Interactive Fuzzy Programming for Random Fuzzy Two-Level Integer Programming Problems through Fractile Criteria with Possibility
Affiliation(s)
Department of System Cybernetics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, Japan.
Department of System Cybernetics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, Japan.
ABSTRACT
This paper considers two-level integer programming problems involving random fuzzy variables with cooperative behavior of the decision makers. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile criteria to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy two-level integer programming problems are reduced to deterministic ones. Through the introduction of genetic algorithms with double strings for nonlinear integer programming problems, interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.
Cite this paper
M. Sakawa and T. Matsui, "Interactive Fuzzy Programming for Random Fuzzy Two-Level Integer Programming Problems through Fractile Criteria with Possibility," Applied Mathematics, Vol. 4 No. 8, 2013, pp. 34-43. doi: 10.4236/am.2013.48A006.
M. Sakawa and T. Matsui, "Interactive Fuzzy Programming for Random Fuzzy Two-Level Integer Programming Problems through Fractile Criteria with Possibility," Applied Mathematics, Vol. 4 No. 8, 2013, pp. 34-43. doi: 10.4236/am.2013.48A006.
References
[1] K. Shimizu, Y. Ishizuka and J. F. Bard, “Nondifferenti able and Two-Level Mathematical Programming,” Klu wer Academic Publishers, Boston, 1997. doi:10.1007/978-1-4615-6305-1 [2] M. Sakawa and I. Nishizaki, “Cooperative and Noncoop erative Multi-Level Programming,” Springer, New York, 2009. [3] W. F. Bialas and M. H. Karwan, “Two-Level Linear Pro gramming,” Management Science, Vol. 30, No. 8, 1984, pp. 1004-1020. doi:10.1287/mnsc.30.8.1004 [4] I. Nishizaki and M. Sakawa, “Computational Methods through Genetic Algorithms for Obtaining Stackelberg Solutions to Two-Level Mixed Zero-One Programming Problems,” Cybernetics and Systems: An International Journal, Vol. 31, No. 2, 2000, pp. 203-221. doi:10.1080/019697200124892 [5] M. Simaan and J. B. Cruz Jr., “On the Stackelberg Strat egy in Nonzero-Sum Games,” Journal of Optimization Theory and Applications, Vol. 11, No. 5, 1973, pp. 533-555. doi:10.1007/BF00935665 [6] Y. J. Lai, “Hierarchical Optimization: A Satisfactory So lution,” Fuzzy Sets and Systems, Vol. 77, No. 3, 1996, pp. 321-325. doi:10.1016/0165-0114(95)00086-0 [7] H. S. Shih, Y. J. Lai and E. S. Lee, “Fuzzy Approach for Multi-Level Programming Problems,” Computers and Operations Research, Vol. 23, No. 1, 1996, pp. 73-91. doi:10.1016/0305-0548(95)00007-9 [8] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Multi-Level Linear Programming Pro blems,” Computers & Mathematics with Applications, Vol. 36, No. 2, 1998, pp. 71-86. doi:10.1016/S0898-1221(98)00118-7 [9] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Multi-Level Linear Fractional Pro gramming Problems with Fuzzy Parameters,” Fuzzy Sets and Systems, Vol. 109, No. 1, 2000, pp. 3-19. doi:10.1016/S0165-0114(98)00130-4 [10] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Two-Level Linear and Linear Fractional Production and Assignment Problems: A Case Study,” Eu ropean Journal of Operational Research, Vol. 135, No. 1, 2001, pp. 142-157. doi:10.1016/S0377-2217(00)00309-X [11] M. Sakawa and I. Nishizaki, “Interactive Fuzzy Program ming for Decentralized Two-Level Linear Programming Problems,” Fuzzy Sets and Systems, Vol. 125, No. 3, 2002, pp. 301-315. doi:10.1016/S0165-0114(01)00042-2 [12] M. Sakawa, I. Nishizaki and Y. Uemura, “A Decentral ized Two-Level Transportation Problem in a Housing Ma terial Manufacturer: Interactive Fuzzy Programming Approach,” European Journal of Operations Research, Vol. 141, No. 1, 2002, pp. 167-185. doi:10.1016/S0377-2217(01)00273-9 [13] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Two-Level Linear Fractional Program ming Problems with Fuzzy Parameters,” Fuzzy Sets and Systems, Vol. 115, No. 1, 2000, pp. 93-103. doi:10.1016/S0165-0114(99)00027-5 [14] M. Sakawa and I. Nishizaki, “Interactive Fuzzy Program ming for Two-Level Nonconvex Programming Problems with Fuzzy Parameters through Genetic Algorithms,” Fuzzy Sets and Systems, Vol. 127, No. 2, 2002, pp. 185-197. doi:10.1016/S0165-0114(01)00134-8 [15] M. Sakawa and H. Katagiri, “Interactive Fuzzy Program ming Based on Fractile Criterion Optimization Model for Two-Level Stochastic Linear Programming Problems,” Cybernetics and Systems, Vol. 41, No. 7, 2010, pp. 508-521. doi:10.1080/01969722.2010.511547 [16] M. Sakawa and K. Kato, “Interactive Fuzzy Programming for Stochastic Two-Level Linear Programming Problems through Probability Maximization,” Interim Report, IR 09-013, International Institute for Applied Systems Ana lysis (IIASA), 2009. [17] M. Sakawa, H. Katagiri and T. Matsui, “Interactive Fuzzy Stochastic Two-Level Integer Programming through Frac tile Criterion Optimization,” Operational Research: An International Journal, Vol. 12, No. 2, 2012, pp. 209-227. doi:10.1016/S0377-2217(01)00273-9 [18] M. Sakawa, H. Katagiri and T. Matsui, “Interactive Fuzzy Random Two-Level Linear Programming through Frac tile Criterion Optimization,” Mathematical and Computer Modelling, Vol. 54, No. 11-12, 2011, pp. 3153-3163. doi:10.1016/j.mcm.2011.08.006 [19] M. Sakawa, I. Nishizaki and H. Katagiri, “Fuzzy Stochas tic Multiobjective Programming,” Springer, New York, 2011. doi:10.1007/978-1-4419-8402-9 [20] H. Kwakernaak, “Fuzzy Random Variables. I. Definitions and Theorems,” Information Sciences, Vol. 15, No. 1, 1978, pp. 1-29. doi:10.1016/0020-0255(78)90019-1 [21] M. L. Puri and D. A. Ralescu, “Fuzzy Random Variables,” Journal of Mathematical Analysis and Applications, Vol. 114, No. 2, 1986, pp. 409-422. doi:10.1016/0022-247X(86)90093-4 [22] G.-Y. Wang and Z. Qiao, “Linear Programming with Fuzzy Random Variable Coefficients,” Fuzzy Sets and Systems, Vol. 57, No. 3, 1993, pp. 295-311. doi:10.1016/0165-0114(93)90025-D [23] M. Sakawa and I. Nishizaki, “Interactive Fuzzy Program ming for Multi-Level Programming Problems: A Re view,” International Journal of Multicriteria Decision Making, Vol. 2, No. 3, 2012, pp. 241-266. doi:10.1504/IJMCDM.2012.047846 [24] B. Liu, “Random Fuzzy Dependent-Chance Programming and Its Hybrid Intelligent Algorithm,” Information Sci ences, Vol. 141, No. 3-4, 2002, pp. 259-271. doi:10.1016/S0020-0255(02)00176-7 [25] M. Sakawa and T. Matsui, “Interactive Fuzzy Program ming for Random Fuzzy Two-Level Programming Prob lems through Possibility-Based Fractile Model,” Expert Systems with Applications, Vol. 39, No. 16, 2012, pp. 12599-12604. doi:10.1016/j.eswa.2012.05.024 [26] A. M. Geoffrion, “Stochastic Programming with Aspira tion or Fractile Criteria,” Management Science, Vol. 13, No. 9, 1967, pp. 672-679. doi:10.1287/mnsc.13.9.672 [27] M. A. Gil, M. Lopez-Diaz and D. A. Ralescu, “Overview on the Development of Fuzzy Random Variables,” Fuzzy Sets and Systems, Vol. 157, No. 19, 2006, pp. 2546-2557. doi:10.1016/j.fss.2006.05.002 [28] S. Nahmias, “Fuzzy Variables,” Fuzzy Sets and Systems, Vol. 1, No. 2, 1978, pp. 97-110. doi:10.1016/0165-0114(78)90011-8 [29] M. Sakawa, “Genetic Algorithms and Fuzzy Multiobjec tive Optimization,” Kluwer Academic Publishers, Boston, 2001. [30] S. Koziel and Z. Michalewicz, “Evolutionary Algorithms, Homomorphous Mapping, and Constrained Parameter Op timization,” Evolutionary Computation, Vol. 7, No. 1, 1999, pp. 19-44. doi:10.1162/evco.1999.7.1.19 [31] Z. Michalewicz and G. Nazhiyath, “GenocopIII: A Co Evolutionary Algorithm for Numerical Optimization Pro blems with Nonlinear Constraints,” Proceedings of the Second IEEE International Conference on Evolutionary Computation, Perth, 29 November-1 December 1995, pp. 647-651. [32] M. Sakawa, K. Kato, M. A. K. Azad and R. Watanabe, “A Genetic Algorithm with Double String for Nonlinear Integer Programming Problems,” Proceedings of 2005 IEEE International Conference on Systems, Man and Cy bernetics, Waikoloa, 10-12 October 2005, pp. 3281-3286.
[1] K. Shimizu, Y. Ishizuka and J. F. Bard, “Nondifferenti able and Two-Level Mathematical Programming,” Klu wer Academic Publishers, Boston, 1997. doi:10.1007/978-1-4615-6305-1 [2] M. Sakawa and I. Nishizaki, “Cooperative and Noncoop erative Multi-Level Programming,” Springer, New York, 2009. [3] W. F. Bialas and M. H. Karwan, “Two-Level Linear Pro gramming,” Management Science, Vol. 30, No. 8, 1984, pp. 1004-1020. doi:10.1287/mnsc.30.8.1004 [4] I. Nishizaki and M. Sakawa, “Computational Methods through Genetic Algorithms for Obtaining Stackelberg Solutions to Two-Level Mixed Zero-One Programming Problems,” Cybernetics and Systems: An International Journal, Vol. 31, No. 2, 2000, pp. 203-221. doi:10.1080/019697200124892 [5] M. Simaan and J. B. Cruz Jr., “On the Stackelberg Strat egy in Nonzero-Sum Games,” Journal of Optimization Theory and Applications, Vol. 11, No. 5, 1973, pp. 533-555. doi:10.1007/BF00935665 [6] Y. J. Lai, “Hierarchical Optimization: A Satisfactory So lution,” Fuzzy Sets and Systems, Vol. 77, No. 3, 1996, pp. 321-325. doi:10.1016/0165-0114(95)00086-0 [7] H. S. Shih, Y. J. Lai and E. S. Lee, “Fuzzy Approach for Multi-Level Programming Problems,” Computers and Operations Research, Vol. 23, No. 1, 1996, pp. 73-91. doi:10.1016/0305-0548(95)00007-9 [8] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Multi-Level Linear Programming Pro blems,” Computers & Mathematics with Applications, Vol. 36, No. 2, 1998, pp. 71-86. doi:10.1016/S0898-1221(98)00118-7 [9] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Multi-Level Linear Fractional Pro gramming Problems with Fuzzy Parameters,” Fuzzy Sets and Systems, Vol. 109, No. 1, 2000, pp. 3-19. doi:10.1016/S0165-0114(98)00130-4 [10] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Two-Level Linear and Linear Fractional Production and Assignment Problems: A Case Study,” Eu ropean Journal of Operational Research, Vol. 135, No. 1, 2001, pp. 142-157. doi:10.1016/S0377-2217(00)00309-X [11] M. Sakawa and I. Nishizaki, “Interactive Fuzzy Program ming for Decentralized Two-Level Linear Programming Problems,” Fuzzy Sets and Systems, Vol. 125, No. 3, 2002, pp. 301-315. doi:10.1016/S0165-0114(01)00042-2 [12] M. Sakawa, I. Nishizaki and Y. Uemura, “A Decentral ized Two-Level Transportation Problem in a Housing Ma terial Manufacturer: Interactive Fuzzy Programming Approach,” European Journal of Operations Research, Vol. 141, No. 1, 2002, pp. 167-185. doi:10.1016/S0377-2217(01)00273-9 [13] M. Sakawa, I. Nishizaki and Y. Uemura, “Interactive Fuzzy Programming for Two-Level Linear Fractional Program ming Problems with Fuzzy Parameters,” Fuzzy Sets and Systems, Vol. 115, No. 1, 2000, pp. 93-103. doi:10.1016/S0165-0114(99)00027-5 [14] M. Sakawa and I. Nishizaki, “Interactive Fuzzy Program ming for Two-Level Nonconvex Programming Problems with Fuzzy Parameters through Genetic Algorithms,” Fuzzy Sets and Systems, Vol. 127, No. 2, 2002, pp. 185-197. doi:10.1016/S0165-0114(01)00134-8 [15] M. Sakawa and H. Katagiri, “Interactive Fuzzy Program ming Based on Fractile Criterion Optimization Model for Two-Level Stochastic Linear Programming Problems,” Cybernetics and Systems, Vol. 41, No. 7, 2010, pp. 508-521. doi:10.1080/01969722.2010.511547 [16] M. Sakawa and K. Kato, “Interactive Fuzzy Programming for Stochastic Two-Level Linear Programming Problems through Probability Maximization,” Interim Report, IR 09-013, International Institute for Applied Systems Ana lysis (IIASA), 2009. [17] M. Sakawa, H. Katagiri and T. Matsui, “Interactive Fuzzy Stochastic Two-Level Integer Programming through Frac tile Criterion Optimization,” Operational Research: An International Journal, Vol. 12, No. 2, 2012, pp. 209-227. doi:10.1016/S0377-2217(01)00273-9 [18] M. Sakawa, H. Katagiri and T. Matsui, “Interactive Fuzzy Random Two-Level Linear Programming through Frac tile Criterion Optimization,” Mathematical and Computer Modelling, Vol. 54, No. 11-12, 2011, pp. 3153-3163. doi:10.1016/j.mcm.2011.08.006 [19] M. Sakawa, I. Nishizaki and H. Katagiri, “Fuzzy Stochas tic Multiobjective Programming,” Springer, New York, 2011. doi:10.1007/978-1-4419-8402-9 [20] H. Kwakernaak, “Fuzzy Random Variables. I. Definitions and Theorems,” Information Sciences, Vol. 15, No. 1, 1978, pp. 1-29. doi:10.1016/0020-0255(78)90019-1 [21] M. L. Puri and D. A. Ralescu, “Fuzzy Random Variables,” Journal of Mathematical Analysis and Applications, Vol. 114, No. 2, 1986, pp. 409-422. doi:10.1016/0022-247X(86)90093-4 [22] G.-Y. Wang and Z. Qiao, “Linear Programming with Fuzzy Random Variable Coefficients,” Fuzzy Sets and Systems, Vol. 57, No. 3, 1993, pp. 295-311. doi:10.1016/0165-0114(93)90025-D [23] M. Sakawa and I. Nishizaki, “Interactive Fuzzy Program ming for Multi-Level Programming Problems: A Re view,” International Journal of Multicriteria Decision Making, Vol. 2, No. 3, 2012, pp. 241-266. doi:10.1504/IJMCDM.2012.047846 [24] B. Liu, “Random Fuzzy Dependent-Chance Programming and Its Hybrid Intelligent Algorithm,” Information Sci ences, Vol. 141, No. 3-4, 2002, pp. 259-271. doi:10.1016/S0020-0255(02)00176-7 [25] M. Sakawa and T. Matsui, “Interactive Fuzzy Program ming for Random Fuzzy Two-Level Programming Prob lems through Possibility-Based Fractile Model,” Expert Systems with Applications, Vol. 39, No. 16, 2012, pp. 12599-12604. doi:10.1016/j.eswa.2012.05.024 [26] A. M. Geoffrion, “Stochastic Programming with Aspira tion or Fractile Criteria,” Management Science, Vol. 13, No. 9, 1967, pp. 672-679. doi:10.1287/mnsc.13.9.672 [27] M. A. Gil, M. Lopez-Diaz and D. A. Ralescu, “Overview on the Development of Fuzzy Random Variables,” Fuzzy Sets and Systems, Vol. 157, No. 19, 2006, pp. 2546-2557. doi:10.1016/j.fss.2006.05.002 [28] S. Nahmias, “Fuzzy Variables,” Fuzzy Sets and Systems, Vol. 1, No. 2, 1978, pp. 97-110. doi:10.1016/0165-0114(78)90011-8 [29] M. Sakawa, “Genetic Algorithms and Fuzzy Multiobjec tive Optimization,” Kluwer Academic Publishers, Boston, 2001. [30] S. Koziel and Z. Michalewicz, “Evolutionary Algorithms, Homomorphous Mapping, and Constrained Parameter Op timization,” Evolutionary Computation, Vol. 7, No. 1, 1999, pp. 19-44. doi:10.1162/evco.1999.7.1.19 [31] Z. Michalewicz and G. Nazhiyath, “GenocopIII: A Co Evolutionary Algorithm for Numerical Optimization Pro blems with Nonlinear Constraints,” Proceedings of the Second IEEE International Conference on Evolutionary Computation, Perth, 29 November-1 December 1995, pp. 647-651. [32] M. Sakawa, K. Kato, M. A. K. Azad and R. Watanabe, “A Genetic Algorithm with Double String for Nonlinear Integer Programming Problems,” Proceedings of 2005 IEEE International Conference on Systems, Man and Cy bernetics, Waikoloa, 10-12 October 2005, pp. 3281-3286.