# Optimal design of multiproduct batch chemical processes using tabu search

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*Corresponding author. Fax: 86 22 23358329; e-mail: cfwang@tju.edu.cn

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

Computers and Chemical Engineering 23 (1999) 427437

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) thateectively 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 diversication 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 eectiveness 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 exibility 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 specied. 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 eectively 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 eectively, tabusearch (TS), an intelligent problem-solving methodthat has demonstrated its eectiveness 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 eectiveness 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 Modis model modied 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 nite 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,

s"(t D semicontinuous substrain belonging to subpro-

cess), s"1, S,

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

tinuous substrain t), t"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

ajaj

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"BisL

is

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

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

Lis" Max

j|Js j|Ts

[ij, h

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

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

ij"hiu#pij#h i (u1)

m0j

, 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, ; t3s ; (9)

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

Bis"Min

j|JsAm

pili

SijB , i"1, I ; t"1, ; t3s ; (10)

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

C. Wang et al. / Computers and Chemical Engineering 23 (1999) 427437428

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

*s"Max

i

[PiS*is(L

is!h

iu#L

i (s1)!h

i (u1))],

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 rst fully described in 1986 (Golver, 1986; Hansen,1986). Since that time, TS has been shown to be a re-markably eective 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 nds 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 moves ( is called tabusize). Those most recent 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 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. Intensication

To improve the eciency of the search procedure, it isnot preferable to explore the whole solution space withequal eort. 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 intensication to achieve this goal.

3.5. Diversication

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 diversication 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. Diversication enables TS to have theability of global optimization.

C. Wang et al. / Computers and Chemical Engineering 23 (1999) 427437 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 onarticial intelligence principles. It is noted that thoseintelligent strategies above are very exible in the TSimplementation. The design of actual mechanism to im-plement these strategies, including discovery of morerened 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 realizedexibly in implementation. Fig. 1 shows the ow chart ofthe algorithm.

4.1. Neighborhood structure

Neighborhood structure plays an important role in TSimplementation, which inuences 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 dierent 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 nd 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 representin...