# OPTIMIZATION OF LINEAR OBJECTIVE FUNCTION SUBJECT TO FUZZY RELATION INEQUALITIES CONSTRAINTS WITH MAX-AVERAGE COMPOSITION

Document Type: Research Paper

Authors

FACULTY OF MATHEMATICS AND COMPUTER SCIENCE, AMIRKABIR UNIVERSITY OF TECHNOLOGY, TEHRAN 15914, IRAN

Abstract

In this paper, the finitely many constraints of a fuzzy relation
inequalities problem are studied and the linear objective function on the region
defined by a fuzzy max-average operator is optimized. A new simplification
technique which accelerates the resolution of the problem by removing the
components having no effect on the solution process is given together with an
algorithm and a numerical example to illustrate the steps of the problem
resolution process.

Keywords

### References

 K. -P. Adlassnig, Fuzzy set theory in medical diagnosis, IEEE Trans. Systems Man
Cybernet., 16 (1986), 260-265.
 M. M. Brouke and D. G. Fisher, Solution algorithms for fuzzy relation equations with
max-product composition, Fuzzy Sets and Systems, 94 (1998), 61-69.
 E. Czogala and W. Pedrycz, Control problems in fuzzy systems, Fuzzy Sets and Systems,
7 (1982), 257-273.
 E. Czogala and W. Predrycz, On identification in fuzzy systems and its applications in
control problem, Fuzzy Sets and Systems, 6, 73-83.
 E. Czogala, J. Drewniak and W. Pedrycz, Fuzzy relation equations on a finite set, Fuzzy
Sets and Systems, 7 (1982), 89-101.
 A. Di Nola, Relational equations in totally ordered lattices and their complete resolution,
J. Math. Appl., 107 (1985), 148-155.
 A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez, Fuzzy relational equations and their
applications in knowledge engineering, Dordrecht: Kluwer Academic Press,1989.
 S. -C. Fang and G. Li, Solving fuzzy relations equations with a linear objective function,
Fuzzy Sets and Systems, 103 (1999), 107-13.
 S. -C. Fang and S. Puthenpura, Linear optimization and extensions: theory and algorithm,
Prentice-Hall, Englewood Cliffs, NJ, 1993.
 S. Z. Guo, P. Z. Wang, A. Di Nola and S. Sessa, Further contributions to the study of
finite fuzzyrelation equations, Fuzzy Sets and Systems, 26 (1988), 93-104.
 F. -F. Guo and 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.
 M. M. Gupta and J. Qi, Design of fuzzy logic controllers based on generalized
t-operators, Fuzzy Sets and Systems, 40 (1991), 473-486.
 M. Guu and Y. K. Wu, Minimizing a linear objective function with fuzzy relation equation
constraints, Fuzzy Optimization and Decision Making, 12 (2002), 1568-4539.
 S. S. Z. Han, A. H. Song, and T. Sekiguchi, Fuzzy inequality relation system
identification via sign matrix method, Proceeding of 1995 IEEE International
Conference, 3 (1995), 1375-1382.

 M. Higashi and G. J. Klir, Resolution of finite fuzzy relation equations, Fuzzy Sets and
Systems, 13 (1984), 65-82.
 C. F. Hu, Generalized Variational inequalities with fuzzy relation, Journal of
Computationaland Applied Mathematics, 146 (1998), 198-203.
 E. Khorram and A.Ghodousian, Linear objective function optimization with fuzzy
relation constraints regarding max-av composition, Applied Mathematics and
Computation, 173 (2006), 827-886.
 G. Li and S. -C. Fang, Resolution of finite fuzzy resolution equations, Report No. 322,
North Carolina State University, Raleigh, NC, May 1996.
 J. Loetamonphong and S. -C. Fang, Optimization of fuzzy relation equations with maxproduct
composition, Fuzzy Sets and Systems, 118 (2001), 509-517.
 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.
 J. Lu and S. -C. Fang, Solving nonlinear optimization problems with fuzzy relation
equation constraints, Fuzzy Sets and Systems, 119 (2001), 1-20.
 W. Pedrycz, On Generalized fuzzy relational equations and their applications, Journal of
Mathematical Analysis and Applications, 107 (1985), 520-536.
 W. Pedrycz, Proceeding in relational structures: fuzzy relational equations, Fuzzy Sets
and Systems, 40 (1991), 77-106.
 M. Prevot, Algorithm for the solution of fuzzy relations, Fuzzy Sets and Systems,
5 (1985), 319-322.
 E. Sanchez, Resolution of composite fuzzy relation equations, Inform. Control,
30 (1976), 38-48.
 W. B. Vasantha Kandasamy and F. Smarandache, Fuzzy relational maps and
neutrosophic relational maps, Hexis Church Rock 2004 (chapter two).
 P. Z. Wang, How many lower solutions of finite fuzzy relation equations, Fuzzy
Mathematics (Chinese), 4 (1984), 67-73.
 P. Z. Wang, Lattecized linear programming and fuzzy relaion inequalies, Journal of
Mathematical Analysis and Applications, 159 (1991), 72-87.
 W. L. Winston, Introduction to mathematical programming: application and algorithms,
Duxbury Press, Belmont, CA, 1995.
 L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.
 H. T. Zhang, H. M. Dong and R. H. Ren, Programming problem with fuzzy relation
inequality constraints, Journal of Liaoning Noramal University, 3 (2003), 231-233.