Abstract
In this paper we consider efficient sets of multiple objective problems, in which the feasible action set is the intersection of two other sets, and where one of these sets has a special structure, such as an assignment or transportation structure. The objective is to find the efficient set of the special structure set, and its intersection with the other set, and to examine how good an approximation this set is to the desired efficient set. The approximation set is called an ε-efficient solution set. Some theoretical partition results are given for a special constraint structure with upper bounds on the objective function levels. For the case of 0-efficient solution sets, and finite explicit sets, a computational cost analysis of two computational sequences is given. We also consider two other 0-efficient solution set cases. Then ε-efficiency is considered for linear problems. Finally, the approach is illustrated by a special multiple objective transportation problem.
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References
Hwang, C. L., Masud, A. S. M., Paidy, S. R. and Yoon, K. (1979), Multiple Objective Decision Making-Methods and Applications. Springer Verlag, Berlin.
Zeleny, M. (1982), Multiple Criteria Decision Making. McGraw-Hill, New York.
Yu, P. L. (1985), Multiple Criteria Decision Making. Plenum Press, New York.
White, D. J. (1982), Optimality and Efficiency. John Wiley & Sons, Chichester.
White, D. J. (1990), A Bibliography on the Applications of Mathematical Programming Multiple-Objective Methods, Journal of the Operational Research Society 41, 669–691.
Benson, H. P. (1995), A Geometrical Analysis of the Efficient Outcome Set in Multiple Objective Convex Programs with Linear Criterion Functions, Journal of Global Optimization 6, 231–251.
White, D. J. (1991), A Characterisation of the Feasible Set of Objective Function Vectors in Linear Multiple Objective Problems, European Journal of Operational Research 52, 261–266.
Dauer, J. P. and Fosnaugh, T. A. (1995), Optimization over the Efficient Set, Journal of Global Optimization 7, 261–277.
Benson, H. P. (1994), Optimizing over the Efficient Set: Four Special Cases, Journal of Optimization Theory and Applications 80, 3–18.
Benson, H. P. (1991), An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set, Journal of Global Optimization 1, 83–104.
Philip, J. (1972), Algorithms for the Vector Maximization Problem, Mathematical Programming 2, 207–229.
White, D. J. (1996), The Maximization of a Function over the Efficient Set via a Penalty Function Approach, European Journal of Operational Research 94, 143–153.
Warburton, A. (1987), Approximation of Pareto Optima in Multiple-Objective, Shortest Path Problems, Operations Research 35, 70–79.
Kung, H. T., Lucio, f. and Preparata, F. P. (1975), On finding the Maxima of a Set of Vectors, Journal of the Association for Computing Machinery 22, 469–476.
White, D. J. (1984), A Special Multi-Objective Assignment Problem, Journal of the Operational Research Society 35, 759–767.
White, D. J. Generalised Efficient Solutions for Sums of Sets, Operations Research 28, 844–846, 1980.
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White, D.J. Epsilon Dominance and Constraint Partitioning in Multiple Objective Problems. Journal of Global Optimization 12, 435–448 (1998). https://doi.org/10.1023/A:1008228830480
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DOI: https://doi.org/10.1023/A:1008228830480