TY - JOUR

T1 - U-Measure

T2 - A Quality Measure for Multiobjective Programming

AU - LEUNG, Yiu Wing

AU - Wang, Yuping

N1 - Funding Information:
Manuscript received March 12, 2001; revised April 8, 2003. This work was supported by the Hong Kong Baptist University under Research Grant FRG II-62. This paper was recommended by Associate Editor J. Lambert. Y.-W. Leung is with the Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong (e-mail: ywleung@comp.hkbu.edu.hk). Y. Wang is with the Department of Mathematics Science, Faculty of Science, Xidian University, Xian, China. Digital Object Identifier 10.1109/TSMCA.2003.817059

PY - 2003/5

Y1 - 2003/5

N2 - 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.

AB - 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.

KW - Multiobjective programming

KW - Nondominated solutions

KW - Pareto-optimality

KW - Quality measures

UR - http://www.scopus.com/inward/record.url?scp=0141638719&partnerID=8YFLogxK

U2 - 10.1109/TSMCA.2003.817059

DO - 10.1109/TSMCA.2003.817059

M3 - Article

AN - SCOPUS:0141638719

VL - 33

SP - 337

EP - 343

JO - IEEE Transactions on Systems, Man, and Cybernetics Part A:Systems and Humans

JF - IEEE Transactions on Systems, Man, and Cybernetics Part A:Systems and Humans

SN - 1083-4427

IS - 3

ER -