U-Measure: A Quality Measure for Multiobjective Programming

Yiu Wing LEUNG*, Yuping Wang

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

55 Citations (Scopus)


A multiobjective programming algorithm may find multiple nondominated solutions. If these solutions are scattered more uniformly over the Pareto frontier in the objective space, they are more different choices and so their quality is better. In this paper, we propose a quality measure called U-measure to measure the uniformity of a given set of nondominated solutions over the Pareto frontier. This frontier is a nonlinear hyper-surface. We measure the uniformity over this hyper-surface in three main steps: 1) determine the domains of the Pareto frontier over which uniformity is measured, 2) determine the nearest neighbors of each solution in the objective space, and 3) compute the discrepancy among the distances between nearest neighbors. The U-measure is equal to this discrepancy where a smaller discrepancy indicates a better uniformity. We can apply the U-measure to complement the other quality measures so that we can evaluate and compare multiobjective programming algorithms from different perspectives.

Original languageEnglish
Pages (from-to)337-343
Number of pages7
JournalIEEE Transactions on Systems, Man, and Cybernetics Part A:Systems and Humans
Issue number3
Publication statusPublished - May 2003

Scopus Subject Areas

  • Software
  • Control and Systems Engineering
  • Human-Computer Interaction
  • Computer Science Applications
  • Electrical and Electronic Engineering

User-Defined Keywords

  • Multiobjective programming
  • Nondominated solutions
  • Pareto-optimality
  • Quality measures

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