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Keywords:
fuzzy optimization; minimizing a linear objective function; maximizing a linear objective function; fuzzy relational equations; system of equations; fuzzy relational inequalities; system of inequalities; $\max -\ast $ composition; solution family; minimal solutions
fuzzy optimization; minimizing a linear objective function; maximizing a linear objective function; fuzzy relational equations; system of equations; fuzzy relational inequalities; system of inequalities; $\max -\ast $ composition; solution family; minimal solutions
Summary:
This paper provides an extension of results connected with the problem of the optimization of a linear objective function subject to $\max-\ast$ fuzzy relational equations and an inequality constraint, where $\ast$ is an operation. This research is important because the knowledge and the algorithms presented in the paper can be used in various optimization processes. Previous articles describe an important problem of minimizing a linear objective function under a fuzzy $\max-\ast$ relational equation and an inequality constraint, where $\ast$ is the $t$-norm or mean. The authors present results that generalize this outcome, so the linear optimization problem can be used with any continuous increasing operation with a zero element where $\ast$ includes in particular the previously studied operations. Moreover, operation $\ast$ does not need to be a t-norm nor a pseudo-$t$-norm. Due to the fact that optimal solutions are constructed from the greatest and minimal solutions of a $\max-\ast$ relational equation or inequalities, this article presents a method to compute them. We note that the linear optimization problem is valid for both minimization and maximization problems. Therefore, for the optimization problem, we present results to find the largest and the smallest value of the objective function. To illustrate this problem a numerical example is provided.
References:
[1] Belohlavek, R.: Fuzzy Relational Systems. Foundations and Principles. Academic Publishers, Kluwer New York 2002.
[2] Czogała, E., Drewniak, J., Pedrycz, W.: Fuzzy relation equations on a finite set. Fuzzy Sets Systems 7 (1982), 89-101. DOI | MR 0635357
[3] Drewniak, J.: Fuzzy relation equations and inequalities. Fuzzy Sets Systems 14 (1984), 237-247. DOI | MR 0768110
[4] Drewniak, J.: Fuzzy Relation Calculus. Silesian University, Katowice 1989. MR 1009161
[5] Drewniak, J., Matusiewicz, Z.: Fuzzy equations $\max-\ast$ with conditionally cancellative operations. Inform. Sci. 206 (2012), 18-29. DOI | MR 2930162
[6] Fang, S. Ch., Li, G.: Solving fuzzy relation equations with a linear objective function. Fuzzy Sets Systems 103 (1999), 107-113 MR 1674026 | Zbl 0933.90069
[7] Guo, F., Pang, L.-P., Meng, D., Xia, Z.-Q.: An algorithm for solving optimization problems with fuzzy relational inequality constraints. Inform. Sci. 252 (2011), 20-31. DOI | MR 3123917
[8] Guu, S.-M., Wu, Y. K.: Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint. Fuzzy Sets Systems 161 (2010), 285-297. DOI | MR 2566245 | Zbl 1190.90297
[9] Han, S. Ch., Li, H.-X., Wang, J.-Y.: Resolution of finite fuzzy relation equations based on strong pseudo-$t$-norms. Appl. Math. Lett. 19 (2006), 752-757. DOI | MR 2232250
[10] Higashi, M., Klir, G. J.: Resolution of finite fuzzy relation equations. Fuzzy Sets Systems 13 (1984), 65-82 DOI 10.1016/0165-0114(84)90026-5 | MR 0747391
[11] Khorram, E., Zarei, H.: Multi-objective optimization problems with fuzzy relation equation constraints regarding max-average composition. Math. Comput. Modell. 5 (2009), 49, 856-867. DOI | MR 2495003
[12] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. MR 1790096 | Zbl 1087.20041
[13] Lee, H.-C., Guu, S.-M.: On the optimal three-tier multimedia streaming services. Fuzzy Optimization and Decision Making 2 (3) (2002), 31-39. DOI
[14] Li, S.-Ch., Fang, P.: A survey on fuzzy relational equations, part I: classification and solvability. Fuzzy Optim. Decision Making 8 (2009), 2, 179-229. DOI | MR 2511474
[15] Liu, Ch.-Ch., Lur, Y.-Y., Wu, Y.-K.: Linear optimization of bipolar fuzzy relational equations with max-Lukasiewicz composition. Inform. Sci. 360 (2016), 149-162. DOI
[16] Matusiewicz, Z., Drewniak, J.: Increasing continuous operations in fuzzy $\max-\ast$ equations and inequalities. Fuzzy Sets Systems 231 (2013), 120-133. DOI | MR 3118539
[17] Molai, A. A.: Fuzzy linear objective function optimization with fuzzy-valued max-product fuzzy relation inequality constraints. Math. Comput. Modell. 51 (2010), 9-10, 1240-1250. DOI | MR 2608910
[18] Peeva, K., Kyosev, Y.: Fuzzy Relational Calculus: Theory, Applications and Software. Advanced Fuzzy Systems - Applications and Theory, World Scientific, Singapore 2004. DOI | MR 2379415 | Zbl 1083.03048
[19] Qin, Z., Liu, X., Cao, B.-Y.: Multi-level linear programming subject to max-product fuzzy relation equalities. In: International Workshop on Mathematics and Decision Science 2018. DOI
[20] Qu, X., Wang, X.-P.: Minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations. Inform. Sci. 178 (2008), 17, 3482-3490. DOI | MR 2436417
[21] Sanchez, E.: Resolution of composite fuzzy relation equations. Inform. Control 30 (1976), 38-48. DOI | MR 0437410
[22] Shieh, B.-S.: Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint. Inform. Sci. 161 (2011), 285-297. DOI | MR 2566245
[23] Xiao, G., Zhu, T.-X., Chen, Y., Yang, X.: Linear Searching Method for Solving Approximate Solution to System of Max-Min Fuzzy Relation Equations With Application in the Instructional Information Resources Allocation. In: IEEE Access 7 (2019), 65019-65028. DOI
[24] Yang, X.-P., Zhou, X.-G., Cao, B.-Y.: Latticized linear programming subject to max-product fuzzy relation inequalities with application in wireless communication. Inform. Sci. 358(C) (2016), 44-55. DOI
[25] Zadeh, L. A.: Similarity relations and fuzzy orderings. Inform. Sci. 3 (1971), 177-200. DOI | MR 0297650
[26] Zhou, X.-G., Yang, X.-P., Cao, B.-Y.: Posynomial geometric programming problem subject to max–min fuzzy relation equations. Inform. Sci. 328 (2016), 15-25. DOI
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