Download - Optimal design of multiproduct batch chemical processes using tabu search

*Corresponding author. Fax: 86 22 23358329; e-mail: [email protected]

1Supported in part by Postdoctoral Science Foundation of Chinaand in part by Nature Science Foundation of Tianjin.

Computers and Chemical Engineering 23 (1999) 427—437

Optimal design of multiproduct batch chemical processes usingtabu search1

Chunfeng Wang, Hongyin Quan, Xien Xu*

Department of Chemical Engineering, Tianjin University, Tianjin 300072, People+s Republic of China

Received 20 May 1998; accepted 23 November 1998

Abstract

In this paper, tabu search (TS), a universal heuristic method besides simulated annealing (SA) and genetic algorithms (GAs) thateffectively overcome local optimum, is adapted to the optimal design problem of multiproduct batch chemical processes successfully.A novel concept of double tabu list is proposed, and the comparison between customized (proposed) algorithm and standard one isgiven, which illustrates the improvement by the use of the double tabu list. The methods of constructing dynamic neighborhood,realizing diversification and changing the step size of continuous variables adaptively are designed. An appropriate empirical equationof tabu size is obtained through computational experimentation. To demonstrate the effectiveness of TS in solving the proposedproblem, four examples adopted from literature, together with the computation results, are presented. Better results are obtained incomparison with the results of mathematical programming (MP) and SA. ( 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction

Batch processes are widely used in the chemical pro-cess industry and are of increasing industrial importancedue to the great importance of low-volume, high-value-added chemicals and the need for flexibility in a market-driven environment. In the optimal design of a multipro-duct batch chemical process, the production requirementof each product and the total production time availablefor all products are specified. The number and size ofparallel equipment units in each stage as well as thelocation and size of intermediate storage are to be deter-mined in order to minimize the investment.

The common approach used by previous research insolving the design problem of batch chemical processeshas been to formulate it as an MINLP problem and thenemploy optimization techniques to solve it. Mathemat-ical programming (MP) (Grossmann and Sargent, 1979;Knopf et al., 1982; Takamatsu et al., 1982) and heuristics(Yeh and Reklaitis, 1987; Mode and Karimi, 1989; Xu etal., 1993) are commonly used. Because of the NP-hard

nature of the design problem of batch chemical processes,a very long impractical computational time will be in-duced by the use of MP when the design problem issomewhat complicated. Severe initial values for the op-timization variables are also necessarily. Moreover, withthe increasing size of the design problem, MP will befutile. Heuristics needs less computational time, and se-vere initial values for optimization variables are notnecessary, but it may end up with a local optimum due toits greedy nature. Also, it is not a general method dueto the fact that special heuristic rules will be neededfor a special problem. Patel et al. (1991), Tricoire andMalone (1991) applied simulated annealing (SA) to solvethe design problem of multiproduct batch chemical pro-cesses. SA performs effectively and gives a solution within0.5% of the global optimum. However, SA has the disad-vantage of long searching time and hence needs moreCPU time than heuristic. In order to speed up the con-vergence of SA, Wang et al. (1996a, b) combined SA withheuristics to solve the design problem of multiproductbatch chemical processes, and satisfactory results wereobtained. Wang et al. (1996a, b) also applied GAs to theproblem successfully.

To solve the proposed problem more effectively, tabusearch (TS), an intelligent problem-solving methodthat has demonstrated its effectiveness in solving the

0098-1354/99/$ — see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S0098-1354(98)00304-4

combinatorial optimization problem and the ‘combina-torial explosion’ associated with it in many areas, isdeveloped in this paper. Some original ideas are pro-posed for the practical implementation of TS, and satis-factory results are obtained.

The rest of this paper is organized as follows. Section 2presents the mathematical model for the design problemof multiproduct batch chemical processes. The basicideas of TS are introduced in Section 3. The adaptationof TS to the proposed optimization problem is given inSection 4. To demonstrate the effectiveness of TS insolving proposed problem, four problems adopted fromliterature, together with their computation results usingC-TS (with double list) and S-TS (standard TS), arepresented in Section 5. Comparisons with MP and SAare also given in Section 5. Finally, Section 6 provides thesummary and conclusions.

2. Mathematical model

The optimal design of multiproduct batch chemicalprocesses can all be introduced to a MINLP model. Thispaper employs Modi’s model modified by Xu et al.(1993). It has the following assumptions:1. The processes operate in the way of overlay;2. The devices in the same production line cannot be

used by the same product;3. The long campaign and the single product campaign

are considered;4. The type and size of parallel items in- or out-of-phase

are the same in one batch stage;5. All intermediate tanks are finite ones;6. The operation between stages can be of zero wait or

no intermediate tank when there is no storage;7. There is no limitation for utility;8. The cleaning time of the batch item can be neglected

or included in processing time;9. The size of the devices can change continuously in its

own range.Assume that there are J batch stages, K semicontinu-

ous stages, and I products to be manufactured; that thereare m

0jout-of-phase groups of parallel units in each

batch stage in which there are m1j

in-phase parallel unitsall of which of sizes »

j; there are R

kparallel units in phase

in each semicontinuous stage, the operating rates ofwhich are all R

k; there are S!1 intermediate tanks that

divide the whole process into S subsystems; and let

Js"( j Dbatch stage belonging to subprocess s), s"1, S,

¹s"(t D semicontinuous substrain belonging to subpro-

cess), s"1, S,

ºt"(k D semicontinuous stage k belonging to semicon-

tinuous substrain t), t"1, ¹

and using the equipment investment as a criterion ofoptimization, which is expressed as a power function ofcharacteristic dimension of equipment, the followingmathematical model could be obtained:

Min f (V, R)"J+i/1

(m0j

mpl

aj»aj

l)#

K+k/1

(nkbkRbk

k)

#

S~1+r/1

(cS»* cs

S) (1)

subject to the following:

(1) dimension constraints: every equipment alters in itsallowable range:

».*/l

)»l)».!9

l, j"1, J, (2)

R.*/k

)Rk)R.!9

k, k"1, K, (3)

(2) time constraint: the summation of available produc-tion time for all products is not more than the totaltime for production:

H*

I+i/1

Hi"

I+i/1

Qi

Pi

, (4)

where all the following true:

(a) the productivity for product i :

pi"

Bis

¹Lis

, i"l, I, s"1, S; (5)

(b) the limiting cycle time for product i in subprocess s:

¹Lis" Max

j|Js j|Ts

[¹ij, h

ij], i"1, I; s"1,S; (6)

(c) the cycling time for product i in batch stage j :

¹ij"

hiu#p

ij#h

i (u`1)m

0j

, i"1, I; j"1,J ; (7)

(d) the processing time for product i in batch stage j :

pij"p0

ij#g

ijAB

ism

piBdn, i"1, I; j"1, J ; j3J

s; (8)

(e) the operating time for product i in substrain t :

hii"Max

k|UtCBisD

ikR

knkD , i"1, I; t"1,¹ ; t3¹

s; (9)

(f ) the batch size for product i is subprocess s:

Bis"Min

j|JsAm

pili

SijB , i"1, I ; t"1,¹ ; t3¹

s; (10)

(3) the constraints of product quantity: the same productin different subprocess posses the same productivity.

C. Wang et al. / Computers and Chemical Engineering 23 (1999) 427—437428

(4) the dimension of intermediate storage is the max-imum value of what is needed by all products:

»*s"Max

i

[PiS*is(¹L

is!h

iu#¹L

i (s`1)!h

i (u`1))],

i"1, I ; s"1, S!1. (11)

Using the mathematical model to optimize a design fora given product demand, the size and number for eachkind of equipment must be calculated to minimize theequipment investment.

3. Tabu search

TS , a general heuristic procedure for global optimiza-tion, saw its seminal beginnings over a decade ago andwas first fully described in 1986 (Golver, 1986; Hansen,1986). Since that time, TS has been shown to be a re-markably effective approach, dominating alternativetechniques, in a wide spectrum of problem areas fromgeneral integer and nonlinear programming to se-quencing and production scheduling problems (Gloverand Laguna, 1993). Together with SA and GAs, TS hasbeen singled out by the Committee on Next Decade ofOperation Research (Condor, 1988) as ‘‘extremely prom-ising’’ for the future treatment of practical applications.

TS is basically a kind of neighborhood search method.Starting from an initial solution, it finds the best solutionin the neighborhood of the given initial solution. Then,taking the best solution as the new initial solution, TSrepeats the above step as long as it seems necessary.Based on the above basic ideas, TS adopts several intelli-gent strategies to improve its search procedure (Glover,1989, 1990, 1993).

3.1. Tabu list

It is obvious that a previously visited solution can bevisited the second time, that is, cycling occurs. To preventcycling, we should forbid moves that would bring us backto a previously visited solution. TS makes use of a tabulist to force the search away from solutions selected forrecent iteration. A tabu list is constructed by the use ofthe so-called recency-based short-term memory (RSM),which records the most recent ¹ moves (¹ is called tabusize). Those most recent ¹ moves are put into the tabulist and the tabu list is changed dynamically as the searchprocedure goes on. Those solutions in the tabu list arecalled tabu, or are said to be in tabu condition.

3.2. Aspiration criteria

Tabu conditions based on selected attributes of movesand solutions can be too drastic in the sense that theymay also forbid moves leading to unvisited solutions, and

in particular to unvisited solutions that may be attract-ive. In addition, too many solutions may be forbiddenduring the search process. It is therefore necessary toallow the tabu status of a solution to be overridden insome circumstances if it seems desirable to do so. Aspira-tion criteria is introduced to determine when the tabustatus of certain solutions can be overridden.

3.3. Recency-based short-term memory (RSM) andfrequency-based long-term memory (FLM)

History information and memory play essentialroles in TS process. There are two kind of memory:recency-based short-term memory (RSM) and frequency-based long-term memory (FLM). RSM belongs toa short time horizon. It provides a recorder of the mostrecently realized ¹ moves in order to avoid coming backto a solution visited earlier. It is not a complete recorderof the history of the previous searching procedure. Mem-ory is also used in TS in a kind of learning process. FLMprovides a type of information that complements theinformation provided by RSM, broadening the founda-tion for selecting preferred moves. It records the numberof times each solution is visited in the previous searchprocedure.

3.4. Intensification

To improve the efficiency of the search procedure, it isnot preferable to explore the whole solution space withequal effort. The solution space that seems attractiveshould be explored more thoroughly in order to improvethe best solution found so far to the greatest extend. TSuses the strategy of intensification to achieve this goal.

3.5. Diversification

An intelligent search technique should not only ex-plore thoroughly a region that contains good solutions, itshould also have a general view of the solution space andtry to make sure that no far region has been entirelyneglected (Werra and Hertz, 1989). A usual way of ap-proaching this goal in an iterative procedure is to repeatthe whole search procedure with a collection of randomlygenerated initial solutions. If the number of initial solu-tion is large enough, probabilistic arguments can beapplied to establish some kinds of convergence proper-ties to a global minimum (as in Monte Carlo methodsfor instance). However, TS realizes diversification bygenerating some initial solutions which are not randomones but precisely the solutions in some regions of thesolution space which have not been explored or exploredintensively earlier. Diversification enables TS to have theability of global optimization.

C. Wang et al. / Computers and Chemical Engineering 23 (1999) 427—437 429

Fig. 1. TS implementation.

4. Implementation

From the previous discussion we know that TS is infact an intelligent search procedure which in some senseimitates a human behavior or apply some rules based onartificial intelligence principles. It is noted that thoseintelligent strategies above are very flexible in the TSimplementation. The design of actual mechanism to im-plement these strategies, including discovery of morerefined ways to exploit the memory and the choice ofmore proper parameters, is the key in applying the TSalgorithm to practical engineering optimization prob-lems. In this section, we adapted TS to the optimal designproblem of multiproduct batch chemical processes inseveral aspects. Those basic ideas of TS are all realizedflexibly in implementation. Fig. 1 shows the flow chart ofthe algorithm.

4.1. Neighborhood structure

Neighborhood structure plays an important role in TSimplementation, which influences solutions’ quality andcomputing speed. A larger neighborhood size providesgenerally high quality solutions but may result in a lon-ger CPU-time. A small one can speed up the convergenceof searching process, but may result in a reduction of thequality of the optimization results, i.e. the algorithm may

be trapped in a local minimum. We have to trade be-tween the computing speed and the solution quality. Forthis reason, the concept of dynamic neighborhood size isintroduced in this paper. We vary the neighborhood sizeover three different ranges (small, medium and large)while the search process goes on. At the beginning,a smaller neighborhood size is preferred for a rapidconvergence and a larger neighborhood size is preferredfor a thorough search at the end. The components ofa neighborhood are selected from the complete-neigh-borhood (a neighborhood that contains all the solutionsthat can be reached from the present solution after exact-ly one move) randomly.

4.2. Double tabu list

In practical implementation we find that those optim-ization variables representing the number of in- andout-phase parallel units are always reduced quickly. Thisresults in an inadequate reduction of those optimizationvariables representing unit size and hence the algorithmis trapped into local optimum. We recognize that theabove result is bound to appear if we deal with all the opti-mization variables in the same way. Because the contri-butions to the objective function of those optimizationvariables representing the number of units and the unitsize are greatly different, it is sure to result in unbalancedchanges of the optimization variables.

To solve the problem, we proposed a novel concept ofdouble tabu list. We adopted to different tabu lists withtwo different tabu size, one for those optimization vari-ables representing the number of units, the other unit size.

4.3. Dynamic tabu size

The key issue of creating a tabu list using RSM is todetermine a proper value of ¹, or tabu size. Rules fordetermining ¹ are classified as static or dynamic. Staticrules choose a value for ¹ that remains fixed throughoutthe search procedure. Dynamic rules allow the valueof ¹ to vary. In this paper dynamic tabu size is chosen.In fact, it is obvious that the tabu size should not bestatic when we vary the neighborhood sizes during thesearching procedure. A proper form of tabu size fora certain kind of optimization problem needs calculationexperience.

4.4. Aspiration criteria

Two aspiration criteria are adopted here: (1) Accepta tabu move if it produces a new best solution. (2) If allavailable moves are classified tabu, and are not renderedadmissible by the first aspiration criteria, then a ‘‘leasttabu’’ move is selected.


Table 1Data for example 1

H"8000 h, J"3, I"3, Q"[100 000, 100 000, 50 000]800)»

i)2400, 300)R

k)1800

SC1 B1 SC2 SC3 B2 SC4 T SC5 SC6 B3

a or b or c 370 592 250 210 582 250 334 250 200 1200a or b or c 0.22 0.65 0.40 0.62 0.39 0.4 0.59 0.40 0.83 0.52i"1S or D 1.2 1.2 1.2 1.2 1.5 1.2 1.1 1.4 1.4 1.1P0 35 1 4g 0.0 0.0 0.0i"2S or D 1.5 1.4 1.5 1.5 1.2 1.5 1.1 1.5 1.5 1.2P0 40 1 8g 0.0 0.0 0.0i"3S or D 1.1 1.0 1.1 1.1 1.0 1.1 1.1 1.2 1.2 1.0P0 30 2 4g 0.0 0.0 0.0

4.5. Intensification

The FLM is used here to achieve the strategy of inten-sification. We choose a neighborhood that is a smallsubset of those elite’s solutions (those solutions that havethe highest frequency in the previous search procedure)and use this neighborhood to replace the present one.Obviously, those solutions spaces that are more attract-ive are given a more thorough search than other solutionspaces.

4.6. Diversification

FLM is also used here to achieve the goal of diversifi-cation. We design a very simple but effective method, wecall it lowest frequency (LF) method to realize diversifi-cation. We just restart the research procedure froma solution that has the lowest frequency in the FLM atsome stages of the search procedure.

4.7. Step size of continuous variables

For an MINLP problem, there exists a problem of stepsize of continuous variables when a neighborhoodsearching method is adopted. The step size can neither betoo large nor too small. A large step size may result ina local optimum while a small step size needs a longersearching time.

A simple but effective dynamic method is designedhere to vary the step size of those continuous variables.We simply let x"x (1$a%) (in which x represents anoptimization variable). The bigger a is the bigger the stepsize and vice versa. We can see that the step size alsochanges with the value of optimization variables. Hence,the step size of continuous variables varies adaptivelyduring the search procedure (at the beginning when theoptimization variables are bigger a bigger step size is

adopted to fasten the search procedure and at the endwhen the optimization variables are smaller a smallerstep size is adopted to improve the preciseness of thesearch procedure). Because the unit size of batch stagespossesses a much greater influence on the objective func-tion than those of semicontinuous variables, we adoptedtwo different a for these two kinds of variables, a smallera for batch stages and a bigger a for semicontinuousstages. But, according to our computing experience,a cannot be larger than 10 in any instance.

4.8. Termination criterion

Terminate the algorithm if the best solution found sofar cannot be improved after certain times of iteration.

5. Examples and analysis

Some examples were computed to demonstrate the ef-fectiveness of the algorithm designed in the above section.Four examples are selected here. The data for examples 1,2, 3, and 4 are presented in Tables 1, 2, 3, and 4, respec-tively. The results are presented in Tables 5, 6, 7, and 8.

5.1. Results and comparison analysis

From the above results we can see that in examples1 and 2 C-TS (customized TS-TS with double tabu list)and S-TS (standard TS) obtained nearly the same resultsas MP but a much faster convergence. We can also seethat better results are obtained in comparison with MPand SA in the somewhat complicated example 3. Forexample 4, Patel et al. (1991) pointed out that, it cannotbe solved using any existing method other than SA due tothe presence of intermediate storage, nonidentical unitsand mixed mode of operation. However, TS (including



H"8000 h, J"3, I"3, Q"[100 000, 100 000, 50 000]800)»

i)2400, 300)R

k)1800

SC1 B1 SC2 SC3 T SC4 B2 SC5 SC6 B3

a or b or c 370 592 250 210 278 250 582 250 200 1200a or b or c 0.22 0.65 0.40 0.62 0.49 0.4 0.39 0.40 0.83 0.52i"1S or D 1.2 1.2 1.2 1.2 1.1 1.2 1.5 1.4 1.4 1.1P0 35 1 4g 0.0 0.0 0.0i"2S or D 1.5 1.4 1.5 1.5 1.1 1.5 1.2 1.5 1.5 1.2P0 40 1 8g 0.0 0.0 0.0i"3S or D 1.1 1.0 1.1 1.1 1.1 1.1 1.0 1.2 1.2 1.0P0 30 2 4g 0.0 0.0 0.0


H"6000 h, J"4, I"3, Q"[437 000, 324 000, 258 000]250)»

i)10 000, 300)R

k)10 000

SC1 B1 SC2 T SC3 B2 SC4 B3 SC5 B4 SC6

a or b or c 370 250 370 278 370 250 370 250 370 250 370a or b or c 0.22 0.60 0.22 0.49 0.22 0.60 0.22 0.60 0.22 0.60 0.22i"1S or D 1.0 8.28 1.0 1.0 1.0 9.7 1.0 2.95 1.0 6.57 1.0P0 1.15 9.86 5.28 1.20g 0.20 0.24 0.40 0.50d 0.40 0.33 0.30 0.20i"2S or D 1.0 5.58 1.0 1.0 1.0 8.09 1.0 3.27 1.0 6.17 1.0P0 5.95 7.01 7.00 1.08g 0.15 0.35 0.70 0.42d 0.40 0.33 0.30 0.20i"3 S or D 1.0 2.34 1.0 1.0 1.0 10.3 1.0 5.70 1.0 5.98 1.0P0 3.96 6.01 5.13 0.66g 0.34 0.50 0.85 0.30d 0.40 0.33 0.30 0.20

C-TS and S-TS) deal with it successfully, and the comput-ing time is less than that of SA.

From the results of example 3, we found that in spite ofthe fact that TS obtained a better result than SA, thepractical production time in TS implementation is small-er. This means that the previously adopted optimizationcriterion (to merely reduce the difference between thepractical total production time and the net total time forproduction) by published literature in MP and heuristicwhen being applied to this problem is in fact not a satis-factory criterion for optimization in this optimal designproblem, or we can say that the smaller the difference thebetter the optimization is not the case.

As is evident from the computation results of examples2—4, C-TS has the advantage over SA in solution qualityand computational time. Also TS (including C-TS andS-TS) seems to be more robust with respect to varying

initial solutions than SA (see later). While using SA,a ‘‘good’’ choice of the control parameters (temperature,annealing schedule, etc.) greatly influences the solutionquality as pointed out by Patel et al. (1991) and Wang etal. (1996). By contrast, TS in its basic version does notrequire such sensitive parameters. Even the refinement ofTS given in this paper may only influence its computa-tion time. These result from the nature of TS and SA. It isknown that flexible memory plays a fundamental role inTS. TS exploits certain forms of flexible memory (historyinformation) to control the search process. For example,TS emphasizes scouting successive neighborhoods toidentify moves of high quality as done by double tabulist, dynamic neighborhood size, and dynamic tabu sizein this paper. This contrasts with the SA approach ofrandom sampling among these moves to apply an accept-ance criterion that disregards the quality of other moves



Q"(m ’000)"[40, 30, 10, 35, 33, 27, 25, 22, 20, 19, 15, 12, 9. 7, 5]H"8000, l"15, 300)»

i)2400, 300)R

m)2400

SC1 B1 SC2 SC3 B2 SC4 T SC5 SC6 B3 SC7

a, or b or c 370 592 250 210 582 250 200 250 200 1200 600a or b or c 0.22 0.65 0.40 0.62 0.39 0.40 0.39 0.40 0.85 0.52 0.40g(i"1, 15) — 0 — — 0 — — — — 0 —i"1 S or D 1.2 1.2 1.2 1.2 1.4 1.4 1.0 1.4 1.4 1.0 1.0P0 — 3.0 — — 1.0 — — — — 4.0 —i"2 S or D 1.5 1.5 1.5 1.5 0.0 0.0 1.0 1.5 1.5 1.0 1.0P0 — 6.0 — — 0.0 — — — — 8.0 —i"3 S or D 1.1 1.1 1.1 1.1 1.2 1.2 1.0 1.2 1.2 1.0 1.0P0 — 2.0 — — 2.0 — — — — 4.0 —i"4 S or D 1.5 1.5 1.5 1.5 1.8 1.8 1.0 1.8 1.8 1.0 1.0P0 — 2.0 — — 1.5 — — — — 3.0 —i"5 S or D 1.3 1.3 1.3 1.3 3.0 3.0 1.0 3.0 3.0 1.0 1.0P0 — 1.0 — — 2.0 — — — — 2.5 —i"6 S or D 1.4 1.4 1.4 1.4 2.1 2.1 1.0 2.1 2.1 1.0 1.0P0 — 2.0 — — 2.5 — — — — 5.0 —i"7 S or D 1.2 1.2 1.2 1.2 5.2 5.2 1.0 5.2 5.2 1.0 1.0P0 — 1.0 — — 0.5 — — — — 7.0 —i"8 S or D 1.1 1.1 1.1 1.1 2.1 2.1 1.0 2.1 2.1 1.0 1.0P0 — 4.0 — — 3.5 — — — — 3.0 —i"9 S or D 1.3 1.3 1.3 1.3 1.1 1.1 1.0 1.1 1.1 1.0 1.0P0 — 2.0 — — 3.0 — — — — 2.0 —i"10 S or D 1.4 1.4 1.4 1.4 1.5 1.5 1.0 1.5 1.5 1.0 1.0P0 — 2.5 — — 2.5 — — — — 4.0 —i"11 S or D 1.5 1.5 1.5 1.5 1.7 1.7 1.0 1.7 1.7 1.0 1.0P0 — 3.0 — — 2.0 — — — — 4.0 —i"12 S or D 1.2 1.2 1.2 1.2 1.9 1.9 1.0 1.9 1.9 1.0 1.0P0 — 3.5 — — 4.5 — — — — 6.5 —i"13 S or D 1.5 1.5 1.5 1.5 3.7 3.7 1.0 3.7 3.7 1.0 1.0P0 — 5.0 — — 7.0 — — — — 9.0 —i"14 S or D 1.8 1.8 1.8 1.8 2.2 2.2 1.0 2.2 2.2 1.0 1.0P0 — 4.5 — — 3.0 — — — — 4.0 —i"15 S or D 1.5 1.5 1.5 1.5 2.7 2.7 1.0 2.7 2.8 1.0 1.0P0 — 3.0 — — 2.0 — — — — 6.0 —

SC indicates batch stage, B indicates semicontinuous stage, T indicates intermediate storage.

Table 5Results of example 1

C-TS (TS with double list) S-TS (standard TS) MP

Objectives function 189679.8 191267.3 189015.7m

01m

11»

11 1 1592.2 1 1 1620.1 1 1 1631.1

m02

m12

»2

1 1 1990.3 1 1 2010.3 1 1 2039.2m

03m

13»

31 1 800.0 1 1 800.0 1 1 800.0

n1

R1

1 1800.0 1 1800.0 1 1800.0n2

R2

1 449.9 1 460.5 1 435.4n3

R3

1 449.9 1 460.5 1 435.4n4

R4

1 300.0 1 300.0 1 300.0n5

R5

1 300.0 1 300.0 1 300.0n6

R6

1 300.0 1 300.0 1 300.0» @

S1659.1 1720.1 1751.1

CPU(s) 9.16 85.5 287.4

IBM PS/2 70 386, OS/2 1.0.



MP objective function 170357.0 (CPU(s): 504)

C-TS (TS with double list) S-TS (standard TS) SA

Objectives function 170394.8 170604.3 170539.8m

01m

11»

11 1 1662.3 1 1 1665.1 2 1 1677.5

m02

m12

»2

1 1 800.0 1 1 800.0 2 1 800.0m

03m

13»

31 1 800.0 1 1 800.0 2 1 800.0

n1

R1

1 1800.0 1 1800.0 1 1800.0n2

R2

1 319.4 1 325.1 1 300.1n3

R3

1 319.4 1 325.1 1 300.1n4

R4

1 300.0 1 300.1 1 300.1n5

R5

1 300.0 1 300.1 1 300.1n6

R6

1 300.0 1 300.1 1 300.1» @

S1723.4 1733 1742.2

CPU(s) 65 185 360

IBM PS/2 70 386, OS/2 1.0.


MP objective function (Patel et al., 1991): 369728

Objective function C-TS S-TS SA (Patel et al., 1991)

362817.3 368131.4 368883m

01m

11»

11 1 6912.3 1 1 7301.3 2 1 4290.0

m02

m12

»2

2 1 9920.2 2 1 9926.3 2 1 9930.0m

03m

13»

32 1 5724.3 2 1 5800.1 2 1 5534.0

m04

m14

»4

1 1 9545.7 1 1 9905.2 1 1 7627.0n1

R1

1 8929.5 1 9006.2 1 9252.0n2

R2

1 2188.8 1 4843.2 1 10000.0n3

R3

1 6537.3 1 7101.4 1 9675.0n4

R4

1 8777.9 1 9104.6 1 10000.0n5

R5

1 9320.3 1 9860.6 1 9000.0n6

R6

1 2604.5 1 3805.1 1 390.0» @

S2745.6 3103.6 1997.0

Production time (h) 5975.2 5988.2 5999.1

available. (Such an acceptance criterion provides the solebasis for sorting the moves selected in the SA method.) InTS method, its neighborhood includes linkages based onhistory, and therefore yield access to information forselecting moves that are not available in heighborhoodsof the type used in SA. In TS, convergence could beobtained by reduction to an enumeration scheme ratherthan by asymptotic arguments of probabilistic flavor (asin SA). In fact, TS may be considered as a kind ofdeterministic version of SA in a broad sense (Werra andHertz, 1989). Besides, TS employs the intelligent strat-egies, i.e. dynamic neighborhood size, dynamic tabu size,and intensification, to speed up its convergence.

Although SA algorithm can be further tuned of itsparameters in practical implementation as done in thework of Wang et al. (1996a, b), the improvement is lim-ited due to its random nature.

5.2. Computational experience

We now comment on some important aspects in ourimplementation of TS and some problems in practice.

(a) Initial solution. In these four examples, we start thesearch procedure from the biggest possible value of alloptimization variables, that is, we adopted the biggestpossible value of each optimization variable as the initialsolution.

The influence of initial solution on the searchingprocedure was checked. We found that different initialsolutions possess no obvious influence on the optimiza-tion result but a better initial solution speeds up theconvergence greatly. For these four examples, it willresult in at least 50% reduction of computing time ifa ‘‘good’’ initial solution is given. This means that thoughTS has no special need for the initial solution, better



C-TS S-TS SA (Patel et al., 1991)

Objective function 401982.3 433329.7 450983.0m

01m

11»

12 1 1070.9 2 1 1090.3 2 1 1590;1780

m02

m12

»2

2 1 2395.8 2 1 2398.1 2 2 2400,896;1934,756m

03m

13»

32 1 1291.9 2 1 1605.2 2 1 1897;1871

n1

R1

2 1973.3 2 2010.2 2 2050;1645n2

R2

1 2098.1 1 2303 1 1512n3

R3

1 1840.5 1 2100.2 1 1512n4

R4

2 2252.0 2 2313.1 2 1564,559n5

R5

1 1830.6 1 2080.5 2 918,300n6

R6

1 1412.9 1 1635.1 1 1185n7

R7

1 1330.8 1 1930.6 1 2046» @

S2032.1 4321.2 5131.0

CPU (min) 2.5 5.6 10

On a Sun Sparc Station.

initial solution, as opposed to a random start, is benefi-cial to the eventual success of the TS implementation(particularly for large problems as example 4 in thispaper). A usual way to obtain an appropriate initialsolution is use of a heuristic procedure as designed byWang et al. (1996a, b).

(b) Dynamic neighborhood. We also found that of allthe strategies explained in Section 4, the strategy ofdynamic neighborhood possesses the greatest contribu-tion in speeding up the searching procedure. Our com-putational experience reveals that this method performsbetter than the method that simply adopts a smallerneighborhood size and not inferior to the method thatadopts a complete-neighborhood, but with at least 20%reduction of the searching time.

(c) Double tabu list. The strategy of double tabu listcontributes the most to the ability of global optimizationof the algorithm. For same design problems includingthese four examples as well as several other design prob-lems, we found that when the traditional single tabu list isadopted it is very difficult for the algorithm to find theglobal optimum, even though N

.!9(maximum number of

iterations during which the last step of procedure isallowed to run without any improvement of the bestsolution) is considerably large (see termination criterion).As stated above, this premature convergence results froman inadequate reduction of those optimization variablesrepresenting unit size. But when the double tabu listmethod is adopted, the algorithm can always find theglobal optimum in an acceptable time period. The compu-tation results in this paper show that C-TS (TS withdouble tabu list) has significant advantage over standardTS in solution quality and computational time. This pro-ved that a double tabu list contributes greatly to theoptimization ability of TS when applied to the optimaldesign problem of multiproduct batch chemical processes.

What is important is that, based on the considerationthat optimization variables of different characteristics

should not be dealt with in the same way in TS imple-mentation, this idea of double tabu list should be univer-sally applicable for those optimization problems whoseoptimization variables give a significantly different con-tribution to the objective function. This discloses thatin-depth study of the problem’s particular ‘‘structure’’can greatly improve the performance of TS.

(d) Dynamic tabu size. Through calculation experi-mentation, we find that ¹"7#N0.5 (in which N is theneighborhood size) is a proper tabu size for the designproblem in this paper. We can see that the tabu sizevaries as the neighborhood size changes over the threedifferent ranges. Computational experiences show thata small tabu size is preferable for exploring the solutionnear a local optimum and a larger tabu size is preferablefor breaking free from the vicinity of this local minimum.Also, practical experience indicates that dynamic tabusize are more robust than static tabu size.

(e) Diversification. Diversification strategy is essentialfor successful TS approaches to practical problems. Theusual method to realize diversification strategy is thepenalty function method. The attractiveness of thosesolutions that have the highest frequency in the FLM isreduced to some extent in this method, the bigger thefrequency the greater the penalty. Some methods to con-struct the penalty function have been developed (Werraand Hertz, 1989; Gendreau et al., 1994). Two penaltyfunction methods were experimented in these four exam-ples. The first one is frequency penalty function ap-proach: we take the objective function as

fM (V, R, a)"f (V, R)#a (ni/N),

where a'0 (ni: the number of the solutions equalling

Xivisited earlier; N: the total number of iterations).

The second one is the shifting penalty approach: a pen-alty function is created by every constraint. The globalobjective function is expressed as a weighted sum of thepenalty functions. At the beginning of the search, every


weight is set to 1, then often every 1 iterations, a weightassociated with a constraint that was always violatedduring the past I iterations is multiplied by 2; a weightassociated with a constraint that was never violatedduring these I iterations is divided by 2, and otherwisethe weights stay unchanged.

The experiments’ results disclosed that the inclusion ofa penalty function made the computation time longerand the search became complicated (specially forexample 4). Moreover, it is difficult to find a properpenalty function for such a complicated optimizationproblems as these four design problems in this paper.And it is difficult to say exactly which penalty function isbetter in implementation. Our computational experienceshows that LF method performs better than the tradi-tional penalty function method, resulting in a somewhatfaster convergence. But the most important aspect is thatthis method is much simpler than the penalty functionmethod.

(f ) Intensification and diversification. Comparison be-tween the strategy of intensification and diversificationshows that intensification has a greater influence in im-proving the best solution found so far, but a globaloptimum cannot be found sometimes without the ad-option of diversification. One possible explanation forthis may be the following: by the adoption of intensifica-tion these solution spaces that are more attractive areexplored more thoroughly and hence it is more possiblethat the present solution can be improved, while diversifi-cation restarts the searching procedure from a less at-tractive solution space it is possible that the best solutionfound so far cannot be improved, but with no diversifica-tion some solution spaces in which the global optimumlocates can possibly be ignored and hence the algorithmis trapped into local optimum.

(g) Step size of continuous variable. For this problem,Patel et al. (1991) designed a method to vary the stepsize of continuous variables dynamically. But ourcoputational experiences show that Patel’s methodinduced unnecessary computation into the searchingprocedure.

(h) ¹ermination. Our computational experience indi-cates that if N

.!9is too low, some good solutions will be

missed. If it is too high, there is a risk that the algorithmwill run a long time without improvement. Sensitivityanalyses performed on these four examples suggest50—100 is a good compromise.

6. Conclusions

For the first time, TS is adapted to the optimal designproblem of multiproducts batch chemical processes. TGuse the algorithm more effectively, several original ideasare proposed in practical implementation. Satisfactoryresults are obtained. TS is a good fit for the proposed

optimization problem and has the following advantagesin application:(1) TS has no special demand for initial values of optim-

ization variables. We can simply take the upper limitof each optimization variable as the initial solution atany instance.

(2) TS has no special demand for the form of the objec-tive function.

(3) As is evident from the computation results, TS yieldsa highly satisfactory global optimum.

(4) Due to its intelligent nature, TS results in a fasterconvergence in comparison with MP and SA.

(5) TS is simple in structure and is convenient for imple-mentation.

Nomenclature

ai

cost coefficient for bath state jbk

cost coefficient for semicontinuous stage kBik

batch size for product i in subprocess s, kgcs

cost coefficient for intermediate storageD

ikduty factor for semicontinuous stage k for prod-uct i

dij

power coefficient for processing time on stagej for product i

gij

coefficient processing time on storage j for prod-uct i

H horizon, hH

iproduction time of product i, h

i index for productI total number of productsj index for bath stageJ total number of bath stagesk index for semicontinuous stageK total number of semicontinuous stagesm

0jnumber of out-of-phase groups in bath stage j

mpj

number of in-phase parallel units in each of theout-of-phase groups in bath stage j

nk

number of parallel units in semicontinuous stage kPi

productivity of product i, kg/hpij

processing time for product i in stage j, hp0ij

constant in processing time equation for producti in stage j

Qi

demand for product i, kgR

kprocessing rate for semicontinuous unit k, l/h

R.!9k

maximum feasible size of semicontinuous stage kR.*/

kminimum feasible size of semicontinuous stage k

S total number of subprocessesSij

size factor for bath stage j for product iSIs

size factor for storage s for product It index for substrain¹ number of substrainstij

recycling time for product i in bath stage jtLis

limiting cycle time of product i in subprocess s»

lsize of bath stage j, l


».!9l

maximum feasible size of bath stage j, l».*/

lminimum feasible size of bath stage j, l

»*s

size of intermediate storage s, laj

cost coefficient for both stage jbk

cost coefficient for semicontinuous stage kcs

cost coefficient for storage shij

operating time for substrain t for product i

References

Committee on the Next Decade of Operation Research (Condor)(1988). Operation research: The next decade, Oper. Res., 36,619—637.

Dammeyer, F., & Voss, S. (1993). Dynamic tabu list management usingthe reverse elimination method. Ann. Oper. Res., 41, 31—46.

Gendreau, M., Hertz, M., & Laporte, G. (1994). A tabu search heuris-tic for the vehicle routing problem. Management Sci., 40,1276—1290.

Glover, F. (1986). Future paths for integer programming and links toartificial intelligence. Comput. Oper. Res., 13, 533—549.

Glover, F. (1989a). Tabu search — Part I. ORSA J. Comput., 1, 190—206.Glover, F. (1989b). New approaches for heuristic search: a bi-

lateral linkage with artificial intelligence, Eur. J. Oper. Res., 39,119—130.

Glover, F. (1990). Tabu search — Part II. ORSA J. Comput., 2, 4—32.Glover, F. (1993). A user’s guide to tabu search. Ann. Oper. Res., 41,

3—28.Glover, F., & Laguna, M. (1993). Tabu search. In: C. Reeves (Ed.),

Modern heuristic techniques for combinatorial problems (pp.70—150). Oxford: Blackwell.

Grossmann, I. E., & Sargent, W. E. (1979). Optimal design of multipur-pose chemical plants. Ind. Eng. Chem. Process Des. Dev., 18, 343—348.

Hansen, P. (1986). The steepest ascent mildest heuristic for combina-torial programming. Congress on Numerical Methods in Combina-torial Optimization, Capri, Italy.

Knopf, F.C., Okos, M. R., & Reklaitis, G. V. (1982). Optimum design ofbatch/semicontinuous processes. Ind. Eng. Chem. Process. Des.Dev., 21, 79—86.

Modi, A. K., & Karimi, I. A. (1989). Design of multiproduct bathprocesses with finite intermediate storage. Comput. Chem. Engng.13, 127—139.

Patel, A. N., Mah, R. S. K., & Karimi, I. A. (1991). Preliminary design ofmultiproduct non-continuous plants using simulating annealing.Comput. Chem. Engng. 15, 451—469.

Takamatsu, T., Hashimoto, I., & Hasebe, S. (1982). Optimal design andoperation of a bath process with intermediate storage tanks. Ind.Eng. Chem. Process. Des. Dev., 21, 431—440.

Tricoire, B., & Malone, M. (1991). A new approach for the design ofmultiproduct bath processes. AIChE Annual Meeting, Los Angeles.

Wang, C., Quan, H., & Xu, X. (1996a). Optimal design of multiproductbatch chemical process — mixed simulated annealing. J. Chem.Engng. 47, 184—191 (in Chinese).

Wang, C., Quan, H., & Xu, X. (1996b). Optimal design of multiproductbatch chemical processes using genetic algorithms. Ind. Eng. Chem.Res. 35, 3560—3566.

Werra, D., & Hertz, A. (1989). Tabu search techniques — a tutorial andan application to neural networks. OR Spektrum, 11, 131—141.

Xu, X., Zheng, G., & Cheng, S. (1993). Optimized design of multi-product bath chemical process — a heuristic approach. J. Chem.Engng. 44, 442—450 (in Chinese).

Yeh, N. C., & Reklaitis, G. V. (1987). Synthesis and sizing of batch/semi-continuous processes: Single product plants. Comput. Chem.Engng. 11, 639—654.


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