Abbasi Molai, A. (2013). Linear Objective Function Optimization with the Max-product Fuzzy Relation Inequality Constraints. Iranian Journal of Fuzzy Systems, 10(5), 47-61. doi: 10.22111/ijfs.2013.1206

Ali Abbasi Molai. "Linear Objective Function Optimization with the Max-product Fuzzy Relation Inequality Constraints". Iranian Journal of Fuzzy Systems, 10, 5, 2013, 47-61. doi: 10.22111/ijfs.2013.1206

Abbasi Molai, A. (2013). 'Linear Objective Function Optimization with the Max-product Fuzzy Relation Inequality Constraints', Iranian Journal of Fuzzy Systems, 10(5), pp. 47-61. doi: 10.22111/ijfs.2013.1206

Abbasi Molai, A. Linear Objective Function Optimization with the Max-product Fuzzy Relation Inequality Constraints. Iranian Journal of Fuzzy Systems, 2013; 10(5): 47-61. doi: 10.22111/ijfs.2013.1206

Linear Objective Function Optimization with the Max-product Fuzzy Relation Inequality Constraints

^{}School of Mathematics and Computer Sciences, Damghan Univer- sity, Damghan, P.O.Box 36715-364, Iran

Abstract

In this paper, an optimization problem with a linear objective function subject to a consistent finite system of fuzzy relation inequalities using the max-product composition is studied. Since its feasible domain is non-convex, traditional linear programming methods cannot be applied to solve it. We study this problem and capture some special characteristics of its feasible domain and optimal solutions. Some procedures are proposed to reduce and decompose the original problem into several sub-problems with smaller dimensions. Combining the procedures, a new algorithm is proposed to solve the original problem. An example is also provided to show the efficiency of the algorithm.

bibitem{AdPa:Srrfre} G. I. Adamopoulos and C. P. Pappis, {it Some results on the resolution of fuzzy relation equations}, Fuzzy Sets and Systems, {bf 60} (1993), 83--88. bibitem{Ad:Fstmd} K. P. Adlassnig, {it Fuzzy set theory in medical diagnosis}, IEEE Trans. Systems Man Cybernet., {bf 16} (1986), 260--265. bibitem{BoFi:Safremc} M. M. Bourke and D. G. Fisher, {it Solution algorithms for fuzzy relational equations with max-product composition}, Fuzzy Sets and Systems, {bf 94} (1998), 61--69. bibitem{CzDrPe:Frefs} E. Czogala, J. Drewniak and W. Pedrycz, {it Fuzzy relation equations on a finite set}, Fuzzy Sets and Systems, {bf 7} (1982), 89--101. bibitem{CzPe:Ifsacp} E. Czogala and W. Pedrycz, {it On Identification in Fuzzy Systems and its applications in control problem}, Fuzzy Sets and Systems, {bf 6} (1981), 73--83. bibitem{Di:Retolcr} A. Di Nola, {it Relational equations in totally ordered lattices and their complete resolution}, J. Math. Anal. Appl., {bf 107} (1985), 148--155. bibitem{DuPr:Nrpsfso} D. Dubois and H. Prade, {it New results about properties and semantics of fuzzy set-theoretic operators}, In: P. P.Wang, S. K. Chang, eds., Fuzzy Sets, Plenum Press, New York, (1986), 59--75. bibitem{FaLi:Sfrelof} S. C. Fang and G. Li, {it Solving fuzzy relation equations with a linear objective function}, Fuzzy Sets and Systems, {bf 103} (1999), 107--113. bibitem{GuWaDiSe:Fcsffre} S. Z. Guo, P. Z. Wang, A. Di Nola and S. Sessa, {it Further contributions to the study of finite fuzzy relation equations}, Fuzzy Sets and Systems, {bf 26} (1988), 93--104. bibitem{GuXi:Asoploffmcfri} F. F. Guo and Z. Q. Xia, {it An algorithm for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities}, Fuzzy optim. Decis. Making, {bf 5} (2006), 33--47. bibitem{GuWu:Mloffrec} S. M. Guu and Y. K. Wu, {it Minimizing a linear objective function with fuzzy relation equation constraints}, Fuzzy Optim. Decis. Making, {bf 12} (2002), 1568--4539. bibitem{HaSoSe:Firsismm} S. Z. Han, A. H. Song and T. Sekiguchi, {it Fuzzy inequality relation system identification via sign matrix method}, Proceeding of 1995 IEEE International Conference 3, (1995), 1375--1382. bibitem{HiKl:Rffre} M. Higashi and G. J. Klir, {it Resolution of finite fuzzy relation equations}, Fuzzy Sets and Systems, {bf 13} (1984), 65--82.

bibitem{Ho:Opdl} M. Hosseinyazdi, {it The optimization problem over a distributive lattice}, J. Glob. Optim., {bf 41} (2008), 283--298. bibitem{Hu:Gvifr} C. F. Hu, {it Generalized variational inequalities with fuzzy relation}, J. Comput. Appl. Math., {bf 146} (1998), 198--203. bibitem{KaSm:Frmnrm} W. B. V. Kandasamy and F. Smarandache, {it Fuzzy relational maps and neutrosophic relational maps}, hexis church rock, 2004.

bibitem{LiFa:Rosfresc} P. Li and S. C. Fang, {it On the resolution and optimization of a system of fuzzy relational equations with sup-T composition}, Fuzzy Optim. Decis. Making, {bf 7} (2008), 169--214. bibitem{LiFa:Rffre} G. Li and S. C. Fang, {it On the resolution of finite fuzzy relation equations}, OR Report No. 322, North Carolina State University, Raleigh, North Carolina, May 1996. bibitem{LoFa:Ofremc} J. Loetamonphong and S. C. Fang, {it Optimization of fuzzy relation equations with max-product composition}, Fuzzy Sets and Systems, {bf 118} (2001), 509--517. bibitem{LoFaYo:Mopfrec} J. Loetamonphong, S. C. Fang and R. E. Young, {it Multi-objective optimization problems with Fuzzy Relation Equation constraints}, Fuzzy Sets and Systems, {bf 127} (2002), 141--164. bibitem{Pe:Prsfre} W. Pedrycz, {it Proceeding in relational structures: fuzzy relational equations}, Fuzzy Sets and Systems, {bf40} (1991), 77--106. bibitem{PeKy:Frctas} K. Peeva and Y. Kyosev, {it Fuzzy relational calculus: theory, applications and software}, World Scientific, New Jersey, 2004. bibitem{Pr:Asfr} M. Prevot, {it Algorithm for the solution of fuzzy relations}, Fuzzy Sets and Systems, {bf 5} (1981), 319--322. bibitem{Sa:Rcfre} E. Sanchez, {it Resolution of composite fuzzy relation equations}, Information and Control, {bf 30} (1976), 38--48.

bibitem{ThZiZy:Smpoi} U. Thole, H. J. Zimmermann and P. Zysno, {it On the suitability of minimum and product operators for intersection of fuzzy sets}, Fuzzy Sets and Systems, {bf 2} (1979), 167--180. bibitem{Wa:Hmlsffre} P. Z. Wang, {it How many lower solutions of finite fuzzy relation equations}, Fuzzy Mathematics (Chinese), {bf 4} (1984), 67--73. bibitem{WaZhSaLe:Llpfri} P. Z. Wang, D. Z. Zhang, E. Sanchez and E. S. Lee, {it Lattecized linear programming and fuzzy relation inequities}, J. Math. Anal. Appl., {bf 159} (1991), 72--87. bibitem{ZhDoRe:Ppfric} H. T. Zhang, H. M. Dong and R. H. Ren, {it Programming problem with fuzzy relation inequality constraints}, J. Liaon. Nor. Univ., {bf 3} (2003), 231--233. bibitem{ZiZy:Lchd} H. J. Zimmermann and P. Zysno, {it Latent connectives in human decision-making}, Fuzzy Sets and Systems, {bf 4} (1980), 37--51.