Alkylation Process Optimization

Alkylation process has been described in Sec. 1.4. Two optimization prob- lems referred in Chap. 1 as Case A (Maximize Profit and Maximize Octane Number) and Case B (Maximize Profit and Minimize Isobutane Recycle) are solved using NSGA-II and SAEA to illustrate the benefits of SAEA. Variable bounds for optimization problem Case A is the same as those

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Surrogate Assisted Evolutionary Algorithm for MOO 147

listed in Table 1.2, while for Case B the variable bound for x₂ was set to

lie between 12,000 and 17,500 barrels/day.

For both optimization problems a population of size 40 is evolved over 31 generations. The results of Case A are shown in Fig. 5.7(a). SAEA performed 426 actual evaluations and NSGA-II was run with 440 evalua- tions. The non-dominated solutions obtained by SAEA have a much better spread as compared to the non-dominated solutions obtained by NSGA-II. The results of the alkylation optimization process Case B are shown in Fig. 5.7(b). SAEA performed 403 actual evaluations and NSGA-II was run with 440 evaluations. It is seen from the overlapping non-dominated solutions that SAEA performance is on par with the NSGA-II.

94.2 94.3 94.4 94.5 94.6 94.7 94.8 94.9 95 1090 1100 1110 1120 1130 1140 1150 1160 Octane Number Profit ($/day) SAEA NSGA-II (a) 12000 13000 14000 15000 16000 17000 18000 850 900 950 1000 1050 1100 1150 1200

Isobutane Recycle (barrels/day)

Profit ($/day) SAEA

NSGA-II

(b)

Fig. 5.7 Results obtained by SAEA and NSGA-II for alkylation process optimization. (a) Case A. (b) Case B.

5.4 Conclusion

In this chapter, an evolutionary algorithm for multi-objective optimiza- tion that is embedded with a surrogate model to reduce the computational cost has been introduced. The surrogate assisted evolutionary algorithm (SAEA) alternates in cycles i.e. SAEA performs actual evaluations once every S generations, trains the surrogate models and employs predictions of the surrogate models in lieu of actual evaluations for the intermediate generations.

In order to maximize the use of information from all actual evalua- tions, the algorithm maintains an external archive that is used to train the RBF model, periodically after every S generations. In order to maintain prediction accuracy, a candidate solution is only approximated using the

148 T. Ray, A. Isaacs and W. Smith

RBF model if at least one solution exists in the archive that is within a distance threshold. This distance threshold plays an important role in the early stages of evolution where more candidate solutions are evaluated us- ing actual computations even during the surrogate phase. Furthermore, a candidate solution is only evaluated using the surrogate model if the MSE of the surrogate on the validation set is below an user defined threshold.

The results of all the test problems support the fact that better non- dominated solutions can be delivered by the SAEA as compared to NSGA-II for the same number of actual function evaluations. Although the algorithm incurs additional computational cost for solution clustering and periodic training of RBF models, such cost is insignificant for problems where the evaluation of a single candidate solution requires expensive analyses like finite element methods or computational fluid dynamics.

In lieu of the RBF model used in this study, other surrogate models such as multilayer perceptron or Kriging could be used. In order to evade the problem of selection of surrogate models a priori, the authors are in- vestigating the performance of surrogate ensembles. The surrogate assisted optimization models are being used by the authors for real-life problems,

e.g. scram-jet design, nose cone design and structural optimization prob-

lems requiring computationally expensive analyses.

References

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Deb, K. and Goyal, M. (1996). A combined genetic adaptive search (GeneAS) for engineering design, Computer Science and Informatics 26, 4, pp. 30–45. Deb, K., Pratap, A., Agarwal, S. and Meyarivan, T. (2002a). A fast and elitist

multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolu-

tionary Computation 6, 2, pp. 182–197.

Deb, K., Pratap, A., Agarwal, S. and Meyarivan, T. (2002b). A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolu-

tionary Computation 6, 2, pp. 182–197.

Farina, M. (2002). A neural network based generalized response surface multiob- jective evolutionary algorithm, in Proceedings of IEEE World Congress on

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Haykin, S. (1999). Neural Networks: A Comprehensive foundation, 2nd edn. (Prentice-Hall Internal Inc., New Jersey).

Jin, Y. (2005). A comprehensive survey of fitness approximation in evolutionary computation, Soft Computing - A Fusion of Foundations, Methodologies

and Applications 9, 1, pp. 3–12.

Knowles, J. (2006). ParEGO: a hybrid algorithm with on-line landscape approx- imation for expensive multiobjective optimization problems, IEEE Trans-

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Nain, P. K. S. and Deb, K. (2002). A computationally effective multiobjective search optimization technique using course to fine grain modeing, Tech. Rep. 2002005, Kangal, IIT Kanpur, India.

Nain, P. K. S. and Deb, K. (2005). A multiobjective optimization procedure with successive approximate models, Tech. Rep. 2005002, Kangal, IIT Kanpur, India.

Ray, T. and Smith, W. (2006). Surrogate assisted evolutionary algorithm for mul- tiobjective optimization, 47th AIAA/ASME/ASCE/AHS/ASC Structures,

Structural Dynamics, and Materials Conference , pp. 1–8.

Santana-Quinter, L. V., Serrano-Hernandez, V. A., Coello Coello, C. A., Hernandez-Diaz, A. G. and Molina, J. (2007). Use of radial basis functions and rough sets for evolutionary multi-objective optimization, in Proceedings

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pp. 107–114.

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Genetic Algorithms, pp. 93–100.

Simpson, T. W., Korte, J. J., Mauery, T. M. and Mistree, F. (1998). Compar- ison of response surface and kriging models for multidisciplinary design optimization, in Proceedings of the 7th AIAA/USAF/NASA/ISSMO Sym-

posium on Multidisciplinary Analysis & Optimization, AIAA-98-4755 (St.

Louis, MO).

Veldhuizen, D. V. and Lamont, G. (2000). On measuring multiobjective evolution- ary algorithm performance, in Proceedings of the Congress on Evolutionary

Computation (CEC 2000), Vol. 1, pp. 204–211.

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ufacture 2006. Proceedings of the Seventh International Conference (The

Institute for People-centred Computation, Bristol, UK), pp. 167–175. Wilson, B., Cappelleri, D., Simpson, T. W. and Frecker, M. (2001). Efficient

Pareto frontier exploration using surrogate approximations, Optimization

and Engineering 2, 1, pp. 31–50.

Won, K. S. and Ray, T. (2004). Performance of kriging and cokriging based surro- gate models within the unified framework for surrogate assisted optimiza- tion, in Proceedings of the IEEE Congress on Evolutionary Computation,

150 T. Ray, A. Isaacs and W. Smith

Won, K. S. and Ray, T. (2005). A framework for design optimization using sur- rogates, Engineering Optimization37, 7, pp. 685–703.

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Computation (CEC’03) (Canberra).

Exercises

(1) For the following optimization problems find the set of non-dominated solutions using NSGA-II and SAEA.

(a) Problem BNH Minimize f₁(x) = 4x2₁+ 4x2₂, Minimize f₂(x) = (x₁− 5)2+ (x₂− 5)2, subject to (x₁− 5)2+ x2₂≤ 25, (x₁− 8)2+ (x₂+ 3)2≥ 7.7, 0≤ x₁≤ 5, 0 ≤ x₂≤ 3. (b) Problem SRN Minimize f₁(x) = 2 + (x₁− 2)2+ (x₂− 1)2, Minimize f₂(x) = 9x₁− (x₂− 1)2, subject to x2₁+ x2₂≤ 225, x₁− 3x₂+ 10≤ 0, − 20 ≤ x1, x₂≤ 20. (c) Problem ZDT4 Minimize f₁(x), Minimize f₂(x) = g(x) h(f₁(x), g(x)), where f₁(x) = x₁, g(x) = 1 + 10(10− 1) + 10  i=2 (x2_i − 10 cos(4πxi)), h(f₁, g) = 1− f₁/g, 0≤ x₁, x₂, . . . , x₁₀≤ 1. (d) Problem SCH1 Minimize f₁(x) = x2, Maximize f₂(x) = (x− 2)2, − 100 ≤ x ≤ 100.

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Surrogate Assisted Evolutionary Algorithm for MOO 151

(2) Observe the performance of NSGA-II on the above problems with popu- lation size varying between 100 and 200, number of generations between 100 and 200, probability of crossover between 0.8 and 1.0, probability of mutation between 0.01 and 0.1, distribution index of crossover between 5 and 20, distribution index of mutation between 10 and 50.

(3) Observe the performance of SAEA on the above problems with varying periodicity of training from every 2 to 10 generations and different user specified distance threshold values.

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Chapter 6

Why Use Interactive Multi-Objective Optimization in Chemical Process Design?

Kaisa Miettinen and Jussi Hakanen

Department of Mathematical Information Technology

P.O. Box 35 (Agora), FI-40014 University of Jyv¨askyl¨a, Finland

[email protected], [email protected] Abstract

Problems in chemical engineering, like most real-world optimization problems, typically, have several conflicting performance criteria or ob- jectives and they often are computationally demanding, which sets spe- cial requirements on the optimization methods used. In this paper, we point out some shortcomings of some widely used basic methods of multi- objective optimization. As an alternative, we suggest using interactive approaches where the role of a decision maker or a designer is empha- sized. Interactive multi-objective optimization has been shown to suit well for chemical process design problems because it takes the prefer- ences of the decision maker into account in an iterative manner that enables a focused search for the best Pareto optimal solution, that is, the best compromise between the conflicting objectives. For this reason, only those solutions that are of interest to the decision maker need to be generated making this kind of an approach computationally efficient. Besides, the decision maker does not have to compare many solutions at a time which makes interactive approaches more usable from the cogni- tive point of view. Furthermore, during the interactive solution process the decision maker can learn about the interrelationships among the objectives. In addition to describing the general philosophy of interac- tive approaches, we discuss the possibilities of interactive multi-objective optimization in chemical process design and give some examples of in- teractive methods to illustrate the ideas. Finally, we demonstrate the usefulness of interactive approaches in chemical process design by sum- marizing some reported studies related to, for example, paper making and sugar industries. Let us emphasize that the approaches described are appropriate for problems with more than two objective functions. Keywords: Multiple criteria decision making (MCDM), interactive methods, scalarization, chemical engineering, Pareto optimality

154 K. Miettinen and J. Hakanen 6.1. Introduction

Problems involving multiple conflicting criteria or objectives are generally known as multiple criteria decision making (MCDM) problems. In such problems, instead of a well-defined single optimal solution, there are many compromise solutions, so-called Pareto optimal solutions, that are mathe- matically incomparable. In the MCDM literature, solving a multi-objective optimization problem is usually understood as helping a human decision maker (DM) in considering the multiple objectives simultaneously and in finding a Pareto optimal solution that pleases him/her the most. In other words, the solution process needs some involvement of the DM and the final solution is determined by his/her preferences.

Examples of surveys of methods available for multi-objective optimiza- tion are Chankong and Haimes (1983); Hwang and Masud (1979); Marler and Arora (2004); Miettinen (1999); Sawaragi et al. (1985); Steuer (1986); Vincke (1992). The methods can be classified in different ways. In Hwang and Masud (1979); Miettinen (1999) they are divided into four classes ac- cording to the role of the DM in the solution process. If there is no DM and his/her preference information available, we can use so-called no-preference methods which find some neutral compromise solution without any addi- tional preference information. In a priori methods, the DM first gives pref- erence information and then the method looks for a Pareto optimal solution satisfying the hopes as well as possible. This is a straightforward approach but the difficulty is that the DM may have too optimistic or pessimistic hopes and then the solution generated may be far from them and, thus, disappointing.

In a posteriori methods, a representative set of Pareto optimal solu- tions is generated and then the DM must select the most preferred one. In this way, the DM gets an overview of the problem but it may be difficult for the DM to analyze a large amount of information. A natural visual- ization on a plane is possible only for problems involving two objectives. Furthermore, generating the set of Pareto optimal solutions may be com- putationally expensive. Evolutionary multi-objective optimization (EMO) algorithms belong to this class but it may happen that the solutions gen- erated are not really Pareto optimal but only nondominated in the current population. The fourth class is that of interactive methods. Many inter- active methods exist but they should become more widely known among people solving real applications. In interactive approaches, a solution pat- tern is formed and then repeated and the DM can specify and adjust one’s

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Interactive Methods for Chemical Process Design 155

In this chapter, we introduce scalarization based approaches and, in par- ticular, interactive methods as alternatives to EMO approaches. Our aim here is to widen the awareness of the readers of the existence of interac- tive methods and the advantages and usefulness of using them. Sometimes multi-objective and bi-objective optimization are regarded as synonyms but this kind of thinking is very limiting. Our approaches are capable of handling genuine multi-objective optimization with more than two ob- jectives. Besides discussing drawbacks of some widely used methods (like the weighting method) that sometimes are regarded as the only nonevolu- tionary multi-objective optimization methods available, we introduce some interactive methods and the NIMBUS method (Miettinen, 1999; Miettinen

and M¨akel¨a, 2006), in particular. In addition, we summarize encouraging

experiences of solving some chemical engineering problems with the interac- tive NIMBUS method. The motivation here is that it is important to get to know that a variety of methods and approaches exists. In this way, people solving different problems are able to use the most appropriate approaches. In this respect, scalarization based and interactive methods complement

evolutionary approaches. More information about bringing the MCDM

and EMO fields closer is available in Branke et al. (2008).

This paper is organized as follows. In Section 6.2, main concepts and the idea of scalarization based methods are introduced. In addition, some basic multi-objective optimization methods and their shortcomings are pre- sented and comparative aspects between scalarization based and evolution- ary approaches are discussed. Section 6.3 concentrates on interactive multi- objective optimization and some methods. Advantages of using interactive approaches in chemical process design are discussed in Section 6.4 and some applications related to sugar and papermaking industries are summarized in Section 6.5. Finally, concluding remarks are given in Section 6.6.

6.2. Concepts, Basic Methods and Some Shortcomings 6.2.1. Concepts

Let us consider multi-objective optimization problems of the form

minimize {f1(x), . . . , fk(x)}

subject to x∈ S, (6.1)

where we have k (≥ 2) conflicting objective functions fi : Rn→ R that we

156 K. Miettinen and J. Hakanen

or design vectors x = (x1, . . . , xn)T belonging to the nonempty feasible

region S ⊂ Rn _{defined by equality, inequality and/or box constraints. In}

multi-objective optimization, we typically are interested in objective vectors

consisting of objective (function) values f (x) = (f1(x), . . . , fk(x))T and the

image of the feasible region is called a feasible objective region Z = f (S).

(Note that if some function fi should be maximized, it is equivalent to

minimize −fi. Thus, without losing any generality, we consider problems

of the form (6.1).)

For multi-objective optimization, theoretical background has been laid, e.g., in Edgeworth (1881); Koopmans (1951); Kuhn and Tucker (1951); Pareto (1896, 1906). Typically, there is no unique optimal solution but a set of mathematically incomparable solutions can be identified. An ob- jective vector can be regarded as optimal if none of its components (i.e., objective values) can be improved without deterioration to at least one of

the other objectives. To be more specific, a decision vector x0∈ S and the

corresponding objective vector f (x0) are called Pareto optimal if there does

not exist another x∈ S such that fi(x) ≤ fi(x0) for all i = 1, . . . , k and

fj(x) < fj(x0) for at least one index j. In the MCDM literature, widely

used synonyms of Pareto optimal solutions are nondominated, efficient, noninferior or Edgeworth-Pareto optimal solutions.

As mentioned in the introduction, we here assume that a DM is able to participate in the solution process. (S)he is expected to know the problem domain and be able to specify preference information related to the objec- tives and/or different solutions. We assume that less is preferred to more in each objective for him/her. (In other words, all the objective functions are to be minimized.) If the problem is correctly formulated, the final solution of a rational DM is always Pareto optimal. Thus, we can restrict our con- sideration to Pareto optimal solutions. For this reason, it is important that the multi-objective optimization method used is able to find any Pareto op- timal solution and produce only Pareto optimal solutions. However, weakly Pareto optimal solutions are sometimes used because they may be easier

to generate than Pareto optimal ones. A decision vector x0 ∈ S (and the

corresponding objective vector) is weakly Pareto optimal if there does not

exist another x_{∈ S such that f}i(x) < fi(x0) for all i = 1, . . . , k. Note that

Pareto optimality implies weak Pareto optimality but not vice versa. The DM may find information about the ranges of feasible Pareto opti- mal objective vectors useful. Lower bounds form a so-called ideal objective

vector z? _{∈ R}k_{. Its components z}?

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Interactive Methods for Chemical Process Design 157

jective function individually subject to the feasible region. Sometimes, for computational reasons, we also need a strictly better utopian objective vec-

tor z?? _{defined as z}??

i = z?i − ε for i = 1, . . . , k, where ε is some small

positive scalar.

The upper bounds of the Pareto optimal set, that is, the components

of a nadir objective vector znad_{, are in practice difficult to obtain. It can}

be estimated using a payoff table but the estimate may be unreliable (see, e.g., Miettinen (1999) and references therein).

Finding a final solution to problem (6.1) is called a solution process. It usually involves the DM and an analyst. An analyst can be a human being or a computer program. The analyst’s role is to support the DM and generate information for the DM. Let us emphasize that the DM is not assumed to know multi-objective optimization theory or methods but (s)he is supposed to be an expert in the problem domain, that is, understand the application considered and have insight into the problem. Based on that, (s)he is supposed to be able to specify preference information related to the objectives considered and different solutions. The DM can be, e.g., a designer . The task of a multi-objective optimization method is to help the DM in finding the most preferred solution as the final one. The most preferred solution is a Pareto optimal solution which is satisfactory for the DM.

Multi-objective optimization problems can be solved by scalarizing the problem, in other words, by forming a problem (or several problems) involv- ing a single objective function (and possibly some additional constraints). Because the scalarized problem has a real-valued objective function (pos- sibly depending on some parameters originating, e.g., from preference in- formation), it can be solved using appropriate (local, global, mixed-integer etc.) single objective optimizers and, thus, we can utilize the theoretical background and large amount of methods developed for single objective optimization. The real-valued objective function can be called a scalariz- ing function. Such scalarizing approaches should be favored that generate Pareto optimal solutions and can find any Pareto optimal solution (as dis-

cussed earlier). Depending on whether a local or a global optimizer is

used we get either locally or globally Pareto optimal solutions (for noncon- vex problems). Because locally Pareto optimal solutions are irrational for DMs, it is important to use appropriate optimizers.

As said in the definition, to move from one Pareto optimal solution to another Pareto optimal solution means trading off. More formally, a trade-off is the ratio of change in objective function values involving the

158 K. Miettinen and J. Hakanen

increment of one objective function that occurs when the value of some other objective function decreases. For details, see, e.g., Chankong and Haimes (1983); Miettinen (1999).

The DM can specify preference information in many ways and the task

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