×

Interactive fuzzy random two-level linear programming through fractile criterion optimization. (English) Zbl 1235.90197

Summary: Assuming cooperative behavior of the decision makers, solution methods for decision making problems in hierarchical organizations under fuzzy random environments are considered. To deal with the formulated two-level linear programming problems involving fuzzy random variables, \(\alpha \)-level sets of fuzzy random variables are introduced and an \(\alpha \)-stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account the vagueness of judgments of decision makers, fuzzy goals are introduced and the \(\alpha \)-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through the use of the fractile criterion optimization model, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. It is shown that all of the problems to be solved in the proposed interactive fuzzy programming can be easily solved by the simplex method, the sequential quadratic programming or the combined use of the bisection method and the sequential quadratic programming. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

MSC:

90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
91A12 Cooperative games
90C05 Linear programming
91A65 Hierarchical games (including Stackelberg games)

References:

[1] Kwakernaak, H., Fuzzy random variables-I. definitions and theorems, Information Sciences, 15, 1-29 (1978) · Zbl 0438.60004
[2] Kruse, R.; Meyer, K. D., Statistics with Vague Data (1987), D. Riedel Publishing Company · Zbl 0663.62010
[3] Puri, M. L.; Ralescu, D. A., Fuzzy random variables, Journal of Mathematical Analysis and Applications, 114, 409-422 (1986) · Zbl 0592.60004
[4] Liu, Y.-K.; Liu, B., Fuzzy random variables: a scalar expected value operator, Fuzzy Optimization and Decision Making, 2, 143-160 (2003) · Zbl 1436.60009
[5] Gil, M. A.; Lopez-Diaz, M.; Ralescu, D. A., Overview on the development of fuzzy random variables, Fuzzy Sets and Systems, 157, 2546-2557 (2006) · Zbl 1108.60006
[6] Wang, G.-Y.; Qiao, Z., Linear programming with fuzzy random variable coefficients, Fuzzy Sets and Systems, 57, 295-311 (1993) · Zbl 0791.90072
[7] Qaio, Z.; Zhang, Y.; Wang, G.-Y., On fuzzy random linear programming, Fuzzy Sets and Systems, 65, 31-49 (1994) · Zbl 0844.90112
[8] Luhandjula, M. K., Fuzziness and randomness in an optimization framework, Fuzzy Sets and Systems, 77, 291-297 (1996) · Zbl 0869.90081
[9] Luhandjula, M. K.; Gupta, M. M., On fuzzy stochastic optimization, Fuzzy Sets and Systems, 81, 47-55 (1996) · Zbl 0879.90187
[10] Liu, B., Fuzzy random chance-constrained programming, IEEE Transaction on Fuzzy Systems, 9, 713-720 (2001)
[11] Liu, B., Fuzzy random dependent-chance programming, IEEE Transaction on Fuzzy Systems, 9, 721-726 (2001)
[12] Rommelfanger, H., A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values, Fuzzy Sets and Systems, 156, 1892-1904 (2007) · Zbl 1137.90766
[13] Luhandjula, M. K., Fuzzy stochastic linear programming: survey and future research directions, European Journal of Operational Research, 174, 1353-1367 (2006) · Zbl 1102.90079
[14] Katagiri, H.; Ishii, H.; Sakawa, M., On fuzzy random linear knapsack problems, Central European Journal of Operations Research, 12, 59-70 (2004) · Zbl 1069.90087
[15] Katagiri, H.; Sakawa, M.; Ishii, H., Fuzzy random bottleneck spanning tree problems, European Journal of Operational Research, 152, 88-95 (2004) · Zbl 1030.90145
[16] Katagiri, H.; Sakawa, M.; Kato, K.; Nishizaki, I., A fuzzy random multiobjective 0-1 programming based on the expectation optimization model using possibility and necessity measures, Mathematical and Computer Modelling, 40, 411-421 (2004) · Zbl 1112.90107
[17] Katagiri, H.; Mermri, E. B.; Sakawa, M.; Kato, K.; Nishizaki, I., A possibilistic and stochastic programming approach to fuzzy random MST problems, IEICE Transaction on Information and Systems, E88-D, 1912-1919 (2005)
[18] Katagiri, H.; Sakawa, M.; Ishii, H., A study on fuzzy random portfolio selection problems using possibility and necessity measures, Scientiae Mathematicae Japonicae, 61, 361-369 (2005) · Zbl 1095.91016
[19] Katagiri, H.; Sakawa, M.; Nishizaki, I., Interactive decision making using possibility and necessity measures for a fuzzy random multiobjective 0-1 programming problem, Cybernetics and Systems, 37, 59-74 (2006) · Zbl 1284.93138
[20] Katagiri, H.; Sakawa, M.; Kato, K.; Nishizaki, I., Interactive multiobjective fuzzy random linear programming: maximization of possibility and probability, European Journal of Operational Research, 188, 530-539 (2008) · Zbl 1149.90400
[21] Ammar, E. E., On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem, Information Sciences, 178, 468-484 (2008) · Zbl 1149.90188
[22] Xu, J.; Liu, Y., Multi-objective decision making model under fuzzy random environment and its application to inventory problems, Information Sciences, 178, 2899-2914 (2008) · Zbl 1146.90067
[23] Sakawa, M.; Nishizaki, I., Cooperative and Noncooperative Multi-Level Programming (2009), Springer: Springer New York
[24] Simaan, M.; Cruz, J. B., On the Stackelberg strategy in nonzero-sum games, Journal of Optimization Theory and Applications, 11, 533-555 (1973) · Zbl 0243.90056
[25] Bialas, W. F.; Karwan, M. H., Two-level linear programming, Management Science, 30, 1004-1020 (1984) · Zbl 0559.90053
[26] Bard, J. F.; Moore, J. T., A branch and bound algorithm for the bi-level programming problem, SIAM Journal on Scientific and Statistical Computing, 11, 281-292 (1990) · Zbl 0702.65060
[27] Hansen, P.; Jaumard, B.; Savard, G., New branch-and-bound rules for liner bi-level programming, SIAM Journal of Scientific and Statistical Computing, 13, 1194-1217 (1992) · Zbl 0760.65063
[28] Júdice, J. J.; Faustino, A. M., A sequential LCP method for bi-level linear programming, Annals of Operations Research, 34, 89-106 (1992) · Zbl 0749.90049
[29] White, D. J.; Anandalingam, G., A penalty function approach for solving bi-level linear programs, Journal of Global Optimization, 3, 397-419 (1993) · Zbl 0791.90047
[30] Shimizu, K.; Ishizuka, Y.; Bard, J. F., Nondifferentiable and Two-Level Mathematical Programming (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0878.90088
[31] Nishizaki, I.; Sakawa, M., Computational methods through genetic algorithms for obtaining Stackelberg solutions to two-level mixed zero-one programming problems, Cybernetics and Systems: An International Journal, 31, 203-221 (2000) · Zbl 1030.90061
[32] Lai, Y. J., Hierarchical optimization: a satisfactory solution, Fuzzy Sets and Systems, 77, 321-325 (1996) · Zbl 0869.90042
[33] Shih, H. S.; Lai, Y. J.; Lee, E. S., Fuzzy approach for multi-level programming problems, Computers & Operations Research, 23, 73-91 (1996) · Zbl 0838.90140
[34] Sakawa, M.; Nishizaki, I.; Uemura, Y., Interactive fuzzy programming for multi-level linear programming problems, Computers & Mathematics with Applications, 36, 71-86 (1998) · Zbl 0937.90123
[35] Sakawa, M.; Nishizaki, I.; Uemura, Y., Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters, Fuzzy Sets and Systems, 115, 93-103 (2000) · Zbl 0978.90112
[36] Lee, E. S., Fuzzy multiple level programming, Applied Mathematics and Computation, 120, 79-90 (2001) · Zbl 1032.90078
[37] Sakawa, M.; Nishizaki, I., Interactive fuzzy programming for decentralized two-level linear programming problems, Fuzzy Sets and Systems, 125, 301-315 (2002) · Zbl 1014.90119
[38] Sakawa, M.; Nishizaki, I., Interactive fuzzy programming for two-level nonconvex programming problems with fuzzy parameters through genetic algorithms, Fuzzy Sets and Systems, 127, 185-197 (2002) · Zbl 0994.90146
[39] Sakawa, M.; Nishizaki, I.; Uemura, Y., Interactive fuzzy programming for two-level linear and linear fractional production and assignment problems: a case study, European Journal of Operational Research, 135, 142-157 (2001) · Zbl 1077.90564
[40] Sakawa, M.; Nishizaki, I.; Uemura, Y., A decentralized two-level transportation problem in a housing material manufacturer—Interactive fuzzy programming approach, European Journal of Operational Research, 141, 167-185 (2002) · Zbl 0998.90091
[41] Sinha, S., Fuzzy programming approach to multi-level programming problems, Fuzzy Sets and Systems, 136, 189-202 (2003) · Zbl 1013.90143
[42] Pramanik, S.; Roy, T. K., Fuzzy goal programming approach to multilevel programming problems, European Journal of Operational Research, 176, 1151-1166 (2007) · Zbl 1110.90084
[43] Abo-Sinna, M. A.; Baky, I. A., Interactive balance space approach for solving multi-level multi-objective programming problems, Information Sciences, 177, 3397-3410 (2007) · Zbl 1278.90350
[44] Roghanian, E.; Sadjadi, S. J.; Aryanezhad, M. B., A probabilistic bi-level linear multi-objective programming problem to supply chain planning, Applied Mathematics and Computation, 188, 786-800 (2007) · Zbl 1137.90659
[45] Sakawa, M., Fuzzy Sets and Interactive Multiobjective Optimization (1993), Plenum Press: Plenum Press New York · Zbl 0842.90070
[46] Kataoka, S., A stochastic programming model, Econometorica, 31, 181-196 (1963) · Zbl 0125.09601
[47] Stancu-Minasian, I. M., Overview of different approaches for solving stochastic programming problems with multiple objective functions, (Slowinski, R.; Teghem, J., Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainty (1990), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London), 71-101 · Zbl 0739.90050
[48] Charnes, A.; Cooper, W. W., Chance constrained programming, Management Science, 6, 73-79 (1959) · Zbl 0995.90600
[49] Charnes, A.; Cooper, W. W., Deterministic equivalents for optimizing and satisficing under chance constraints, Operations Research, 11, 18-39 (1963) · Zbl 0117.15403
[50] Stancu-Minasian, I. M., Stochastic Programming with Multiple Objective Functions (1984), D. Reidel Publishing Company: D. Reidel Publishing Company Dordrecht · Zbl 0554.90069
[51] Wets, R. J.B., Challenges in stochastic programming, Mathematical Programming, 75, 115-135 (1996) · Zbl 0874.90151
[52] Birge, J. R.; Louveaux, F., Introduction to Stochastic Programming (1997), Springer: Springer London · Zbl 0892.90142
[53] Zimmermann, H.-J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1, 45-55 (1978) · Zbl 0364.90065
[54] Bard, J. F., An investigation of the linear three-level programming problem, IEEE Transactions on Systems, Man and Cybernetics, 14, 711-717 (1984) · Zbl 0552.90081
[55] Wen, U.-P.; Bialas, W. F., The hybrid algorithm for solving the three-level linear programming problem, Computer & Operations Research, 13, 367-377 (1986) · Zbl 0643.90058
[56] Bard, J. F.; Falk, J. E., An explicit solution to the multi-level programming problem, Computer & Operations Research, 9, 77-100 (1982)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.