On Fuzzy Random-Valued Optimization

Monga K. Luhandjula
DOI: 10.4236/ajor.2011.14030                

Abstract

In this paper, we propose a novel approach for Fuzzy random-valued Optimization. The main idea behind our approach consists of taking advantage of interplays between fuzzy random variables and random sets in a way to get an equivalent stochastic program. This helps avoiding pitfalls due to severe oversimplification of the reality. We consider a numerical example that shows the efficiency of the proposed method.

Keywords

Fuzzy random variables, random sets, fuzzy stochastic Optimization under.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	H. Bandemer and W. Gerlach, “Evaluating Implicit Functional Relationships from Fuzzy Observations,” Freiberger Forschungshefter, Vol. D170, 1985, pp. 101-118.
[2]	C. M. Hwang, “A Theorem of Renewal Process for Fuzzy Random Variables and Its Application,” Fuzzy Sets and Systems, Vol. 116, No. 2, 2000, pp. 237-244. doi:10.1016/S0165-0114(98)00143-2
[3]	M. K. Luhandjula, “Fuzzy Stochastic Linear Programming: Survey and Future Research Directions,” European Journal of Operational Research, Vol. 174, No. 3, 2006, pp. 1353-1367. doi:10.1016/j.ejor.2005.07.019
[4]	H. Katagiri, E. B. Mermri, M. Sakawa, K. Kato and I. Nishizaki “A Possibilistic and Stochastic Programming Approach to Fuzzy Random MST Problems,” IEICE- Transactions on Information Systems, Vol. E88-D, No. 8, 2008, pp. 1912-1919.
[5]	H. Katagiri, M. Sakawa, K. Kato and I. Nishizaki, “Interactive Multiobjective Fuzzy Random Linear Programming: Maximization of Possibility and Probability,” European Journal of Operational Research, Vol. 188, No. 2, 2008, pp. 530-539. doi:10.1016/j.ejor.2007.02.050
[6]	B. Liu, “Fuzzy Roandom Chance-Constrained Programming,” IEEE Transactions on Fuzzy Systems, Vol. 9, No. 5, 2001, pp. 713-720. doi:10.1109/91.963757
[7]	Y. K. Liu and B. Liu, “A Class of Fuzzy Random Optimization: Expected Value Models,” Information Sciences, Vol. 155, No. 1-2, 2002, pp. 89-102. doi:10.1016/S0020-0255(03)00079-3
[8]	Y. K. Liu and B. Liu, “Fuzzy Random Programming with Equilibrium Chance Constraints,” Information Sciences, Vol. 170, No. 2-4, 2005, pp. 363-395. doi:10.1016/j.ins.2004.03.010
[9]	E. E. Ammar, “On Fuzzy Random Multiobjective Quadratic Programming,” European Journal of Operational Research, Vol. 193, No. 2, 2009, pp. 329-341. doi:10.1016/j.ejor.2007.11.031
[10]	Z. Zmeskel, “Value at Risk Methodology under Soft Conditions Approach (Fuzzy-Stochastic Approach),” European Journal of Operational Research, Vol. 161, No. 2, 2005, pp. 337-347. doi:10.1016/j.ejor.2003.08.048
[11]	P. Dutta, D. Chakraborty and A. R. Roy, “A Single-Pe- riod Inventory Model with Fuzzy Random Variable Demand,” Mathematical and Computer Modelling, Vol. 41, No. 8-9, 2005, pp. 915-922. doi:10.1016/j.mcm.2004.08.007
[12]	F. Ben Abdelaziz, L. Enneifar and J. M. Martel, “A Multiobjective Fuzzy Stochastic Program for Water Resource Optimization: The Case of Lake Management,” 2005. http://www.sharjah.ac.ae/academi/
[13]	M. K. Luhandjula, “Optimization under Hybrid Uncertainty,” Fuzzy Sets and Systems, Vol. 146, No. 2, 2004, pp. 187-203. doi:10.1016/j.fss.2004.01.002
[14]	C. Mohan and H. T. Nguyen, “An Interactive Satisfying Method for Solving Multiobjective Mixed Fuzzy Stochastic Programming Problems,” Fuzzy Sets and Systems, Vol. 117, No. 1, 2001, pp. 61-79. doi:10.1016/S0165-0114(98)00269-3
[15]	S. Nanda, G. Panda and J. Dash, “A New Solution Method for Fuzzy Chance Constrained Programming Problem,” Fuzzy Optimization and Decision Making, Vol. 5, No. 4, 2006, pp. 355-370. doi:10.1007/s10700-006-0018-8
[16]	W. M. Kirby, “Paradigm Change in Operations Research: Thirty Years of Debate,” Operations Research, Vol. 55, No. 1, 2008, pp. 1-13. doi:10.1287/opre.1060.0310
[17]	R. Kruze and K. D. Meyer, “Statistics with Vague Data,” Reidel, Dordrecht, 1987. doi:10.1007/978-94-009-3943-1
[18]	I. Moklanov, “Theory of Random Sets,” Springer, New York, 2005.
[19]	G. Matheron, “Random Sets and Integral Geometry,” John Wiley & Sons, New York, 1975.
[20]	D. Dubois and H. Prade, “Fuzzy Sets and Systems: Theory and Applications,” Academic Press, New York, 1980.
[21]	J. Bán, “Radon-Nikodyym Theorem and Conditional Expectation of Fuzzy-Valued Measures and Variables,” Fuzzy Sets and Systems, Vol. 34, No. 3, 1990, pp. 383- 392. doi:10.1016/0165-0114(90)90223-S
[22]	E. P. Klement, M. L. Puri and D. A. Ralescu, “Limit Theorems for Fuzzy Random Variables,” Proceedings of the Royal Society A, Vol. 407, No. 1832, 1986, pp. 171- 182. doi:10.1098/rspa.1986.0091
[23]	C. X. Wu and M. Ma, “Embedding Problem of Fuzzy Number Space,” Fuzzy Sets and Systems, Vol. 44, No. 1, 1991, pp. 33-38. doi:10.1016/0165-0114(91)90030-T
[24]	H. C. Wu, “Evaluate Fuzzy Optimization Problems Based on Biobjective Programming Problems,” Computer and Mathematics with Applications, Vol. 47, No. 6-7, 2004, pp. 893-902. doi:10.1016/S0898-1221(04)90073-9
[25]	P. Kall, “Stochastic Linear Programming,” Springer, New York. doi:10.1007/978-3-642-66252-2
[26]	S. Vajda, “Probabilistic Programming,” Academic Press, New York, 1972.
[27]	G. J. Klir, “Principles of Uncertainty: What Are They? Why do We Need Them?” Fuzzy Sets and Systems, Vol. 74, No. 1, 1995, pp. 13-31. doi:10.1016/0165-0114(95)00032-G
[28]	J. Li, J. Xu and M. Gen, “A Class of Multiobjective Linear Programming Model with Fuzzy Random Coefficients,” Mathematical and Computer Modelling, Vol. 44, No. 11-12, 2008, pp. 1097-1113. doi:10.1016/j.mcm.2006.03.013
[29]	V. H. Nguyen, “Solving Linear Programming Problems under Fuzziness and Randomness Environment Using Attainment Values,” Information Sciences, Vol. 177, No. 14, 2007, pp. 2971-2984. doi:10.1016/j.ins.2007.01.032
[30]	K. Glashoff and S. A. Gustafson, “Linear Optimization and Approximation,” Springer-Verlag, Berlin, 1983. doi:10.1007/978-1-4612-1142-6