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Computers and Chemical Engineering 31 (2007) 760–772

A study of differential evolution and tabu search for benchmark,phase equilibrium and phase stability problems

Mekapati Srinivas, G.P. Rangaiah ∗Department of Chemical and Biomolecular Engineering, National University of Singapore,

4 Engineering Drive 4, Singapore 117576, Singapore

Received 10 October 2005; received in revised form 28 July 2006; accepted 31 July 2006Available online 7 September 2006

bstract

Phase equilibrium calculations (PEC) and phase stability (PS) problems play a crucial role in the simulation, design and optimization ofeparation processes such as distillation and extraction. The former involve the global minimization of Gibbs free energy function whereas theatter requires the global minimization of tangent plane distance function (TPDF). In this work, two promising global optimization techniques:ifferential evolution (DE) and tabu search (TS) have been evaluated and compared, for the first time, for benchmark, PEC and PS problems. A
ocal optimization technique is used at the end of both TS and DE to improve the accuracy of the final solution. Benchmark problems involve–20 variables with a few to hundreds of local minima whereas PEC and PS problems consist of multiple components with comparable minima.EC involves both vapor–liquid, liquid–liquid and vapor–liquid–liquid equilibria with popular thermodynamic models. The results show that DE
s more reliable but computationally less efficient compared to TS for benchmark, PEC and PS problems tested.2006 Elsevier Ltd. All rights reserved.

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eywords: Differential evolution; Tabu search; Benchmark problems; Phase eq

. Introduction

Phase equilibrium calculations (PEC) and phase stability (PS)roblems have to be solved a very large number of times in theesign and analysis of chemical processes. For a system withpecified components, composition, temperature and pressure,EC involves the calculation of number of moles of each phasend its composition at equilibrium whereas PS analysis deter-ines the stability of the system. There are mainly two different

pproaches for PEC: equation solving approach and Gibbs freenergy minimization approach. The former involves solving aystem of non-linear equations resulting from mass balances andquilibrium relationships whereas the latter involves the mini-ization of highly non-linear Gibbs free energy function. Even

hough equation solving approach seems to be faster and simple,he solution obtained in this method may not correspond to the
rue minimum of Gibbs free energy function. Also, it needs a pri-ri knowledge of phases existing at equilibrium. Hence, Gibbsree energy minimization is the desirable approach for PEC.
∗ Corresponding author. Tel.: +65 6516 2187; fax: +65 6779 1936.E-mail address: [email protected] (G.P. Rangaiah).

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098-1354/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2006.07.015

um calculations; Phase stability problems; Gibbs free energy minimization

The concept of Gibbs free energy minimization for chemi-al equilibrium was proposed by White, Johnson, and Dantzig1958) stating that a necessary condition for a given system toe at equilibrium is that the total Gibbs free energy must be athe global minimum. The objective function in this approachs highly non-linear and non-convex necessitating reliable andfficient global optimization. Several researchers have appliedifferent global optimization methods using this approach andreview of these studies can be found in Teh and Rangaiah

2003). Recently, Nichita, Gomez, and Luna (2002a) employedtunneling method for PEC. In this method, the calculations

re organized in a stepwise manner, combining PS analysisy minimization of tangent plane distance function (TPDF)or phase splitting calculations. The tunneling method has twohases: a local bounded minimization using a limited memoryuasi-Newton method and a tunneling phase to find a betternitial estimate for the next phase of local minimization. Theesults show that the tunneling method is efficient and reliableor solving multiphase equilibrium problems. Iglesias-Silva,
onilla-Petriciolet, Eubank, Holste, and Hall (2003) proposedn algebraic method that includes Gibbs free energy mini-ization for PEC. The method uses orthogonal derivatives,
he tangent plane condition and mass balances to reduce the

mailto:[email protected]

dx.doi.org/10.1016/j.compchemeng.2006.07.015

M. Srinivas, G.P. Rangaiah / Computers and Chemical Engineering 31 (2007) 760–772 761

Nomenclature

A amplification factorCR crossover constantDE differential evolutionES2 Easom functionF total moles of feedG partial molar Gibbs free energyGen generationsGP2 Goldstein and Price functionG/RT dimensionless Gibbs free energyH dimensionless tangent plane distance functionH3 Hartmann functionhn length of the hyper-rectangleIter iterationsLLE liquid–liquid equilibriumnc number of componentsnp number of phasesnk total number of moles in phase knk

i number of moles of component i in phase kN dimension of the problemNneigh number of neighborsNp promising list sizeNt tabu list sizeNFE number of function evaluationsNP population sizeNRTL non-random two liquidP system pressurePEC phase equilibrium calculationsPS phase stabilityQN quasi-Newton methodran random numberROSN Rosenbrock functions centroid of the hyper-rectangleSc successive iterations/generationsSC stopping criterionSR success ratet tangent planeT system temperatureTPDF tangent plane distance functionTS tabu searchTS-M TS with mixed generation of neighborsTS-R TS with random generation of neighborsTS-S TS with systematic generation of neighbors using

hyper-rectanglesU trial vectorUNIQUAC universal quasi-chemicalV mutated vectorVLE vapor–liquid equilibriumVLLE vapor–liquid–liquid equilibriumx mole fraction of the given phaseX target vectorzi moles of ith component in the feedZAKN Zakharov function

Greek lettersβi ith component of the decision variableγ i activity coefficient of component iε radiusφi fugacity coefficient of component i

SubscriptsG generationinit initial stagemax maximum numberneigh neighbor pointp promising pointt tabu point

Superscripts

GttBbtttrheol

oGinaapHdopMatttTwit(

i

L liquid phaseV vapor phase

ibbs minimization procedure to a task of finding the solu-ion of a system of non-linear equations. The results showhat the method has good convergence rate. Burgos-Solorzano,rennecke, and Stadtherr (2004) solved the problem by com-ining reliable deterministic techniques such as interval Newtonechnique with local optimization procedure. The deterministicechniques validate the results obtained from the local optimiza-ion technique and provide corrective feedback until the rightesult is found. The results show that the procedure is good forigh pressure chemical and multiphase equilibrium using cubicquation of state models. In these cited works, there is no rig-rous comparison of the proposed methods with others in theiterature.

Most often PEC needs a priori information about the numberf phases existing at equilibrium. Some of the methods (e.g.,autam & Seider, 1979) explore all possible number of phases

n a systematic manner. Initially the method assumes a smallumber of phases and checks for the stability of phases. If theyre stable the method retains the phases else a new phase will bedded. A PS problem can be formulated as either minimizationroblem or an equivalent non-linear equation solving problem.owever, the conventional solution techniques are initializationependent and may converge to a local or trivial solution basedn the initial guess. Baker, Pierce, and Luks (1982) stated androved the tangent plane criterion often used for PS analysis.ichelsen (1982) described several numerical methods for PS

nalysis based on this tangent plane criterion. The criterion stateshat a hypothetical phase is thermodynamically stable providedhe tangent plane generated at the given composition lies belowhe molar Gibbs free energy surface for all the compositions.he problem can be formulated as the minimization of TPDF,hich is a non-linear and non-convex objective function requir-

ng global optimization. A review of several works using the
angent plane criterion for PS problems can be found in Rangaiah2001).
Tessier, Brennecke, and Stadtherr (2000) implementednterval Newton technique for PS analysis. The examples are

7 nd Chemical Engineering 31 (2007) 760–772

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62 M. Srinivas, G.P. Rangaiah / Computers a

odeled by non-random two liquid (NRTL) and universaluasi-chemical (UNIQUAC) thermodynamic models. Theylso proposed two enhancements for interval Newton method.he results indicate that the computational efficiency of thenhanced methods is better compared to the original one.ichita, Gomez, and Luna (2002b) used a global optimizationethod namely, tunneling method for PS analysis. The problem

as been formulated both in conventional approach (i.e., com-osition space) and in reduced variable approach. The resultshow that the method is reliable in solving the PS problems.alogh, Csendes, and Stateva (2003) used a modified TPDF

uch that the zeros of the objective function become its minima,ince it is advantageous to search for minima with known zeroinimum value. They employed a method namely, stochastic

ampling and clustering to locate the minima of the modifiedPDF. The results show that the method is able to solve small

o moderate size problems in an efficient and reliable way.However, most of the methods employed for PEC and PS

roblems are local in nature and relatively few stochastic globalptimization techniques have been explored for these problems.tochastic methods are usually quite simple to implement andse, and they do not require transformation of the original prob-em. Furthermore, these techniques can locate the vicinity oflobal solutions with relative efficiency compared to determin-stic techniques (Moles, Mendes, & Banga, 2003). Among the

any, differential evolution (DE) (Storn & Price, 1997) and tabuearch (TS) (Chelouah & Siarry, 2000) are some of the mostromising methods reported in the literature. Even though theyave been tested for several applications in chemical engineer-ng and other fields (e.g., Bingul, 2004; Lin & Miller, 2004a,b;

ayer, Kinghorn, & Archer, 2005), they have not been appliedo PEC and PS problems except Teh and Rangaiah (2003) whotudied PEC problems by TS. Also, DE and TS have not beenomprehensively compared for benchmark problems. Hence, inhis work, both DE and TS are first evaluated and comparedor benchmark problems with 2–20 variables but involving aew to hundreds of local minima. The methods are then testedor PEC and PS problems involving multiple components, mul-iple phases and popular thermodynamic models. The evalua-ion includes both reliability and computational efficiency usingractical stopping criteria.

. Differential evolution

DE (Storn & Price, 1997) is a population based direct searchethod. The algorithm implemented in this study (Fig. 1) startsith specifying the parameters, namely, amplification factor

A), crossover constant (CR), type of strategy, population sizeNP), maximum number of successive iterations (Scmax) withoutmprovement in the best function value and maximum numberf generations (Genmax). The initial population is randomly gen-rated using the uniformly distributed random numbers to coverhe entire solution space. The individuals are checked for the
oundary violation to see if any individual is generated in thenfeasible region; the infeasible points are replaced by gener-ting new individuals. The objective function values of all thendividuals are calculated and the best point is determined. Then
tttp

Fig. 1. Flow chart of DE-QN.

he three main steps: mutation, crossover and selection on theopulation, are carried out. Mutation and crossover operationsre performed to diversify the search thus escaping from theocal minima. The mutant vector is generated for each randomlyhosen target vector Xi,G by

i,G+1 = Xr1,G + A(Xr2,G − Xr3,G), i = 1, 2, 3, . . . , NP

(1)

here r1, r2 and r3 belongs to the set {1, 2, 3, . . . ,NP} andr1,G, Xr2,G and Xr3,G represents the three random individuals

hosen in the current generation, G, to produce the mutant vectoror the next generation, Vi,G+1. The random numbers r1, r2 and3 should be different from the running index, i, and hence NPhould be ≥4 to allow mutation. A is a real value between 0 andwhich controls the amplification of the differential variation

etween the two random vectors.In the crossover step, the trial vector, Ui,G+1 is produced by

opying some elements of the mutant vector, Vi,G+1 to the tar-et vector, Xi,G with probability equal to CR. As illustrated inig. 2, a random number (ran) is generated for each elementf the target vector. If ran ≤ CR, the element of mutant vectors copied else the target vector element is copied. After muta-ion and cross-over operations, the trial vector competes withhe target vector for selection into the next generation. A greedyriterion based on objective function value is used to screen
he trial vector. If the trial vector has a better value comparedo the target vector, it replaces the target vector in the popula-ion thus allowing the best solution into further generations. Therocess of mutation, crossover and selection is repeated until a


Fr

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ig. 2. Schematic diagram of crossover operation; for continuous lines,an ≤ CR, and for dotted lines, ran > CR.

ermination criterion such as maximum number of generations isatisfied. The algorithm then terminates providing the best pointhat has been explored over all the generations. The best points further refined using a fast convergent quasi-Newton methodo achieve the best minimum which is declared to be the global

inimum.

. Tabu search

TS, first developed by Glover (1989,1990), has been widelysed for combinatorial optimization (Youssef, Sait, & Adiche,001) but its use is very limited in continuous optimizationChelouah & Siarry, 2000; Lin & Miller, 2004a,b; Teh &angaiah, 2003). TS is a meta heuristic that guides the heuris-

ics to escape from the local minima. The main concepts ofS include diversification and identifying the most promising

egion. The diversification step performs an exhaustive searchn the entire solution space by generating solutions that areot seen before. To implement this, TS maintains both tabuist (consisting of unpromising points) and promising list tovoid repeated visits to the same place in the search region,hich in turn improves the computational efficiency. Afterspecified maximum number of iterations, in-depth search

nown as intensification is performed from the most promisingoint.

The TS algorithm (Fig. 3) starts with the selection of val-es for the parameters: tabu list size (Nt), promising list sizeNp), tabu and promising radii (εt and εp), length of the hyper-ectangle (hn), initial population size (NPinit), number of neigh-ors (Nneigh), maximum number of successive iterations (Scmax)ithout improvement in the best function value and maximumumber of iterations (Itermax). The algorithm then randomly gen-rates a population of specified size and evaluates the objectiveunction value at each individual. The best point is filled intohe promising list and the remaining will be sent to the tabuist. The best point found is selected as the current centroids) of the hyper-rectangle, which is used to generate neighborso explore for better points in the neighborhood. The genera-ion of neighbors can be executed in many ways, i.e., either
y using hyper-circles or hyper-rectangles, etc. In this study,yper-rectangles have been used to generate the neighbors. Aetailed explanation about the generation of neighbors usingyper-rectangles is available in Teh and Rangaiah (2003). The
ecag

Fig. 3. Flow chart of TS-QN.

eighbors are then compared with the points in tabu and promis-ng lists, and only those points away from the latter are evaluated.he rejection of the neighbors which are nearer to the points in

abu and promising lists improves the computational efficiencyf TS avoiding repeated visits to the same place during theearch. The algorithm selects the best point found in the cur-ent iteration as the centroid of the hyper-rectangle to generateeighbors for the next iteration. The best point in the currentteration is accepted even if it is worse than that of the previousterations to avoid entrapment in the local minima. The processf generating neighbors is repeated and the tabu and promis-ng lists are updated in each iteration. Once the tabu/promisingist is filled, the next tabu/promising point will be placed in therst position of tabu/promising list and subsequent positions areccupied by the remaining points. Thus, both tabu and promisingists are updated dynamically during the search to keep the lat-st point(s) in the list by replacing the earliest-entered point(s).fter a specified number of iterations, most promising area is

dentified and is further investigated by intensification step. Gen-
rally, a local optimization technique is used in this step; a fastonvergent quasi-Newton technique is used in this study. Thelgorithm then terminates by declaring the final solution as thelobal minimum.

764 M. Srinivas, G.P. Rangaiah / Computers and Chemical Engineering 31 (2007) 760–772

Table 1Details of the benchmark problems

Function Number of variables (N) Global minimum Remarks

Goldstein and Price function (GP2) 2 3 at x = {0, −1} Four local minimaEasom function (ES2) 2 −1 at x = {π, π} Several local minimaShubert function (SH2) 2 −186.7309 at x = {−0.8427, −0.1889} 18 global minima; 760 local minimaHartmann function (H3) 3 −3.86278 at x = {0.114614, 0.555649, 0.852547} Four local minimaR {1, . .Z {0, . .

4

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5

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bfieNimaGcSofaompcSdthe size range (up to 20 variables) of the problems consideredin this study.

Table 2Optimal parameter values for TS-QN and DE-QN

Parameters Benchmarkproblems

Phase equilibriumcalculations

Phase stabilityproblems

TS-QNNt and Np 10 10 10εt and εp 0.01 0.02 0.02NPinit 20N 20N 20NNneigh

a 2N 2N 2Nhn 0.5 0.5 0.5Itermax 50N 100N min {50N, 100}Scmax 6N 2N 2N

DE-QNA 0.5 0.3 0.3CR 0.5 0.9 0.9NP 50 min {50N, 200} min {50N, 100}

osenbrock function (ROSN) 2, 5, 10 and 20 0 at x =akharov function (ZAKN) 2, 5, 10 and 20 0 at x =

. Implementation of DE and TS

A Matlab code for DE is taken from the websitettp://www.icsi.berkeley.edu/∼storn/code.html, and a boundaryiolation check is implemented in the code. For the local mini-ization step, an in-built subroutine from the Matlab optimiza-

ion tool box namely, FMINCON is used. The objective functionor DE code is written in FORTAN and simple gateway functionsre used to call it from the Matlab. This is adopted as all our pro-rams for PEC and PS are in FORTRAN. For TS, the FORTRANode developed by Teh and Rangaiah (2003) is used; it uses theMSL subroutine namely, DBCONF for the local minimizationtep. Both FMINCON and DBCONF employ the fast conver-ent quasi-Newton method with BFGS update for the Hessianatrix. For the first time, a local optimization technique is usedith DE in this study, and a similar work is done for TS by Teh

nd Rangaiah (2003). The minimization technique at the end ofhese methods is executed to find the final solution accuratelynd efficiently.

. Benchmark problems

Several benchmark problems having 2–20 variables and a fewo several hundreds of local minima are used to evaluate both DEollowed by quasi-Newton method (DE-QN) and TS followedy quasi-Newton method (TS-QN). A brief description of theunctions and the global minima are given in Table 1. Two typesf stopping criteria are used in this study. They are maximumumber of iterations/generations (Itermax in TS-QN and Genmaxn DE-QN) (referred as stopping criterion 1 (SC1)) and max-mum number of iterations/generations or maximum numberf successive iterations/generations (Scmax) without improve-ent in the best function value (referred as stopping criterion(SC2)). Several published studies (e.g., Cai & Shao, 2002)

mployed convergence to the global minimum as a stoppingriterion. On the contrary, we used SC1 and SC2 because, ineality, global minimum of application problems is unknown ariori. The performance of the two methods is evaluated basedn both reliability (measured in terms of how many times thelgorithm located the global minimum out of 100 trials, referreds success rate (SR)) and computational efficiency (measured in
erms of average number of function evaluations (NFE) in allhe 100 trials). The gradient is calculated numerically and theFE includes the function calls for evaluating both the objec-
ive function and the numerical gradient for the quasi-Newtonethod.

ac

. ,1} Several local minima

. ,0} Several local minima

.1. Parameter tuning

Test functions GP2, ES2, SH2, ROS5, ROS10 and ROS20 haveeen selected to tune the parameters of TS-QN and DE-QN tond the global minimum with good reliability and computationalfficiency. The nominal parameter values chosen for TS-QN aret and Np = 10; εt and εp = 0.01; hn = 0.5; NPinit = 20N, where N

s the dimension of the problem; Nneigh = 2N (subject to a mini-um of 10 and a maximum of 30); Scmax = 5N and Itermax = 50N,

nd for DE-QN, A = 0.4; CR = 0.1; NP = 50; Scmax = 5N andenmax = 50. The nominal values for TS-QN and DE-QN are

hosen based on the optimum values available in Chelouah andiarry (2000), and preliminary numerical experience with somef the benchmark problems, respectively. The tuning is per-ormed by varying one parameter at a time while the rest are fixedt their nominal/recent optimum values. The optimal parametersbtained for TS-QN and DE-QN are given in Table 2. The opti-al parameters found for TS-QN are the same as its nominal

arameters. This may be because the nominal parameters arehosen based on the optimal settings given in Chelouah andiarry (2000). The optimal NP obtained for DE-QN is indepen-ent of N because the effect of N is accounted via Genmax and

Genmax 20N 75 50Scmax 10N 12N 6N

a Nneigh is restricted to a minimum of 10 and a maximum of 30 for benchmarknd PEC problems, and to 20–30 for PS problems to have good reliability andomputational efficiency.

http://www.icsi.berkeley.edu/~storn/code.html

M. Srinivas, G.P. Rangaiah / Computers and C

Table 3aSuccess rate (SR) and number of function evaluations (NFE) for solving bench-mark problems by DE-QN and TS-QN

Function TS-QN DE-QN

SC1 SC2 SC1 SC2

SR NFE SR NFE SR NFE SR NFE

GP2 100 918 99 301 100 2026 100 1998ES2 90 1040 85 433 100 2072 76 1747SH2 100 1033 92 355 99 3051 100 1790ROS2 100 1059 100 475 100 2065 100 2065ZAK2 100 1009 100 343 100 2077 100 2048H3 100 987 100 386 100 3071 100 3071ROS5 78 2799 79 2081 100 5177 100 5171ZAK5 100 2629 100 1294 100 5093 100 5093ROS10 78 8578 78 8541 100 10213 100 10210ZAK10 100 8491 100 8473 100 10143 100 10125RZ

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OS20 75 22074 75 22074 98 20394 95 20338AK20 100 19157 100 19157 100 20238 100 20234

.2. Results and discussion

The results for solving the benchmark problems by TS-QNnd DE-QN are given in Tables 3a and 3b. Each benchmarkroblem is solved 100 times, each time by generating a randomnitial estimate. The results are compared in terms of SR andFE, which is the average of all 100 trials. It is evident fromable 3a that the reliability of DE-QN is better compared to TS-N, for both SC1 and SC2. This is perhaps due to the different

scaping mechanisms associated with these two methods. DEerforms mutation and crossover over a set of individuals (i.e.,opulation), whereas TS accepts the best point in each iterations the new centroid of the hyper-rectangle for generating neigh-ors even though it is worse than the previous best points inrder to escape from the local minima. The reliability of TS-QNs less for ES2 function because the function is flat everywheren the feasible region except near the center (global minimum
egion). As the function is flat, all the neighbors generated inS-QN will have the same value trapping the search in that
egion, whereas DE-QN explored the global minimum region by

able 3bR and NFE using DE-QN and TS-QN with same Nneigh/NP (=20) andtermax/Genmax (=50N)

unction TS-QN DE-QN

SR NFE SR NFE

P2 100 1384 100 2036S2 83 1683 100 2027H2 100 1621 100 2056OS2 100 1603 100 2124AK2 100 1542 100 2027

3 100 2021 100 3039OS5 76 4002 100 5021AK5 100 4142 100 5061OS10 87 8395 99 10185AK10 100 8268 100 10092OS20 81 18499 96 20308AK20 100 15603 100 20020

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hemical Engineering 31 (2007) 760–772 765

enerating different new individuals by the process of mutationnd crossover. The reliability of both TS-QN and DE-QN forhubert function is high even though it has 760 local minima.his may be because locating one of the several global minima

around 18) in this example is sufficient to achieve the bestunction value. The reliability of TS-QN is less for Rosenbrockunctions because of narrow global minimum region in theseunctions.

Even though reliability of DE-QN is more than TS-QN, itsomputational efficiency is less compared to TS-QN (Table 3a).FE for DE-QN is 1.05 (ZAK20) to 3.11 times (H3) more than

hat for TS-QN using SC1, and is 1.05 (ZAK20) to 7.95 timesH3) more than that of TS-QN using SC2. This could be becausef avoiding repeated visits to the same place in TS by keepingrack (i.e., by maintaining tabu and promising lists) of the pre-ious search points which in turn improves the computationalfficiency. NFE for both TS-QN and DE-QN increases with theumber of variables due the increase in the size of the solutionpace which makes both the algorithms to generate more points.

DE-QN and TS-QN have also been evaluated using SC2. Theomputational efficiency and reliability (Table 3a) are better andomparable using SC2 compared to that of SC1 for both DE-N and TS-QN. NFE of DE-QN using SC1 is 1.01 (ZAK2)

o 1.7 times (H3) more compared to SC2; similarly, NFE ofS-QN is 1.34 (ROS5) to 3.05 times (GP2) more compared toC2 for TS-QN. This is because the algorithms will terminate

f the best function value does not change successively after thepecified Scmax iterations/generations resulting in good compu-ational efficiency with SC2.

The performance of TS-QN and DE-QN is also compared byeeping the similar parameters (i.e., Nneigh and Itermax in TS-N; NP and Genmax in DE-QN) to the same value. The values

hosen are: Nneigh and NP = 20; and Itermax and Genmax = 50N,nd the remaining (algorithm-specific) parameters are chosenrom Table 2. The algorithms are compared using SC1 so thatnother parameter Scmax need not be included. From the resultsiven in Table 3b, it is clear that the reliability (i.e., SR) ofE-QN is better compared to that of TS-QN, whereas the com-utational efficiency (NFE) of the latter is better than that ofE-QN for all the functions tested. Thus, the relative perfor-ance of TS-QN and DE-QN with similar parameter values

s the same as that with optimal parameters. Tables 3a and 3bhow that SR and NFE for TS-QN with similar parameter val-es are inferior to those with optimal parameters for smallerroblems. On the other hand, SR and NFE for DE-QN withimilar parameter values (Table 3b) are comparable to thoseith optimal parameters (Table 3a). These findings highlight the

hallenges in tuning parameters and the possibility of differentets of optimal parameters. In this study, parameters are tunedarefully and systematically to ensure a fair comparison of theethods.

. Phase equilibrium problems

The expression for total free energy, G can be simplifiedor different situations by eliminating constant terms for aarticular system (Rangaiah, 2001). If vapor and liquid phases

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re described by different thermodynamic models, then theimensionless free energy for a non-reacting system is given by

GI

RT=

∑k ∈ np (liquid phase only)

nc∑i=1

nL(k)i [ln (xL(k)

i γL(k)i P sat

i )]

+nc∑i=1

nVi [ln (yiϕ

Vi P)] (2)

here the superscript, L(k) and V refer to liquid phase k andapor phase, respectively. The first term in the above equationefers to only liquid phases assuming activity coefficient modelsor them. For equation of state models describing all liquid andapor phases, the above equation can be written as

GII

RT=

∑k ∈ np (liquid phase only)

nc∑i=1

nL(k)i [ln (xL(k)

i ϕL(k)i )]

+nc∑i=1

nVi [ln (yiϕ

Vi )] (3)

f only liquid phases exist, then Eq. (2) can be simplified bygnoring P sat

i (which does not change for a particular example)nd the second term for vapor phase can be eliminated. Thenhe equation becomes:

GIII

RT=

np∑k=1

nc∑i=1

nL(k)i [ln (xL(k)

i γL(k)i )] (4)

In Eqs. (2)–(4), nki are the number of moles of component i

n phase k, np is the number of phases at equilibrium, nc is theotal number of components in the system, γk

i and ϕki are the

ctivity and partial fugacity coefficient of component i in phase, respectively. P sat

i and yi are the saturated vapor pressure andole fraction in vapor phase of component i.At physical equilibrium (i.e., without any chemical reaction),

oles of each component should be conserved and the num-er of moles of each component (nk

i ) should be non-negative.ence, free energy minimization should satisfy the following

onstraints and bounds:

np

k=1

nki = ziF, i = 1, 2, . . . , nc (5)

≤ nki ≤ ziF, i = 1, 2, . . . , nc (6)

ecision variables for minimizing the free energy are nki . The

onstrained minimization problem can be simplified to annconstrained problem by introducing the variables βk

i (for= 1, 2, . . . ,nc; k = 1, 2, . . . ,np − 1) instead of mole numbers nk

i

for i = 1, 2, . . . , nc; k = 1, 2, . . . , np) as the decision variables.he new variables, βk

i are bounded between 0 and 1 and areelated to nk

i by

1i = β1

i ziF, i = 1, 2, . . . , nc (7)

dBsaeTpa

hemical Engineering 31 (2007) 760–772

ki = βk

i

⎛⎝ziF −

k−1∑j=1

nji

⎞⎠ ,

= 1, 2, . . . , nc; k = 2, . . . , np − 1 (8)

npi =

⎛⎝ziF −

np−1∑j=1

nji

⎞⎠ , i = 1, 2, . . . , nc (9)

he equality constraints (Eq. (5)) are eliminated by the intro-uction of Eqs. (7)–(9) thus reducing the number of decisionariables from nc × np to nc × (np − 1). The bounds on the deci-ion variables are set as

0−15 ≤ βki ≤ 1.0, i = 1, 2, . . . , nc (10)

he lower bounds of the variables are taken as 10−15 instead of.0 to avoid numerical difficulties associated with the Gibbs freenergy function when the number of moles of a component in ahase is equal to zero.

Phase equilibrium examples considered in this study includeapor–liquid equilibrium (VLE), liquid–liquid equilibriumLLE) and vapor–liquid–liquid equilibrium (VLLE) examplesnvolving multiple components (two to eight components) andopular thermodynamic models. The feed composition, oper-ting conditions and the thermodynamic models used for eachxample, and global and local minima for all these examplesre given in Teh and Rangaiah (2003). Along with local min-ma, there are trivial solutions for several examples at whichquilibrium composition equals to the feed composition (Teh &angaiah, 2002).


Examples 5, 9 and 10 are selected for tuning the parameters ofS-QN and DE-QN, which are shown to be difficult in Teh andangaiah (2002). The parameters are tuned one at a time whileeeping others fixed at their nominal/recent optimum values.he optimal parameters obtained for TS-QN and DE-QN areiven in Table 1. The difference in the values of the parametersA, CR and Genmax for DE-QN, and εt, εp and Itermax for TS)ompared to those for benchmark problems is due to the presencef a few but comparable minima (i.e., function values at last localinimum and the global minimum are close to each other) in

he PEC examples.


All the examples are solved 100 times each, starting from aifferent point randomly chosen each time in the feasible region.oth DE-QN and TS-QN are evaluated based on two types of

topping criteria: SC1 and SC2. The results are shown in Table 4and are given in terms of reliability (i.e., SR) and computational
fficiency (i.e., NFE), and are average over all the 100 trials.he study here is limited to the calculation of equilibrium com-ositions and the number of phases existing at equilibrium isssumed to be known a priori.


Table 4aSR and NFE for solving phase equilibrium problems by DE-QN and TS-QN

Example numbera TS-QN DE-QN

SC1 SC2 SC1 SC2

SR NFE SR NFE SR NFE SR NFE

Vapor–liquid equilibrium problems1 (2) 99 1348 99 1348 100 7608 88 63452 (6a) 96 1618 96 1618 100 11421 100 114213 (6b) 96 1639 96 1639 100 11424 100 114244 (9) – – – – – – – –

Liquid–liquid equilibrium problems5 (11a) 98 1432 98 1432 100 7600 100 67886 (11b) 100 1359 100 1359 100 7625 100 70767 (12) 100 1367 100 1367 100 7621 100 75108 (13) 94 1575 94 1575 100 11417 99 11359

Vapor–liquid–liquid equilibrium problems9 (17) 100 5648 100 5648 100 15215 100 1521510 (18) 68 5486 68 5486 100 15226 100 15226

iah (2w

praemrf(apfovtttfimvmv

tt−OcT7tisi

td

ccbgtnnNnm

wf(c(MchcaHt2ttTp

a The number in the brackets refers to the example number in Teh and Rangaork.

Table 4a shows that DE-QN is able to solve all but one exam-le tested with 100% success rate with both SC1 and SC2. Theeliability of TS-QN is comparable to DE-QN for all the VLEnd LLE examples except for example 8. This is due to the pres-nce of comparable minima (function values at the global mini-um and at the trivial solution are −0.360353 and −0.354340,

espectively) in this example. The reliability of TS-QN is lowor example 10 (VLLE) also because of its comparable minimafunction value at the local and global minima are −1.235896nd −1.233294, respectively). The SR of TS-QN for this exam-le is lower compared to example 8 because of the very closeunction values at the local and global minima and more numberf variables (6 for example 10 and 3 for 8). When the functionalue at a local minimum is close to that at the global minimum,he search traps into the local minimum because of either prema-ure convergence (i.e., all members of the population are closeo one another) or narrow better regions (i.e., regions where theunction value is less than that at the local minimum) resultingn low SR. The former is more likely if both the local and global

inima are close enough in terms of both location and functionalue whereas the latter is more likely if both the local and globalinima are located far away and also have comparable function

alues.For example 4, both DE-QN and TS-QN failed to locate

he global minimum because of the comparable minima (func-ion value at the local and global minima are −161.5364 and

161.5416) and the increase in dimensionality (10 variables).verall, reliability of DE-QN is better compared to TS-QN indi-

ating that the escaping mechanism of DE is better than that ofS-QN using SC1. The NFE of DE-QN is 2.69 (example 9) to.24 (example 8) times more than that of TS-QN. Even though
he reliability of DE-QN is more, its computational efficiencys inferior to TS-QN. This is because of avoiding revisits to theame place during the search in TS-QN. For all examples, NFEncreases with number of variables (example 2 with 2 variables
ff(t

003). This study considers only the more difficult examples from the previous

o example 10 with 6 variables) for both TS-QN and DE-QNue to the associated large solution space.

As shown in Table 4a, the reliability and computational effi-iency of DE-QN using SC2 is comparable and slightly betterompared to SC1. This is because the algorithm terminates if theest function value does not change even after Scmax number ofenerations in SC2. For TS-QN, there is no improvement inhe computational efficiency using SC2 because the maximumumber of iterations (Itermax) is reached before the specifiedumber of Scmax iterations, causing the algorithm to terminate.ote that both the methods can be used with different thermody-amic models (such as activity coefficient and equation of stateodels) for describing the physical equilibrium.The CPU times of DE-QN and TS-QN are also compared

ith interval Newton and branch and bound methods for aew examples available in Hua, Brennecke, and Stadtherr1998) and McDonald and Floudas (1997), respectively. Theomputer system used in the present study is Pentium 4CPU 2.8 GHz, 512MB RAM). The average (over 10 trials)

Flops (million floating point operations per second) of thisomputer for the LINPACK benchmark program (available atttp://www.netlib.org) for a matrix of order 500 is 211. Theomputer systems used in Hua et al. (1998) and McDonaldnd Floudas (1997) are Sun Ultra 1/170 workstation andP9000/730, respectively, and the corresponding MFlops for

he LINPACK benchmark program for a matrix of order 100 are4 and 76–75, respectively (Teh & Rangaiah, 2003). Therefore,he computer system used in this study is about 2.8 and 8.8imes faster than Sun Ultra 1/170 and HP9000/730, respectively.he order of the matrix in the present LINPACK benchmarkrogram (at http://www.netlib.org) is higher (500), and is used
or the computer in this study compared to the earlier (100) usedor computers of Hua et al. (1998) and McDonald and Floudas1997), to cope with current high-speed computers. Althoughhe difference in the matrix orders may have some effect on
http://www.netlib.org/

http://www.netlib.org/


Table 4bComparison of CPU times for DE-QN and TS-QN with those in the literature

Example number and details Computation time (s) Reference

TS-QN DE-QN Reported (estimate)a

1 (VLE, Peng–Robinson Model) 0.022 0.122 0.643 (0.232) Hua et al. (1998)5 (LLE, NRTL) 0.018 0.090 0.234 (0.026)

McDonald and Floudas (1997)6 (LLE, UNIFAC) 0.024 0.156 0.370 (0.042)9

ing L5

MfmrfsmtCQdeow

7

xst(

g

T{

t

wT

H

DHa

H

wiub

H

wtf

∑

a

TD

C

1

2

(VLLE, UNIFAC) 0.201 0.201

a The number in the brackets is the estimated CPU time, based on MFlops us12MB RAM) used in the present study.

Flops, this is not considered here. The CPU times (in seconds)or DE-QN, TS-QN, interval Newton, and branch and boundethods are compared in Table 4b; the numbers in the brackets

epresent the corresponding CPU time (estimated using theactors based on MFlops) for the computer system used in thistudy. CPU time taken by TS-QN is less compared to all otherethods because of its lesser NFE; CPU time of DE-QN is less

han that of interval Newton method for example 1. ThoughPU time of branch and bound method is less than that of DE-N for examples 5 and 6, it is higher for example 9, probablyue to the number of variables in them (number of variables forxamples 5, 6 and 9 is 2, 2 and 6, respectively). The numberf nodes in branch and bound method increases exponentiallyith number of variables, resulting in more computational time.

. Phase stability problems

For a given temperature (T), pressure (P) and composition= (x1, x2, x3, . . . ,xnc), the molar Gibbs free energy, g of the

ystem is given as the summation of the product of mole frac-ion and partial molar Gibbs free energy, Gi for all componentsRangaiah, 2001):

=nc∑i=1

xiGi (11)

he tangent plane, t at a specified composition x∗ =x∗, x∗, x∗, . . . , x∗ } is given as
1 2 3 nc
=nc∑i=1

xiG∗i (12)

0

Tns

able 5etails of the phase stability example 5—toluene (1), water (2) and aniline (3) at 298

omposition Component Feed composition, x∗i

Liquid 1 L

1 0.29989 –2 0.20006 –3 0.50005 –

1 0.34673 0.2 0.07584 0.3 0.57742 0.

8.800 (1.001)

INPACK benchmark problem, for the computer (Pentium 4, CPU 2.8 GHz and

here superscript * represents evaluation at composition x*. ThePDF can be expressed as

= g − t =nc∑i=1

xi(Gi − G∗i ) (13)

epending on the expressions of Gi and G∗i , different forms of

exist. If the non-ideality of the phase is described by fugacitypproach, then the dimensionless H can be expressed as

=nc∑i=1

xi[ln (φixi) − ln (φ∗i x

∗i )] (14)

here φi represents the fugacity coefficient of the component in the given phase. If the excess Gibbs free energy approach issed to represent the non-ideality, then the dimensionless F cane expressed as

=nc∑i=1

xi[ln (xi) − ln (x∗i γ

∗iL)] (15)

here γ iL represents the activity coefficient of component i inhe liquid phase L. Depending upon the approach, the objectiveunction is either Eqs. (14) or (15) and the constraints are

nc

i=1

xi = 1 (16)

nd

≤ xi ≤ 1 (17)

he decision variables in PS problems are xi for i = 1, 2, . . . ,c. The constrained problem can be transformed into an uncon-trained problem by introducing the variables,βi (for i = 1, 2, . . . ,

K and 1.0 atm (Castillo & Grossmann, 1981)

Global solution

iquid 2 Objective function value x

−0.29454012 0.0000670.9968650.003068

00009 0.0 0.34674099495 0.07584000496 0.577420


Table 6Global and comparable minima for PS problems

Composition Function value

Global minimum Comparable minimum

Example 1 (1)1 −0.03246624 0 at {0.5; 0.5}2 −0.21418620 No local minimum3 −0.07427426 −6.06283 × 10−3 at {0.39221; 0.60778}4 −0.00671171 0 at {0.65; 0.35}5 −0.00070557 0 at {0.93514; 0.06486}6 0 1.11127 × 10−3 at {0.93476; 6.5230 × 10−2}

Example 2 (2)1 −0.11395074 0 at {0.4; 0.3; 0.3}2 −0.05876117 −4.11636 × 10−6 at {0.61986; 0.00562; 0.37452}3 −0.22827470 −2.71678 × 10−3 at {0.20145; 0.43018; 0.368351}4 −0.02700214 −3.09637 × 10−6 at {0.03001; 0.00211; 0.96788}5 0.0 1.323587 × 10−6 at {0.69280; 0.00399; 0.30321}

Example 3 (3)1 −0.00395983 1.08905 × 10−2 at {0.11520; 0.88479}2 −0.08252179 −5.68944 × 10−2 at {0.11813; 0.88186}3 −0.00246629 0 at {0.112; 0.888}

Example 4 (4)1 −1.48621570 −1.48554 at {0.94672; 4.35930e−2; 7.85484 × 10−3;

1 × 10−15; 1.2478 × 10−3; 1.96839 × 10−4;2.63986 × 10−4; 1.20802 × 10−4}

Example 50.299cal m

N 01).

ntb(tftiCbccTpi

7

tpdirenap

mβ

atihyper-rectangles to generate neighbors. As the local minimumis close to the boundary, the distribution of neighbors (‘o’ pointsin Fig. 4) is not spread to the global minimum region (i.e., one

1 −0.29454012 0 at {2 0 No lo

ote: The number in the brackets refers to the example number in Rangaiah (20

c − 1) instead of mole fractions xi (for i = 1, 2, . . . , nc) similaro Eqs. (7)–(9). To avoid the computational difficulties, the lowerounds are taken as 10−15 instead of 0. The examples consideredRangaiah, 2001) include multiple components and differenthermodynamic models. Several compositions are consideredor each example. The number of components, feed composi-ion, temperature and pressure of all these examples can be foundn Rangaiah (2001) except for example 5 (Table 5) taken fromastillo and Grossmann (1981). The global minimum, compara-le minimum (i.e., the local minimum with function value verylose to that of the global minimum) and the composition at theomparable minimum for all these cases are given in Table 6.he composition at the global minimum for the first four exam-les can be found in Rangaiah (2001) while that for fifth examplen Table 5.


Compositions 2, 4 and 5 of example 2, which are foundo be difficult in the preliminary trials, are chosen to tune thearameters of TS-QN and DE-QN. For PS examples, a ran-om way of generating neighbors from the centroid in TS-QNs also studied along with the systematic way (using hyper-ectangles) of generating neighbors. This is because, for some
xamples, generation of neighbors using hyper-rectangles didot give good reliability, perhaps due to the distribution of localnd global minima in these examples. As shown in Fig. 4, theroblem (second example, fifth composition) has a local mini-
Fr(

89; 0.20006; 0.50005}inimum

um at β = (0.692780, 0.0129963) and the global minimum at= (0.278990, 0.682251) with function values 1.323587 × 10−6

nd 0, respectively. Initially the TS-QN using hyper-rectangleso generate neighbors found a best point (‘+’ point in Fig. 4)n the local minimum region and set it as a new centroid of

ig. 4. Generation of neighbors for example 2 (composition 5) using hyper-ectangles (points with ‘o’) and randomly (points with ‘*’) from the best pointpoint with ‘+’) at β = {0.69280; 0.01298}.


Table 7SR and NFE for solving phase stability problems by DE-QN, TS-S-QN, TS-M-QN and TS-R-QN using SC1

Composition NFE for TS-S-QN NFE for TS-M-QN NFE for TS-R-QN NFE for DE-QN

Example 11 567 (99%) 360 684 25682 563 330 645 25623 465 361 681 25664 571 332 642 25695 583 324 640 25576 581 325 639 2567

Example 21 1196 1890 1989 51122 1273 (64%) 1768 2023 51153 1272 1956 2035 51164 1240 (91%) 1810 1987 51115 1277 (84%) 1878 (85%) 1984 (76%) 5114

Example 31 569 645 757 25842 577 523 759 25823 558 646 758 2582

Example 41 2100 2203 2767 5143

Example 51 1306 1095 2048 51202 1280 1026 1981 5108

N

svtimor

oIipc

7

dtorQfu(QrQwl

fsiciCcww

bbtrTmtQk

t(ebtt

ote: SR is 100% for all cases except for those given in the brackets.

ide of the hyper-rectangle becomes the lower boundary of theariables forcing many points near the boundary). To circumventhis difficulty, a random way of generating neighbors (‘*’ pointsn Fig. 4) is implemented to explore better points in the global

inimum region for these problems. A mixed (generating halff the total number of neighbors using hyper-rectangles and theest randomly) way of generating neighbors is also studied.

For PS problems, the optimal values of parameters (Table 2)btained for TS-QN are the same as those for PEC excepttermax = 50N. The optimal parameters of DE obtained are givenn Table 2. The parameters Itermax and Genmax are less com-ared to PEC because the number of decision variables is lessompared to PEC for these problems.


All the PS examples are solved 100 times, each time from aifferent randomly chosen point in the feasible region. Initially,he examples are solved with SC1 to study the performancef TS-QN with different ways of generating neighbors and theesults (averaged over all the 100 trials) of TS-QN and DE-N are given in Table 7. The results for TS-QN are given

or three types of generating neighbors: TS-S-QN (systematicsing hyper-rectangles), TS-R-QN (randomly) and TS-M-QNmixed). The results in Table 7 show that both DE-QN and TS-N have high reliability in locating the global minimum. The

eliability of DE-QN is comparable to TS-M-QN and TS-R-N, and is better than TS-S-QN. This shows that the systematicay of generating neighbors has less reliability for these prob-

ems. The SR of TS-M-QN, TS-R-QN and DE-QN are 100%

ttri

or all examples except for second example with fifth compo-ition for which TS-QN with all types of generating neighborss low compared to DE-QN. This is because of the presence ofomparable minima (function value at the local and global min-ma are 1.32358 × 10−6 and 0, respectively) in this example.onsequently, the better region becomes narrower and narrowerausing failure of TS-QN to locate the global minimum region,hereas DE-QN is able to explore the global minimum regionith its escaping mechanism (mutation and crossover).The computational efficiency of TS-M-QN and TS-S-QN is

etter compared to TS-R-QN. This may be because some neigh-ors in the mixed type of generation may be close enough suchhat they are near to the points in the tabu list which in turn avoidsepeated evaluations. Even though DE-QN is more reliable thanS-QN, the latter is computationally more efficient than the for-er. The NFE of DE-QN is around 2.3 times (fourth example)

o 7.1 times (first example, first composition) more than TS-M-N. This is due to avoiding repeated visits to the same place byeeping track of the previous points during the search.

The examples are also solved with SC2 for TS-M-QN (foundo be the best among all TS-QN tried) and DE-QN. The resultsTable 8) show that there is no improvement in the computationalfficiency of TS-M-QN using SC2 compared to that of SC1,ecause the maximum number of iterations is reached beforehe specified Scmax number of iterations. This also indicates thathe parameter Itermax is fine tuned. The results (Table 8) show
hat the computational efficiency of DE-QN using SC2 is betterhan that of SC1, and is due to the termination of the algo-ithm once the specified Scmax number of iterations is reachedrrespective of maximum number of iterations. NFE of DE-QN


Table 8Comparison of SR and NFE for solving phase stability problems by DE-QN and TS-M-QN using SC1 and SC2

Composition TS-M-QN DE-QN

SC1 SC2 SC1 SC2

Example 11 360 360 2568 7552 330 330 2562 7793 361 361 2566 8694 332 332 2569 7275 324 324 2557 6476 325 325 2567 643

Example 21 1890 1890 5112 3808 (99%)2 1768 1768 5115 4624 (96%)3 1956 1956 5116 48244 1810 1810 5111 40245 1878 (85%) 1878 (85%) 5114 3642 (99%)

Example 31 645 645 2584 7162 523 523 2582 7643 646 646 2582 698

Example 41 2203 2203 5143 5143

Example 5982910

N

wtobva(nco

TCa

F

GESRZHRZRZRZ

h

bam

way is also studied for the benchmark problems and PEC usingSC1. The results averaged over 100 trials (Tables 9a and 9b)show that the performance of generation of neighbors using

1 9822 910

ote: SR is 100% for all the cases except for those given in the brackets.

ith SC1 is 1.1 (second example, second composition) to 4.0imes (first example, sixth composition) more compared to thatf SC2. For example 4, the NFE of DE-QN is the same withoth SC1 and SC2. This is because here the optimum Scmaxalue (i.e., 6N ∼= 48) is high because of more number of vari-bles and the number of iterations reaches its maximum numberi.e., Itermax = 50 for these problems) earlier than Scmax, termi-
ating the algorithm. For example 5, the NFE of DE-QN foromposition 1 is higher than that of composition 2 even thoughptimum Scmax value is the same for both of them. This may
able 9aomparison of performance of TS-QN with different types of neighbor gener-tion for benchmark problems

unction TS-S-QNa TS-M-QNa TS-R-QNa

SR NFE SR NFE SR NFE

P2 100 918 100 571 100 1060S2 90 1040 15 580 33 1065H2 100 1033 95 569 99 1059OS2 100 1059 100 611 100 1096AK2 100 1009 100 561 100 1050

3 100 987 100 548 100 994OS5 78 2799 86 1656 84 2901AK5 100 2629 100 1409 100 2649OS10 78 8578 80 6326 82 11218AK10 100 8491 100 5323 100 10298OS20 75 22074 83 36141 78 36259AK20 100 19157 100 30668 100 30662

a TS-S-QN, TS-M-QN and TS-R-QN represent respectively systematic (usingyper-rectangles), mixed and random way of generation of neighbors.

h

TCa

E

V

L

V

1

h

5120 50185108 3566

e because the first composition has a comparable minimum,nd whereas the second composition does not have any localinimum.The effect of generation of neighbors by random and mixed

yper-rectangles is better compared to mixed and random way

able 9bomparison of performance of TS-QN with different types of neighbors gener-tion for phase equilibrium calculations

xample number TS-S-QNa TS-M-QNa TS-R-QNa

SR NFE SR NFE SR NFE

apor–liquid equilibrium1 99 1348 94 1018 100 19472 96 1618 95 1545 95 20293 96 1639 92 1094 90 19854 – – – – – –

iquid–liquid equilibrium5 98 1432 97 1051 99 20026 100 1359 100 1072 100 20237 100 1367 100 1091 100 20398 94 1574 87 1058 99 1943

apor–liquid–liquid equilibrium9 100 5648 100 5097 100 76390 81 5486 9 3985 24 7601

a TS-S-QN, TS-M-QN and TS-R-QN represent respectively systematic (usingyper-rectangles), mixed and random way of generation of neighbors.

7 nd C

omopwanla

8

itaotapsraptPnhtpigaioaaps

R

B

B

B

B

C

C

C

G

GGH

H

I

L

L

M

M

M

M

N

N

R

S

T

T

T

W


f generations even though the mixed way is computationallyore efficient than the others. They also show that the reliability

f TS-M-QN for Rosenbrock functions is slightly better com-ared to TS-S-QN. These are due to the distribution of minima,hich depends on the problem type, and hence generalizing for

ll types of functions is difficult. The mixed way of generatingeighbors seems to be suitable for problems such as PS prob-ems where the local minimum is near the boundary and also farway from the global minimum.

. Conclusions

Two most promising methods namely, DE and TS have beenmplemented along with a local minimization method (QN) athe end to refine the solution, and evaluated for benchmark, PECnd PS problems. Initially both DE-QN and TS-QN are testedn benchmark problems comprising of 2–20 variables and a fewo hundreds of local minima. The algorithms are then evalu-ted and compared for PEC involving multiple components andhases with popular thermodynamic models. Both the methodsuccessfully located the global minima with DE-QN being moreeliable compared to TS-QN and the latter being computation-lly more efficient than the former for both benchmark and PECroblems. For example 4 of PEC, both DE-QN and TS-QN failedo locate the global minimum. The methods are then tested forS problems involving multiple components. The generation ofeighbors in TS is implemented in three different ways: usingyper-rectangles, mixed and random way of generation, to studyheir effectiveness. The reliability of DE-QN and TS-QN for PSroblems is high and the computational efficiency of the latters better than the former for these problems. The mixed way ofenerating neighbors is also studied for benchmark problemsnd PEC, and the results show that systematic way of generat-ng neighbors is suitable for these problems. In summary, resultsf this study show that the escaping mechanism (via mutationnd crossover) in DE-QN is more effective than that of TS-QN,nd that TS-QN is computationally more efficient than DE-QN,erhaps due to avoiding revisits to the same place during theearch.

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a study of differential evolution and tabu search for benchmark, phase equilibrium and phase...

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