Abstract
Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems. To compare and evaluate the accuracy and efficiency of various solution methods proposed for solving systems of fuzzy relational equations as well as the associated optimization problems, a test problem random generator for systems of fuzzy relational equations is needed. In this paper, procedures for generating test problems of fuzzy relational equations with the sup-\({\mathcal{T}}\) composition are proposed for the cases of sup-\({\mathcal{T}_M}\), sup-\({\mathcal{T}_P}\), and sup-\({\mathcal{T}_L }\) compositions. It is shown that the test problems generated by the proposed procedures are consistent. Some properties are discussed to show that the proposed procedures randomly generate systems of fuzzy relational equations with various number of minimal solutions. Numerical examples are included to illustrate the proposed procedures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbasi Molai A., Khorram E. (2007) A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian. Applied Mathematics and Computation 190: 1161–1167
Abbasi Molai A., Khorram E. (2008) An algorithm for solving fuzzy relation equations with max-\({\mathcal{T}}\) composition operator. Information Sciences 178(5): 1293–1308
Blyth T. S., Janowitz M. F. (1972) Residuation theory. Pergamon Press, Oxford
Di Nola A., Sessa S., Pedrycz W., Sanchez E. (1989) Fuzzy relation equations and their applications to knowledge engineering. Kluwer, Dordrecht, The Netherlands
Fang S.-C., Li G. (1999) Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems 103(1): 107–113
Ghodousian A., Khorram E. (2006) An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition. Applied Mathematics and Computation 178(2): 502–509
Guo F. F., Xia Z. Q. (2006) 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(1): 33–47
Guu S. M., Wu Y. K. (2002) Minimizing a linear objective function with fuzzy relation equation constraints. Fuzzy Optimization and Decision Making 1(4): 347–360
Hu C.-F., Fang S.-C. (2011) Set covering-based surrogate approach for solving sup-\({\mathcal{T}}\) equation constrained optimization problems. Fuzzy Optimization and Decision Making 10(1): 125–152
Khorram E., Ghodousian A. (2006) Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition. Applied Mathematics and Computation 173(2): 872–886
Klement E. P., Mesiar R., Pap E. (2000) Triangular norms. Kluwer, Dordrecht
Klir G. J., Yuan B. (1995) Fuzzy sets and fuzzy logic: Theory and applications. Prentice-Hall, New Jersey
Li, P. (2009). Fuzzy relational equations: Resolution and optimization. North Carolina State University, PhD Dissertaion.
Li, P., Fang, S.-C., & Zhang, X. (2008). Nonlinear optimization subject to a system of fuzzy relational equations with max-min composition. In Proceedings of the seventh international symposium of operations research and its applications, Lijiang, China, pp. 1–9.
Li P., Fang S.-C. (2008) On the resolution and optimization of a system of fuzzy relational equations with sup-\({\mathcal{T}}\) composition. Fuzzy Optimization and Decision Making 7(2): 169–214
Li P., Fang S.-C. (2009) A survey on fuzzy relational equations, Part I: classification and solvability. Fuzzy Optimization and Decision Making 8: 179–229
Li P., Fang S.-C. (2009) Latticized Linear Optimization on the Unit Interval. IEEE Transactions on Fuzzy Systems 17: 1353–1365
Li, P., & Fang, S.-C. (2011) On the unique solvability of fuzzy relational equations, Fuzzy Optimization and Decision Making.
Loetamonphong J., Fang S.-C. (2001) Optimization of fuzzy relation equations with max-product composition. Fuzzy Sets and Systems 118(3): 509–517
Markovskii A. (2005) On the relation between equations with max-product composition and the covering problem. Fuzzy Sets and Systems 153(2): 261–273
Pedrycz W. (1985) On generalized fuzzy relational equations and their applications. Journal of Mathematical Analysis and Applications 107: 520–536
Pedrycz W. (1991) Processing in relational structures: Fuzzy relational equations. Fuzzy Sets and Systems 40: 77–106
Peeva K., Kyosev Y. (2004) Fuzzy relational calculus: Theory, applications, and software. World Scientific, New Jersey
Sanchez E. (1976) Resolution of composite fuzzy relation equations. Information and Control 30(1): 38–48
Sanchez E. (1977) Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic. In: Gupta M. M., Saridis G. N., Gaines B. R. (eds) Fuzzy automata and decision processes. North-Holland, Amsterdam, pp 221–234
Wu Y. K., Guu S. M., Liu J. Y. C. (2002) An accelerated approach for solving fuzzy relation equations with a linear objective function. IEEE Transactions on Fuzzy Systems 10(4): 552–558
Wu Y. K., Guu S. M. (2004) A note on fuzzy relation programming problems with max-strict-t-norm composition. Fuzzy Optimization and Decision Making 3(3): 271–278
Wu Y. K., Guu S. M. (2005) Minimizing a linear function under a fuzzy max-min relational equation constraint. Fuzzy Sets and Systems 150(1): 147–162
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Science Council (NSC) of R.O.C., under Grant NSC 99-2918-I-214-001.
Rights and permissions
About this article
Cite this article
Hu, CF., Fang, SC. Randomly generating test problems for fuzzy relational equations. Fuzzy Optim Decis Making 11, 1–28 (2012). https://doi.org/10.1007/s10700-011-9115-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-011-9115-4