multicriteria optimization of multiproduct batch chemical process using genetic algorithm

MULTICRITERIA OPTIMIZATION OF MULTIPRODUCT BATCHCHEMICAL PROCESS USING GENETIC ALGORITHM

D. MOKEDDEM1 and A. KHELLAF

Department of ElectronicsFaculty of Engineering

University of Setif19000 Setif, Algeria

Accepted for Publication May 19, 2008

ABSTRACT

Optimal design problems are widely known by their multiple performancemeasures that are often competing with each other. In this paper, an optimalmultiproduct batch chemical plant design is presented. The design is firstformulated as a multiobjective optimization problem, to be solved using thewell-suited nondominating sorting genetic algorithm (NSGA-II). The NSGA-IIhas the capability to achieve fine tuning of variables in determining a set ofnondominating solutions distributed along the Pareto front in a single runof the algorithm. The NSGA-II ability to identify a set of optimal solutionsprovides the decision maker with a complete picture of the optimal solutionspace to gain better and appropriate choices. The effectiveness of NSGA-IImethod with multiobjective optimization problem is illustrated through acarefully referenced example.

PRACTICAL APPLICATIONS

The inherent dynamic nature of batch processes allows for their ability tohandle variations in feedstock and product specifications, and provides theflexibility required for multiproduct or multipurpose facilities. They are thusbest suited for the manufacture of low-volume, high-value products, such asspecialty chemicals, pharmaceuticals, agricultural, food and consumer prod-ucts, and most recently the constantly growing spectrum of biotechnology-enabled products. Batch processing in the food industry has some unitoperations, such as panning, whipping (emulsifying), and the pulling of taffy,which are not too often used in other industries, as well as other operations,

1 Corresponding author. TEL: 00 213 774 875 892; EMAIL: [email protected]

Journal of Food Process Engineering 33 (2010) 979–991. All Rights Reserved.© Copyright the AuthorsJournal Compilation © 2009 Wiley Periodicals, Inc.DOI: 10.1111/j.1745-4530.2008.00319.x

979

such as blending, tabletting (including granulation) and lyophilization, whichare common in other industries. Reduced time to market, lower productioncosts and improved flexibility are all critical success factors for batchprocesses.

INTRODUCTION

Batch processes are used in production of many low-volume but highvalue-added products (such as specialty chemicals, health care, food, agro-chemicals, etc.) because of operation flexibility in today’s market-drivenenvironment. Manufactory of these products generally involves multistepsynthesis. In addition, if two or more products require similar processingsteps, the same set of equipment is considered for at least economicalreasons. A batch plant producing multiple products is categorized either as amultiproduct plant or as a multipurpose plant. Multiproduct plants producemultiple products following a sequential similar recipe. In such a plant, allthe products follow the same path through the process and only one productis manufactured at a time. Each step is carried out on single equipment or onseveral parallel equipment units. Processing of other products is carried outusing the same equipment in successive production runs or campaigns. Ina multipurpose plant, each product follows one or more distinct processingpaths; so more than one product may be produced simultaneously in suchplants. The present work is directed toward the optimal design problems ofmultiproduct batch plants.

In conventional optimal design of a multiproduct plant, productionrequirements of each product and a total production time for all products areavailable and specified. The number, the required volume and size of parallelequipment units in each stage are then determined to minimize the invest-ment. It should be emphasized that the batch plant’s design has long beenidentified as a key problem in chemical engineering as reported in the literature(Montagna et al. 2000; Cao and Yuan 2002; Chunfeng and Xin 2002; Heoet al. 2003; Cavin et al. 2004; Chakraborty et al. 2004; Goel et al. 2004; Pintoet al. 2005). Formulation of a batch plant design generally involves math-ematical programming methods, such as linear programming (LP), nonlinearprogramming (NLP), mixed-integer LP or mixed-integer NLP (MINLP).Mathematical programming or different optimization techniques, such asbranch and bound, heuristics, genetic algorithm, simulated annealing, arethoroughly used to derive optimal solutions. However, in reality the multi-product design problem can be formulated as a multiobjective design optimi-zation problem in which one seeks to minimize investment, operation cost andtotal production time, and, simultaneously, to maximize the revenue. Recall

980 D. MOKEDDEM and A. KHELLAF

that not much work has been reported in the literature on the multiobjectiveoptimal design of a multiproduct batch plant. Huang and Wang (2002) intro-duced a fuzzy decision-making approach for multiobjective optimal designproblem of a multiproduct batch plant. A monotonic increasing or decreasingmembership function is used to define the degree of satisfaction for eachobjective function and the problem is then represented as an augmented min/max problem formulated as MINLP models. To obtain a unique solution, theMINLP problem is solved using a hybrid differential evolution technique.Dedieu et al. (2003) presented the development of a two-stage methodologyfor a multiobjective batch plant design and retrofit according to multiplecriteria. The authors used a multiobjective genetic algorithm (MOGA) basedon the combination of a single-objective genetic algorithm and a Pareto-sortprocedure for proposing several plant structures and a discrete event simulatorfor evaluating the technical feasibility of the proposed configurations.

In the case of multiple objectives, an optimum solution with respect toall objectives may not exist. In most cases, the objective functions are inconflict, because in order to decrease any of the objective functions, we needto increase other objective functions. The presence of multiple objectives ina problem usually gives rise to a family of nondominated solutions, largelyknown as Pareto-optimal solutions, where each objective component of anysolution along the Pareto front can only be improved by degrading at leastone of its other objective components. Since none of the solutions in thenondominated set is absolutely better than any other, any one of them is thenan acceptable solution. As it is difficult to choose any particular solution fora multiobjective optimization problem without iterative interaction with thedecision maker (DM) (Brans and Mareschal 1994), one general approachis to establish first the entire set of Pareto-optimal solutions, where anexternal DM direct intervention gives interactive information in the multi-objective optimization loop (Bhaskar et al. 2000). So, a satisfactory solutionof the problem is found as soon as the knowledge is acquired (Fonseca andFleming 1995).

A more recent algorithm, based on scalarization with a weighted sumfunction, is proposed in Ishibuchi and Murata (1998) where the weights arerandomly chosen. Many successful evolutionary multiobjective optimizationalgorithms were developed based on the two ideas suggested by Goldberg(1989): Pareto dominance and niching. Pareto dominance is used to exploit thesearch space in the direction of the Pareto front and niching technique exploresthe search space along the front to keep diversity. The well-known algorithmsin this category include MOGA (Fonseca and Fleming 1995); niched Paretogenetic algorithm (Horn et al. 1994); strength Pareto evolutionary algorithm(Zitzler and Thiele 1999); multiobjective evolutionary algorithm (Tan et al.2001); the nondominated sorting genetic algorithm (NSGA) proposed by

OPTIMIZATION OF MULTIPRODUCT BATCH CHEMICAL PROCESS 981

Srinivas and Deb (1994) was one of the first evolutionary algorithm for solvingmultiobjective optimization problems. Although NSGA has been successfullyapplied, the main criticisms of this approach has been its high computationalcomplexity of nondominated sorting, lack of elitism and the need for specify-ing a tuneable parameter called sharing parameter. Recently, Deb et al. (2002)reported an improved version of NSGA, which they called NSGA-II, toaddress all these issues.

The purpose of this study is to extend this methodology for solution ofmultiobjective optimal control problems under the framework of NSGA-II.

FORMULATION OF THE MULTIOBJECTIVE PROBLEM

The problem of a multiproduct batch plant covered in this paper can bedefined by assuming that the plant consists of a sequence of M batch process-ing stages that are used to manufacture N different products. At each stage jthere are Nj identical units in parallel operating out of phase, each with a sizeVj. Each product i follows the same general processing sequence. Batches aretransferred from one stage to the next without any delay, i.e., we consider azero-wait operating policy.

In the conventional design of a multiproduct batch plant, one seeks tominimize the investment cost by determining the optimal number, requiredvolume and size of parallel equipment units in each stage for a specifiedproduction requirement of each product and the total production time.However, in reality the designer considers not only minimizing the investmentbut also minimizing the operation cost and total production time at the sametime maximizing the revenue, simultaneously.

Max Revenuej j i i i

i iN V B T Q H

i

N

L

f Cp Q, , , , ,

= ==∑1

1

(1)

Min Investment costj j i i i

j

j j jN V B T Q H

j

N

L

f N a V, , , , ,

= ==

∑21

β(2)

Min Operation costj j i i i

j

j

iN V B T Q H

Ej

M

i

N

oL

f CQ

BC

, , , , ,= = +

==∑∑3

11ii iQ (3)

Min Total production timej j i i iN V B T Q HL

f H, , , , ,

= =4 (4)


So, the multiobjective problem consists of determining the parameters Nj, Vj,Bi, TLi, Qi and H, at the same time satisfying certain constraints such asvolume, time, etc.

The constraints are expressed as follows:

(1) Volume constraints: volume Vj has to be able to process all the products i.

S B V i N j Mij i j≤ ∀ = ∀ =, , . . . , ; , . . . ,1 1 (5)

(2) Time constraint: the summation of available production time for allproducts is not more than the net total time for production.

Q

BT HL

i

Ni

ii

=∑ ≤

1

(6)

(3) The limiting cycle time for product i

τ ij

jiN

T i N j ML≤ ∀ = ∀ =, , . . . , ; , . . . ,1 1 (7)

The time required to process one batch of product i in stage j is expressedas:

τ τ γij ij ij i

j= + ∀ ∀c B i j, , (8)

Where tij � 0, cij � 0 and gj are constants and Bi is the batch size forproduct i. Thus, the processing time is not a constant, but depends on thedecision parameters of the batch size.

(4) Dimension constraints: every unit has restricted allowable ranges.

V V V j M

B B B j N

jL

j jU

jL

j jU

≤ ≤ ∀ =

≤ ≤ ∀ =

, , . . . ,

, , . . . ,

1

1(9)

ELITIST NSGA-II

The NSGA-II Pareto ranking algorithm is an elitist system (Debet al. 2002) and maintains an external archive of the Pareto solutions.


Fundamentally, the nondominated sorting segregates the population into“layers” by first finding the nondominated solutions in the population andlabels these points as the first front. These points are removed and the non-dominated solutions in the remaining population are then identified andremoved as well. This process continues until the entire population has beenclassified into layers. The algorithm updates the current archive by identifyingall the nondominated solutions in the union of the old archive and currentpopulation. The layers are selected in turn until the maximum size of thearchive is reached (often the population size). The last layer to be added istruncated if necessary by employing a crowding distance operator and select-ing solutions with the smallest crowding distance for removal, leading to themost diverse set. The crowding distance is the average of the distances inobjective space between the point under consideration and the nearest objec-tive values above and below (calculated independently for each objective axis).

New solutions are implemented by selecting parents from the archiveset. The classic NSGA II algorithm uses a tournament selection systembased on crowding distance to break ties. The algorithm is described asfollows:

(1) randomly initialize population (designs in the variable space) of size npop;(2) compute objectives and constraints for each design;(3) rank the population using nondomination criteria (many individuals can

have same rank and rank 1 is the best);(4) compute crowding distance (this distance finds the relative closeness of a

solution to other solutions in the function space and is used to differentiatebetween the solutions on same rank);

(5) employ genetic operators – selection, crossover and mutation – to createintermediate population of size npop;

(6) evaluate objectives and constraints for this intermediate population;(7) combine the two (parent and intermediate) populations, rank them and

compute the crowding distance;(8) select new population of npop best individuals based on the rank and

crowding distance;(9) go to step 3 and repeat until termination criteria is reached, which is

chosen to be the number of generations in the current study.

DESCRIPTION OF THE PROCESS

To demonstrate the effectiveness of NSGA-II on batch plant processes,an example consisting of four processing stages (mixer, reactor, extractionand centrifuge) to manufacture three products: A, B and C. The data for the


problem is illustrated in Table 1 (processing times, size factor for the units andcost for each product).

RESULTS AND DISCUSSION

The batch plant in this case consists of four processing stages tomanufacture three products – A, B and C – with four-objective optimizationproblem as expressed in Eqs. (1) to (4). The set of decision variables consistsof the batch size, total production time, number of parallel units at eachstage, cycle time for each product and the required volume of a unit in eachstage. Since the number of parallel units at each stage is an integer decisionvariable, we code this variable as a binary variable. All other decision vari-ables are coded as real numbers. Thus, there are four integer variables and14 real variables. In addition to the constraints expressed by Eqs. (5) to (9),we consider bounds on objective functions as additional constraints to

TABLE 1.DATA OF THE PROCESS

Processing times, τ ij (h) Unit price for the product ($/kg)

Product Mixer Reactor Extractor Centrifuge Product Cp C0

A 1.15 9.86 0.4 0.5 A 0.27 0.08B 5.95 7.01 0.7 0.42 B 0.29 0.10C 3.96 6.01 0.85 0.3 C 0.32 0.12g ij 0.4 0.33 0.3 0.2

Minimum size = 250 LMaximum size = 10,000 L

Product size factors (L/kg)

A 8.28 9.7 6.57 2.95B 5.58 8.09 6.17 3.27C 2.34 10.3 5.98 5.7

Product Coefficients Cij

A 0.2 0.24 0.4 0.5B 0.15 0.35 0.7 0.42C 0.34 0.5 0.85 0.3

Cost of equipment ($, V in liters)

250V0.6 250V0.6 250V0.6 250V0.6

Operating cost factor (CE)

20 30 15 30


generate feasible nondominated solutions in the range desired by the DM, tohave 31 constraints in all.

f f f iiL

i iU . . .≤ ≤ = 1 4, , (10)

Then NSGA-II is employed to solve the optimization problem with thefollowing parameters: maximum number of generation up to 200; populationsize 500; probability of crossover 0.85; probability of mutation 0.05; distribu-tion index for the simulated crossover operation 10; and distribution index forthe simulated mutation operation 20.

The revenue ( f1) increases as operation cost ( f3) increases, whereas theinvestment cost ( f2) decreases following operation cost ( f3). The Pareto-optimal solutions are presented in Fig. 1. The relationships of the variousdecision variables are shown in Fig. 2. The large set of multiple optimalsolutions provides the DM with immediate information about the relationshipamong the several objective criteria and a set of feasible solutions. Thus, ithelps the DM to select a highly confident choice of solution. Let it be men-tioned that when all four objective functions are considered simultaneously,the solutions obtained in the present study improve significantly, as the resultspresented by Huang and Wang (2002) for the same problem. For example, thesolution presented by Huang and Wang (2002) with unit reference membershiplevel for all objectives ( f1 = 274,312, f2 = 375,688, f3 = 175,688, f4 = 5,639) thesolution 1 presented in Table 2 of the present study improves the abovesolution f1, f3, f4 whereas f2 is comparable.

The fixed optimal plant structure as 2,221 corresponds to a two mixers,two reactors, two extractors and one centrifuge design as presented in Fig. 3.

Implementation of a trade-off analysis is dependent upon the availabilityof the DM’s preferences.

CONCLUSION

A multiobjective decision in a batch plant process design is consideredand NSGA-II is developed to get an optimal zone containing solutions underthe concept of Pareto set. NSGA-II capability have been proved in evolving theentire set of nondominating solutions along the Pareto front in a single run ofthe algorithm. Thus, the DM is provided with the best trade-off operating zone.Furthermore, a better confident choice of design among several compromisesthe DM if the decision variables effects on the objective functions are known.

Finally, the large set of solutions presents a useful base for further alter-native approaches to fulfill the DM targets.


260000 265000 270000 275000 280000 285000 290000 295000 300000 305000145000

150000

155000

160000

165000

170000

175000

f 3

f1

260000 265000 270000 275000 280000 285000 290000 295000 300000 305000365000

370000

375000

380000

385000

390000

395000

f 2

f1

FIG. 1. PARETO OPTIMAL SOLUTIONS OF THE PROCESS


80008200

84008600

88009000

7400

7500

7600

7700

7800

7900

8000

97009750

98009850

99009950

10000

Reactor

volum

e (L

)

Cen

trifu

ge v

olum

e (L

)

Mixer volume (L)

920940

960980

10001020

1040

840

860

880

900

920

940

960

1200

1210

12201230

1240

Batch s

ize B

Bat

ch s

ize

C

Batch size A

80008200

84008600

88009000

5000

5200

5400

5600

5800

6000

97009750

98009850

99009950

10000

Reactor

volum

e (L

)

Ext

ract

or

volu

me

(L)

Mixer volume (L)

400000420000

440000460000

480000

200000

210000

220000

230000

240000

250000

260000

270000

320000

340000360000

380000400000

Product

require

mentB

Pro

duc

t req

uire

men

t C

Product requirement A

FIG. 2. THREE-DIMENSION PLOT OF THE RELATIONSHIP BETWEEN SOMEDECISION VARIABLES

TABLE 2.OPTIMAL OBJECTIVE FUNCTION VALUES OF THE PROCESS

Case Optimal objective function values

f1($) f2($) f3($) f4($)

1 275,766.10 388,262.10 156,449.20 5,505.402 276,096.30 369,977.30 161,392.30 5,710.903 281,818.70 369,843.48 163,552.80 5,718.00

Bounds for objective function: [ f f1 1L U, ] = [250,000, 300,000]; [ f f2 2

L U, ] = [350,000, 400,000];[ f f3 3

L U, ] = [150,000, 200,000]; [ f f4 4L U, ] = [5,500, 6,000].


NOMENCLATURE

Bi size of the batch of product i at the end of the M stages (kg)CEJ

operation cost in stage j ($)COi

operation cost of product i to be produced ($/kg)CPi

price of product i ($/kg)H total production time (h)M number of stages in the batch processN number of products to be producedNj number of parallel units in stage jQi production requirement of product i (kg)Sij size factor of product i in stage j (L/kg)TLi cycle time for product i (h)Vj required volume of a unit in stage j (L)aj cost coefficient for unit jbj cost exponent for unit jtij processing time of product i in stage j (h)

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Product A B C Batch size (kg) 1024.18 1233.61 903.29 Cycle time (h) 6.30 6.12 5.48 Production requirement (kg) 41067.64 347391.86 200866.3 Total production time (h) = 5505.4

Mixer CentrifugeV=8946.32L V=7851.67L V=5839.16L

Reactor ExtractionV=9990.68L

FIG. 3. OPTIMAL DESIGN OF BATCH PLANT


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multicriteria optimization of multiproduct batch chemical process using genetic algorithm

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