using GA & Evolution Strategies (ESs) and com-pared the results with M-SIMPSA algorithm. They found thatthe performance of M-SIMPSA with penalty is better thanthat of M-SIMPSA without penalty. Also the performanceof M-SIMPSA is comparable to GA. Evolution Strategiesemerged as the best algorithms in most of the problems stud-ied. However, ESs exhibited difculties in highly constrainedproblems but in general, they are found most efcient in termsof function evaluations. Also, all the algorithms (GA, ESs, andM-SIMPSA) are found to have great difculties with multi-product batch plant problem (Grossmann and Sargent, 1979),which is highly constrained; the global optimum correspondsto a point where a very small variation in any of the continuousvariables produces infeasibility.

Previous studies (Storn, 1995; Storn and Price, 1997; Wangand Chiou, 1997; Babu and Sastry, 1999; Babu and Angira,2002; Angira and Babu, 2003; Chakraborti and Kumar, 2003;Chakraborti et al., 2004; Colaco et al., 2004, 2005; Babuet al., 2005; Angira, 2005; Angira and Babu, 2005b etc. toname a few) have shown that differential evolution (DE) isan efcient, effective and robust evolutionary optimizationmethod. The details on DE algorithm, various strategies ofDE and wide range of applications in various engineering,manufacturing and management areas are well documentedin literature (Corne et al., 1999; Babu, 2004; Onwubolu andBabu, 2004). In the present study, a novel modied differen-tial evolution (MDE) algorithm (Angira and Babu, 2005a) isused for solving seven test problems on process synthesis anddesign. These problems are difcult non-convex optimizationproblems with continuous and discrete variables. Also a newapproach for handling binary (discrete) variables is proposedand compared with the nonlinear transformation for model-ing binary variables as continuous variables proposed by Li(1992). The performance of MDE is compared with that ofDE, GA, ESs, and M-SIMPSA algorithms.

2. Modied differential evolution (MDE)

The principle of modied DE is same as DE. The majordifference between DE and MDE is that MDE maintains onlyone array. The array is updated as and when a better solutionis found. Also, these newly found better solutions can take partin mutation and crossover operation in the current generationitself as opposed to DE (where another array is maintainedand these better solutions take part in mutation and crossoveroperations in next generation). Updating the single array

R. Angira, B.V. Babu / Chemical Engineering Science 61 (2006) 47074721 4709

continuously enhances the convergence speed leading to lessfunction evaluations as compared to DE (Angira, 2005; An-gira and Babu, 2005a; Babu and Angira, 2006). However, DEmaintains two arrays consuming extra memory and CPU-time(more function evaluations). This modication enables thealgorithm to get a better trade-off between the convergencerate and the robustness. By choosing the key parameters (NP,CR, and F) wisely/appropriately, the problem of prematureconvergence can be avoided to a large extent.

Such an improvement can be advantageous in many real-world problems where the evaluation of a candidate solutionis a computationally expensive operation and consequentlynding the global optimum or a good sub-optimal solutionwith the original differential evolution algorithm is too time-consuming, or even impossible within the time available (Fanand Lampinen, 2003; Babu and Angira, 2006). This has beenfound to be very true in examples such as optimization inthe eld of computational mechanics, computational magnet-ics, computational uid dynamics and unsteady solidication(Rogalsky and Derksen, 2000; Stumberger et al., 2000; Colacoet al., 2005). The pseudo code of MDE used in the presentstudy is given below:

Let P a population of size NP,and xj the jth individual of dimension D in population P,and CR denotes the crossover probabilityinput D,NP4; F (0, 1+); CR [0, 1], and initialbounds: lower(xi); upper(xi); i = 1, . . . ..Dinitialize P = {x1, . . . .., xNP} asFor each individual j P

xji = lower(xi) + randi[0, 1] (upper(xi) lower(xi)); i = 1, . . . . . . D

end For eachEvaluate P

while the stopping criterion is not satised doforall jNP

Randomly select r1, r2, r3 (1, . . . . . . .NP),j = r1 = r2 = r3randomly select irand (1, . . . . . . ..D)forall iD,

xi =

xr3i + F (xr1i xr2i )if (random[0, 1)


function, equations, and inequalities, respectively. Finally,Ax=a represents the subset of linear equations, while By + Cxdlinear equalities or inequalities that involve the continuous andbinary variables.

In the context of the synthesis problem, the continuousvariables x include ows, pressures, temperatures, and sizeswhile the binary variables y denote the potential existence ofprocess units which are embedded in the superstructure. Theequations h(x) = 0 and Ax = a, correspond to material andenergy balances and design equations. Process specicationsare represented by g(x)0 and by lower and upper boundson the variables in x. Logical constraints that must hold fora ow sheet conguration to be selected from within thesuperstructure are represented by By + Cxd and Eye.The cost function involves xed cost charges in the termcTy for the investment, while revenues, operating costs, andsize-dependent costs for the investment are included in thefunction f(x).

4. Test problems on process synthesis and design

To illustrate the applicability and efciency of MDE and DEto the nonconvexMINLP problems, seven test problems on pro-cess synthesis and design proposed by different authors havebeen solved. These problems arise from the area of chemical en-gineering, and represent difcult nonconvex optimization prob-lems, with continuous and discrete variables. Problems 2 and4, with equality constraints are reformulated (as Problems 2and 4) by eliminating the equality constraints and incorporat-ing them in inequality constraints and/or in objective functionthereby reducing the number of constraints and decision vari-ables. Comparisons are made with GA, ESs, and M-SIMPSA(a algorithm based on the combination of the nonlinear simplexmethod of Nelder & Mead and Simulated Annealing). Table 1shows characteristics of the seven test problems considered inthe present study.

Problem 1 (Process synthesis problem). This example has anon-linear constraint and has been proposed by Kocis andGrossmann (1988). It has also been solved by other authors(Floudas et al., 1989; Ryoo and Sahinidis, 1995; Cardoso et al.,1997; Costa and Oliviera, 2001)

Min f (x, y) = 2x + ys.t. 1.25 x2 y0,

x + y1.6,0x1.6,y {0, 1}.

The global optimum is (x, y; f ) = (0.5, 1; 2).

Problem 2 (Process synthesis and design problem). Thisproblem, with a non-linear constraint is proposed by Kocisand Grossmann (1988) and is also studied by Salcedo (1992),

Table 1Characteristics of the test problems

Problem No. of variables No. of constraints

Real Integer Binary Total Equality Inequality

1 1 1 2 22 2 1 3 1 12 1 1 2 13 2 1 3 34 7 2 9 6 44 4 1 5 65 3 4 7 96 3 2 5 37 7 3 10 18

Cardoso et al. (1997), Costa and Oliviera (2001)Min f (x1, x2, y) = y + 2x1 + x2s.t. x1 2 exp(x2) = 0,

x1 + x2 + y0,0.5x11.4,y {0, 1}.

The global optimum is (x1, x2, y; f )=(1.375, 0.375, 1; 2.124).

Problem 2 (Process synthesis and design problem). Problem2 can be reformulated, by eliminating the nonlinear equalityconstraint, as follows:

Min f (x1, y) = y + 2x1 ln(x1/2)s.t. x1 ln(x1/2) + y0,

0.5x11.4,y {0, 1}.

The global optimum is same as in Problem 2.

Problem 3 (Process owsheeting problem). This problem wasrst studied by Floudas (1995) and is nonconvex because ofthe rst constraint. It has also been solved by Cardoso et al.(1997), and Costa and Oliviera (2001)Min f (x1, x2, y) = 0.7y + 5(x1 0.5)2 + 0.8s.t. exp(x1 0.2) x20,

x2 + 1.1y 1.0,x1 y0.2,0.2x11, 2.22554x2 1,y {0, 1}.

The global optimum is (x1, x2, y; f ) = (0.94194,2.1, 1;1.07654).

Problem 4 (Two-reactor problem). It has been taken fromKocis and Grossmann (1989). The objective here is to select


y1 v1x1 z1

x 10

y2 v2x2 z2

Fig. 1. Superstructure for two-reactor problem.

one between two candidate reactors (as shown in Fig. 1) inorder to minimize the production cost. Also, it has been solvedby Diwekar et al. (1992), Diwekar and Rubin (1993), Cardosoet al. (1997), and Costa and Oliviera (2001)

Min f (x, y1, y2, v1, v2) = 7.5y1 + 5.5y2 + 7v1 + 6v2 + 5xs.t. y1 + y2 = 1,

z1 = 0.9[1 exp(0.5v1)]x1,z2 = 0.8[1 exp(0.4v2)]x2,z1 + z2 = 10,x1 + x2 = x,z1y1 + z2y2 = 10,v110y1,

v210y2,

x120y1,

x220y2,

x1, x2, z1, z2, v1, v20,

y1, y2 {0, 1}.The binary variables y1 and y2 denote the existence (nonex-

istence) of reactor 1 and 2 when their value is 1 (0). In theobjective function, there are xed charges for purchasing reac-tor 1 (7.5) or reactor 2 (5.5), linear terms in v1 and v2 (reactorvolumes) and the purchase price for raw material x. The twononlinear equations are the inputoutput relations for the re-actors which dene the output ows (z1 and z2) in terms ofthe input ows (x1 and x2) and the volumes. The raw materialx is split into the reactor input ows x1 and x2; a total de-mand of 10 units must be met by the output ows z1, z2. Thenext four inequalities are logical constraints which insure thatif a given reactor does not exist (e.g. y1 = 0), then the corre-sponding volume and feed stream are zero. The last constraintrequires that either reactor 1 or 2 be selected. The suboptimalsolution corresponding to (y1, y2)=(0, 1) has an objective func-tion value of 107.376 at (x1, x2) = (0.0, 15.0) and (v1, v2) =(0.0, 4.479). The global optimum is: (x, y1, y2, v1, v2; f ) =(13.36227, 1, 0, 3.514237, 0; 99.245209).

Problem 4 (Two-reactor problem). This can be reformulatedwithout equality constraints as follows:

Min f (y1, v1, v2) = 7.5y1 + 55(1 y1) + 7v1 + 6v2

+ 50 1 y10.8[1 exp(0.4v2)]

+ 50 y10.9[1 exp(0.5v1)]

s.t. 0.9[1 exp(0.5v1)] 2y10,0.8[1 exp(0.4v2)] 2(1 y1)0,v110y1,

v210(1 y1),v1, v20,

y1 {0, 1}.The global optimum is same as in Problem 4.

Problem 5 (Process synthesis problem). This problem wasstudied by Floudas et al. (1989), Salcedo (1992), Ryoo andSahinidis (1995), Cardoso et al. (1997), and Costa and Oliviera(2001). This problem features nonlinearities in both continuousand binary variables and has seven degrees of freedom.

Min f (x1, x2, x3, y1, y2, y3, y4)

= (y1 1)2 + (y2 1)2 + (y3 1)2 ln(y4+1) + (x11)2 + (x22)2 + (x33)2

s.t. y1 + y2 + y3 + x1 + x2 + x35,y23 + x21 + x22 + x235.5,y1 + x11.2,y2 + x21.8,y3 + x32.5,y4 + x11.2,y22 + x221.64,y23 + x234.25,y22 + x234.64,x1, x2, x30,

y1, y2, y3, y4 {0, 1}.

The global optimum is (x1, x2, x3, y1, y2, y3, y4; f ) =(0.2, 1.28062, 1.95448, 1, 0, 0, 1; 3.557473).

Problem 6 (Process design problem). It is a maximi-zation problem studied by Cardoso et al. (1997), and


Table 2Constants for Problem 6

a1 = 85.334407 a5 = 80.51249 a9 = 9.300961a2 = 0.0056858 a6 = 0.0071317 a10 = 0.0047026a3 = 0.0006262 a7 = 0.0029955 a11 = 0.0012547a4 = 0.0022053 a8 = 0.0021813 a12 = 0.0019085

Costa and Oliviera (2001)Max f (x1, x2, x3, y1, y2)

= 5.357854x21 0.835689y1x3 37.29329y1 + 40792.141

s.t. a1 + a2y2x3 + a3y1x2 a4x1x392,a5 + a6y2x3 + a7y1y2 + a8x21 9020,a9 + a10x1x3 + a11y1x1 + a12x1x2 205,27x1, x2, x345,y1 {78, . . . , 102}, integer,y2 {33, . . . , 45}, integer,

where a1 to a12 are constants the values of which are given inTable 2. The global optimum (for any combination of x2, y2)is: (x1, x3, y1; f ) = (27, 27, 78; 32217.4).

Problem 7 (Multi-product batch plant (MPBP)). This is amulti-product batch plant problem with M serial processingstages, where xed amounts Qi from N products must beproduced. Many researchers (Grossmann and Sargent, 1979;Kocis and Grossmann, 1988; Salcedo, 1992; Cardoso et al.,1997; Costa and Oliviera, 2001) studied this problem.

Min f =Mj=1

NjVj

s.t.Ni=1

QiTLi

BiH ,

Vj SijBi ,NJTLi tij ,1Nj Nuj ,

V lj Vj V uj ,

T lLiTLiT uLi ,

Blj Bj Buj ,

where, for the specic problem considered, M =3, N =2, H =6000, j = 250, j = 0.6, Nuj = 3, V lj = 250 and V uj = 2500.The values of T lLi, T

uLi, B

lj and B

uj are given by

T lLi = max tij /Nuj ,T uLi = max tij ,

Table 3Values of Sij and tij of Problem 7

Sij tij

2 3 4 8 20 84 6 3 16 4 4

Blj = Qi TLi/H ,

Buj = min(Qi min

jV uj /Sij

).

The values of Sij and tij [i = 1.2 (rows); and j = 13(columns)] are given in Table 3. The global optimum is:(N1, N2, N3, V1, V2, V3, B1, B2, T1, T2; f )=(1, 1, 1, 480, 720,960, 240, 120, 20, 16; 38499.8).

5. Handling of integer and binary variables

5.1. Integer variables

Original DE algorithm is only capable of handling con-tinuous variables. Extending it to optimize integer variables,however, is very easy and requires only couple of simple mod-ications (Corne et al., 1999). First, integer values should beused to evaluate the objective function, even though DE itselfstill works internally with continuous oating-point values.Therefore

yi ={xi for continuous variables,INT(xi) for integer variables.

INT() is a function for converting a real value to an integervalue by truncation. Additionally, truncation is preformed herefor evaluating trial vectors and for handling boundary con-straints. Truncated values are not assigned elsewhere. Hence,DE works with a population of continuous variables regard-less of the corresponding object variable type which is alsoessential for maintaining diversity of the population and therobustness of the algorithm.

5.2. Binary or discrete variables

In the present study, integer variables are handled in thesame way as described above however, a new procedure isproposed and evaluated to handle binary (discrete) variables.This procedure is called Approach 1 and is compared withanother procedure (Li, 1992), called Approach 2 in the presentstudy. These are described below:

Approach 1. In this procedure, for the function evaluation,binary variables are handled as follows:

yi ={0 if xi0.5,1 otherwise,

where xi is a continuous variable 0xi1. However bound-ary constraint is handled in the same way as for continuous


variables. The only difference is that lower and upper boundare set to zero and one respectively.

Approach 2. It is a nonlinear transformation for modeling bi-nary variables as continuous variables proposed by Li (1992).A binary variable y {0, 1} can be modeled as a continuousvariable x [0, 1], simply by the addition of the followingconstraint in the problem:

x(1 x) = 0, 0x1

which forces x to take either 0 or 1. Hence with this trans-formation any MINLP model can be converted into an equiv-alent NLP model. The function x(1 x) is a non-convexnonlinear function. Li (1992) referred to this as the binarycondition to model binary variables and found this procedureto be more convenient than current approaches as branch-and-bound method and implicit enumeration method. The resultingNLP was solved using a modied penalty function method,however only local optimal solutions were found due to the ad-dition of non-convexities. Also, the use of standard NLP solverlike SQP (sequential quadratic programming) and GRG (gen-eralized reduced gradient) is ruled out as they could be sensi-tive to initial guess and hence get stuck at local optima. Ryooand Sahinidis (1995) proposed a specialized branch and reducealgorithm for the global optimization of NLPs and MINLPswherein they refer to the usage of such procedure for handlingbinary and discrete variables. The present study evaluates theapplication of DE and MDE algorithms for solving the result-ing nonconvex NLP problems to global optimality.

6. Handling of bound violations and constraints

Through manipulating its own scaling factor, one can controlthe scaling of difference of two solutions/vectors. However,it does not guarantee that the random noisy vector/solutiongenerated after adding this weighted difference to the thirdvector/solution would be lying within the given bounds. Forunconstrained optimization problems, when the variables areout of bounds, their objective function values are also worse,and hence they are eliminated automatically via selection bycomparison of the function values in subsequent generations,but denitely not by controlling the scaling factor. But, in thecase of constrained optimization problems, the points outsidethe bounds may have a better function value but may not comeunder the feasible region. In such cases, a method for bringingthe solutions to within the bounds of feasible region is required.

Most of the process synthesis and design problems areconstrained. The difculty of using EAs in the constrained op-timization is that the evolutionary operators used to manipulatethe individuals of the population often produce solutions whichare unfeasible. There may be many ways of doing it. Two suchmethods (prescribing externally) are discussed, applied andcompared in the present study, which are discussed below.

Bound violations (whether upper or lower) may occur aftermutation step of DE and MDE. This can be repaired by one

of the following methods: (1) If there is bound violation for aparameter, then assign the upper or lower bound value if upperor lower bound is violated (forced bound), (2) if there is boundviolation for a parameter, then that parameter is again gener-ated randomly between given lower and upper bound (withoutforcing) using the following equation:

xji = lower(xi) + randi[0, 1] (upper(xi) lower(xi));i = 1 . . . D,

where D is the number of decision variables. In the presentpaper, the rst method is called forced bound method (FBM)while the second method is called method without forcing thebound (MWFB).

The penalty function methods are one of the most populartechniques in EAs to handle constraints. The techniques trans-form the constrained problem into an unconstrained problemby penalizing unfeasible solutions. In addition, the penaltyfunction methods are easy to implement and considered ef-cient. In the present study, an absolute value of constraint vi-olation is multiplied with a high penalty and added/subtractedto objective function depending upon the type of problem,i.e., minimization/maximization. In case of more than oneconstraint, all such absolute violations are rst multipliedwith high penalty and then added or subtracted from objectivefunction value (for minimization or maximization problem,respectively).

7. Results and discussion

7.1. Effect of key parameters (CR, F, and NP)

The performance of DE and MDE algorithms depends onkey parameters, namely, CR, F, and NP. By choosing the keyparameters (NP, CR, and F) wisely/appropriately, the problemof premature convergence can be avoided to a large extent. Tostudy the effect of key parameters, two highly multimodal mul-tidimensional test functions, discussed in Fan and Lampinen(2003), are used. Also, a basic key parameter set (CR = 0.85,F = 99, and NP = 30) is rst chosen as mentioned in Fanand Lampinen (2003). Then, all the numerical simulations arecarried out around this basic set taking maximum number ofgeneration to be 250. A systematic numerical simulation isperformed keeping two key parameters constant while vary-ing the third at a time. The two test functions are denedbelow:

Ackleys function: This is a continuous, highly nonlinear func-tion that causes the search with moderate complications.

f1 = 20 exp0.2

1n

ni=1

x2i

exp

(1n

ni=1

cos(2xi)

)

+ 20 + e, 20xi30, n = 30.

The global minimum is: f1 = 0 with xi = 0, i = 1, 2, . . . ., n.


01020304050607080

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1CR

Func

tion

Valu

e

Rastrigin(MDE) Ackley(MDE)Ackley(DE) Rastrigin(DE)

Fig. 2. Effect of Crossover constant (CR).

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2F

Func

tion

Valu

e

Rastrigin(DE) Ackley(DE)Rastrigins(MDE) Ackley(MDE)

Fig. 3. Effect of scaling factor (F).

Rastrigins function: This function is also considered rel-atively difcult to minimize because the number of locallyoptimum points is high.

f2 = 2n +n

i=1(x2i 2 cos(2xi)),

5.12xi5.12, n = 20.The global minimum is: f2 = 0 with xi = 0, i = 1, 2, . . . ., n.

Figs. 24 show the effect of CR, F, and NP, respectively,on the performance of MDE and DE using the two functionsas mentioned above. Each point on the gures represents theaverage of 100 independent experiments. It is evident fromthese gures that both DE and MDE are affected in a similarway, i.e., the variation of function value with CR/F/NP is samequalitatively.

It is important to note that variation of function value withCR, F, and NP is different for the same problem but it is samefor DE and MDE.Also, the variation of function value with CRand F is problem dependent, i.e., different problems may have

0102030405060708090

0 50 100 150 200 250 300NP

Func

tion

Valu

e

Ackley (MDE) Ackley (DE)Rastrigin (MDE) Rastrigin (DE)

Fig. 4. Effect of Population Size (NP).

different trends of function value vs. CR/F/NP. Therefore, forcomparison purposes, it is essential to keep the same setting ofkey parameters for both DE and MDE although this setting canbe different for different problems.

Babu and Angira (2006) evaluated the performance of MDEusing several multimodal test functions followed by selectednonlinear chemical processes. In the present study, the perfor-mance of MDE algorithm is further examined using test prob-lems on process synthesis and design. The obtained numericalsimulation results are providing empirical evidences on the ef-ciency and effectiveness of the proposed modied differentialevolution algorithm.

7.2. Test problems on process synthesis and design

For each problem ten experiments are carried out with dif-ferent seed values. NFE, NRC and CPU-time in the subsequenttables are the mean values of the ten experiments. The stoppingcriteria adopted for MDE and DE is to terminate the search pro-cess when one of the following conditions is satised: (1) themaximum number of generations is reached (assumed 10,000generations for Problem 7 and 5000 generations for other prob-lems). (2) |f kmax f kmin|< 105, where f is the value of objec-tive function for kth generation.

7.2.1. Approach 1Tables 4 and 5 show the results obtained using DE and MDE

(with/without forcing the bound on variables), respectively, us-ing Approach 1. From Table 4, it is clear that in all the sevenproblems, NFE for forced bound method is less than that formethod without forcing the bound but the difference is quitesignicant for Problems 3, 4, 5, 6, and 7. NFE for method with-out forcing the bound is about 3.4 times for Problem 3, 2.03times for Problem 4, 1.12 times for Problem 5, 6.15 times forProblem 6, and 1.94 times for Problem 7, more than that offorced bound method.

It is important to note that NRC is not 100% for all theproblems using either forced bound method or method withoutforcing the bound. Excepting for Problem 7, the NRC is almost


Table 4Results of DE using Approach 1

ProblemNo.

NFE/NRC/CPU-time Key parameters(NP/CR/F)

FBM MWFB

1 802/90/0.022 812/100/0.0 20/0.8/0.52 610/100/0.011 626/100/0.011 20/0.8/0.53 801/10/0.011 2727/100/0.071 30/0.8/0.74 1080/100/0.040 2196/100/0.083 30/0.8/0.55 11739/100/0.610 13265/90/0.696 30/0.9/0.66 1045/100/0.039 6430/100/0.280 20/0.8/0.57 46090/100/3.225 89490/0/6.291 100/0.8/0.5Converged to a non-optimal solution.

Table 5Results of MDE using Approach 1

ProblemNo.


FBM MWFB

1 690/100/0.000 705/100/0.000 20/0.8/0.52 490/100/0.011 490/100/0.000 20/0.8/0.53 1062/0/0.027 1974/100/0.055 30/0.8/0.54 1179/80/0.033 1797/100/0.055 30/0.8/0.55 10129/100/0.527 11914/100/0.621 30/0.9/0.66 865/100/0.030 5495/100/0.220 20/0.8/0.57 40550/100/2.846 79380/0/5.544 100/0.8/0.5Converged to a non-optimal solution.

100% using method without forcing the bound. Similarly, forforced bound method NRC is almost 100% excepting Problem3. It is because when upper limit of bound is violated, thevalue of variable is forced to the upper limit that resulted inconvergence to non-optimal solution. However, for Problem 7,the NRC is 100% for forced boundmethod and 0.0% for methodwithout forcing the bound, as global optimum is located on thebound of decision variables. CPU-time is found to be more incase of method without forcing the bound.

It is observed from Table 5 that NRC is 100% for all theproblems excepting Problem 7 using method without forcingthe bound while NRC is zero and 80% for Problems 3 and 5,respectively, using forced bound method. NFE and hece CPU-time is found to be more for method without forcing the boundas compared to forced bound method. In all the seven problems,NFE for forced bound is less than that for without forcingbut the difference is quite signicant for Problems 3, 4, 5, 6,and 7 than for Problems 1 and 2. NFE for without forcingis about 1.86 times for Problem 3, 1.52 times for Problem 4,1.17 times for Problem 5, 6.35 times for Problem 6, and 1.96times for Problem 7, more than that of forced bound method.This observation is similar to that seen for results reported inTable 4 using DE.

7.2.1.1. Comparison of DE and MDE. Figs. 5 and 6 showthe comparison of DE and MDE in terms of NFE andNRC, respectively, using method without forcing the bound.As can be seen from Fig. 5. NFE using MDE is less as

0100002000030000400005000060000700008000090000

100000

1 2* 3 4* 5 6 7Problem No.

NFE

DE MDE

Fig. 5. NFE variation using DE and MDE (MWFB, Approach 1).

84

86

88

90

92

94

96

98

100

1 2* 3 4* 5 6 7Problem No.

NR

CDE MDE

Fig. 6. NRC variation using DE and MDE (MWFB, Approach 1).

compared to DE in all the seven problems. NFE is about13.17%, 21.73%, 27.61%, 18.17%, 10.18%, 14.54%, and11.3% more, respectively, for Problems 1, 2, 3, 4, 5, 6, and 7 incase of DE as compared to MDE. NRC for MDE, as shown inFig. 6, is 100% in all the problems except Problem 7 (for whichNRC is zero). For DE too, it is 100% for all the problemsexcept Problem 5 (for which NRC is 90).

Figs. 7 and 8 show the comparison of DE and MDE interms of NFE and NRC, respectively, using forced boundmethod. As can be seen from Fig. 7. NFE using MDE is lessas compared to DE for Problems 1, 2, 5, 6, and 7. However,NFE using DE is less for Problems 3 and 4. NFE is about13.97%, 19.67%, 13.71%, 17.22%, and 12.02% more, respec-tively, for Problems 1, 2, 5, 6 and 7 in case of DE as comparedto MDE. As shown in Fig. 8, NRC for MDE is 100% in all theproblems except Problem 5 (for which NRC is 90). For DEtoo, it is 100% for all the problems except Problems 1 and 3(for which NRC is 90 and 10, respectively).


05000

100001500020000250003000035000400004500050000

1 2* 3 4* 5 6 7Problem No.

NFE

DE MDE

Fig. 7. NFE variation using DE and MDE (FBM, Approach 1).

0102030405060708090

100

1 2* 3 4* 5 6 7Problem No.

NR

C

DE MDE

Fig. 8. NRC variation using DE and MDE (FBM, Approach 1).

Table 6Results of DE using Approach 2

ProblemNo.


FBM MWFB

1 964/90/0.011 1910/60/0.055 20/0.8/0.52 654/100/0.011 1332/100/0.011 20/0.8/0.53 801/10/0.027 2754/100/0.083 30/0.8/0.74 1071/100/0.055 5328/100/0.187 30/0.8/0.55 16114/80/0.852 53158/90/3.016 30/0.9/0.67 57330/100/4.022 122170/0/8.681 100/0.8/0.5Converged to a non-optimal solution.

7.2.2. Approach 2Tables 6 and 7 show the results obtained using DE and MDE

(with/without forcing the bound on variables), respectively, us-ing Approach 2. It is clear that in all the seven problems, NFEfor forced bound method is signicantly less than that for with-out forcing. NFE for without forcing is about 49.5%, 50.9%,70.9%, 79.9%, 69.7%, and 53.07%, respectively, for Problems1, 2, 3, 4, 5, and 7, more than that for forced bound method.It is important to note that NRC is not 100% for all the prob-

Table 7Results of MDE using Approach 2

ProblemNo.


FBM MWFB

1 742/60/0.011 1408/90/0.022 20/0.8/0.52 560/100/0.016 1218/100/0.055 20/0.8/0.53 882/20/0.030 3297/100/0.110 30/0.8/0.54 927/100/0.027 4602/100/0.176 30/0.8/0.55 13566/60/0.703 45801/90/2.511 30/0.9/0.67 52350/100/3.621 110970/0/7.786 100/0.8/0.5Converged to a non-optimal solution.

0

20000

40000

60000

80000

100000

120000

140000

1 2* 3 4* 5 7Problem No.

NFE

DE MDE


lems using either forced bound method or without forcing themethod. It is important to note that for Problem 7, the NRCis 100% using forced bound method and 0.0% using withoutforcing method as found for Approach 1. CPU-time is foundto be more in case of without forcing the bound method.

It is observed from Table 7 that NRC is not 100% for all theproblems using either forced bound method or method withoutforcing the bound but certainly it is better or equal in case ofmethod without forcing the bound for all the problems except-ing Problem 7 (where NRC is zero). NFE and hence CPU-timeis found to be twice or more for without forcing the boundmethod as compared to force bound method. To be precise, NFEfor without forcing is about 47.3%, 54.0%, 73.25%, 79.9%,70.4%, and 52.8%, respectively, for Problems 1, 2, 3, 4, 5,and 7, more than that of forced bound method. This observationis similar to that seen for results reported in Table 6 using DE.

7.2.2.1. Comparison of DE and MDE. Figs. 9 and 10 showthe comparison of DE and MDE in terms of NFE and NRC,respectively, using method without forcing the bound and Ap-proach 2. As can be seen from Fig. 9, NFE using MDE is lessas compared to using DE for all the problems except Prob-lem 3, where NFE using DE is less than that of MDE. NFE isabout 26.28%, 8.56%, 13.62%, 13.84%, and 9.17% more, re-spectively, for Problems 1, 2, 4, 5, and 7 in case of DE ascompared to MDE. As shown in Fig. 10, NRC for MDE and


0102030405060708090

100

1 2* 3 4* 5 7Problem No.

NR

C

DE MDE


0

10000

20000

30000

40000

50000

60000

70000

1 2* 3 4* 5 7Problem No.

NFE

DE MDE

Fig. 11. NFE variation using DE and MDE (FBM, Approach 2).

DE is same for all the problems except Problem 1 where NRCfor DE is 60% as compared to 90% for MDE. It is to be notedthat NRC is zero for Problem 7 using both DE and MDE.

Figs. 11 and 12 show the comparison of DE and MDE interms of NFE and NRC respectively using forced bound methodand Approach 2. As can be seen from Fig. 11, NFE usingMDE is less as compared to DE for all the problems except forProblem 3 where NFE using DE is slightly less (about 9.0%)than that of MDE. NFE is about 23.03%, 14.37%, 13.44%,15.81%, and 8.68% more, respectively, for Problems 1, 2,4, 5, and 7 in case of DE as compared to MDE. As shownin Fig. 12, NRC for MDE and DE is 100% for Problems 2,4, and 7. For Problems 1 and 5, NRC is higher for DE ascompared to MDE but for Problem 3, NRC for DE is only10% as compared to 20% for MDE. It is to be noted thatNRC is not 100% for Problems 1,3 and 5 using both DE andMDE. As compared to without forcing, the NRC is less for allthe problems using forced bound method. The savings in NFEusing MDE is almost same in both the methods (forced boundmethod and method without forcing the bound). This indicatesthat the method without forcing the bound is better than theforced bound method.

0

10

20

30

40

50

60

70

80

90

100

1 2* 3 4* 5 7Problem No.

NR

C

DE MDE

Fig. 12. NRC variation using DE and MDE (FBM, Approach 2).

0

20000

40000

60000

80000

100000

120000

140000

1 2* 3 4* 5 7Problem No.

NFE

DE (Approach-1) DE (Approach-2)

Fig. 13. NFE variation for DE using Approaches 1 and 2 (MWFB).

7.2.3. Comparison of Approaches 1 and 2Figs. 13 and 14 show the comparison of Approaches 1 and 2

for DE in terms of NFE and NRC, respectively, using methodwithout forcing the bound. As can be seen from Fig. 13, NFEusing Approach 2 is signicantly more as compared to Ap-proach 1 for all the problems except for Problem 3 where NFEusing is almost same (2727 using Approach 1 and 2754 usingApproach 2, respectively) in both the approaches. NFE is about57.49%, 53.00%, 58.78%, 75.04%, and 26.75% more, respec-tively, for Problems 1, 2, 4, 5 and 7 in case of Approach 2 ascompared to Approach 1. This is because of addition of con-straints (as binary or discrete variable is modeled as continuousvariable) and addition of nonconvexities to the problem in caseof Approach 2.

As shown in Fig. 14, NRC for Approaches 1 and 2 is samei.e. 100%, 100%, 100%, 90%, and zero, respectively, for theProblems 2, 3, 4, 5, and 7. For Problem 1, NRC is 100% forApproach 1 as compared to 60% for Approach 2. It is to benoted that NRC is zero for Problem 7 using both the approaches.


0

10

20

30

40

50

60

70

80

90

100

1 2* 3 4* 5 7Problem No.

NR

C


Fig. 14. NRC variation for DE using Approaches 1 and 2 (MWFB).

0

20000

40000

60000

80000

100000

120000

1 2* 3 4* 5 7Problem No.

NFE

MDE (Approach-1) MDE (Approach-2)

Fig. 15. NFE variation for MDE using Approaches 1 and 2 (MWFB).

Therefore, using DE, the Approach 1 is found to be better thanApproach 2 in terms of both NFE and NRC.

Figs. 15 and 16 show the comparison of Approaches 1 and 2for MDE in terms of NFE and NRC, respectively, using methodwithout forcing the bound.As can be seen from Fig. 15, NFE us-ing Approach 2 is signicantly more as compared to Approach1 for all the problems. NFE is about 50.0%, 59.77%, 40.13%,60.95%, 74%, and 28.47% more, respectively, for Problems 1,2, 3, 4, 5 and 7 in case of Approach 2 as compared to Ap-proach 1.

As shown in Fig. 16, NRC for Approaches 1 and 2 is same,i.e., 100%, 100%, 100%, and zero respectively for the Problems2, 3, 4, and 7. For Problem 1, NRC is 100% for Approach 1as compared to 90% for Approach 2. It is to be noted that NRCis zero for Problem 7 using both approaches. Therefore, usingMDE, the Approach 1 is found to be better than Approach 2in terms of both NFE and NRC. Also, it is important to notethat NRC using MDE is 100% for all the problems except forProblem 7. But using DE it is 90% for Problem 5 and zero for

0102030405060708090

100

1 2* 3 4* 5 7Problem No.

NR

C


Fig. 16. NRC variation for MDE using Approaches 1 and 2 (MWFB).

0

10000

20000

30000

40000

50000

60000

70000

1 2* 3 4* 5 7Problem No.

NFE


Fig. 17. NFE variation for DE using Approaches 1 and 2 (FBM).

Problem 7. Hence, MDE using Approach 1 seems to be a betterstrategy for solving such types of problems.

Figs. 17 and 18 show the comparison of Approaches 1 and2 for DE in terms of NFE and NRC, respectively, using forcedbound method. As can be seen from Fig. 17, NFE using Ap-proach 2 is more as compared to Approach 1 for the Problems1, 2, 5, and 7. For Problems 3 and 4, where NFE is almostsame (801 and 1080 using Approach 1 and 801 and 1071 us-ing Approach 2, respectively) in both the approaches using DE.NFE is about 16.8%, 6.73%, 27.15%, and 19.61% more, re-spectively, for Problems 1, 2, 5, and 7 in case of Approach 2as compared to Approach 1.

As shown in Fig. 18, NRC for Approaches 1 and 2 is samei.e. 90%, 100%, 10%, 100%, and 100%, respectively, for theProblems 1, 2, 3, 4, and 7. For Problem 5, NRC is 100% forApproach 1 as compared to 80% for Approach 2.

Figs. 19 and 20 show the comparison of Approaches 1 and 2for MDE in terms of NFE and NRC, respectively, using forced


0

10

20

30

40

50

60

70

80

90

100

1 2* 3 4* 5 7Problem No.

NR

C


Fig. 18. NRC variation for DE using Approaches 1 and 2 (FBM).

0

10000

20000

30000

40000

50000

60000

1 2* 3 4* 5 7Problem No.

NFE


Fig. 19. NFE variation for MDE using Approaches 1 and 2 (FBM).

bound method. As can be seen from Fig. 19, NFE using Ap-proach 2 is more as compared toApproach 1 for all the problemsexcept for Problems 3 and 4 (where NFE is about 16.95% and21.37% more using Approach 1 than that of Approach 2, re-spectively, for Problems 3 and 4). NFE is about 7.0%, 12.50%,25.33%, and 22.54% more, respectively, for Problems 1, 2, 5and 7 in case of Approach 2 as compared to Approach 1.

As shown in Fig. 20, NRC for Approaches 1 and 2 is samei.e. 100%, 100%, and 100%, respectively, for the Problems2, 4, and 7. For Problems 1, 3, and 5, NRC for Approach 1is higher as compared to Approach 2. Therefore, using MDE,the Approach 1 is found to be better than Approach 2 interms of both NFE and NRC. This indicates that nonconvexity(induced due to modeling of binary variable as continuousvariable) affects the NRC (Figs. 14, 16, 18, and 20). Also, itis important to note that NRC using MDE is higher than thatof using DE (Figs. 18 and 20). Therefore, MDE using Ap-

0

10

20

30

40

50

60

70

80

90

100

1 2* 3 4* 5 7Problem No.

NR

C


Fig. 20. NRC variation for MDE using Approaches 1 and 2 (FBM).

proach 1 seems to be a better strategy for solving such types ofproblems.

7.2.4. Comparison of MDE, GA, M-SIMPSA, and (+ )-ESIn the present study, DE and MDE are used to solve process

synthesis and design problems and their performance is com-pared. Further, the performance of MDE is evaluated and com-pared with that of GA, M-SIMPSA, and (+)-ES algorithms.Table 8 shows the comparison of MDE with GA, M-SIMPSA,M-SIMPSA-pen, and ( + )-ES. The NFE in MDE is about82% less than of that in GA for all the problems (The range is82.098.16% to be precise).

Similarly, the NFE in MDE is about 8196.0% less thanthat of M-SIMPSA for various test problems. While comparing( + )-ES and MDE it is found that for Problems 1 and 2NFE for MDE is about 53.56% and 78.27% less than that of( + )-ES but for Problems 3, 5, and 6, NFE for ( + )-ESis, respectively, about 11.48%, 43.68%, and 53.85% less thanthat of MDE. It is important to note that ( + )-ES is foundto converge to a non-optimal solution for Problems 4 and 7,whereas MDE is able to locate the global optimum for all theproblems solved in the present study.

It may be noted that GA could not converge to global opti-mum, while execution was reported to be halted in M-SIMPSAfor Problem 7. However, for Problem 7, NRC is 100% onlyfor MDE, and 92% for MIMPSA-pen, while GA, M-SIMPSAand ( + )-ES were not able to converge to global optimum.The performance of (+ )-ES and MDE is found to be com-parable in terms of function evaluations but MDE has the ad-vantage over ES in terms of convergence to optimal solution.The performance of MDE is found to be better than that ofGA & M-SIMPSA in optimizing the mixed integer nonlinearprogramming problems considered in the present study. Also,results clearly show the potential of ( + )-ES and MDE forsolving such process synthesis problems. We have already seenthat MDE is better than DE in terms of functions evaluations(NFE) and convergence to optimal solution (NRC).


Table 8Comparison of MDE, GA, ES, M-SIMPSA & M-SIMPSA-pen

Problem No. NFE/NRC

GA M-SIMPSA M-SIMPSA-pen ES MDE

1 6787/100 607/99 16282/100 1518/100 705/1002 13939/100 10582/83 14440/100 2255/100 490/1003 107046/90 #/0 38042/100 1749/100 1974/1004 22489/100 14738/100 42295/100 /0 1797/1005 102778/60 22309/60 63751/97 6710/100 11914/1006 37167/100 27410/87 33956/95 2536/100 5495/1007 225176/0 #/0 257536/92 /0 40550/100

# Execution halted Converged to a non-optimal solution.

8. Conclusions

Seven test problems on process synthesis and design fromchemical engineering have been solved using DE and MDE inthe present work. Performance of various algorithms (e.g. GA,M-SIMPSA, M-SIMPSA-pen, ( + )-ES, DE, and MDE) iscompared in terms of NFE and NRC. Also, a new approach forhandling binary variables is studied and compared with the ap-proach of Li (1992) along with two methods for handling boundviolations. Results indicate that the method without forcing thebound is robust however the CPU-time is more as compared tothe forced bound method. It is important to note that for prob-lems where global optimum is located on the bound of deci-sion variables forced bound method is found to outperform themethod without forcing the bound (e.g. Problem 7) whereas forproblems where local optimum is located on the bound of de-cision variables the method without forcing the bound is foundto be robust (e.g. Problem 3). Also, the Approach 1 for han-dling binary variables is found to be better than that of Ap-proach 2. This is because of addition of constraints (as binaryor discrete variable is modeled as continuous variable) and ad-dition of nonconvexities to the problem in case of Approach 2.It is found that NFE is the least and NRC is highest in MDE ascompared to GA & M-SIMPSA-pen. The NRC in M-SIMPSAis better than that of GA, M-SIMPSA, and ( + )-ES. The(+ )-ES and MDE are comparable in terms of NFE (rather( + )-ES perform slightly better than MDE) but MDE isfound to be robust than (+)-ES being higher NRC for Prob-lems 4 and 7 (Table 8). The performance of MDE is found tobe the best among the methods compared for all the problemsstudied.

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Optimization of process synthesis and design problems:A modified differential evolution approachIntroductionModified differential evolution (MDE)Problem formulationTest problems on process synthesis and designHandling of integer and binary variablesInteger variablesBinary or discrete variables

Handling of bound violations and constraintsResults and discussionEffect of key parameters (CR, F, and NP)Test problems on process synthesis and designApproach 1Approach 2Comparison of Approaches 1 and 2Comparison of MDE, GA, M-SIMPSA, and (mu+lambda)-ES

ConclusionsReferences

optimization of process synthesis and design problems: a modified differential evolution approach

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