Document Type: Research Paper

**Authors**

School of Mathematics and Computer Sciences, Damghan University, Damghan, Iran

**Abstract**

This paper studies the nonlinear optimization problems subject to bipolar max-min fuzzy relation equation constraints. The feasible solution set of the problems is non-convex, in a general case. Therefore, conventional nonlinear optimization methods cannot be ideal for resolution of such problems. Hence, a Genetic Algorithm (GA) is proposed to find their optimal solution. This algorithm uses the structure of the feasible domain of the problems and lower and upper bound of the feasible solution set to choose the initial population. The GA employs two different crossover operations: 1- N-points crossover and 2- Arithmetic crossover. We run the GA with two crossover operations for some test problems and compare their results and performance to each other. Also, their results are compared with the results of other authors' works.

**Keywords**

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Applied Mathematics and Computation, 175 (2006), 269-276.

[3] B. De Baets, Analytical solution methods for fuzzy relational equations, in: D. Dubois, H.

Prade (Eds.), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, Kluwer

Academic Publishers, Dordrecht, (2000), 291-340.

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straints, Information Sciences, 234 (2013), 3-15.

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optimization problems with fuzzy relation constraints using max-product composition, Applied

Soft Computing, 11 (2011), 551-560.

[8] W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Lecture

Notes in Economics and Mathematical Systems, vol. 187, Springer, New York, 1981.

[9] J. Holland, Adaptation in Natural and Artificial Systems, The University of Michigan Press,

Ann Arbor, MI, 1975.

[10] A. Homaifar, S. Lai and X. Qi, Constrained optimization via genetic algorithms, Simulation,

62 (1994), 242-254.

[11] J. A. Joines and C. Houck, On the Use of Non-stationary Penalty Function to Solve Nonlinear

Constrained Optimization Problems with GAs, In: Z. Michalewicz (Ed.), Proc. 1st IEEE

Internat. Conf. on Evolutionary Computation, IEEE Service Center, Piscataway, NJ, (1994),

579-584.

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star composition equation constraints, Applied Mathematics and Computation, 178 (2006),

654-661.

[13] E. Khorram and R. Hassanzadeh, Solving nonlinear optimization problems subjected to fuzzy

relation equation constraints with max-average composition using a modified genetic algo-

rithm, Computers and Industrial Engineering, 55 (2008), 1-14.

[14] P. Li and Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using

the lukasiewicz triangular norm, Soft Computing, 18 (2014), 1399-1404.

[15] C. Lichun and P. Boxing, The fuzzy relation equation with union or intersection preserving

operator, Fuzzy Sets and Systems, 25 (1988), 191-204.

[16] J. Loetamonphong and S. C. Fang, An efficient solution procedure for fuzzy relation equations

with max-product composition, IEEE Transactions on Fuzzy Systems, 7 (1999), 441-445.

[17] J. Loetamonphong and S. C. Fang, Optimization of fuzzy relation equations with max-product

composition, Fuzzy Sets and Systems, 118 (2001), 509-517.

[18] J. Loetamonphong, S. C. Fang and R. E. Young, Multi-objective optimization problems with

fuzzy relation equation constraints, Fuzzy Sets and Systems, 127 (2002), 141-164.

[19] J. Lu and Sh. Fang, Solving nonlinear optimization problems with fuzzy relation equation

constraints, Fuzzy Sets and Systems, 119 (2001), 1-20.

[20] L. Luoh, W. J. Wang and Y. K. Liaw, New algorithms for solving fuzzy relation equations,

Mathematics and Computers in Simulation, 59 (2002), 329-333.

[21] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer, New

York, 1994.

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Sciences, 177 (2007), 4208-4215.

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Sciences, 177 (2007), 4216-4229.

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t-norm composition, Fuzzy Optimization and Decision Making, 3 (2004), 271-278.

[33] Y. K. Wu and S. M. Guu, Minimizing a linear function under a fuzzy max-min relational

equation constraint, Fuzzy Sets and Systems, 150 (2005), 147-162.

[34] Y. K. Wu, S. M. Guu and J. Y. C. Liu, An accelerated approach for solving fuzzy rela-

tion equations with a linear objective function, IEEE Transactions on Fuzzy Systems, 10(4)

(2002), 552-558.

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Systems, 159 (2008), 23-39.

[36] K. Zimmerman, Disjunctive optimization, max-separable problems and extremal algebras,

Theoretical Computer Science, 293 (2003), 45-54.

and Applied Mathematics, 19 (2006), 169-188.

[25] K. Peeva, Composite Fuzzy Relational Equations in Decision Making: Chemistry, In: Cheshankov

B, Todorov M (eds) Proceedings of the 26th summer school applications of mathematics

in engineering and economics, Sozopol (2000), Heron Press, (2001), 260-264.

[26] K. Peeva and Y. Kyosev, Fuzzy Relational Calculus: Theory, Applications and Software,

World Scientific, New Jersey, 2004.

[27] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control, 30

(1976), 38-48.

[28] M. Schoenauer and S. Xanthakis, Constrained GA Optimization, in: S. Forrest (Ed.), Proc.

5th Internat. Conf. on Genetic Algorithms, Morgan Kaufmann, San Mateo, CA, 1993.

[29] B. S. Shieh, Solutions of fuzzy relation equations based on continuoust-norms, Information

Sciences, 177 (2007), 4208-4215.

[30] W. B. Vasantha Kandasamy and F. Smarandache, Fuzzy Relational Maps and Neutrosophic

Relational Maps, Hexis Church Rock, 2004.

[31] Y. K. Wu, Optimization of fuzzy relational equations with max-av composition, Information

Sciences, 177 (2007), 4216-4229.

[32] Y. K. Wu and 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.

[33] Y. K. Wu and S. M. Guu, Minimizing a linear function under a fuzzy max-min relational

equation constraint, Fuzzy Sets and Systems, 150 (2005), 147-162.

[34] Y. K. Wu, S. M. Guu and J. Y. C. Liu, An accelerated approach for solving fuzzy rela-

tion equations with a linear objective function, IEEE Transactions on Fuzzy Systems, 10(4)

(2002), 552-558.

[35] C. T. Yeh, On the minimal solutions of max-min fuzzy relational equations, Fuzzy Sets and

Systems, 159 (2008), 23-39.

[36] K. Zimmerman, Disjunctive optimization, max-separable problems and extremal algebras,

Theoretical Computer Science, 293 (2003), 45-54.

Volume 15, Issue 2

March and April 2018

Pages 109-131