Resolution of bipolar fuzzy relation equations with max-Łukasiewicz composition

Abstract

The bipolar system of fuzzy relation equations with the max-Łukasiewicz composition is investigated in this work. A bipolar path approach is proposed for such a system. It is found that the complete solution set of the bipolar system is fully determined by its conservative bipolar paths, which are finite. Our proposed resolution approach is performed using the so-called path-based algorithm, step by step, and illustrated with numerical examples. Moreover, the global and local minimal (or maximal) solutions are discussed in this paper with a comparison to those of a classical unipolar system.

Introduction

E. Sanchez [19] first opened the research direction on fuzzy relation equation (FRE). Because of its important application in various practical fields, many scholars have focused on the theoretical research of FRE, including its resolution approach and some specific optimization problems with FRE constraints. The solution set of a consistent system of FREs with the max-t-norm composition could be written as a finite number of closed intervals, whose begin point is one of the minimal solutions, and the end point is the unique maximum solution. Hence, for solving the FREs system, one should find out all its minimal solutions. However, solving all the minimal solutions has been proved to be an NP-hard problem [3], as it is equivalent to the set covering problem [13], [18]. There is no polynomial-time algorithm. Resolution of an FREs system remains a hot research topic in the fuzzy set theory [17].

In addition to the resolution of the FREs, the optimization problem with an FREs constraint is another relevant research topic [9], [21], [22]. Most of the optimization models have been found and established in the practical application background. In these problems, the authors must provide an effective algorithm to compute the optimal solution [10]. In general, the solution algorithm is related to the feasible domain, i.e., the complete solution set of a group of FREs. The properties of the FRE greatly contribute to the solution algorithm.

Bipolarity exists widely in human understanding of information and preference [6]. Bipolar representation seems to be useful in the development of intelligent technologies. In [6] the authors tried to show how the possibility theory framework is convenient for handling bipolar representations. There are three forms of bipolarity: symmetric univariate, dual bivariate (or symmetric bivariate unipolarity), and asymmetric (or heterogeneous) bipolarity [6]. D. Dubois and H. Prade shown an overview of the asymmetric bipolar representation of positive and negative information in possibility theory [7]. The bipolar representation was applied to distinguish between negative and positive information in preference modeling [4], [7].

Recently, the concept of FRE was generalized to a so-called bipolar fuzzy relation equation. S. Freson and B. De Baets et al. [8] proposed the concept of bipolar fuzzy relation equation for the first time, with application in public awareness of the products for a supplier. The bipolar fuzzy relation equation with max-min composition could be expressed as [8]

$$\underset{j=1,2,\cdots ,n}{\max }\max \{\min \{a_{ij}^{+},x_j\},\min \{a_{ij}^{-},{\overline{x}}_j\}\},i=1,2,\cdots ,m,$$

where $a_{ij}^{+},a_{ij}^{-},x_j,{\overline{x}}_j\in [0,1]$ and ${\overline{x}}_j=1-x_j$. To optimize (increase) the suppliers' benefits, which are related to the public awareness, the authors introduced the corresponding optimization problem, maximizing a linear objective function with the bipolar fuzzy relation equations constraint. Until now, few studies have reported the relevant results on the bipolar fuzzy relation equation. Most of them focused on the optimization problems subject to bipolar fuzzy relation equations [1], [2], [16].

As noted in [11], [14], there are three commonly used compositions in the bipolar fuzzy relation equation: max-min [12], [15], max-product [5] and max-Łukasiewicz [23]. The bipolar max-Łukasiewicz equation constrained linear optimization problem has been studied in [11], [14]. After investigation of the structure and properties of the feasible domain (solution set of system of bipolar max-Łukasiewicz fuzzy relation equations), the authors searched the optimal solution form the potential maximal and minimal solutions of the constraints. Some basic properties of bipolar fuzzy relation equations [5], [12], [15], as well as its corresponding optimization problem [1], [2], [16] have been investigated.

Similar to the unipolar fuzzy relation system, the resolution of a bipolar fuzzy relation system is also an important issue in the relevant research. However resolution method for obtaining the complete solution set of bipolar max-Łukasiewicz fuzzy relation equations (see system (6) in next section) has not been found in the existing works. It was shown in [11], [14] that the optimal solution of the linear optimization problem with bipolar max-Łukasiewicz fuzzy relation equations constraint could be selected from the maximal solutions or the minimal solutions of system (6). Hence, resolution of system (6), especially all the maximal and minimal solutions, appears to be important. The purpose of this work is to propose effective method for obtaining all the solutions of system (6).

In this paper, we aim to investigate the resolution of the bipolar fuzzy relation equation with the max-Łukasiewicz composition. The remainder of this paper is organized as follows. Sec. 2 shows the necessary definitions and basic properties of the bipolar fuzzy relation equations with the max-Łukasiewicz composition. The major result is presented in Sec. 3, where we develop the bipolar path approach in detail. Based on the bipolar path approach, the structure of the complete solution set is obtained. The path-based algorithm and illustrative examples are shown in Sec. 4. Finally, Secs. 5 and 6 show a simple discussion and the conclusion, respectively.