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

B. Kim, Esma Gel, John Fowler, W. M. Carlyle, J. Wallenius

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 ({greater than or slanted equal to}3) dimensional integration. In this paper, we suggest a theoretical framework for obtaining IPF for k ({greater than or slanted equal to}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.

Original languageEnglish (US)
Pages (from-to)565-582
Number of pages18
JournalEuropean Journal of Operational Research
Volume173
Issue number2
DOIs
StatePublished - Sep 1 2006

Fingerprint

Nondominated Solutions
Exact Method
Solution Set
Bicriteria Optimization
Multiple Objective Optimization
Optimization Problem
Evaluation
Approximation
evaluation
Set of points
Computational geometry
Convex Polytopes
Bicriteria
Computational Geometry
Triangulation
Approximation Methods
Convex Hull
Point Sets
Monte Carlo method
Assign

Keywords

  • Combinatorial optimization
  • Metaheuristics
  • Multiple objective programming

ASJC Scopus subject areas

  • Information Systems and Management
  • Management Science and Operations Research
  • Statistics, Probability and Uncertainty
  • Applied Mathematics
  • Modeling and Simulation
  • Transportation

Cite this

Evaluation of nondominated solution sets for k-objective optimization problems : An exact method and approximations. / Kim, B.; Gel, Esma; Fowler, John; Carlyle, W. M.; Wallenius, J.

In: European Journal of Operational Research, Vol. 173, No. 2, 01.09.2006, p. 565-582.

Research output: Contribution to journalArticle

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