Fuzzy Relation Inequalities Research Articles

Resolution of the sec-max-product fuzzy relation inequalities system and its lexicographic minimum solution

Resolution of the sec-max-product fuzzy relation inequalities system and its lexicographic minimum solution

Tri-composed fuzzy relation inequality with weighted-max-min composition and the relevant min-max optimization problem

In the existing works, the classical fuzzy relation inequality (FRI) with double-composed operators has been introduced to describe the P2P network system. In this work, considering the successful rate of the line connection within the P2P network system, we establish the so-called tri-composed FRI, with a weighted-max-min composition, for the first time. We study the solutions induced through paths and further construct the complete solution set of the tri-composed FRI using the path-induced solutions. Moreover, the whole solution set could be fully determined by all the path-induced solutions and its unique maximum solution. Furthermore, considering a specific optimal management objective in the P2P network, we further study the min-max optimization problem with the tri-composed FRI constraints. A path-based algorithm is proposed for handling the problem we studied, accompanied by an illustrative example.

Max-min fuzzy bi-level programming: resource sharing system with application

This study delves into the application of min-max fuzzy relation systems, specifically focusing on equality and inequality, in the examination of educational institution networks and their resource-sharing dynamics, akin to a P2P network structure. Expanding upon this framework, this article intricately explores a mathematical system featuring three or more terminals within an education resource system, each comprising distinct sharing capabilities. The terminals exchange information on various educational facets, encompassing variations and expenditures per share data, within the framework of a programming-type fuzzy objective function, constrained by the parameters of an educational information system. In these scenarios, the transfer of resources from one terminal to another is interlinked, introducing complexities that warrant a comprehensive examination. Notably, the consideration of greater downloading resources in specific cases is proposed for enhanced practicality. The primary objective of this article is to minimize network congestion within different educational networks, given fixed priority grades assigned to the terminals. To achieve this, the concept of a Lexicographic minimum solution is introduced to address max-min fuzzy relation inequalities, aligning with the defined objectives. The article presents a detailed and effective scenario for implementing the Lexicographic minimal solution. For validation purposes, the proposed scenario is supported by illustrative examples, offering tangible insights into its applicability and efficacy.

Weighted minimax programming subject to the max-min fuzzy relation inequalities

<abstract><p>Recently, max-min fuzzy relation inequalities (FRIs) have been used to model a (peer-to-peer) P2P network system. Any feasible scheme in the P2P network system is reflected by a solution of the max-min FRIs. One of the objectives of system managers is to decrease network congestion. To satisfy this objective, we attempt to minimize a weighted minimax function motivated by existing research. As a consequence, we establish a weighted minimax programming model in which the constraint is the max-min FRIs. Our goal in this work is to develop an effective algorithm to obtain the optimal solution of the optimization model. The so-called SCP-based algorithm is proposed to find the optimal solution. A numerical example shows the efficiency of our proposed SCP-based algorithm.</p></abstract>

On the minimal solution for max-product fuzzy relation inequalities

<p>Minimal solutions play a crucial role in constructing the complete solution set of the max-product fuzzy relation inequalities, as well as in solving the corresponding fuzzy relation optimization problems. In this work, we propose a sufficient and necessary condition for checking whether a given solution is minimal in the max-product system. Our proposed approach is useful for eliminating non-minimal solutions from the set of all quasi-minimal solutions. Our proposed checking approach helps reduce computational complexity when solving the max-product system or related optimization problems.</p>

Optimal symmetric interval solution of fuzzy relation inequality considering the stability in P2P educational information resources sharing system

The max-min fuzzy relation inequalities have been recently introduced for modeling the Peer-to-Peer (P2P) educational information resources sharing system. Each solution of the max-min system is indeed a feasible scheme in the corresponding P2P network system. For a given solution, the allowed fluctuation tolerant for all the components, reflects the stability of its corresponding feasible scheme. Considering the stability, we are interested in the maximum fluctuation for a given solution in this work. In order to differentiate the absolute fluctuation and the relative fluctuation, we define and investigate the MAF-based and the MRF-based optimal symmetric interval solutions, respectively. Two independent algorithms are developed for solving these optimal symmetric interval solutions. The numerical examples show that our proposed algorithms are valid and efficient.

Weighted-minimax programming subject to the RTA max-product fuzzy relation inequality system and its optimal strong solution

Recently the max-product fuzzy relation inequalities have been applied to describe the wireless communication basic-station system. In this paper we further define the so-called random-term-absent (RTA) max-product fuzzy relation inequalities, which enable the system to bear a random communication line fault. Moreover, in order to make all basic-stations take part in the communication activity and avoid waste of resources, we investigate the strong solution to the RTA system. In general case, the strong solution set is composed by a unique maximum strong solution and a finite number of minimal strong solutions. Based on the practical application consideration, we propose the corresponding weighted minimax optimization problem. Detailed algorithm is designed for obtaining the optimal strong solution of such an optimization problem, with an illustrative example.

Fixed-Index-Set Based Approach for Solving a Bilevel Fuzzy Relation Programming With Addition-Min Composition

In recent years, the addition-min fuzzy relation inequalities have been adopted to describe the flow constraint in a P2P network system. Each solution of the inequalities represents a feasible flow control scheme. Motivated by some different managerial objectives, several optimization problems subject to the addition-min inequalities have been recently studied. For example, considering the total efficiency for decreasing the network congestion, the linear objective function was adopted. While considering the fairness among the terminals, the min-max objective function was employed. In this work, combining these two objectives, we establish a corresponding bi-level fuzzy relation programming subject to the addition-min inequalities. For solving the bi-level programming, we first investigate some properties of the first-level programming, using the concept of fixed index set. Based on the properties of the first-level programming, our studied bi-level programming could be equivalently converted into a single-level programming and then solved by the existing linear programming approach. Our resolution approach is carried out by the fixed-index-set based algorithm and illustrated by some numerical examples.

A non-linear generalization of optimization problems subjected to continuous max-t-norm fuzzy relational inequalities

A non-linear generalization of optimization problems subjected to continuous max-t-norm fuzzy relational inequalities

Mathematical insights into interval solutions of fuzzy relation inequalities for effective Information resource sharing in business education

To examine the connectivity of educational and business institution networks and assess their resource accessibility, we have employed min-max fuzzy relation systems. This piece aims to delve into the outcomes resulting from various factors within a three-compartmental financial system. The research undergoes the dynamical max-min fuzzy solutions, which are expressed in an interval format. These solutions take into consideration variables such as the price index, interest rate, investment demand, fluctuations, cost per investment derived from saved funds, and the elasticity of market demand within the commercial sector. These variables are essential for comprehending the dynamics of the three-compartment financial system. It is difficult to obtain the entire set of solutions for the max-min fuzzy relation system using only minimal solutions. Some of these systems can be solved using optimized fuzzy relation models for both maximal and minimal solutions. Therefore, by converting the business market's financial system into a max-min fuzzy relation inequality, the strategy presented in this article tries to get the broadest and changing solution fluctuating between minimal and maximal solutions. With this suggested strategy for solving the problem, the limiting upper and lower values within which the given quantities fluctuate are identified. By measuring the breadth of the resulting interval, the range of fluctuation may be identified, giving a thorough insight into the system's behaviour. Due to the potential instability of market agents and their respective influences, fluctuations may occur in the demand, supply, and prices of indices in the financial system. Consequently, presenting the financial system in the format of the widest fuzzy interval solution can provide easily accessible information to the general public by illustrating the max-min fuzzy relation system of financial dynamics, outlining the necessary steps, and including a numerical example.

Minimal Solutions of Fuzzy Relation Inequalities With Addition-Min Composition and Their Applications

This paper deals with the set of minimal solutions for fuzzy relation inequalities with addition-min composition. We first show a necessary and sufficient condition for a solution of the fuzzy relation inequalities being a minimal one, then we propose an algorithm to find all minimal solutions of the fuzzy relation inequalities. Finally, we use all minimal solutions of the fuzzy relation inequalities to solve the related optimization problems, which will be illustrated by a series of numeral examples.

Optimal Pricing With Weighted Factors in a Product Transportation System Based on the Min-Plus Fuzzy Relation Inequality

In this work, we introduce the fuzzy relation inequalities with min-plus composition for describing the pricing problem in a product transportation system. Some basic properties of the min-plus inequalities system are investigated. We define the concept of feasible index chain and further investigate the corresponding quasi-maximal solution. Moreover, it is found that the solution set of a min-plus system could be generated by a unique minimum solution and a finite number of quasi-maximal solutions. As a consequence, one is able to find the complete solution set of a min-plus system. Besides, motivated by the optimal managerial objective, we establish an optimization model with the min-plus inequalities constraints. A so-called FIS-based algorithm is developed for searching an optimal solution of our studied optimization model. We have provided a simple numerical example for checking the effectiveness of our proposed FIS-based algorithm. The obtained optimal solution would be helpful for the system manager, as an optimal pricing scheme.

Two-sided fuzzy relation inequalities with addition-min composition

Considering the bilateral requirements of the terminals in a P2P network system, we aim to study the two-sided fuzzy relation inequalities with addition-min composition in this work. Each solution of such a two-sided fuzzy relation system is indeed a feasible flow control scheme for the corresponding P2P network system. The major content includes three aspects: (i) finding a minimal solution less than or equal to a given solution; (ii) finding a maximal solution more than or equal to a given solution; (iii) constructing the structure of the solution set to the fuzzy relation system. The purpose of (i) or (ii) is to find some specific minimal or maximal solutions of the two-sided system. We develop two resolution algorithms, i.e., Algorithm I and II, to find the specific minimal and maximal solutions. The computation complexities of both Algorithm I and II are polynomial. Their effectiveness is illustrated by some numerical examples. It is found that the complete solution set of the two-sided system could be fully determined by all the minimal solutions and maximal solutions. Moreover, the solution set might be non-convex.

On Min–Max Optimization for Systems with Addition-Min-Product Fuzzy Relational Inequalities

On Min–Max Optimization for Systems with Addition-Min-Product Fuzzy Relational Inequalities

On the solvability of weakly linear systems of fuzzy relation equations☆

Systems of fuzzy relation equations and inequalities in which an unknown fuzzy relation is on the one side of the equation or inequality are linear systems. They are the most studied ones, and a vast literature on linear systems focuses on finding solutions and solvability criteria for such systems. The situation is quite different with the so-called weakly linear systems, in which an unknown fuzzy relation is on both sides of the equation or inequality. Precisely, the scholars have only given the characterization of the set of exact solutions to such systems. This paper describes the set of fuzzy relations that solve weakly linear systems to a certain degree and provides ways to compute them. We pay special attention to developing the algorithms for computing fuzzy preorders and fuzzy equivalences that are solutions to some extent to weakly linear systems. We establish additional properties for the set of such approximate solutions over some particular types of complete residuated lattices. We demonstrate the advantage of this approach via many examples that arise from the problem of aggregation of fuzzy networks.

Evaluation and its derived classification in a Server-to-Client architecture based on the fuzzy relation inequality

Evaluation and its derived classification in a Server-to-Client architecture based on the fuzzy relation inequality

Generalized min-max programming problems subject to addition-min fuzzy relational inequalities

In this paper, we explore a new generalized min-max programming problem with constraints of addition-min fuzzy relational inequalities. This new generalized min-max programming model provides unified settings that enable the system manager to understand the system congestion level or the worst individual cost in the BitTorrent-like peer-to-peer file-sharing system. Theoretical results are presented to illustrate how the optimal value of generalized min-max programming problem can be obtained by solving a single-variable optimization model. Two approaches (an analytic method and an iterative approach) are provided to solve this single-variable optimization model. The complexity analyses of these two approaches are provided. Numerical examples demonstrate our proposed approaches.

An active-set approach to finding a minimal-optimal solution to the min-max programming problem with addition-min fuzzy relational inequalities

In the literature, a BitTorrent-like peer-to-peer (BT-P2P) file-sharing system has been modeled as a system of fuzzy relational inequalities (FRI) with addition-min composition. And a min-max programming problem has been proposed to study its system congestion. From a cost-saving viewpoint, optimal solutions to the min-max programming problem may not be the minimal-optimal solution. The “minimal” solution gives better cost performance while the “optimal” solution gives the least system congestion. Such a drawback has been studied in the literature. In this paper, we propose a simple active-set approach to finding a cost-saving optimal solution, i.e. a minimal-optimal solution to the min-max programming problem. The complexity of our approach is O(m2n), where m is the number of decision variables and n is the number of constraints. Numerical examples are given to illustrate our procedures. Since our active-set approach depends on the given order of decision variables, by using a different sequence of decision variables, it may be able to find other minimal-optimal solutions. Therefore, our method may be able to provide more choices for the manager to use in decision-making.

Bi-level energy optimization model in smart integrated engineering systems using WSN

Based on fuzzy relational inequality, a bi-level linear programme optimizes the visible light brightness and operating costs of access points in a wireless transmission station system. Consider the first computing problem utilizing a minimum solution matrix. A convex infinite set is generated by a restricted number of closed intervals. Second, computing is an objective-domain nonlinear mathematical optimization problem. A multi-objective optimization problem is used to solve the second programming challenge. The constraint set must be used. Use discrete optimization techniques and branch-and-bound procedures for “digital integer linear programming”. Our technique has been shown to be both practical and successful. The programming complexity increases as the organization expands.

Optimization problems subject to addition-Łukasiewicz-product fuzzy relational inequalities with applications in urban sewage treatment systems

In this paper, fuzzy relational inequalities with addition-Łukasiewicz-product compositions are first introduced for describing the urban sewage treatment systems. It’s well known that a classical fuzzy relational system usually consists of two composed operations. The most commonly used composed operations consists of max–min and max-product. But in our introduced fuzzy relational system, there exist three composed operations, i.e., addition-Łukasiewicz-product. Due to the specific composed operations, the construction for the corresponding solution set is much different. For a consistent fuzzy relation inequality system, composed by addition-Łukasiewicz-product, there exist some maximal solutions and a unique minimum solution. Especially, the number of the maximal solutions might be infinite. In fact, the solution set can be constructed by means of these maximal solutions and the sole minimum solution. Properties of the solutions to an addition-Łukasiewicz-product system is investigated. Following the presented properties, we further study the corresponding single-level and bilevel optimization problems. Two effective resolution approaches are developed and demonstrated by some numerical experiments.