A Linear Programming Approach for Minimizing a Linear Function Subject to Fuzzy Relational Inequalities With Addition–Min Composition

In this paper, we study an optimization problem of minimizing a linear function subject to fuzzy relational inequalities with the addition–min composition. This optimization setting has recently been proposed to model the network cogestion issue when a BitTorrent-like peer-to-peer file-sharing system is used for data transmission. In a 2014 paper, a pseudominimal index (PMI)-based approach was proposed to search for an optimal solution. It turns out that the PMI-based approach may require to solve several to many linear programming problems in order to get an optimal solution. In this paper, we point out that the feasible domain is indeed convex. And we only need to solve a single linear programming problem to generate an optimal solution for the original optimization problem. Furthermore, our approach could be extended to the case with a nonlinear continuous objective function.

[1]  Amin Ghodousian,et al.  Solving linear optimization problems with max-star composition equation constraints , 2006, Appl. Math. Comput..

[2]  Jian-Xin Li,et al.  Fuzzy relation inequalities about the data transmission mechanism in BitTorrent-like Peer-to-Peer file sharing systems , 2012, 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery.

[3]  Yan-Kuen Wu,et al.  Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint , 2010, Fuzzy Sets Syst..

[4]  Robert M. Freund,et al.  Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function , 1991, Math. Program..

[5]  Bing-Yuan Cao,et al.  Min-Max Programming Problem Subject to Addition-Min Fuzzy Relation Inequalities , 2016, IEEE Transactions on Fuzzy Systems.

[6]  Ranjit Biswas,et al.  An application of intuitionistic fuzzy sets in medical diagnosis , 2001, Fuzzy Sets Syst..

[7]  B. De Baets,et al.  Linear optimization with bipolar max-min constraints , 2013, Inf. Sci..

[8]  Yan-Kuen Wu,et al.  Optimization of fuzzy relational equations with max-av composition , 2007, Inf. Sci..

[9]  Shuang Feng,et al.  A Kind of Nonlinear and Non-convex Optimization Problems under Mixed Fuzzy Relational Equations Constraints with Max-min and Max-average Composition , 2012, 2012 Eighth International Conference on Computational Intelligence and Security.

[10]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[11]  Amin Ghodousian,et al.  Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition , 2006, Appl. Math. Comput..

[12]  Elie Sanchez,et al.  Truth-qualification and fuzzy relations in natural languages, application to medical diagnosis , 1996, Fuzzy Sets Syst..

[13]  Bih-Sheue Shieh,et al.  Linear optimization problem constrained by fuzzy max-min relation equations , 2013, Inf. Sci..

[14]  Bing-Yuan Cao,et al.  Multi-level linear programming subject to addition-min fuzzy relation inequalities with application in Peer-to-Peer file sharing system , 2015, J. Intell. Fuzzy Syst..

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

[16]  Jun-Lin Lin,et al.  On fuzzy relational equations and the covering problem , 2011, Inf. Sci..

[17]  Amin Ghodousian,et al.  Solving a linear programming problem with the convex combination of the max-min and the max-average fuzzy relation equations , 2006, Appl. Math. Comput..

[18]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

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

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

[21]  Shao-Jun Yang,et al.  An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition , 2014, Fuzzy Sets Syst..

[22]  Bih-Sheue Shieh,et al.  Minimizing a linear objective function under a fuzzy max-t norm relation equation constraint , 2011, Inf. Sci..

[23]  Paul P. Wang,et al.  Fuzzy Relation Equations (II): The Branch-point-solutions and the Categorized Minimal Solutions , 2006, Soft Comput..

[24]  Witold Pedrycz,et al.  Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction , 2000, IEEE Trans. Fuzzy Syst..

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

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

[27]  A. V. Markovskii,et al.  On the relation between equations with max-product composition and the covering problem , 2005, Fuzzy Sets Syst..

[28]  Yan-Kuen Wu,et al.  An Efficient Procedure for Solving a Fuzzy Relational Equation With Max–Archimedean t-Norm Composition , 2008, IEEE Transactions on Fuzzy Systems.

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

[30]  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..

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