Discrete Optimization
Evaluation of nondominated solution sets for k-objective optimization problems: An exact method and approximations

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Abstract

Integrated Preference Functional (IPF) is a set functional that, given a discrete set of points for a multiple objective optimization problem, assigns a numerical value to that point set. This value provides a quantitative measure for comparing different sets of points generated by solution procedures for difficult multiple objective optimization problems. We introduced the IPF for bi-criteria optimization problems in [Carlyle, W.M., Fowler, J.W., Gel, E., Kim, B., 2003. Quantitative comparison of approximate solution sets for bi-criteria optimization problems. Decision Sciences 34 (1), 63–82]. As indicated in that paper, the computational effort to obtain IPF is negligible for bi-criteria problems. For three or more objective function cases, however, the exact calculation of IPF is computationally demanding, since this requires k (⩾3) dimensional integration.

In this paper, we suggest a theoretical framework for obtaining IPF for k (⩾3) objectives. The exact method includes solving two main sub-problems: (1) finding the optimality region of weights for all potentially optimal points, and (2) computing volumes of k dimensional convex polytopes. Several different algorithms for both sub-problems can be found in the literature. We use existing methods from computational geometry (i.e., triangulation and convex hull algorithms) to develop a reasonable exact method for obtaining IPF. We have also experimented with a Monte Carlo approximation method and compared the results to those with the exact IPF method.

Introduction

Many multiple objective a posteriori heuristics have been developed and successfully applied for solving real world multiple objective optimization problems (Aksoy et al., 1996, Coello, 1999, Czyzak and Jaszkiewicz, 1998, Viana and De Sousa, 2000, Zitzler, 1999). These a posteriori heuristics provide a set of nondominated solutions for the decision maker’s evaluation rather than a single final solution. One of the key issues is how to evaluate the quality of approximate solution sets generated by different heuristics or different parameter settings for a heuristic (Coello, 1999). For this purpose, various measures have been suggested in the literature (a detailed review can be found in Carlyle et al., 2003). A measure called Integrated Preference Functional (IPF) for two objective function problems was recently developed in Carlyle et al. (2003) and applied in the comparison of two competing bi-criteria genetic algorithms for a parallel machine scheduling problem (Fowler et al., 2005). As shown in these two papers and Kim et al. (2001), the IPF possesses many good qualities and provides a robust, quantitative measure for comparing different solution sets for difficult bi-criteria optimization problems.

In this paper, we extend the exact IPF method presented in Carlyle et al. (2003) to three or more objectives, when the implicit value function is a convex combination of objectives and the weight density function has a uniform distribution. This extension is important because many difficult multiple objective optimization problems have more than two objectives. We have also conducted a computational experiment to test the performance of the exact IPF method for three, four, and five objective functions and compared it against approximation methods. Furthermore, we discuss two research topics in numerically approximating IPF.

The organization of the rest of the paper is as follows. In Section 2, we review the IPF measure and illustrate it for the case of two objectives. In Section 3, an exact IPF method is described for the case of three or more objectives. In Section 4, a numerical example to illustrate the steps of the exact method is provided. Experimental results on both exact and approximate methods are discussed in Section 5. Section 6 provides concluding remarks and suggestions for future research. In Appendix A we present the basic terminology, and state (and prove where appropriate) several theorems related to the IPF measure. We note that the computer code used for the experimentation in this paper is available upon request from the authors.

Section snippets

Review of IPF with an illustration to the bi-criteria case

A review of the IPF measure is provided for the reader’s convenience. Further details can be found in Carlyle et al. (2003).

Consider a set of finite nondominated solution vectors (x  X) in Rn, the corresponding objective function vector z = f(x) in Rk, and a parameterized family of implicit value functions g(z; α), where a given value of the parameter vector α in its domain A produces a specific scalar-valued value function to be optimized. Note that throughout the paper, we consider minimization

Exact IPF method for k  3 objectives

Consider n nondominated points in a set Z in Rk. Then

  • If the value function g(z; α) is a convex combination of objective functions and h(α) follows a uniform distribution, then the optimality region of weights for supported point z is a (k  1)-convex polytope in Rk−1 as can be seen from Theorem 2 in Appendix A.

  • The value surface g(z; α), over the optimality region of weights, is clearly a linear function of α by the assumptions we have made. Hence, for a supported point z, integration of the value

A numerical example

A numerical example of calculating IPF(Z) for a set of nondominated points is provided to illustrate the calculation procedures and to show the input and output data structures throughout the steps of the method.

Consider a three objective function case and assume that there are six points in set Z: z1(1, 2, 3), z2(2, 5, 1), z3(3, 2, 2), z4(4, 1, 5), z5(5, 3, 1), z6(2, 4, 2) in the objective space. Notice that these six points are all nondominated.
[Step-1] Enumerating all supported points and their adjacent

Computational tests

In this section, the computational efficiency of the exact IPF method is analyzed and compared to that of the approximation methods. To generate a number of appropriate solution sets with a desired number of extreme points quickly, we adopted the following simplistic approach. Throughout this experimentation, the points in the solution sets were randomly generated from the uniform distribution U(0, 1) for each objective, and only the nondominated points are included in the set. We continue

Conclusions and future research

Many multiple criteria a posteriori heuristics have appeared in the literature to solve difficult multiple criteria combinatorial optimization problems. Accordingly, a robust and efficient measure is needed to compare alternative heuristics and to tune parameters employed in them. For this purpose, the Integrated Preference Functional (IPF) for bi-criteria optimization problems was suggested in Carlyle et al. (2003). The computational effort of obtaining IPF for the bi-criteria case is trivial.

Acknowledgment

The authors would like to acknowledge the support of the National Science Foundation under contract DMII-0121815.

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