MULTI-OBJECTIVE OPTIMIZATION WITH PREEMPTIVE PRIORITY SUBJECT TO FUZZY RELATION EQUATION CONSTRAINTS

Document Type: Research Paper

Authors

1 Faculty of Mathematics and Computer Science, Amirkabir Uni- versity of Technology, 424,Hafez Ave.,15914,Tehran, Iran

2 Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424,Hafez Ave.,15914,Tehran, Iran

Abstract

This paper studies a new multi-objective fuzzy optimization prob-
lem. The objective function of this study has di
erent levels. Therefore, a
suitable optimized solution for this problem would be an optimized solution
with preemptive priority. Since, the feasible domain is non-convex; the tra-
ditional methods cannot be applied. We study this problem and determine
some special structures related to the feasible domain, and using them some
methods are proposed to reduce the size of the problem. Therefore, the prob-
lem is being transferred to a similar 0-1 integer programming and it may be
solved by a branch and bound algorithm. After this step the problem changes
to solve some consecutive optimized problem with linear objective function on
discrete region. Finally, we give some examples to clarify the subject.

Keywords


[1] R. E. Bellman and L. A. Zadeh, Decision making in fuzzy environment, Management Sci.,
17 (1970), 141-164.
[2] M. M. Bourke and D. G. Fisher, Solution algorithms for fuzzy relational equations with max
product composition, Fuzzy Sets and Systems, 94 (1998), 61-69.
[3] L. Chen and P. P. Wang, Fuzzy relation equations (I): the general and specialized solving
algorithm, Soft Computing, 6 (2002), 428-435.
[4] E. Czogala, J. Drewniak and W. Pedrycz, Fuzzy relation equations on a nite set, Fuzzy Sets
and Systems, 7 (1982), 89-101.
[5] F. Di Martino and S. Sessa, Digital watermarking in coding / decoding processes with fuzzy
relation equations, Soft Computing, 10 (2006), 238-243.
[6] F. Di Martino, V. Loia and S. Sessa, A method in the compression / decompression of images
using fuzzy equations and fuzzy similarities, In: Proceedings of Conference IFSA 2003, 29/6-
2/7/2003, Istanbul, Turkey, (2003), 524-527.
[7] A. Di Nola, W. Pedrycz and S. Sessa, Some theoretical aspects of fuzzy relation equations
describing fuzzy systems, Information Sciences, 34 (1984), 241-264.
[8] A. Di Nola, W. Pedrycz and S. Sessa, Fuzzy relation equations and algorithms of inference
mechanism in expert systems, In: M. M. Gupta, A. Kandel, W. Bandler and J. B. Kiszka,
eds, Approximate Reasoning in Expert System, Elsevier Science Publishers (North Holland),
Amsterdam, (1985), 355-367
[9] A. Di Nola, W. Pedrycz, S. Sessa and E. Sanchez, Fuzzy relation equations theory as a basis
of fuzzy modeling: an overview, Fuzzy Sets and Systems, 40 (1991), 415-429.
[10] A. Di Nola and C. Russo, Lukasiewicz transform and its application to compression and
reconstruction of digital images, Information Sciences, 177 (2007), 1481-1498.
[11] A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez, Fuzzy relation equations and their appli-
cations to knowledge engineering, Kluwer, Dordrecht, 1989.
[12] D. Dubois and H. Prade, Fuzzy sets and systems: theory and applications, Academic Press,
New York, 1980.
[13] P. Y. Ekel and F. H. Schu er Neto, Algorithms of discrete optimization and their application
to problems with fuzzy coecients, Information Sciences, 176(19) (2006), 2846-2868.
[14] S. C. Fang and G. Li, Solving fuzzy relational equations with a linear objective function,
Fuzzy Sets and Systems, 103 (1999), 107-113.
[15] S. C. Fang and S. Puthenpura, Linear optimization and extensions: theory and algorithms,
Prentice Hall, Engelwood Cli s, NJ, 1993.
[16] M. J. Fernandez and P. Gil, Some speci c types of fuzzy relation equations, Information
Sciences, 164(1-4) (2004), 189-195.
[17] A. Ghodousian and E. Khorram, Fuzzy linear optimization in presence of fuzzy relation
inequality constraint with max-min composition, Information sciences, 178 (2008), 501-519.
[18] F. F. Gue and Z. Q. Xia, An algorithm for solving optimization problems with one linear
objective function and nitely many constraints of fuzzy relation inequalities, Fuzzy Opti-
mization and Decision Making, 5 (2006), 33-47.
[19] S. M. Guu and Y. K. Wu, Minimizing a linear function with fuzzy relation equation con-
straints, Fuzzy Optimization and Decision Making, 12 (2002), 1568-4539.

[20] S. C. Han and H. X. Li, Notes on pseudo-t-norms and implication operators on a complete
brouwerian lattice and Pseudo-t-norms and implication operators: direct products and direct
product decompositions, Fuzzy Sets and Systems, 13 (1984), 65-82.
[21] S. C. Han, H. X. Li and J. Y. Wang, Resolution of nite fuzzy relation equations based on
strong pseudo-t-norms, Applied Mathematics Letters, 19 (2006), 752-757.
[22] H. Higashi and G. J. Klir, Resolution of nite fuzzy relation equations, Fuzzy Sets and
Systems, 13 (1984), 65-82.
[23] K. Hirota, H. Nobuhara, K. Kawamoto and S. I. Yoshida, On a lossy compres-
sion/reconstruction method based on fuzzy relational equations, Iranian Journal of Fuzzy
Systems, 1(1) (2004), 33-42.
[24] K. Hirota and W. Pedrycz, Fuzzy relational compression, IEEE Transaction on Systems, Man
and Cybernetics, Part B, 29(3) (1999), 407-415.
[25] K. Hirota and W. Pedrycz, Speci city shift in solving, Fuzzy Sets and Systems, 106 (1999),
211-220.
[26] A. F. Jajou and K. Zimmermann, A note on optimization problems with additively separable
objective function and max-separable constraints, International Journal of Pure and Applied
Mathematics, 45(4) (2008), 525-532.
[27] E. P. Klement, R. Mesiar and E. Pap, Triangular norms. Position I: basic analytical and
algebraic propertied, Fuzzy Sets and Systems, 143 (2004), 5-26.
[28] E. P. Klement, R. Mesiar and E. Pap, Triangulr norms. Position paper II, Fuzzy Sets and
Systems, 145 (2004), 411-438.
[29] E. P. Klements, R. Mesiar and E. Pap, Triangular norms. Position paper III, Fuzzy Sets and
Systems, 145 (2004), 439-454.
[30] G. J. Klir and B. Yuan, Fuzzy sets and fuzzy logic: theory and applications, Prentice hell,
PTR, USA, 1995.
[31] C. Lai Hwang and A. S. M. Masud, Multiple objective decision making method and Applica-
tions, Springer verlag Berlin Heidelberg, 1979.
[32] Y. J. Lai and C. L. Hwang, Fuzzy mathematical programming: method and application,
Sprringer-verlag, 1992.
[33] H. C. Lee and S. M. Guu, On the optimal three-tier multimedia streaming services, Fuzzy
Optimization and Decision Making, 2(1) (2002), 31-39.
[34] 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.
[35] J. Loetamonphong and S. C. Fang, Optimization of fuzzy relation equations with max-product
composition, Fuzzy Sets and Systems, 118 (2001), 509-517
[36] V. Loia and S. Sessa, Fuzzy relation equations for coding/decoding processes of images and
videos, Information Sciences, 171 (2005), 145-172.
[37] V. Loia and S. Sessa, Fuzzy relation equations for coding/ decoding processes of images and
videos, Information sciences, 171 (2005), 145-172.
[38] J. Lu and S. C. Fang, Solving nonlinear optimization problems with fuzzy relation equations
constraints, Fuzzy Sets and Systems, 119 (2001), 1-20.
[39] A. V. Markovskii, On the relation between equations with max-product composition and the
covering problem, Fuzzy Sets and Systems, 153 (2005), 261-273.
[40] S. Ming Guu and Y. Kuen Wu, Minimizing a linear objective function with fuzzy relation
equation constraints, Fuzzy Optimization and Decision Making, 1 (2002), 347-360.
[41] A. A. Molai and E. Khorram, An algorithm for solving fuzzy relation equations with max-T
composition operator, Information Sciences, 178(5) (2008), 1293-1308.
[42] H. Nobuhara, B. Bede and K. Hirota, On various Eigen fuzzy sets and their application to
image reconstruction, Information Sciences, 176 (2006), 2988-3010.
[43] H. Nobuhara, K. Hitora, F. Di Martino, W. Pedrycz and S. Sessa, Fuzzy relation equations
for compression/ decompression processes of colure images, In the RGB and YUV Colour
Spaces, 2005.
[44] H. Nobuhara, K. Hitora and W. Pedrycz, Relational image compression: optimizations
through the design of fuzzy coders and YUV colour space, Soft Computing, 9 (2005), 471-479.

[45] H. Nobuhara, W. Pedrycz and K. Hitora, A digital watermarking algorithm using image
compression method based on fuzzy relational equations, In Proceedings of FUZZ-IEEE, IEEE
Press, 2 (2002), 1568-1573.
[46] H. Nobuhara, W. Pedrycz and K. Hitora, Fast solving method of fuzzy relational equation
and its application to lossy image compression, IEEE Trans Fuzzy Sys, 8 (2000), 325-334.
[47] W. Pedrycz, An approach to the analysis of fuzzy systems, Int. J. Control, 34 (1981), 403-421.
[48] W. Pedrycz, Fuzzy relational equations with generalized connectives and their applications,
Fuzzy Sets and Systems, 5 (1983), 185-201.
[49] W. Pedrycz, Inverse problem in fuzzy relational equations, Fuzzy Sets and Systems, 36 (1990),
277-291.
[50] W. Pedrycz, On generalized fuzzy relational equations and their applications, Journal of
Mathematical Analysis and Applications, 107 (1985), 520-536.
[51] W. Pedrycz, s-t fuzzy relational equations, Fuzzy Sets and Systems, 59 (1993), 189-195.
[52] W. Pedrycez and A. V. Vasilakos, Modularization of fuzzy relational equations, Soft Com-
puting, 6 (2002), 33-37.
[53] K. Peeva and Y. Kyosev, Fuzzy relational calculus, theory, applications and software, Fuzzy
Sets and Systems, 147 (2004), 363-383.
[54] I. Per lieva, Fuzzy function as an approximate solution to a system of fuzzy relation equa-
tions, Fuzzy Sets and Systems, 147 (2004), 263-383.
[55] I. Per lieva and V. Novak, System of fuzzy relation equations as a continuous model of
IF-THEN rules, Information Sciences, 177(16) (2007), 3218-3227.
[56] X. B. Qu and X. P. Wang, Minimization of linear objective functions under the constraints
expressed by a system of fuzzy relation equations, Information Sciences, 178 (2008), 3482-
3490.
[57] J. L. Ringues, Multiobjective optimization: Behavioral and computational comsiderations,
Kluwer Academic publishers, 1992.
[58] M. Sakawa, Fuzzy sets and interactive multi - objective optimization, Plenum Press, 1993.
[59] E. Sanchez, Resolution of composite fuzzy relation equation, Information and Control, 30
(1978), 38-48.
[60] E. Sanchez, Solutuon in composite fuzzy relation equations: application to medical diagnosis
in Brouwerian logic, In Fuzzy Automata and Decision Processes, (Edited by M. M. Gupta,
G. N. Saridis and B R Games), North-Holland, New York, (1077), 221-234.
[61] B. S Shieh, Deriving minimal solutions for fuzzy relation equations with max-product com-
position, Information Sciences, 178(19) (2008), 3766-3774.
[62] E. Shivanian, E. Khorram and A. Ghodousian, Optimization of linear objective function sub-
ject to fuzzy relation inequalities constraints with max-average composition, Iranian Journal
of Fuzzy Systems, 4(2) (2007), 15-29.
[63] E. Shivanian and E. Khorram, Optimization of linear objective function subject to fuzzy rela-
tion inequalities constraints with max-product composition, Iranian Journal of Fuzzy Systems,
7(3) (2010), 51-71.
[64] T. Tsabadze, The reduction of binary fuzzy relations and its applications, Information Sci-
ences, 178(2) (2008), 562-572.
[65] S. Wang, S. C. Fang and H. L. W. Nuttle, Solution sets of interval - valued fuzzy relational
equations, Fuzzy Optimization and Decision Making, 2 (2003), 23-35.
[66] H. F. Wang, A multi - objective mathematical programming problem with fuzzy relation
consteaints, Journal of Multi - Criteria Decision Analysis, 4 (1995), 23-35.
[67] P. Z. Wang, Lattecized linear programming and fuzzy relation inequalities, Journal of Math-
ematical Analysis and Applications, 159 (1991), 72-87.
[68] G. J. Wang, On the logic foundation of fuzzy reasoning, Information Sciences, 117(1) (1999),
47-88.
[69] F.Wenstop, Deductive verbal models of organizations, Int. J. Man- Machine Studies, 8 (1976),
293-311.
[70] W. L. Winston, Introduction to mathematical programming: application and algorithms,
Duxbury Press. Belmont. Ca, 1995.

[71] Y. K. Wu, S. M. Guu and Y. C. Liu, An accelerated approach for solving fuzzy relation
equations with a linear objective function, LEEE Transactions on Fuzzy Systems, 10(4)
(2002), 552-558.
[72] Y. K. Wu and S. M. Guu, A note on fuzzy relation equation programming problems with
max-strict-t-norm composition, Fuzzy Optimization and Decision Making, 3 (2004), 271-278.
[73] 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.
[74] Q. Q. Xiong and X. P. Wang, Some properties of sup-min fuzzy relational equations on
in nite domains, Fuzzy Sets and Systems, 151 (2005), 393-402.
[75] L. A. Zadeh, Is there a need for fuzzy logic, Information Sciences, 178(10) (2008), 2751-2779.
[76] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outline, Information
Sciences, 172(1-2) (2005), 1-40.
[77] H. J. Zimmerman, Fuzzy set theory, and its applications, Kluwer Academic Publisher, Third
Edition, 1996.