×
Benson, H. P.; Sayin, S.

Optimization over the efficient set: Four special cases. (English) Zbl 0797.90058

J. Optimization Theory Appl. 80, No. 1, 3-18 (1994).
Summary: Recently, researchers and practitioners have been increasingly interested in the problem (P) of maximizing a linear function over the efficient set of a multiple objective linear program. Problem (P) is generally a difficult global optimization problem which requires numerically intensive procedures for its solution. In this paper, simple linear programming procedures are described for detecting and solving four special cases of problem (P). When solving instances of problem (P), these procedures can be used as screening devices to detect and solve these four special cases.

Cited in 13 Documents

MSC:

90C05 Linear programming
90C29 Multi-objective and goal programming

Keywords:

efficient set; multiple objective linear program; global optimization
PDF BibTeX XML Cite
\textit{H. P. Benson} and \textit{S. Sayin}, J. Optim. Theory Appl. 80, No. 1, 3--18 (1994; Zbl 0797.90058)
Full Text: DOI

References:

[1] Evans, G. W.,An Overview of Techniques for Solving Multiobjective Mathematical Programs, Management Science, Vol. 30, pp. 1268–1282, 1984. · Zbl 0551.90090
[2] Goicoechea, A., Hansen, D. R., andDuckstein, L.,Multiobjective Decision Analysis with Engineering and Business Applications, John Wiley and Sons, New York, New York, 1982. · Zbl 0584.90045
[3] Rosenthal, R. E.,Principles of Multiobjective Optimization, Decision Sciences, Vol. 16, pp. 133–152, 1985.
[4] Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, Orlando, Florida, 1985. · Zbl 0566.90053
[5] Stadler, W.,A Survey of Multicriteria Optimization or the Vector Maximum Problem, Part I: 1776–1960, Journal of Optimization Theory and Applications, Vol. 29, pp. 1–52, 1979. · Zbl 0388.90001
[6] Steuer, R. E.,Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley and Sons, New York, New York, 1986. · Zbl 0663.90085
[7] Yu, P. L.,Multiple Criteria Decision Making, Plenum, New York, New York, 1985. · Zbl 0643.90045
[8] Yu, P. L.,Multiple Criteria Decision Making: Five Basic Concepts, Optimization, Edited by G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, North-Holland, Amsterdam, Holland, 1989.
[9] Zeleny, M.,Multiple Criteria Decision Making, McGraw-Hill, New York, New York, 1982. · Zbl 0588.90019
[10] Benson, H. P.,Optimization over the Efficient Set, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 562–580, 1984. · Zbl 0534.90077
[11] Benson, H. P.,An Algorithm for Optimizing over the Weakly-Efficient Set, European Journal of Operational Research, Vol. 25, pp. 192–199, 1986. · Zbl 0594.90082
[12] Philip, J.,Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207–229, 1972. · Zbl 0288.90052
[13] Dessouky, M. I., Ghiassi, M., andDavis, W. J.,Estimates of the Minimum Nondominated Criterion Values in Multiple-Criteria Decision Making, Engineering Costs and Production Economics, Vol. 10, pp. 95–104, 1986.
[14] Isermann, H., andSteuer, R. E.,Computational Experience Concerning Payoff Tables and Minimum Criteria Values over the Efficient Set, European Journal of Operational Research, Vol. 33, pp. 91–97, 1987. · Zbl 0632.90074
[15] Reeves, G. R., andReid, R. C.,Minimum Values over the Efficient Set in Multiple Objective Decision Making, European Journal of Operational Research, Vol. 36, pp. 334–338, 1988.
[16] Weistroffer, H. R.,Careful Use of Pessimistic Values Is Needed in Multiple Objectives Optimization, Operations Research Letters, Vol. 4, pp. 23–25, 1985. · Zbl 0569.90087
[17] Horst, R., andTuy, H.,Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, Germany, Second Edition, 1993. · Zbl 0704.90057
[18] Pardalos, P. M., andRosen, J. B.,Constrained Global Optimization: Algorithms and Applications, Springer-Verlag, Berlin, Germany, 1987. · Zbl 0638.90064
[19] Bolintineanu, S.,Minimization of a Quasi-Concave Function over an Efficient Set, Mathematics Research Paper No. 90-15, La Trobe University, 1990.
[20] Benson, H. P.,An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set, Journal of Global Optimization, Vol. 1, pp. 83–104, 1991. · Zbl 0739.90056
[21] Benson, H. P.,A Finite, Nonadjacent Extreme Point Search Algorithm for Optimization over the Efficient Set, Journal of Optimization Theory and Applications, Vol. 73, pp. 47–64, 1992. · Zbl 0794.90048
[22] Dauer, J. P.,Optimization over the Efficient Set Using an Active Constraint Approach, Zeitschrift für Operations Research, Vol. 35, pp. 185–195, 1991. · Zbl 0734.90081
[23] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401
[24] Benson, H. P.,Complete Efficiency and the Initialization of Algorithms for Multiple Objective Programming, Operations Research Letters, Vol. 10, pp. 481–487, 1991. · Zbl 0748.90060
[25] Geoffrion, A. M.,Solving Bicriterion Mathematical Programs, Operations Research, Vol. 15, pp. 39–54, 1967. · Zbl 0173.21602
[26] Benson, H. P.,Vector Maximization with Two Objective Functions, Journal of Optimization Theory and Applications, Vol. 28, pp. 253–257, 1979. · Zbl 0372.90126
[27] Wendell, R. E., andLee, D. N.,Efficiency in Multiple Objective Optimization Problems, Mathematical Programming, Vol. 12, pp. 406–414, 1977. · Zbl 0362.90092
[28] Aneja, Y. P., andNair, K. P. K.,Bicriteria Transportation Problem, Management Science, Vol. 25, pp. 73–78, 1979. · Zbl 0442.90056
[29] Ecker, J. G., andKouada, I. A.,Finding Efficient Points for Linear Multiple Objective Programs, Mathematical Programming, Vol. 18, pp. 375–377, 1975.
[30] Yu, P. L., andZeleny, M.,The Set of All Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method, Journal of Mathematical Analysis and Applications, Vol. 49, pp. 430–468, 1975. · Zbl 0313.65047
[31] Shachtman, R.,Generation of the Admissible Boundary of a Convex Polytope, Operations Research, Vol. 22, pp. 151–159, 1974. · Zbl 0277.52006
[32] Ecker, J. G., andKouada, I. A.,Generating Maximal Efficient Faces for Multiple Objective Linear Programs, CORE Discussion Paper No. 7617, Université Catholique de Louvain, 1976.
[33] Ecker, J. G., Hegner, N. S., andKouada, I. A.,Generating All Maximal Efficient Faces for Multiple Objective Linear Programs, Journal of Optimization Theory and Applications, Vol. 30, pp. 353–381, 1980. · Zbl 0393.90087
[34] Benveniste, M.,Testing for Complete Efficiency in a Vector Maximization Problem, Mathematical Programming, Vol. 12, pp. 285–288, 1977. · Zbl 0362.90057
[35] Murty, K. G.,Linear Programming, John Wiley and Sons, New York, New York, 1983.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.