Multicriteria Optimization in Engineering and in the Sciences

In any decision or design process, one attempts to make the best decision within a specified set of possible ones. The notion of “best” is in the eye of the beholder.

The topic of numerical methods in multicriteria optimization lends itself to many interpretations, and, of course, much has been written on the subject. This chapter will not be a survey of the numerical techniques of multicriteria optimization (MCO). Those interested in such a work should see, for example, the book of Hwang and Masud (Ref. 1). Nor will this chapter contain comparisons of the numerical efficiency of a variety of MCO algorithms. Instead, this work will be on my views and experience in numerically analyzing “real” linear MCO problems and the mathematics necessary for such an analysis. Of course, “real” means MCO problems as I have encountered them in applications. Imaginary (or unreal) must therefore refer to all the rest.

In this chapter we investigate certain vector approximation problems, which are approximation problems where a vectorial norm is used instead of a usual (real-valued) norm. About 50 years ago vectorial norms were first introduced by Kantorovitch (Ref. 1), who developed a mathematical theory of linear spaces equipped with a vectorial norm. Many important results known from approximation theory (e.g., see Refs. 2, 3) can be extended to this vector-valued case (compare Ref. 4). In this chapter we present an application-oriented approach to vector approximation and we do not intend to formulate the results in the most general way. Therefore, we develop the proofs also in this special setting, although several results could be deduced from a general theory of vector approximation.

It is probably well known to everyone working in the field of multi-criteria optimization that the roots of this field can easily be traced back to welfare economics, more precisely to the contributions of Vilfredo Pareto (Ref. 1). As a matter of fact, Stadler’s survey on multicriteria optimization (Ref. 2) gives a fairly detailed review of the historical development of multicriteria optimization in the context of welfare theory. There is obviously no point in duplicating his effort. There are also many standard textbooks on welfare economics (e.g., Refs. 3–6). These texts are written for economists, but just translating and summarizing them for noneconomists could certainly not be adequate in this volume.

Resource planning problems present many excellent examples of why multicriterion optimization (MCO) can be so useful in practice. These problems virtually always involve a public decision-making process, and they virtually never can be characterized as having a single criterion. The protection of the environment—a particular kind of resource planning problem—is by its very nature a multicriterion problem: the environment is being “protected from” economic activities. Thus, problems in environmental control are born out of conflict between criteria: economic development and environmental preservation. Resource problems, in general, exhibit these criteria and others, such as equity in the distribution of benefits and costs—the classic upstream-downstream conflict in water resource problems—and risk to human health.

Consider a multispecies ecosystem (e.g., predator-prey) that is exploited by different groups of harvesters. Each group will concentrate on a single species. The operation is to be directed by a manager who must set rules for the maximum level of harvesting by each group of harvesters. The manager must set these limits without knowing the specific details of how the harvesters may actually operate under these rules, except that it is assumed that the harvesters will not violate the maximum limits. The manager’s objective is to “maximize” the harvested yield for each species without having any of them become endangered by being driven to unacceptably low population levels.

In his investigations on animal fighting behavior, John Maynard Smith (Ref. 1) coined the term “evolutionarily stable strategy” (ESS) to denote a behavioral strategy that is stable against invasion by a small number of individuals who employ a “mutant,” or deviant strategy. The notion of an ESS is quite similar—but not identical to—the concept of a Nash equilibrium in game theory. Several authors had previously attempted to apply game theoretic formalisms to evolutionary problems (e.g., Lewontin, Ref. 2; Slobodkin and Rappoport, Ref. 3; Rocklin and Oster, Ref. 4). With the exception of Maynard Smith’s analyses, however, few empirically verifiable predictions were generated. Moreover, with the exception of Stewart (Ref. 5), the models were mostly restricted to static games. In this study we shall present a number of models that treat ESSs from a dynamic viewpoint. In particular, we shall attempt to generalize conventional competition theory by permitting the competing parties to adjust their strategies. Rather than seeking dynamically stable equilibria, as in Volterra-Lotka theory, we shall look for strategically stable solutions, or ESSs (cf. Maynard Smith, Ref. 6). Ultimately, one must extend competitive models to the level of the genetic loci influencing the strategies. Unfortunately, the efforts in this direction usually lead to models that are mathematically intractible (cf. Rocklin and Oster, Ref. 4); therefore, there is some justification for taking a phenomeno-logical approach such as game theory.

In the design of airplane control systems, many disparate objectives must be considered. The pilot desires rapid, precise, and decoupled response to his control inputs, so that natural objective functions for computer-aided design (CAD) are computable functions that are useful measures of the speed, stability, and coupling of the responses. These response properties are often referred to as the handling qualities or flying qualities of the airplane. The military has developed a set of specifications for a number of handling quality functions, and the CAD research described in this paper uses objective functions based on these military handling qualities criteria. Additional design objective functions have been developed to avoid control limiting, since there are always limits on available control in any real system, and limiting can be destabilizing in an automatic control system. Another important property of a good design is that it be “robust”; that is, the design objectives should be insensitive to significant uncertainties in system parameters. In fact, such insensitivity is an essential property of any well-designed feedback system. Therefore, a vector of “stochastic sensitivity” functions is defined as the vector of probabilities that each “deterministic” objective violate specified requirement limits, and decreasing sensitivity is considered a design objective. If both the deterministic objectives (the nominal or expected values) and their sensitivities are considered in the design process, the number of objective functions is doubled. Moreover, modern airplanes operate over a wide range of speed and altitude, and the linearized differential equations that are used to describe the response to controls (the plant dynamic models) are different at each flight condition.

The origin of structural optimization can be traced back several centuries (Ref. 1), but it is only during the last two decades or so, with the advent of modern computers, that it has evolved into a mature discipline in engineering. The literature published in this field is extensive and it can be reasonably discussed here only by referring to some recently written articles and textbooks found in Refs. 2–4. The major part of the articles deal with such numerical optimization techniques in finite-dimensional problems as optimality criteria or mathematical programming methods, but considerable efforts have also been made in applying the control theory approach to distributed parameter structural systems. The finite element method is commonly used in analyzing load supporting structures and there is usually a finite-dimensional optimization problem associated with it. In this chapter truss design problems, which by nature belong to this class, are considered. Various mainly nonlinear programming approaches have been developed to numerically solve scalar problems where the number of design variables and constraints is constantly increasing.

The following considerations show the necessity of introducing optimization procedures into the practical construction phase:

Good design is based on a thorough understanding of the limitations imposed by natural law as well as the existent technology. In 1775 the Parisian Academy of Sciences ceased to accept papers concerning perpeda mobilae based on the universal observation that all motion within our experience eventually attenuates unless some sort of driving force sustains it. Such machines were later recognized to be in conflict with the second law of thermodynamics in that they implied entropy generation. The design of substances and materials is limited by the fact that there are numerous chemical reactions that cannot take place and chemical bonds that cannot be sustained. In mechanical behavior, the amount of force available implies clear limitations on the speed that a particle can achieve in a given amount of time. On a more subtle level, there are motions in particle dynamics that cannot be sustained by noncentral forces, and so on. What is clear is that all design is subject to the limitations of natural law or, more precisely, natural law as now understood. A clear understanding of natural phenomena can overcome perceived limitations of false theories. Therefore, in order to free ourselves from the shackles of such false limitations, our primary efforts must be directed toward an understanding of natural law. Our designs then will reflect this understanding.