Comparison of Two Multiobjective Optimization Techniques with and within Genetic Algorithms

Engineering decision making involving multiple competing objectives relies on choosing a design solution from an optimal set of solutions. This optimal set of solutions, referred to as the Pareto set, represents the tradeoffs that exist between the competing objectives for different design solutions. Generation of this Pareto set is the main focus of multiple objective optimization. There are many methods to solve this type of problem. Some of these methods generate solutions that cannot be applied to problems with a combination of discrete and continuous variables. Often such solutions are obtained by an optimization technique that can only guarantee local Pareto solutions or is applied to convex problems. The main focus of this paper is to demonstrate two methods of using genetic algorithms to overcome these problems. The first method uses a genetic algorithm with some external modifications to handle multiple objective optimization, while the second method operates within the genetic algorithm with some significant internal modifications. The fact that the first method operates with the genetic algorithm and the second method within the genetic algorithm is the main difference between these two techniques. Each method has its strengths and weaknesses, and it is the objective of this paper to compare and contrast the two methods quantitatively as well as qualitatively. Two multiobjective design optimization examples are used for the purpose of this comparison. INTRODUCTION It is common in engineering decision making problems to have multiple design objectives (see, for example, Eschenauer et al., 1990). There are many methods for solving such problems, each of which has some advantages and disadvantages, and most have evolved or been refined from some earlier version, as discussed in the survey b Stadler (1979) and Hwang and Masud (1979), Palli et al.(1998), and more recently b Miettinen (1999). Most of these methods generate solutions that cannot be applied to problems with a combination of discrete and continuous variables. Often, such solutions can only be guaranteed to be locall optimum. However, optimization with Genetic Algorithms (hereafter referred to as GAs) can obtain discrete, global, and non-convex solutions. In this paper, we will look at two different ways of utilizing GAs to solve multiobjective design optimization problems. The first method uses an existing GA with some modifications external to the GA, and the second method operates within the GA with more significant modifications to it. This brings up the concept of multiobjective design optimization with and within GAs. The multiobjective optimization with genetic algorithms just uses an existing GA (Goldberg, 1989) with some modifications external to the GA. The GA basically operates independently of the optimization problem formulation with only some minimal data passed to and from it. However, the multiobjective optimization within the GA works on the inside of the GA making some significant modifications to the algorithm. Each method has its advantages and disadvantages with respect to one another. In general, as it will be shown in the paper, the multiobjective optimization with GA requires more computational time but is easier to 1 Copyright © 1999 by ASME apply, and within GA requires less computational time but can encounter some difficulties in generating the Pareto set. So, the overall objective of this paper is to compare and contrast quantitatively as well as qualitatively two different methods of multiobective optimization that utilize GAs, highlighting the strengths and weaknesses of each. The remainder of the paper is organized as follows. The formulation of a general multiobjective optimization problem and some terminology are given in the next section. Next, an overview of GAs is. The second and third sections explain the two multiobjective optimization techniques that are to be compared. The fourth section includes the application of these methods to two different engineering design problems for the purpose of the comparison. The paper is concluded with the remarks in the last section. OVERVIEW OF MULTIOBJECTIVE OPTIMIZATION PROBLEMS The formulation of a typical multiobjective optimization problem with m objective functions is shown below in Eq.(1). } ,... 1 , 0 ) ( ; ,..., 1 , 0 ) ( : { : )} ( ),..., ( ),..., ( { ) ( : 1 K k h J j g D D to subject f f f Minimize