The addition-min-product fuzzy relation inequalities were recently introduced to model peer-to-peer file-sharing systems, considering scenarios where the terminals transmit their local file data at varying quality levels. Previous research primarily focused on min–max fuzzy relation programming, which assumed equal treatment for all terminals. However, this approach is inadequate when the terminals need to be differentiated or assigned fixed priority grades. To address network congestion while considering terminal priority, this study explores the lexicographic minimum solution for the addition-min-product FRIs. We propose a detailed algorithm to compute the lexicographic minimum solution, associated with a numerical illustrative example.

Considering the download traffic requirements of the terminals within a certain range, the two-sided fuzzy relation inequalities with addition-min have been recently introduced for modeling the P2P network system. In such a model, the solutions of the FRIs correspond the feasible flow schemes in the P2P network system. To characterize the stability of a given flow scheme, we define the amplitude of the corresponding solution, in this work. If a solution has a bigger amplitude, then it is able to bear a larger perturbation among its components. Accordingly, the corresponding flow scheme is considered to be more stable. For obtaining the amplitude of a given solution, we propose some effective algorithms to compute the upper and lower amplitudes. Following our proposed algorithms, the stability of any pair of solutions could be compared, based on the obtained amplitudes. Our provided results are demonstrated by some numerical examples

Fuzzy relation system with max-min composition could be applied to describe the three-tier multimedia streaming architecture. In such an architecture, the regional servers supply their local multimedia streaming services to the clients. Random fault has not been considered in the relevant existing researches. However, the random fault of a regional server is possible to occur, or even unavoidable, due to some unpredictable reasons. In this article, we consider a random fault occurring at either one of the regional servers. With such a consideration, the corresponding architecture is characterized by the so-called variable-absent fuzzy relation inequalities system with max-min composition. Some properties of our proposed variable-absent fuzzy relation inequalities system are investigated. Based on such properties, we further propose a 2-D path approach for obtaining the complete solution of the variable-absent system. In addition, we also study a corresponding weighted minimax optimization problem motivated by the practical management objective. An effective algorithm is developed for searching the optimal solution, with an illustrative example.

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.

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 a 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 feasible index set (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.

Maximum amplitude and its corresponding maximum-amplitude solution are defined in an addition-min fuzzy relation inequalities system. The maximum-amplitude solution could be viewed as a stable solution, since it is able to survive the biggest fluctuation or perturbation appearing in its components. For solving the maximum amplitude and maximum-amplitude solution, we construct a relevant optimization problem and then convert it into several subproblems. Some auxiliary functions and index sets are introduced to deal with these subproblems. Algorithm I is developed to find their optimal solutions. Based on the optimal solutions of the subproblems, we further propose Algorithm II for obtaining the maximum amplitude and maximum-amplitude solution of an addition-min fuzzy relation inequalities system. Numerical examples illustrate the validness and efficiency of our proposed algorithms.

The addition-min fuzzy relation inequalities could be applied to characterize the peer-to-peer (P2P) network system. This article focuses on the problem that how many arbitrary (or specific) line faults a P2P network system is able to bear at most. In order to find the maximum number of tolerant line faults, we first study the consistency checking of the P2P network system with a given number of arbitrary line faults. Corresponding properties and illustrative example are also presented. Furthermore, the so-called reordered-vector-based (RVB) algorithms I and II are proposed for searching the maximum number of arbitrary and specific line faults in a given P2P network system, respectively. The validness of our proposed RVB algorithms is verified by formally proving and some numerical examples.

The max-product fuzzy relation inequalities have been applied to describe the quantitative relation in a wireless communication base-station system in the existing work. However, the potential line fault between a base station and a signal testing point has not been considered. In this article, we define the so-called random-term-absent max-product fuzzy relation inequalities for describing the wireless communication base-station system with a random line fault. Based on the practical application consideration, we further construct a bilevel optimization problem subject to random-term-absent max-product fuzzy relation inequalities. For solving such bilevel optimization problem, properties of the random-term-absent system, as well as the common single-level fuzzy relation programming, are investigated. At last, a detailed algorithm is developed to find the optimal solution of our proposed bilevel optimization problem.

Max-product fuzzy relation inequalities have been widely applied in foodstuff management, wireless communication base-station system, and wireless server-to-client network system. Under such application background, any solution of the max-product fuzzy relation system represents a feasible scheme. In order to reflect the stability of a given feasible scheme in this article, the concepts of allowable deviation and maximum deviation are defined for a given solution. Moreover, the centralized solution is further defined as the solution, which owns the biggest maximum deviation in a given max-product system. The centralized solution could be viewed as the most stable solution in the system. Two algorithms with polynomial complexities are proposed for obtaining the maximum deviation of a given solution and the centralized solution of a given max-product system, respectively. Numerical examples illustrate the feasibility and efficiency of our proposed algorithms.

In this paper, supereigenvalue and constrained supereigenvalue problems of an addition-min fuzzy matrix are investigated. In a peer-to-peer file sharing system, the download requirement of the terminals could be described by a constant constraint system with addition-min fuzzy relation inequalities. Without considering the constant constraint system, we first study the supereigenvalue problem. While considering the constant constraint system, the so-called constrained supereigenvalue is further proposed and investigated. A constrained supereigenvalue represents the ratio of the data-download quality to the data-sending quality under a constant constraint. In order to arouse the enthusiasm of the terminals to share their local file data, we aim to maximize the constrained supereigenvalue. A nonlinear programming approach is developed to find the unique maximum constrained supereigenvalue, with some illustrative examples.

As an extension of the fuzzy relation linear programming, geometric programming with single-variable term subject to max–min fuzzy relation equations is a nonlinear and non-convex problem. To deal with this problem, we develop the variable substitution method to find out all of the optimal solutions (if not unique). The variable substitution method is carried out by the resolution procedures with an illustrative example. In addition, the proposed method is applied to the optimization management in laboratory Peer-to-Peer network system.

Optimal management model in BitTorrent-like peer-to-peer (P2P) file sharing system can be reduced into a min-max programming problem subject to addition-min fuzzy relation inequality. To deal with such problem, we introduce the concept of optimal vector. Besides, a novel optimal-vector-based (OVB) algorithm is developed to find a minimal optimal solution. Based on practical application consideration, solution obtained by the OVB algorithm is theoretically demonstrated to be better than or equal to that obtained by the existing method. Advantage of our proposed OVB algorithm is illustrated by detailed numerical examples.

A BitTorrent-like peer-to-peer file-sharing system can be reduced into a system of addition-min fuzzy relation inequalities. Addition-min is a new composition, and the solution set of addition-min fuzzy relation inequalities differs much from that of general max-t-norm fuzzy relation inequalities or equations. In order to avoid network congestion and improve the stability of data transmission, a min-max programming problem is proposed subject to addition-min fuzzy relation inequalities in this paper. Based on some relevant theorems on resolution, a novel algorithm is developed step by step to find an optimal solution of the proposed problem. Moreover, an application example is given to illustrate the feasibility and efficiency of the algorithm. Finally, some further discussions are made concerning the optimal solution of the proposed problem.

Motivated by application in BitTorrent-like Peer-to-Peer resource sharing system, we introduce the maximization and minimization semi-latticized fuzzy relation geometric programming problems with max-min composition in this paper. The objective function in the proposed problem in nonlinear and the feasible domain is non-convex. The maximization problem is converted into a fuzzy relation monomial geometric programming and then solved. However, this approach is not effective for the minimization one. The linear programming method is applied to deal with the minimization problem. A step-to-step algorithm is develop to carried out the linear programming method and tow illustrative examples are provided at last.