A new achievement scalarizing function based on parameterization in multiobjective optimization |
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Authors: | Yury Nikulin Kaisa Miettinen Marko M M?kel? |
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Affiliation: | (1) Department of Mathematical Information Technology, P.O. Box 35 (Agora), 40014 University of Jyv?skyl?, Finland;(2) Aalto University School of Economics, Department of Business Technology, P.O. Box 21220, 00076 Aalto, Finland;(3) Present address: Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India |
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Abstract: | This paper addresses a general multiobjective optimization problem. One of the most widely used methods of dealing with multiple
conflicting objectives consists of constructing and optimizing a so-called achievement scalarizing function (ASF) which has
an ability to produce any Pareto optimal or weakly/properly Pareto optimal solution. The ASF minimizes the distance from the
reference point to the feasible region, if the reference point is unattainable, or maximizes the distance otherwise. The distance
is defined by means of some specific kind of a metric introduced in the objective space. The reference point is usually specified
by a decision maker and contains her/his aspirations about desirable objective values. The classical approach to constructing
an ASF is based on using the Chebyshev metric L
∞. Another possibility is to use an additive ASF based on a modified linear metric L
1. In this paper, we propose a parameterized version of an ASF. We introduce an integer parameter in order to control the degree
of metric flexibility varying from L
1 to L
∞. We prove that the parameterized ASF supports all the Pareto optimal solutions. Moreover, we specify conditions under which
the Pareto optimality of each solution is guaranteed. An illustrative example for the case of three objectives and comparative
analysis of parameterized ASFs with different values of the parameter are given. We show that the parameterized ASF provides
the decision maker with flexible and advanced tools to detect Pareto optimal points, especially those whose detection with
other ASFs is not straightforward since it may require changing essentially the reference point or weighting coefficients
as well as some other extra computational efforts. |
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