Solving max-Archimedean t-norm interval-valued fuzzy relation equations

doi:10.1016/j.fss.2021.07.014

Fuzzy Sets and Systems

Volume 440, 30 July 2022, Pages 62-76

https://doi.org/10.1016/j.fss.2021.07.014 Get rights and content

Abstract

This paper discusses a new method for finding the complete set of tolerable solutions of max-Archimedean interval-valued fuzzy relation equations. According to the literature, three types of solution sets, namely; tolerable solution set, united solution set and controllable solution set can be identified with interval-valued fuzzy relation equations. The structure and the properties of the tolerable solution set are studied. The complete set of tolerable solutions can be characterized by one maximum solution and finitely many minimal solutions. An efficient method based on the concept of covering is proposed which computes all minimal solutions. The concept of covering is useful for large size problems in terms of computation. The proposed method is illustrated with some examples.

Introduction

The system of fuzzy relation equations (FRE) was first investigated by Sanchez [32], considering max-min composition. Since, the system of FRE is useful in modelling many systems such as medical diagnosis, image processing, neural networks, fuzzy control, knowledge engineering, decision making (See Di Nola et al. [2], Klir and Yuan [13], Dubois and Prade [3], Pedrycz and Gomide [28], for the various applications of FRE), therefore it is intensively studied by many researchers taking different types of compositions. A general representation of the system of FRE with sup–⊗ composition is: $x \circ A = b$ where $A = {[a_{i j}]}_{m \times n}, 0 \leq a_{i j} \leq 1$ , is an $m \times n$ dimensional real matrix, $b = {[b_{j}]}_{1 \times n}, 0 \leq b_{j} \leq 1$ , is an n-dimensional real vector and ∘ denotes the sup–⊗ composition of x and A, ⊗ being a continuous t-norm. The resolution problem of FRE (1) is to determine an m-dimensional real vector $x = {[x_{i}]}_{1 \times m}, 0 \leq x_{i} \leq 1$ , such that (1) holds.

Determining the solutions of the problem of the type (1) is one of the most important and widely studied problems in the field of fuzzy sets and systems. The general method for solving max-min FRE is given by Higashi and Klir [11]. Di Nola et al. [2] worked on the system of FRE with sup–⊗ composition and proved that the solution set can be completely described by a unique maximum solution and finitely many minimal solutions. After this, numerous methods have been established to identify the minimal solutions of the system of FRE (1) with sup–⊗ composition. The analytical behaviour of FRE and methods for the complete solution set of the system of polynomial lattice equations in distributive lattices is studied by De Baets [1]. The system of FRE with max-product composition is studied by Markovskii [25]. It was shown that solving such type of system is closely related with the covering problem, which is an NP-hard problem. Lin [18] considered the generalization of this problem by taking sup–t FRE, t being any Archimedean t-norm and established a one to one correspondence between the minimal solutions and the irredundant coverings. More work regarding searching the minimal solutions can be found in Han et al. [8], Qu and Wang [30], Wu and Guu [41], Yeh [43], Díaz-Moreno et al. [26]. Sun et al. [34] studied the concept of pre-solution matrices for finding the solution of the system of FRE over a complete Brouwerian lattice. They explored the properties and constructions of pre-solutions.

Fang and Li [5] first introduced a linear optimization problem subject to the system of FRE with max-min composition. They studied the nature of solution set, converted the system to an equivalent 0-1 integer programming problem and solved it using branch and bound technique. Loetamonphong and Fang [22] studied a linear optimization problem with max-product FRE. They derived special characteristics of the feasible domain, optimal solutions and proposed some methods to reduce the original problem. More work in this regard can be found in Qu and Wang [31], Wu and Guu [40], Thapar et al. [35].

A nonlinear optimization problem subject to the system of max-min FRE was first considered by Lu and Fang [24]. They designed a domain specific genetic algorithm by taking advantage of the structure of the solution set of FRE. The individuals from the initial population were chosen from the feasible solution set and were kept within the feasible region during the mutation and crossover operations. Lin et al. [19] applied genetic algorithm for solving a nonlinear optimization model subject to max–t FRE as constraints. Ghodousian and Babalhavaeji [6] proposed an efficient genetic algorithm for solving nonlinear optimization problem subject to FRE with max-Łukasiewicz composition. The proposed algorithm is applied on some test problems and compared with the related algorithms.

Loetamonphong et al. [23] studied the class of optimization problems with multiple objective functions subject to max-min FRE. They proposed a genetic algorithm to find the Pareto optimal solutions. More work on multi-objective optimization problem subject to max-t FRE and max-product FRE can be seen in Guu et al. [7] and Thapar et al. [36] respectively.

When a system is modelled as a system of FRE, then sometimes a transition from FRE to interval-valued FRE becomes crucial. Interval-valued FRE come into existence in following cases:

(i)
When the system of FRE (1) has no solution: Currently, the theory of FRE relies, by and large, on the assumption that the solution set is not empty. However, this is often not the case in practical applications. Since the consistency of the system of FRE is very sensitive to the given information, so the incorrect or incomplete information may result in inconsistency of the system of FRE. Moreover, from a practical point of view, one might not be satisfied with a negative answer when no solution exists.
(ii)
When the membership grades are not real numbers in [0,1], but rather closed intervals in [0,1]: In the system of FRE, the membership grades are real numbers in [0,1]. On the other hand, if the membership grades are closed intervals in [0,1], as in the case of interval-valued fuzzy sets, then the system is modelled as interval-valued FRE. The transition of membership grades from a real number in [0,1] to a closed interval in [0,1] also arises when due to rapid variations in the parameters of a system it becomes difficult to assign membership grade a real number in [0,1].

In both the above cases, FRE (1) are replaced by an interval-valued fuzzy relation equation. The study of interval-valued FRE has been an area of interest among many researchers. Being an extension of the constant-valued FRE, the interval-valued FRE are more flexible in terms of handling impreciseness and uncertainties. Each element of an interval-valued fuzzy relational matrix is a sub-interval in the unit interval [0,1]. This nature enhances the flexibility of applications of interval-valued FRE in real-life problems. The interval-valued FRE has important application in fuzzy control and medical diagnosis.

Wang and Chang [37] gave method for resolving the composite interval-valued FRE. They explored the properties of interval-valued FRE with max-min composition and proposed an algorithm to resolve it. Li and Fang [15] analyzed the properties of the solution set of interval-valued max-min FRE and converted the problem into a system of fuzzy relational inequalities. They suggested more efficient method in terms of computational work for finding the complete solution set. Wang et al. [38] discussed three types of solution sets namely: tolerable solution set, united solution set and controllable solution set for interval-valued max–t FRE. They studied the properties of each solution set and gave relationship between them. For further research in this direction, see Li and Fang [16]. Wang et al. [39] first discussed the solution sets of interval-valued min-s-norm FRE. More work in this regard can be found in Xiong and Wang [42], Li et al. [17]. Pȩkala [29] has discussed interval-valued fuzzy relations as an important tool that may be used to model imperfect information in a better way, especially when facts are imperfectly defined and knowledge is imprecise.

The tolerable solutions of interval-valued FRE are important in fuzzy control problems. Thus, in this paper our aim is to find the complete solution set of tolerable solutions of max-Archimedean interval-valued FRE. In Wang et al. [38], a method is proposed for finding the complete solution set of tolerable solutions of max-Archimedean interval-valued FRE. Since this method finds potential minimal solutions which consist of both minimal and nonminimal solutions, thus it may involve more computational work for large size problems. This motivates us to propose an efficient method, which directly computes all minimal solutions. In Section 2, some basic concepts of the system of interval-valued linear equations are recalled. Section 3 introduces the system of interval-valued FRE, their properties and the structure of the solution set. In Section 4, the concept of covering is proposed for finding the tolerable solution set of interval-valued FRE. In Section 5, the method for finding irredundant coverings for the covering problem is explained. The proposed method is illustrated using some examples in Section 6.

Section snippets

Preliminary properties: system of interval-valued linear equations

The system of interval-valued linear equations is a direct extension of the system of linear equations where the scalar coefficients are generalized by intervals. The detailed discussion about the generalization of usual linear systems to interval-valued linear systems can be found in Neumaier [27], Kulpa and Roslaniec [14], Lodwick and Dubois [21]. A real-valued system of linear equations is given as: $x A = b$ where A is an $n \times n$ dimensional real-valued matrix and b is an n-dimensional real-valued

System of interval-valued FRE

Let $I = {1, 2, \dots, m}$ and $J = {1, 2, \dots, n}$ be the index sets, $A^{-} = {[a_{i j}^{-}]}_{m \times n}$ and $A^{+} = {[a_{i j}^{+}]}_{m \times n}$ with $0 \leq a_{i j}^{-} \leq a_{i j}^{+} \leq 1$ , $\forall i \in I, \forall j \in J$ , be $m \times n$ dimensional real matrices; $b^{-} = {[b_{j}^{-}]}_{1 \times n}$ and $b^{+} = {[b_{j}^{+}]}_{1 \times n}$ with $0 \leq b_{j}^{-} \leq b_{j}^{+} \leq 1$ , $\forall j \in J$ , be n-dimensional real vectors. Consider an $m \times n$ interval-valued matrix $A^{I} = [A^{-}, A^{+}] = {[[a_{i j}^{-}, a_{i j}^{+}]]}_{m \times n}$ , formed by $A^{-}$ and $A^{+}$ and an n-dimensional interval-valued vector $b^{I} = [b^{-}, b^{+}] = {[[b_{j}^{-}, b_{j}^{+}]]}_{1 \times n}$ , formed by $b^{-}$ and $b^{+}$ . An interval-valued FRE is represented as: $x \circ A^{I} \approx b^{I}$ where ≈ denotes the relation between

The covering problem

In this section, we propose some necessary and sufficient conditions to convert the system of interval-valued FRE (4) into an equivalent covering problem. The equivalent covering problem is solved to find the minimal solutions of the original problem.

Definition 3

For $x = {[x_{i}]}_{i \in I} \in X (A^{I}, b^{I})$ , $x_{i}$ is called a binding variable if $x_{i} \otimes a_{i j}^{-} \geq b_{j}^{-}$ holds for some equation $j \in J$ and a constraint $j \in J$ is said to be a binding constraint if $x_{i} \otimes a_{i j}^{-} \geq b_{j}^{-}$ holds for some $i \in I$ .

Let $X^{T} (A^{I}, b^{I}) \neq \emptyset$ , then define $I_{j} = {i \in I | {\hat{x}}_{i} \otimes a_{i j}^{-} \geq b_{j}^{-}}, \forall j \in J$ and

Minimal solutions of the problem (4)

Let us consider that after the simplification through redundant coverings, the covering table is simplified as much as possible. Consider some methods of finding the irredundant coverings with the help of the simplified covering table.

The simplified covering table after removing redundant coverings is shown in Table 2.

Table 2 consists of two irredundant coverings ${r_{1}}$ and ${r_{2}, r_{3}, \dots, r_{m}}$ . We consider an algebraic method of searching irredundant coverings by considering the simplified covering

Numerical illustrations

Example 1

Consider the following system of max-product interval-valued FRE with $x \otimes a = x \cdot a$ , $x \circ A^{I} \approx b^{I}$ where

A^{I} = [\begin{matrix} [0.51, 0.80] & [0.34, 0.72] & [0.17, 0.39] \\ [0.42, 0.58] & [0.60, 0.96] & [0.30, 0.80] \\ [0.60, 0.65] & [0.12, 0.52] & [0.20, 0.70] \\ [0.35, 0.82] & [0.25, 0.91] & [0.36, 0.60] \end{matrix}]

b^{I} = [\begin{matrix} [0.30, 0.60] & [0.20, 0.60] & [0.10, 0.40] \end{matrix}]

In this problem, we have $A^{-} = [\begin{matrix} 0.51 & 0.34 & 0.17 \\ 0.42 & 0.60 & 0.30 \\ 0.60 & 0.12 & 0.20 \\ 0.35 & 0.25 & 0.36 \end{matrix}]$ $A^{+} = [\begin{matrix} 0.80 & 0.72 & 0.39 \\ 0.58 & 0.96 & 0.80 \\ 0.65 & 0.52 & 0.70 \\ 0.82 & 0.91 & 0.60 \end{matrix}]$ $b^{-} = [\begin{matrix} 0.30 & 0.20 & 0.10 \end{matrix}]$ $b^{+} = [\begin{matrix} 0.60 & 0.60 & 0.40 \end{matrix}]$

Step 1

Compute the maximum solution $\hat{x}$ , according to (6). We get ${\hat{x}}_{1} = 0.75, {\hat{x}}_{2}$

Conclusion

In this study, the system of interval-valued FRE with max-Archimedean composition is considered. Some basic properties related to max-Archimedean interval-valued FRE are studied. The complete solution set of interval-valued FRE contains one maximum solution and finitely many minimal solutions. The maximum solution can be computed explicitly using a formula. For minimal solutions, the considered problem is converted to an equivalent covering problem. This is then solved to determine all

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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Solving max-Archimedean t-norm interval-valued fuzzy relation equations

Abstract

Introduction

Section snippets

Preliminary properties: system of interval-valued linear equations

System of interval-valued FRE

The covering problem

Minimal solutions of the problem (4)

Numerical illustrations

Conclusion

Declaration of Competing Interest

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Appl. Soft Comput.

Comput. Math. Appl.

Appl. Math. Lett.

Fuzzy Sets Syst.

Inf. Sci.

Fuzzy Sets Syst.

Expert Syst. Appl.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Inf. Control

Fuzzy Sets Syst.

Appl. Soft Comput.