scholarly journals Resolution of Max-Product Fuzzy Relation Equation with Interval-Valued Parameter

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-16
Author(s):  
Xiaobin Yang ◽  
Haitao Lin ◽  
Gang Xiao ◽  
Huanbin Xue ◽  
Xiaopeng Yang
Keyword(s):  
Complete Solution ◽  
Order Relation ◽  
Fuzzy Relation ◽  
Solution Set ◽  
Network System ◽  
Resolution Method ◽  
Index Set ◽  
Fuzzy Relation Equation ◽  
Fuzzy Relation Equations ◽  
Interval Valued

Considering the application background on P2P network system, we investigate the max-product fuzzy relation equation with interval-valued parameter in this paper. Order relation on the set of all interval-valued numbers plays key role in the construction and resolution of the interval-valued-parameter fuzzy relation equation (IPFRE). The basic operations supremum (a∨b) and infimum (a∧b) in the IPFRE should be defined depending on the order relation. A novel total order is introduced for establishing the IPFRE. We also discuss some properties of the IPFRE system, including the consistency and structure of the complete solution set. Concepts of close index set and open index set are defined, helping us to construct the resolution method of the IPFRE system. We further provide a detailed algorithm for obtaining the complete solution set. Besides, the solution set is compared to that of the classical max-T fuzzy relation equations system.

Covering Problem for Solutions of Max-Archimedean Bipolar Fuzzy Relation Equations

2020 ◽  
Vol 28 (04) ◽  
pp. 613-634
Author(s):  
Vijay Lakshmi Tiwari ◽  
Antika Thapar
Keyword(s):  
Integer Programming ◽  
Programming Problem ◽  
Complete Solution ◽  
Fuzzy Relation ◽  
Solution Set ◽  
Covering Problem ◽  
Integer Programming Problem ◽  
Fuzzy Relation Equations ◽  
Reduction Methods

This paper discusses the resolution of max-Archimedean bipolar fuzzy relation equations. In the literature, many methods have been proposed based on 0-1 integer programming problem or reduction methods for the optimization with bipolar fuzzy relation equations. A new concept based on the idea of covering and the notions of leading, non-leading variables are introduced in the present paper for finding the solutions of max-Archimedean bipolar fuzzy relation equations. It is shown that the problem of finding the complete solution set of the system of max-Archimedean bipolar fuzzy relation equations is equivalent to solving a covering problem and the solutions of such equations correspond to irredundant coverings of the covering problem. The proposed method is illustrated with some examples.

Complete solution sets of inf-→ interval-valued fuzzy relation equations

2013 ◽  
Vol 219 ◽  
pp. 111-123 ◽  
Author(s):  
De-chao Li ◽  
Yong-jian Xie ◽  
Sheng-ling Geng
Keyword(s):  
Complete Solution ◽  
Fuzzy Relation ◽  
Solution Sets ◽  
Fuzzy Relation Equations ◽  
Interval Valued

NEW RESOLUTION OF FINITE FUZZY RELATION EQUATIONS WITH MAX-MIN COMPOSITION

2008 ◽  
Vol 16 (01) ◽  
pp. 19-33 ◽  
Author(s):  
BIH-SHEUE SHIEH
Keyword(s):  
Fast Algorithm ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Fuzzy Relation ◽  
Necessary And Sufficient Conditions ◽  
Fuzzy Relation Equation ◽  
Fuzzy Relation Equations ◽  
Necessary And Sufficient

The work considers the problem of solvability of a fuzzy relation equation with max-min composition. It presents the necessary and sufficient conditions for the existence of solutions, then derives a fast algorithm for finding all minimal solutions. The results are compared with those of previous publications regarding this subject.

MINIMIZING A LINEAR OBJECTIVE FUNCTION SUBJECT TO FUZZY RELATION EQUATIONS CONSTRAINTS WITH MAX-HAMACHER PRODUCT COMPOSITION

2008 ◽  
Vol 25 (02) ◽  
pp. 243-266
Author(s):  
ALI ABBASI MOLAI ◽  
ESMAILE KHORRAM
Keyword(s):  
Objective Function ◽  
Efficient Algorithm ◽  
Original Problem ◽  
Feasible Solution ◽  
Optimal Solution ◽  
Fuzzy Relation ◽  
Solution Set ◽  
Fuzzy Relation Equations ◽  
Product Composition ◽  
Linear Objective Function

In this paper, an optimization model with a linear objective function subject to a system of fuzzy relation equations, using max-Hamacher product composition operator, is presented. Since its nonempty feasible solution set is in general a nonconvex set, conventional linear programming methods are not suitable to solve such a problem, so an efficient solution procedure for such problems is necessary. In this paper, the feasible solution set of this problem is studied at first. Then, one efficient algorithm (i.e. tabular method algorithm) is proposed in order to solve the problem. Some procedures are also presented to reduce the original problem. Then, the reduced problem is decomposed (if possible) into several sub-problems with smaller dimensions, so solving them becomes very easier by the algorithm. By combining the algorithm and these procedures, another more efficient algorithm is suggested in order to obtain the optimal solution of the original problem. Some numerical examples are also given to illustrate the algorithms.

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