Elsevier

Fuzzy Sets and Systems

Volume 447, 15 October 2022, Pages 22-38
Fuzzy Sets and Systems

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

https://doi.org/10.1016/j.fss.2022.03.017Get rights and content

Abstract

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.

Introduction

In the literature, fuzzy relational equations based on various compositions have been investigated. The fuzzy relational equation with max-min composition was first proposed by Sanchez [36], in which a fundamental theorem was given for existence and determination of solutions. Later, Pedrycz [34] studied the commonly used fuzzy relational equations with max-product composition. The investigation of fuzzy relational equations or inequalities has been extended to max-Archimedean t-norm [26], [38], [41], max-t-norm [23], [33], [37], and interval valued with max-t-norm compositions [40], [43]. Bipolar fuzzy relational equations based on max-min or max-product compositions have also been considered in relevant studies [5], [6], [24]. Di Nola et al. [8] proposed that the solution set of max-t-norm fuzzy relational equations can be fully determined from a unique maximum solution and all minimal solutions, and the number of minimal solutions is always finite [16]. An analytic formula can easily be used to obtain the maximum solution. However, it is much more difficult and challenging to find a complete set of minimal solutions, which is an NP problem [31]. This challenge has encouraged many researchers to consider and propose various resolution methods to enrich the theoretical results of max-t-norm fuzzy relational equations [1], [2], [21], [30], [35], [50].

Also many studies have considered dealing with the applicable perspective of fuzzy relational equations or inequalities and all of them relate to optimization problems. Some scholars were interested in the relevant linear optimization problem [9], [10], [11], [13], [18]. The nonlinear optimization problem involving various objective functions with fuzzy relational equations or inequalities constraints has also been studied [22], [29], [32], [51].

Furthermore, fuzzy relational equations and/or inequalities have been applied in different fields, for instance, knowledge engineering [8], fuzzy inference systems [19], medical diagnosis [7], media streaming systems [20], the coding/decoding of images and videos [17], [28], [33], fuzzy control [4], wireless communication management models [47], and fuzzy social network analysis [39].

In the past, the network file-transfer system was mainly based on the client-server model. This model transfers files from the server to the client device when downloading files, such as FTP, HTTP, PUB, etc. However, this file-transmission model may have problems in some special situations. In particular, when there are more clients, the bandwidth requirements will increase accordingly. Furthermore, if there are too many clients simultaneously time, network transmission bottlenecks result. To overcome this limitation, many peer-to-peer (P2P) file-sharing application systems are explored to dramatically change large-scale file transfers. The BitTorrent (BT) system is one of the most important P2P file-sharing applications. This type of file-sharing system is different from the client-server model. Roughly speaking, BT will divide the file into many parts on the server where it is first uploaded. When user A downloads some data (called M part) and user B downloads N part from the server, user A's BT will go to user B's device to download the N part that B has obtained. And user B's BT will download the M part that A has obtained. This peer-to-peer (point-to-point) file-sharing system not only reduces the load on the server, but also speeds up the download speed of the client.

Recently, a system of fuzzy relational inequalities with addition-min composition was first proposed by Li and Yang [25] to model a BitTorrent-like peer-to-peer (BT-P2P) file-sharing transmission system. To represent the file-sharing in BT-P2P systems, this system models that user Aj downloads some parts of the file from m1 users, and user Ai sends the file data with transmission level xi to Aj. The bandwidth between Ai and Aj is aij. Due to the bandwidth limitation, what Aj is receiving from Ai is actually min{xi,aij}. If the aggregated data amount (which may view as a downloaded quality) downloaded by Aj is at least level bj, then the transmission level of the data user Aj receives from m1 users can be denoted by ijmin{xi,aij}bj.

In this paper, we use the following slightly generalized mathematical expression of fuzzy relational inequalities with addition-min composition:i=1mmin{xi,aij}bj,jJ, where xi,aij[0,1], bj(0,) for all iI={1,2,,m} and jJ={1,2,,n}.

Li and Yang [25] discussed some properties to the solution of system (1) and presented the necessary and sufficient conditions for the minimal solution. Based on the results, they presented an algorithm to find a minimal solution. Yang et al. [44] investigated some properties and structure of the solution set of addition-min fuzzy relational inequalities. They pointed out that the minimal solution set of a consistent system (1) may be infinite. Furthermore, when the solution set is non-empty, it is completely determined by a unique maximum solution and an infinite number of minimal solutions.

In the literature, several papers have investigated the cost-management issue for system (1). Yang [42] proposed a linear cost objective function for a minimization problem, i.e., minZ(x)=iIcixi, subject to system (1). It could be shown that an optimal solution in Yang [42] is also a minimal solution of system (1). Yang presented a pseudo-minimal indexes (PMI) algorithm to yield an optimal solution. To consider the priority issue of transmission level in a BT-P2P file-sharing system, Yang et al. [49] proposed the multi-level linear programming problem with the constraints in the system (1). The optimization objective with priority rank is defined such that the first objective is to minimize x1, and the second objective is to minimize x2, and so on, until the last objective is to minimize xm.

Guu and Wu [12] also considered a minimization problem with a linear cost objective function subject to system (1). They transformed this minimization problem into a classic linear programming problem. Their approach can also be used to handle some nonlinear cost objective functions. Interestingly, some objective functions can have an optimal solution but not a minimal solution to system (1). Therefore, Guu et al. [15] focused on finding an optimal solution that also yielded better cost-saving properties. A two-phase approach was proposed to consider the network congestion and transmission cost together. Later, because in reality the system manager may consider several managerial issues simultaneously, a multiple objective programming problem subject to system (1) proposed by Guu and Wu [14] that enables the system manager to consider system congestion, cost and penalty simultaneously.

To improve the stability of data transmission to avoid network congestion, a min-max programming problem subject to a system of fuzzy relational inequalities constraints with addition-min composition was proposed by Yang et al. [48] as follows:MinimizeZ(x)=max{x1,x2,,xm}subject togj(x)=i=1mmin{xi,aij}bj,jJ,0xi1,iI. An algorithm was developed by Yang et al. [48] to convert problem (2) into subproblems according to the number of inequalities. In their algorithm, an optimization problem that contains a single decision variable subject to one inequality constraint can first be transformed by each subproblem. In order to generate an optimal solution to problem (2), Yang et al. then proposed another algorithm to select the largest objective value that derived from the optimal solutions of all the subproblems. Their method always yields an optimal solution with equal value for all decision variables.

By investigating the fundamental structure of problem (2), Chiu et al. [3] proved that when this problem is feasible, there is always an optimal solution with the same value for all its components. Furthermore, this optimal value can be found by solving a single-variable optimization model as following:MinimizeZ(y)=ysubject togj(y)=i=1mmin{y,aij}bj,jJ,y[0,1]. To solve this single-variable optimization model, Chiu et al. [3] proposed two methods (an analytical method and an iterative method) to yield an optimal solution y. The (y,,y) is an optimal solution to problem (2), which is exactly the optimal solution obtained by Yang et al. [48]. Chiu et al. [3] called this optimal value y as the system congestion of system (1) when fulfilling the demands bjs.

Although the problem of single-variable optimization model easily gives an optimal value y and an optimal solution (y,,y) to problem (2), the drawback of this equal-component optimal solution is that it is usually not a minimal solution of system (1). Therefore, it may be “improved” if the system manager wants to seek other optimal solutions (when they exist) with lower-cost performance. To overcome this drawback, Guu et al. [15] proposed a two-phase approach to finding an optimal system congestion solution that also minimizes the cost of transmission. Recently, Yang [45] also noticed this drawback and developed an optimal-vector-based (OVB) algorithm to find a so-called “minimal optimal solution” for problem (2).

The generalized min-max programming problem subject to a system of fuzzy relational inequalities with addition-min composition is studied in this paper as follows:MinimizeZ(x)=max{c1x1,c2x2,,cmxm}subject toi=1mmin{xi,aij}bj,jJ,xi[0,1],iI, where given parameters with aij[0,1],bj(0,), and ci>0 for all iI={1,2,,m}, jJ={1,2,,n}. The ci>0 may represent the unit cost of transmission level xi. The optimal value of problem (3) can be regarded as the system cost to meet the demand bjs of user in the network system. To distinguish the different importance of variables, Lin et al. [27] and Yang et al. [46] proposed and studied the weighted min-max programming problem for problem (3).

The motivation for this generalization is twofold. First, the question is why problem (2) always (when feasible) has the equal-component optimal solution (y,,y)? This phenomenon never occurs when another objective function (say, Z(x)=i=1mcixi) is used. To solve this puzzle, we use Z(x)=max{c1x1,c2x2,,cmxm}. We will show in this paper that when the optimal value is y, an optimal solution isxi=min{1,yci}, i.e., xi={1 if yci>1,yci if yci1,iI. This explains why the min-max programming problem (that corresponds to c1=c2==cm=1) (when feasible) will have an equal-component optimal solution. Furthermore, we will also show that this optimal solution is the maximum optimal solution.

Second, recall that the objective function Z(x)=max{x1,x2,,xm} represents the emphasis on controlling the transmission amounts in the system. Now, in the generalized setting, cixi may represent some cost incurred by client i. Therefore, the Z(x)=max{c1x1,c2x2,,cmxm} may represent the cost management of controlling the worst individual amount among all clients, instead of considering the conventional aggregate cost function Z(x)=i=1mcixi. Furthermore, since model (3) only requires ci>0, the generalized min-max programming problem provides a unified framework for the manager to study system congestion or the worst individual cost when fulfilling all clients' demands bjs.

This paper is organized as follows. In Section 2, we present some preliminaries and how the optimal objective value of problem (3) can be obtained by solving a single-variable optimization model. An analytic method is proposed to find this optimal value. We also show how the maximum optimal solution to the generalized min-max programming problem can be generated from the obtained optimal objective value. A numerical example is provided to illustrate our method. In Section 3, we propose an iterative algorithm for finding an optimal solution to problem (3). Its complexity analysis is also provided. Section 4 contains a straightforward extension of problem (3). Precisely, the Z(x)=max{C1(x1),C2(x2),,Cm(xm)} where Ci(xi) is a nonlinear cost function associated with the decision level xi. Here we assume that Ci(xi) is a continuous and strictly increasing function with the existence of its inverse. Several important results of problem (3) will be presented. Conclusions are in Section 5.

Section snippets

A single-variable optimization model and an analytic method

Some basic definitions and properties of fuzzy relational inequalities with addition-min composition in (1) are first given in this section. We then present new theoretical results to illustrate why a single-variable optimization model can be used to finding the optimal objective value to problem (3). Using these provided properties, we propose an analytic method to finding the optimal value for the generalized min-max programming problem (3).

Definition 1

Let x1=(xi1)iI and x2=(xi2)iI be two vectors. For

An iterative approach

In the previous section, the concept of the analytic method used to find the optimal value for problem (3) is to first solve each constraint Gj(y)=bj for jJ. Then, the optimal value y of problem (3) can be obtained by using the intersection of solution sets of all Gj(y)bj. Obviously, when there are many constraints in problem (3), the analytic method may require more computations to find the optimal value. An iterative approach is developed by using an approximate value to improve this

The extended cost structure

In our problem (3), we assume Z(x)=max{c1x1,,cmxm} as its objective function. In this section, we show that many results obtained in previous sections can be extended naturally for more general cost settings where Z(x)=max{C1(x1),,Cm(xm)}. Here, we assume that all cost functions, Ci(t)>0,iI, are continuous and strictly increasing for t0. Since the proofs of these results are similar to the cases with Z(x)=max{c1x1,,cmxm}, we only present the results. For our convenience, we assume the

Conclusions

In the literature, a system of fuzzy relational inequalities with addition-min composition was applied to model the data transmission in BitTorrent-like peer-to-peer (BT-P2P) file-sharing systems. In order to study some managerial aspects of BT-P2P file-sharing systems, various optimization problems subject to addition-min fuzzy relational inequalities were proposed. To avoid the network congestion and improve the stability of data transmission, Yang et al. [48] thought that minimizing system

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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