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 journalArticlepeer-review

6 Scopus citations


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
Issue number2
StatePublished - Sep 1 2006


  • Combinatorial optimization
  • Metaheuristics
  • Multiple objective programming

ASJC Scopus subject areas

  • Computer Science(all)
  • Modeling and Simulation
  • Management Science and Operations Research
  • Information Systems and Management


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