Linear optimization with an arbitrary fuzzy relational inequality

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated whereby an arbitrary function is considered as fuzzy composition. The feasible solution set is determined and compared with some common results in the literature. A necessary and sufficient condition and three other necessary conditions are presented to conceptualize the feasibility of the problem. It is shown that in general a lower bound is always attainable for the optimal objective value. Moreover, we prove that the optimal solution of the problem is obtained if the problem is defined by a non-decreasing or non-increasing function. An algorithm is presented to summarize the process and an example is described to illustrate the algorithm.

[1]  Hong-Xing Li,et al.  Resolution of finite fuzzy relation equations based on strong pseudo-t-norms , 2006, Appl. Math. Lett..

[2]  W. Pedrycz,et al.  Fuzzy Relation Equations and Their Applications to Knowledge Engineering , 1989, Theory and Decision Library.

[3]  Shu-Cherng Fang,et al.  On the resolution and optimization of a system of fuzzy relational equations with sup-T composition , 2008, Fuzzy Optim. Decis. Mak..

[4]  Shu-Cherng Fang,et al.  Solving fuzzy relation equations with a linear objective function , 1999, Fuzzy Sets Syst..

[5]  Witold Pedrycz,et al.  Fuzzy Relation Equations for Compression/Decompression Processes of Colour Images in the RGB and YUV Colour Spaces , 2005, Fuzzy Optim. Decis. Mak..

[6]  Jiranut Loetamonphong,et al.  Multi-objective optimization problems with fuzzy relation equation constraints , 2002, Fuzzy Sets Syst..

[7]  Hsiao-Fan Wang.  A multi‐objective mathematical programming problem with fuzzy relation constraints , 1995 .

[8]  Jiranut Loetamonphong,et al.  Optimization of fuzzy relation equations with max-product composition , 2001, Fuzzy Sets Syst..

[9]  W. Pedrycz,et al.  Fuzzy relation equations on a finite set , 1982 .

[10]  Salvatore Sessa,et al.  Digital watermarking in coding/decoding processes with fuzzy relation equations , 2006, Soft Comput..

[11]  Shu-Cherng Fang,et al.  Solving nonlinear optimization problems with fuzzy relation equation constraints , 2001, Fuzzy Sets Syst..

[12]  W. Pedrycz.  An approach to the analysis of fuzzy systems , 1981 .

[13]  Hans-Jürgen Zimmermann,et al.  Decision Making in Fuzzy Environment , 1985 .

[14]  Spyros G. Tzafestas,et al.  Resolution of composite fuzzy relation equations based on Archimedean triangular norms , 2001, Fuzzy Sets Syst..

[15]  Elie Sanchez,et al.  Resolution of Composite Fuzzy Relation Equations , 1976, Inf. Control..

[16]  Amin Ghodousian,et al.  Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition , 2008, Inf. Sci..

[17]  G. Klir,et al.  Resolution of finite fuzzy relation equations , 1984 .

[18]  Paul P. Wang,et al.  Fuzzy relation equations (I): the general and specialized solving algorithms , 2002, Soft Comput..

[19]  Jiranut Loetamonphong,et al.  An efficient solution procedure for fuzzy relation equations with max-product composition , 1999, IEEE Trans. Fuzzy Syst..

[20]  Esmaile Khorram,et al.  On optimizing a linear objective function subjected to fuzzy relation inequalities , 2009, Fuzzy Optim. Decis. Mak..

[21]  Esmaile Khorram,et al.  An algorithm for solving fuzzy relation equations with max-T composition operator , 2008, Inf. Sci..

[22]  Hsi-Chieh Lee,et al.  On the Optimal Three-tier Multimedia Streaming Services , 2003, Fuzzy Optim. Decis. Mak..

[23]  Yordan Kyosev,et al.  Fuzzy Relational Calculus - Theory, Applications and Software , 2004, Advances in Fuzzy Systems - Applications and Theory.

[24]  Shu-Cherng Fang,et al.  A survey on fuzzy relational equations, part I: classification and solvability , 2009, Fuzzy Optim. Decis. Mak..

[25]  Kaoru Hirota,et al.  On various eigen fuzzy sets and their application to image reconstruction , 2006, Inf. Sci..

[26]  E. Stanley Lee,et al.  Multi-objective optimization with a max-t-norm fuzzy relational equation constraint , 2011, Comput. Math. Appl..

[27]  Shu-Cherng Fang,et al.  On the unique solvability of fuzzy relational equations , 2011, Fuzzy Optim. Decis. Mak..

[28]  Yan-Kuen Wu,et al.  Minimizing a Linear Objective Function with Fuzzy Relation Equation Constraints , 2002, Fuzzy Optim. Decis. Mak..

[29]  Alessio Ishizaka,et al.  Multi-criteria decision analysis , 2013 .

[30]  Amin Ghodousian,et al.  An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition , 2006, Appl. Math. Comput..

[31]  Yan-Kuen Wu,et al.  Minimizing a linear function under a fuzzy max-min relational equation constraint , 2005, Fuzzy Sets Syst..

[32]  Yan-Kuen Wu,et al.  An accelerated approach for solving fuzzy relation equations with a linear objective function , 2002, IEEE Trans. Fuzzy Syst..

[33]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[34]  Zun-Quan Xia,et al.  An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities , 2006, Fuzzy Optim. Decis. Mak..

[35]  E. Sanchez.  SOLUTIONS IN COMPOSITE FUZZY RELATION EQUATIONS: APPLICATION TO MEDICAL DIAGNOSIS IN BROUWERIAN LOGIC , 1993 .

[36]  E. Khorram,et al.  OPTIMIZATION OF LINEAR OBJECTIVE FUNCTION SUBJECT TO FUZZY RELATION INEQUALITIES CONSTRAINTS WITH MAX-AVERAGE COMPOSITION , 2007 .

[37]  Salvatore Sessa,et al.  Fuzzy relation equations for coding/decoding processes of images and videos , 2005, Inf. Sci..

[38]  Antonio di Nola,et al.  Lukasiewicz transform and its application to compression and reconstruction of digital images , 2007, Inf. Sci..

[39]  Witold Pedrycz,et al.  Modularization of fuzzy relational equations , 2002, Soft Comput..

[40]  H. Zimmermann,et al.  Comparison of fuzzy reasoning methods , 1982 .

[41]  Pei-Zhuang Wang,et al.  Latticized linear programming and fuzzy relation inequalities , 1991 .

[42]  Fred Wenstøp,et al.  Deductive verbal models of organizations , 1976 .