American Journal of Computation, Communication and Control
Manuscript Information

Solving a Non-Linear Optimization Problem Constrained by a Non-Convex Region Defined by Fuzzy Relational Equations and Schweizer-Sklar Family of T-Norms
American Journal of Computation, Communication and Control
Vol.5 , No. 2, Publication Date: May 31, 2018, Page: 68-87
484 Views Since May 31, 2018, 211 Downloads Since May 31, 2018

Authors

 [1] Amin Ghodousian, Faculty of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran. [2] Maryam Raeisian Parvari, Department of Algorithms and Computation, University of Tehran, Tehran, Iran. [3] Raziyeh Rabie, Department of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran. [4] Tarane Azarnejad, Department of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran.

Abstract

Schweizer-Sklar family of t-norms is a parametric family of continuous t-norms, which covers the whole spectrum of t-norms when the parameter is changed from zero to infinity. In this paper, we study a nonlinear optimization problem where the feasible region is formed as a system of fuzzy relational equations (FRE) defined by the Schweizer-Sklar t-norm. We firstly investigate the resolution of the feasible region and present some necessary and sufficient conditions for determining the feasibility of the problem. Moreover, two procedures are presented for simplifying the problem. Since the feasible solutions set of FREs is non-convex, conventional nonlinear programming methods may not be directly employed. For this reason, a genetic algorithm is presented to solve the nonlinear non-convex problems. The proposed GA preserves the feasibility of new generated solutions. Additionally, we propose a method to generate feasible max-Schweizer-Sklar FREs as test problems for evaluating the performance of our algorithm. The proposed method has been compared with some related works. The obtained results confirm the high performance of the proposed method in solving such nonlinear problems.

Keywords

Fuzzy Relational Equations, Nonlinear Optimization, Genetic Algorithm

Reference

 [01] D. Dubois, H. Prade, Fundamentals of Fuzzy Sets, Kluwer, Boston, 2000. [02] Min-Xia Luo, Ze Cheng, Robustness of Fuzzy Reasoning Based on Schweizer-Sklar Interval-valued t-Norms, Fuzzy Inf. Eng. (2016) 8: 183-198. [03] Minxia Lua, Ning Yao, Triple I algorithms based on Schweizer-Sklar operator in fuzzy reasoning, International Journal of Approximate Reasoning 54 (2013) 640-652. [04] Zhang Xiaohong, HE Huacan, XU Yang, A fuzzy logic system based on Schweizer-Sklar t-norm, Science in China: Series F Information Sciences (2006) Vol. 49 No. 2 175-188. [05] E. Sanchez, Resolution of composite fuzzy relation equations, Inf. Control 30 (1976) 38-48. [06] F. Di Martino, V. Loia, S. Sessa, Digital watermarking in coding/decoding processes with fuzzy relation equations, Soft Computing 10 (2006) 238-243. [07] M. J. Fernandez, P. Gil, Some specific types of fuzzy relation equations, Information Sciences 164 (2004) 189-195. [08] F. F. Guo, Z. Q. Xia, An algorithm for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities, Fuzzy Optimization and Decision Making 5 (2006) 33-47. [09] S. C. Han, H. X. Li, Notes on pseudo-t-norms and implication operators on a complete Brouwerian lattice and pseudo-t-norms and implication operators: direct products and direct product decompositions, Fuzzy Sets and Systems 153 (2005) 289-294. [10] V. Loia, S. Sessa, Fuzzy relation equations for coding/decoding processes of images and videos, Information Sciences 171 (2005) 145-172. [11] H. Nobuhara, K. Hirota, W. Pedrycz, Relational image compression: optimizations through the design of fuzzy coders and YUV colors space, Soft Computing 9 (2005) 471-479. [12] H. Nobuhara, K. Hirota, F. Di Martino, W. Pedrycz, S. Sessa, Fuzzy relation equations for compression/decompression processes of color images in the RGB and YUV color spaces, Fuzzy Optimization and Decision Making 4 (2005) 235-246. [13] W. Pedrycz, A. V. Vasilakos, Modularization of fuzzy relational equations, Soft Computing 6 (2002) 3-37. [14] I. Perfilieva, V. Novak, System of fuzzy relation equations model of IF-THEN rules, Information Sciences 177 (16) (2007) 3218-3227. [15] S. Wang, S. C. Fang, H. L. W. Nuttle, Solution sets of interval-valued fuzzy relational equations, Fuzzy Optimization and Decision Making 2 (1) (2003) 41-60. [16] Y. K. Wu, S. M. Guu, A note on fuzzy relation programming problems with max-strict-t-norm composition, Fuzzy Optimization and Decision Making 3 (2004) 271-278. [17] Q. Q. Xiong, X. P. Wang, Some properties of sup-min fuzzy relational equations on infinite domains, Fuzzy Sets and Systems 151 (2005) 393-402. [18] E. P. Klement, R. Mesiar, E. Pap, Triangular norms. Position paper I: Basic analytical and algebraic properties, Fuzzy Sets and Systems 143 (2004) 5-26. [19] I. Perfilieva, Fuzzy function as an approximate solution to a system of fuzzy relation equations, Fuzzy Sets and Systems 147 (2004) 363-383. [20] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy relational equations and their applications in knowledge engineering, Dordrecht: Kluwer academic press, 1989. [21] L. Chen, P. P. Wang, Fuzzy relation equations (i): the general and specialized solving algorithms, Soft Computing 6 (5) (2002) 428-435. [22] L. Chen, P. P. Wang, Fuzzy relation equations (ii): the branch-point-solutions and the categorized minimal solutions, Soft Computing 11 (2) (2007) 33-40. [23] A. V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy Sets and Systems 153 (2005) 261-273. [24] J. L. Lin, On the relation between fuzzy max-archimedean t-norm relational equations and the covering problem, Fuzzy Sets and Systems 160 (2009) 2328-2344. [25] J. L. Lin, Y. K. Wu, S. M. Guu, On fuzzy relational equations and the covering problem, Information Sciences 181 (2011) 2951-2963. [26] K. Peeva, Resolution of fuzzy relational equations-methods, algorithm and software with applications, Information Sciences 234 (2013) 44-63. [27] I. Perfilieva, Finitary solvability conditions for systems of fuzzy relation equations, Information Sciences 234 (2013) 29-43. [28] X. B. Qu, X. P. Wang, Man-hua. H. Lei, Conditions under which the solution sets of fuzzy relational equations over complete Brouwerian lattices form lattices, Fuzzy Sets and Systems 234 (2014) 34-45. [29] B. S. Shieh, Infinite fuzzy relation equations with continuous t-norms, Information Sciences 178 (2008) 1961-1967. [30] F. Sun, Conditions for the existence of the least solution and minimal solutions to fuzzy relation equations over complete Brouwerian lattices, Information Sciences 205 (2012) 86-92. [31] F. Sun, X. P. Wang, x. B. Qu, Minimal join decompositions and their applications to fuzzy relation equations over complete Brouwerian lattices, Information Sciences 224 (2013) 143-151. [32] Y. K. Wu, S. M. Guu, An efficient procedure for solving a fuzzy relation equation with max-Archimedean t-norm composition, IEEE Transactions on Fuzzy Systems 16 (2008) 73-84. [33] Q. Q. Xiong, X. P. Wang, Fuzzy relational equations on complete Brouwerian lattices, Information Sciences 193 (2012) 141-152. [34] C. T. Yeh, On the minimal solutions of max-min fuzzy relation equations, Fuzzy Sets and Systems 159 (2008) 23-39. [35] P. Li, S. C. Fang, A survey on fuzzy relational equations, part i: classification and solvability, Fuzzy Optimization and Decision Making 8 (2009) 179-229. [36] C. W. Chang, B. S. Shieh, Linear optimization problem constrained by fuzzy max–min relation equations, Information Sciences 234 (2013) 71–79. [37] Y. R. Fan, G. H. Huang, A. L. Yang, Generalized fuzzy linear programming for decision making under uncertainty: Feasibility of fuzzy solutions and solving approach, Information Sciences 241 (2013) 12-27. [38] A. Ghodousian, E. Khorram, An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition, Applied Mathematics and Computation 178 (2006) 502-509. [39] A. Ghodousian, E. Khorram, Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition, Information Sciences 178 (2008) 501-519. [40] A. Ghodousian, E. Khorram, Linear optimization with an arbitrary fuzzy relational inequality, Fuzzy Sets and Systems 206 (2012) 89-102. [41] A. Ghodousian, E. Khorram, Solving a linear programming problem with the convex combination of the max-min and the max-average fuzzy relation equations, Applied Mathematics and computation 180 (2006) 411-418. [42] F. F. Guo, L. P. Pang, D. Meng, Z. Q. Xia, An algorithm for solving optimization problems with fuzzy relational inequality constraints, Information Sciences 252 (2013) 20-31. [43] S. M. Guu, Y. K. Wu, Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint, Fuzzy Sets and Systems 161 (2010) 285-297. [44] S. M. Guu, Y. K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy Optimization and Decision Making 12 (2002) 1568-4539. [45] S. M. Guu, Y. K. Wu, Minimizing an linear objective function under a max-t-norm fuzzy relational equation constraint, Fuzzy Sets and Systems 161 (2010) 285-297. [46] E. Khorram, E. Shivanian, A. Ghodousian, Optimization of linear objective function subject to fuzzy relation inequalities constraints with max-average composition, Iranian Journal of Fuzzy Systems 4 (2) (2007) 15-29. [47] E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Applied Mathematics and Computation 173 (2006) 872-886. [48] E. Khorram, A. Ghodousian, A. A. Molai, Solving linear optimization problems with max-star composition equation constraints, Applied Mathematic and Computation 178 (2006) 654-661. [49] P. K. Li, S. C. Fang, On the resolution and optimization of a system of fuzzy relational equations with sup-t composition, Fuzzy Optimization and Decision Making 7 (2008) 169-214. [50] C. C. Liu, Y. Y. Lur, Y. K. Wu, Linear optimization of bipolar fuzzy relational equations with max-Łukasiewicz composition, Information Sciences 360 (2016) 149–162. [51] B. S. Shieh, Minimizing a linear objective function under a fuzzy max-t-norm relation equation constraint, Information Sciences 181 (2011) 832-841. [52] Y. K. Wu, Optimization of fuzzy relational equations with max-av composition, Information Sciences 177 (2007) 4216-4229. [53] X. P. Yang, X. G. Zhou, B. Y. Cao, Latticized linear programming subject to max-product fuzzy relation inequalities with application in wireless communication, Information Sciences 358–359 (2016) 44–55. [54] S. C. Fang, G. Li, Solving fuzzy relation equations with a linear objective function, Fuzzy Sets and Systems 103 (1999) 107-113. [55] Y. K. Wu, S. M. Guu, Minimizing a linear function under a fuzzy max-min relational equation constraints, Fuzzy Sets and Systems 150 (2005) 147-162. [56] H. C. Lee, S. M. Guu, On the optimal three-tier multimedia streaming services, Fuzzy Optimization and Decision Making 2 (1) (2002) 31-39. [57] J. Loetamonphong, S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and Systems 118 (2001) 509-517. [58] S. Dempe, A. Ruziyeva, On the calculation of a membership function for the solution of a fuzzy linear optimization problem, Fuzzy Sets and Systems 188 (2012) 58-67. [59] D. Dubey, S. Chandra, A. Mehra, Fuzzy linear programming under interval uncertainty based on IFS representation, Fuzzy Sets and Systems 188 (2012) 68-87. [60] S. Freson, B. De Baets, H. De Meyer, Linear optimization with bipolar max–min constraints, Information Sciences 234 (2013) 3–15. [61] P. Li, Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using lukasiewicz triangular norm, Soft Computing 18 (2014) 1399-1404. [62] Y. K. Wu, S. M. Guu, J. Y. Liu, Reducing the search space of a linear fractional programming problem under fuzzy relational equations with max-Archimedean t-norm composition, Fuzzy Sets and Systems 159 (2008) 3347-3359. [63] S. J. Yang, An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition, Fuzzy Sets and Systems 255 (2014) 41-51. [64] P. Z. Wang, Latticized linear programming and fuzzy relation inequalities, Journal of Mathematical Analysis and Applications 159 (1991) 72-87. [65] J. Lu, S. C. Fang, Solving nonlinear optimization problems with fuzzy relation equations constraints, Fuzzy Sets and Systems 119 (2001) 1-20. [66] R. Hassanzadeh, E. Khorram, I. Mahdavi, N. Mahdavi-Amiri, A genetic algorithm for optimization problems with fuzzy relation constraints using max-product composition, Applied Soft Computing 11 (2011) 551-560. [67] W. Hock, K. Schittkowski, Test examples for nonlinear programming codes Lecture Notes in Economics and Mathematical Systems, vol. 187, Springer, New York, 1981. [68] A. Ghodousian, M. Naeeimi, A. Babalhavaeji, Nonlinear optimization problem subjected to fuzzy relational equations deﬁned by Dubois-Prade family of t-norms, Computers & Industrial Engineering 119 (2018) 167–180. [69] P. K. Singh, m-polar fuzzy graph representation of concept lattice, Engineering Applications of Artificial Intelligence 67 (2018) 52-62. [70] A. Ghodousian, R. Zarghani, Linear optimization on the intersection of two fuzzy relational inequalities deﬁned with Yager family of t-norms, Journal of Algorithms and Computation 49 (1) (2017) 55-82. [71] P. K. Singh, C. A. Kumar, J. Li, Knowledge presentation using interval-valued fuzzy formal concept lattice, Soft Computing 20 (2016) 1485-1502. [72] P. Li, Q. Jin, On the resolution of bipolar max-min equations, Kybernetika 52 (2016) 514-530.