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A method for constructing the polar cone of a polyhedral cone, with applications to linear multicriteria decision problems

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Abstract

The necessary and sufficient conditions for solution sets of linear multicriteria decision problems are given in the first part of this paper. In order to find the solution sets by applying the theorem describing the conditions, the constructions of the open polar cone and the semi-open polar cone of a given polyhedral cone are required.

A method of construction of the polar cone, open polar cone, and semi-open polar cone is presented. For this purpose, edge vectors of the polar cone are introduced and characterized in terms of the generating vectors of a given polyhedral cone. It is shown that these polar cones are represented by the edge vectors.

Numerical examples of linear multicriteria decision problems are solved to illustrate the construction of the polar cones and to explain the application of the theorem to obtain the solution sets.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sophia University, Tokyo, Japan

    K. Tamura (Associate Professor)

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  1. K. Tamura
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Communicated by G. Leitmann

The author is grateful to Professor P. L. Yu for helpful comments concerning the development of Theorem 2.1.

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Tamura, K. A method for constructing the polar cone of a polyhedral cone, with applications to linear multicriteria decision problems. J Optim Theory Appl 19, 547–564 (1976). https://doi.org/10.1007/BF00934654

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