Solutions and strong solutions of min-product fuzzy relation inequalities with application in supply chain

Abstract

Min-product fuzzy relation inequalities are introduced to describe the price requirements in the supply chain system. We first investigate the resolution method and structure of the complete solution set to min-product fuzzy relation inequalities. Motivated by the practical application, we further define concept of strong solution. A strong solution represents a feasible pricing scheduling, enabling all the suppliers and retailers profitable in the trade activity. Relevant results about the strong solution are obtained. Based on such results, a step-by-step algorithm is developed for finding all the strong solutions to a system of min-product fuzzy relation inequalities. Numerical examples are given to illustrate the algorithm.

Introduction

Fuzzy relation equation (FRE) has become an important researching branch since it appeared in Sanchez's paper [25] for the first time in 1976. Both theoretical and practical aspects on FRE were developed rapidly and deeply. Classical FRE, with max-min or max-product composition, has been completely solved. It's non-convex solution set (when the solution set is nonempty) has special structure. It could be written as a union of a finite number of closed intervals, who share the same right endpoint, but own different left endpoints. Obviously the same right endpoint is the maximum solution, while left endpoint the minimal solution. Computing all the minimal solutions is important in solving the classical max-T FRE, where T is a left continuous triangular norm. There exist various methods for this [11], [13], [18], [23], [28]. Besides, the equivalent problem and hardness of finding all the minimal solutions were also investigated [3], [6], [16], [17], [19]. In most existing works, the set of all minimal solutions is computed before obtaining the entire solution set. However, Bartl and Prochazka proposed a new representing method of the solution set [4]. In the infinite case, the max-T FRE turns out to be the sup-T FRE. Moreover, the commonly used unit interval $[0,1]$ could be extended to the general complete Brouwerian lattice [24], [27], [29], [30].

As the dual form of the above-mentioned max-T FRE, min-S FRE was also introduced [21], where S represented an s-norm. On the lattice $L^U=[0,1]$, a t-norm (triangular norm) is an associative, commutative and monotone in its first component operation $T:L^U\times L^U\to L^U$, satisfying $T(x,1)=x$ for any $x\in L^U$. While a s-norm (co-triangular norm) is an associative, commutative and monotone in its first component operation $S:L^U\times L^U\to L^U$, satisfying $S(x,0)=x$ for any $x\in L^U$. In the existing literatures, most researchers concentrated on the resolution of max-T FRE. However, as pointed out in [7], the min-S composition is “equally interesting and important, but less well known” than the max-T composition. Min-S FRE could be dealt with by the similar method used in the max-T FRE [22]. A consistent system of min-S FREs has a unique minimum solution and a finite number of maximal solutions. The superiority of the min-max composition rule, over the classical max-min one, was demonstrated by Kundu [10], especially in studying the non-logical fuzzy equivalence relationships. Interval-valued min-S fuzzy relation equations [14], [26] were also studied by Wang and Li et al. Analogous to the concepts in max-T system, the authors introduced three types of solutions in the interval-valued min-S FRE system, i.e. tolerable solution, united solution, and controllable solution.

It was checked in [2], [5] that the sup-T and the inf-residuum are one type of relational composition. Inf-→ fuzzy relation equations with interval-valued parameters and variables were also studied [12]. Necessary and sufficient conditions were given for checking the consistency of the system. The complete solution set of such system was determined by a smallest solution and a finite number of maximal solutions. The authors proposed method for obtaining all the maximal solution and then generated the complete solution set.

In recent years sup-preserving [5] and inf-preserving [8] aggregation structures were considered in the fuzzy relation equations. Motivated by the sup-preserving and inf-preserving aggregation operators, general fuzzy relation equations were introduced [1], [8], in which the classical max-T composition was extended to a general one. In [1], the aggregation operator was assumed to be commutative with infima [20]. J. Medina [20] further weaken this restriction and obtained similar results. Similar procedures were proposed to find the minimal solution set of fuzzy relation equations in a more general setting [20].

Recently fuzzy relation inequalities with addition-min composition was introduced to describe the data transmission mechanism in the BitTorrent-like Peer-to-Peer file sharing systems [15], [33], [35]. Analogous to the classical fuzzy relation system, the consistency of such system could be checked by the existence of its maximum solution. However, different to the classical max-T fuzzy relation system, an addition-min system might have infinite number of minimal solutions. And moreover, its complete solution set is always convex when nonempty [33]. Optimization problems with addition-min fuzzy relation inequalities constraint were also considered [9], [31], [32], [34], in order to reduce the degree of network congestion.

In this paper we introduce another kind of fuzzy relation system, namely min-product fuzzy relation inequalities. The min-product composition is different from the classical max-t-norm or min-s-norm one, since product is exactly a triangular norm. The min-product system own its application in the multi-suppliers and multi-retailers supply chain (see next section). Moreover, we define the new concept of strong solution and provide its resolution method in this paper.

The rest of this paper is organized as follows. Application background and problem statement of the min-product fuzzy relation inequalities is introduced in Section 2. In Section 3 we provide the resolution method and structure of the complete solution set of the min-product system. Strong solution of such system is defined and investigated in Section 4, while its resolution algorithm is further studied in Section 5. Sections 6 and 7 are simple discussion and conclusion.