RESOLUTION OF NONLINEAR OPTIMIZATION PROBLEMS SUBJECT TO BIPOLAR MAX-MIN FUZZY RELATION EQUATION CONSTRAINTS USING GENETIC ALGORITHM

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


[1] S. Abbasbandy, E. Babolian and M. Allame, Numerical solution of fuzzy max-min systems,
Applied Mathematics and Computation, 174 (2006), 1321-1328.
[2] M. Allame and B. Vatankhahan, Iteration algorithm for solving Ax=b in max-min algebra,
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.
[4] S. C. Fang and G. Li, Solving fuzzy relation equations with a linear objective function, Fuzzy
Sets and Systems, 103 (1999), 107-113.
[5] D.B. Fogel, Evolving Artifi cial Intelligence, Ph.D. Thesis, University of California, 1992.
[6] S. Freson, B. De Baets and H. De Meyer, Linear optimization with bipolar max-min con-
straints, Information Sciences, 234 (2013), 3-15.
[7] R. Hassanzadeh, E. Khorram, I. Mahdavi and N. Mahdavi-Amiri, A genetic algorithm for
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 Arti ficial 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.
[12] E. Khorram, A. Ghodousian and A. A. Molai, Solving linear optimization problems with max-
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 modi fied 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.
[22] Z. Michalewicz and C. Janikow, Handling Constraints in Genetic Algorithms, Proc. 4th
Internat. Conf. on Genetic Algorithms, Morgan Kaufmann Publishers, Los Altos, CA, (1991),
151-157.
[23] Z. Michalewicz and C. Janikow, Genetic algorithms for numerical optimization, Statistics
and Computing, 1 (1991), 75-91.

[24] K. Peeva, Universal algorithm for solving fuzzy relational equations, Italian Journal of Pure
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 Scienti fic, 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.