Abstract
The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients.
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References
Bellman, R. E. and L. A. Zadeh. (1970). “Decision making in a fuzzy environment,” Management Science 17, 141–164.
Delgado, M., J. Kacprzyk, J.-L. Verdegay, and M. A. Vila (eds.) (1994). Fuzzy Optimization: Recent Advances. New York: Physica-Verlag.
Jahn, J. (1986). Mathematical Vector Optimization in Partially Ordered Linear Spaces. New York: Verlag Peter Lang GmbH, Frankfurt am Main.
Kall, P. (1976). Stochastic Linear Programming. New York: Springer-Verlag.
Lai, Y.-J. and C.-L. Hwang. (1992). Fuzzy Mathematical Programming: Methods and Applications, Lecture Notes in Economics and Mathematical Systems 394. New York: Springer-Verlag.
Lai, Y.-J. and C.-L. Hwang. (1994). Fuzzy Multiple Objective Decision Making: Methods and Applications, Lecture Notes in Economics and Mathematical Systems 404. New York: Springer-Verlag.
Prékopa, A. (1995). Stochastic Programming. Boston: Kluwer Academic Publishers.
Slowiński, R. (ed) (1998). Fuzzy Sets in Decision Analysis, Operations Research and Statistics. Boston: Kluwer Academic Publishers.
Stancu-Minasian, I. M. (1984). Stochastic Programming with Multiple Objective Functions. Bucharest, Romania: D. Reidel Publishing Company.
Vajda, S. (1972). Probabilistic Programming. New York: Academic Press.
Zadeh, L. A. (1975). “The concept of liguistic variable and its application to approximate reasoning I, II and III,” Information Sciences 8, 199–249; 8, 301–357; 9, 43–80.
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Wu, HC. Fuzzy Optimization Problems Based on Ordering Cones. Fuzzy Optimization and Decision Making 2, 13–29 (2003). https://doi.org/10.1023/A:1022896029935
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DOI: https://doi.org/10.1023/A:1022896029935