Multiobjective Two-Level Fuzzy Random Programming Problems with Simple Recourses and Estimated Pareto Stackelberg Solutions
Graduate School of Humanities and Social Sciences, Nagoya City University
1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya, Aichi 467-8501, Japan
In this paper, we focus on multiobjective two-level fuzzy random programming problems with simple recourses, in which multiple objective functions are involved in each level, shortages and excesses resulting from the violation of the constraints with fuzzy random variables are penalized, and sums of the objective functions and expectation of the amount of the penalties are minimized. To deal with such problems, the concept of estimated Pareto Stackelberg solutions of the leader is introduced under the assumption that the leader can estimate the preference of the follower as a weighting vector of the weighting problems. Employing the possibility measure for fuzzy numbers, weighting method for multiobjective programming problems, and Kuhn-Tucker approach for two-level programming problems, a nonlinear optimization problem under complementarity conditions is formulated to obtain the estimated Pareto Stackelberg solutions for the leader. A numerical example illustrates the proposed method for a multiobjective two-level fuzzy random programming problem with simple recourses. Several types of estimated Pareto Stackelberg solutions are derived corresponding to the weighting vectors and permissible possibility levels specified by the leader.
-  H.-J. Zimmermann, “Fuzzy programming and linear programming with several objective functions,” Fuzzy Sets and Systems, Vol.1, pp. 45-55, 1978.
-  R. E. Bellman and L. A. Zadeh, “Decision-making in a fuzzy environment,” Management Science, Vol.17, No.4, pp. 141-164, 1970.
-  M. Sakawa, “Fuzzy Sets and Interactive Multiobjective Optimization,” Plenum Press, 1993.
-  M. Sakawa and H. Yano, “An interactive fuzzy satisficing method using augmented minimax problems and its application to environmental systems,” IEEE Trans. on Systems, Man and Cybernetics, Vol.6, pp. 720-729, 1985.
-  M. Sakawa and H. Yano, “Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters,” Fuzzy Sets and Systems, Vol.29, pp. 315-326, 1989.
-  M. Sakawa and H. Yano, “An interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters,” Fuzzy Sets and Systems, Vol.30, pp. 221-238, 1989.
-  M. Sakawa and H. Yano, “An interactive fuzzy satisficing method for generalized multiobjective linear programming problems with fuzzy parameters,” Fuzzy Sets and Systems, Vol.35, pp. 125-142, 1990.
-  H. Yano and M. Sakawa, “Interactive fuzzy decision making for generalized multiobjective linear fractional programming problems with fuzzy parameters,” Fuzzy Sets and Systems, Vol.32, pp. 245-261, 1989.
-  D. Dubois and H. Prade, “Fuzzy Sets and Systems: Theory and Applications,” Academic Press, 1980.
-  J. R. Birge and F. Louveaux, “Introduction to Stochastic Programming,” Springer, 1997.
-  G. B. Danzig, “Linear programming under uncertainty,” Management Science, Vol.1, No.3/4, pp. 197-206, 1955.
-  P. Kall and J. Mayer, “Stochastic Linear Programming: Models, Theory, and Computation,” Springer, 2005.
-  D. W. Walkup and R. Wets, “Stochastic programs with recourse,” SIAM J. on Applied Mathematics, Vol.15, pp. 139-162, 1967.
-  R. Wets, “Stochastic programs with fixed recourse: the equivalent deterministic program,” SIAM Review, Vol.16, pp. 309-339, 1974.
-  A. Charnes and W. W. Cooper, “Chance-constrained programming,” Management Science, Vol.6, No.1, pp. 73-79, 1959.
-  M. Sakawa, I. Nishizaki, and H. Katagiri, “Fuzzy Stochastic Multiobjective Programming,” Springer, 2011.
-  M. Sakawa, H. Yano, and I. Nishizaki, “Linear and Multiobjective Programming with Fuzzy Stochastic Extensions,” Springer, 2013.
-  M. Sakawa, K. Kato, and H. Katagiri, “An interactive fuzzy satisficing method through a variance minimization model for multiobjective linear programming problems involving random variables,” Knowledge-Based Intelligent Information Engineering Systems & Allied Technologies KES2002, pp. 1222-1226, 2002.
-  M. Sakawa, H. Katagiri, and K. Kato, “An interactive fuzzy satisficing method for multiobjective stochastic programming problems using fractile optimization model,” The 10th IEEE Int. Conf. on Fuzzy Systems, pp. 25-31, 2001.
-  M. Sakawa, K. Kato, and H. Katagiri, “An interactive fuzzy satisficing method for multiobjective linear programming problems with random variable coefficients through a probability maximization model,” Fuzzy Sets and Systems, Vol.146, pp. 205-220, 2004.
-  M. Sakawa, K. Kato, and I. Nishizaki, “An interactive fuzzy satisficing method for multiobjective stochastic linear programming problems through an expectation model,” European J. of Operational Research, Vol.145, pp. 665-672, 2003.
-  S. Sekizaki, I. Nishizaki, and T. Hayashida, “Electricity retail market model with flexible price settings and elastic price-based demand responses by consumers in distribution network,” Int. J. of Electrical Power and Energy Systems, Vol.81, pp. 371-386, 2016.
-  W. Sun, C. An, G. Li, and Y. Lv, “Applications of inexact programming methods to waste management under uncertainty: current status and future directions,” Environmental Systems Research, Vol.3, pp. 1-15, 2014.
-  R. Wang, Y. Li, and Q. Tan, “A review of inexact optimization modeling and its application to integrated water resources management,” Frontiers of Earth Science, Vol.9, pp. 51-64, 2015.
-  M. Zugno, J. M. Morales, P. Pinson, and H. Madsen, “A bilevel model for electricity retailers’ participation in a demand response market environment,” Energy Economics, Vol.36, pp. 182-197, 2013.
-  M. Sakawa and T. Matsui, “Interactive fuzzy multiobjective stochastic programming with simple recourse,” Int. J. of Multicriteria Decision Making, Vol.4, pp. 31-46, 2014.
-  H. Kwakernaak, “Fuzzy random variables-I,” Information Sciences, Vol.15, pp. 1-29, 1978.
-  M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” J. of mathematical analysis and applications, Vol.114, pp. 409-422, 1986.
-  H. Katagiri, M. Sakawa, K. Kato, and I. Nishizaki, “Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability,” European J. of Operational Research, Vol.188, pp. 530-539, 2008.
-  H. Katagiri, M. Sakawa, K. Kato, and S. Ohsaki, “An interactive fuzzy satisficing method for fuzzy random multiobjective linear programming problems through the fractile optimization model using possibility and necessity measures,” Proc. of the 9th Asia Pacific Management Conf., pp. 795-802, 2003.
-  J. F. Bard, “Some properties of the bilevel programming problem,” J. of Optimization Theory and Applications, Vol.68, No.2, pp. 371-378, 1991.
-  R. G. Jeroslow, “The polynomial hierarchy and a simple model for competitive analysis,” Mathematical Programming, Vol.32, No.2, pp. 146-164, 1985.
-  M. Sakawa and I. Nishizaki, “Cooperative and Noncooperative Multi-Level Programming,” Springer, 2009.
-  W. F. Bialas and M. H. Karwan, “Two-level linear programming,” Management Science, Vol.30, No.8, pp. 1004-1020, 1984.
-  J. J. Júdice and A. M. Faustino, “A sequential LCP method for bilevel linear programming,” Annals of Operations Research, Vol.34, No.1, pp. 89-106, 1992.
-  D. J. White, and G. Anandalingam, “A penalty function approach for solving bi-level linear programs,” J. of Global Optimization, Vol.3, pp. 397-419, 1993.
-  V. P. Singh and D. Chakraborty, “Solving bi-level programming problem with fuzzy random variable coefficients,” J. of Intelligent & Fuzzy Systems, Vol.32, pp. 521-528, 2017.
-  N. Ranarahu, J. K. Dash, and S. Acharya, “Multi-objective bilevel fuzzy probabilistic programming problem,” OPSEARCH, Vol.54, pp. 475-504, 2017.
-  H. Katagiri, K. Kato, and T. Uno, “Possibilistic Stackelberg solutions to bilevel linear programming problems with fuzzy parameters,” J. of Intelligent & Fuzzy Systems, Vol.32, pp. 4485-4501, 2017.
-  H. Yano and R. Zhang, “Interactive Decision Making for Multiobjective Fuzzy Random Programming Problems with Simple Recourse through a Fractile Model,” IAENG Int. J. of Applied Mathematics, Vol.46, pp. 1-9, 2015.
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