Monomial geometric programming with fuzzy relation inequality constraints with max-product composition

doi:10.1016/j.cie.2008.08.015

Computers & Industrial Engineering

Volume 56, Issue 4, May 2009, Pages 1386-1392

https://doi.org/10.1016/j.cie.2008.08.015 Get rights and content

Abstract

Monomials function has always been considered as a significant and most extensively used function in real living. Resource allocation, structure optimization and technology management can often apply these functions. In optimization problems the objective functions can be considered by monomials. In this paper, we present monomials geometric programming with fuzzy relation inequalities constraint with max-product composition. Simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. Also, an algorithm and a few practical examples are presented to abbreviate and illustrate the steps of the problem resolution.

Introduction

Fuzzy relation equations (FRE), fuzzy relation inequalities (FRI), and their connected problems have been investigated by many researchers in both theoretical and applied areas (Czogala et al., 1982, Czogala and Predrycz, 1982, Di Nola et al., 1984, Fang and Puthenpura, 1993, Guo et al., 1988, Gupta and Qi, 1991, Higashi and Klir, 1984, Han et al., 1995, Hu, 1998, Li and Fang, 1996, Perfilieva and Novák, 2007, Prevot, 1985, Sheue Shieh, 2007, Wang, 1984, Wang, 1991, Zadeh, 1965, Zener, 1971). Sanchez (1977) started the development of the theory and applications of FRE treated as a formalized model for non-precise concepts. Generally, FRE and FRI have a number of properties that make them suitable for formulizing the uncertain information upon which many applied concepts are usually based. The application of FRE and FRI can be seen in many areas, for instance, fuzzy control, fuzzy decision making, systems analysis, fuzzy modeling, fuzzy arithmetic, fuzzy symptom diagnosis and especially fuzzy medical diagnosis, and so on (see Adlassnig, 1986, Berrached et al., 2002, Czogala and Predrycz, 1982, Di Nola and Russo, 2007, Dubois and Prade, 1980, Loia and Sessa, 2005, Nobuhara et al., 2006, Pedrycz, 1981, Pedrycz, 1985, Perfilieva and Novák, 2007, Vasantha et al., 2004, Wenstop, 1976, Zener, 1971).

An interesting extensively investigated kind of such problems is the optimization of the objective functions on the region whose set of feasible solutions have been defined as FRE or FRI constraints (Brouke and Fisher, 1998, Fang and Li, 1999, Fernandez and Gil, 2004, Guo and Xia, 2006, Guu and Wu, 2002, Higashi and Klir, 1984, Khorram and Hassanzadeh, 2008, Loetamonphong and Fang, 2001; Loetamonphong et al., 2002; Zadeh, 2005). Fang and Li solved the linear optimization problem with respect to FRE constraints by considering the max–min composition (Fang & Li, 1999). The max–min composition is commonly used when a system requires conservative solutions, in the sense that the goodness of one value cannot compensate for the badness of another value (Loetamonphong & Fang, 2001). Recent results in the literature, however, show that the min operator is not always the best choice for the intersection operation. Instead, the max-product composition has provided results better than or equivalent to the max–min composition in some applications (Adlassnig, 1986).

The fundamental result for fuzzy relation equations with max-product composition goes back to Pedrycz (1985). A recent study in this regard can be found in Brouke and Fisher (1998). They extended the study of an inverse solution of a system of fuzzy relation equations with max-product composition. They provided theoretical results for determining the complete sets of solutions as well as the conditions for the existence of resolutions. Their results showed that such complete sets of solutions can be characterized by one maximum solution and a number of minimal solutions. An optimization problem was studied by Loetamonfong and Fang with max-product composition (Loetamonphong & Fang, 2001) which was improved by Guu and Wu by shrinking the search region (Guu & Wu, 2002). The linear objective optimization problem with FRI was investigated by Zhang, Dong, and Ren (2003), where the fuzzy operator is considered as the max–min composition. Also, Guo and Xia presented an algorithm to accelerate the resolution of this problem (Guo & Xia, 2006).

The geometric programming (GP) theory proposed in 1961 by Zeneret al. for first time (Duffin et al., 1967, Peterson, 1976). Business administration, economic analysis, resource allocation and environmental engineering have a large number of applications in GP (Zener, 1971). The fuzzy geometric programming problem proposed by Cao (2001). He considered a few number of power systems problems (Cao, 1999) and also Liu applied it in economic management (Liu, 2004). The fuzzy geometric programming with multi-objective functions has studied by Biswal, 1992, Verma, 1990. In order to show importance of geometric programming and the fuzzy relation equation in theory and applications a fuzzy relation geometric programming problem has proposed by Yang and Cao, 2005a, Yang and Cao, 2005b. Furthermore, they discussed optimal solutions with two kinds of objective functions based on the fuzzy max-product operator. Also, they consider monomial geometric programming with fuzzy relation equation (FRI) constraints with max–min composition (Yang & Cao, 2007).

In this paper, we consider the monomial geometric programming of the FRI with the max-product operator. This problem can be formulated as follows: $\begin{matrix} \min c \prod_{j = 1}^{n} x_{j}^{α_{j}} \\ s . t . A \cdot x ⩾ d^{1} \\ B \cdot x ⩽ d^{2} \\ x \in [0, 1]^{n} \end{matrix}$ where $c, α_{j} \in R, c > 0, A = (a_{ij})_{m \times n}, a_{ij} \in [0, 1], B = (b_{ij})_{l \times n}, b_{ij} \in [0, 1]$ , are fuzzy matrices, $d^{1} = (d_{i}^{1})_{m \times 1} \in [0, 1]^{m}, d^{2} = (d_{i}^{2})_{l \times 1} \in [0, 1]^{l}$ are fuzzy vectors, $x = (x_{j})_{n \times 1} \in [0, 1]^{n} 0$ is an unknown vector, and “ $\cdot$ ” denotes the fuzzy max-product operator as defined below. Problem (1) can be rewritten as the following problem in detail: $\begin{matrix} \min c \prod_{j = 1}^{n} x_{j}^{α_{j}} \\ s . t . a_{i} \cdot x ⩾ d_{i}^{1} i \in I^{1} = {1, 2, \dots, m} \\ b_{i} \cdot x ⩽ d_{i}^{2} i \in I^{2} = {1, 2, \dots, l} \\ 0 ⩽ x_{j} ⩽ 1 j \in J = {1, 2, \dots, n} \end{matrix}$ where $a_{i}$ and $b_{i}$ are the ith row of the matrices A and B, respectively, and the constraints are expressed by the max-product operator definition as: $\begin{matrix} a_{i} \cdot x = \max_{j \in J} {a_{ij} \cdot x_{j}} ⩾ d_{i}^{1} \forall i \in I^{1} \\ b_{i} \cdot x = \max_{j \in J} {b_{ij} \cdot x_{j}} ⩽ d_{i}^{2} \forall i \in I^{2} \end{matrix}$

In Section 2, the set of the feasible solutions of Problem (2) and its properties are studied. A necessary condition and a sufficient condition are given to realize the feasibility of Problem (2). In Section 3, some simplification operations are presented to accelerate the resolution process. Then, in Section 4 an algorithm is introduced to solve the problem by using the results of the previous sections, and a numerical example is also given to illustrate the algorithm in this section. Finally, the conclusion is stated in Section 5.

Section snippets

The characteristics of the set of feasible solution

Notation

We shall use, during the paper, these notations as follows: $\begin{matrix} S (A, d^{1})_{i} = \{x \in [0, 1]^{n} : a_{i} \cdot x ⩾ d_{i}^{1}\} for each i \in I^{1} \\ S (B, d^{2})_{i} = \{x \in [0, 1]^{n} : b_{i} \cdot x ⩽ d_{i}^{2}\} for each i \in I^{2} \\ S (A, d^{1}) = \underset{i \in I^{1}}{\cap} S (A, d^{1})_{i} = {x \in [0, 1]^{n} : A \cdot x ⩾ d^{1}} \\ S (B, d^{2}) = \underset{i \in I^{2}}{\cap} S (B, d^{2})_{i} = {x \in [0, 1]^{n} : B \cdot x ⩽ d^{2}} \\ S (A, B, d^{1}, d^{2}) = S (A, d^{1}) \cap S (B, d^{2}) = {x \in [0, 1]^{n} : A \cdot x ⩾ d^{1}, B \cdot x ⩽ d^{2}} . \end{matrix}$

Corollary 1

$x \in S (A, d^{1})_{i}$ for each $i \in I^{1}$ if and only if there exists some $j_{i} \in J$ such that $x_{j_{i}} ⩾ \frac{d_{i}^{1}}{a_{{ij}_{i}}}$ ; similarly, $x \in S (B, d^{2})_{i}$ for each $i \in I^{2}$ if and only if $x_{j} ⩽ \frac{d_{i}^{2}}{b_{ij}}$ , $\forall j \in J$ .

Proof

This clearly results from relations (3). □

Lemma 1

(a)
$S (A, d^{1}) \neq ϕ$ if and only if for each $i \in I^{1}$

Simplification operations and the resolution algorithm

In order to solve Problem (1), we first convert it into the two sub-problems below: $\begin{matrix} \begin{matrix} \min & c \prod_{j \in R^{+}} x_{j}^{α_{j}} \end{matrix} \\ \begin{matrix} s . t . & A \cdot x = b \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} x \in [0, 1]^{n} \end{matrix} \\ \begin{matrix} \min & c \prod_{j \in R^{-}} x_{j}^{α_{j}} \end{matrix} \\ \begin{matrix} s . t . & A \cdot x = b \end{matrix} \end{matrix}$ $\begin{matrix} x \in [0, 1]^{n} \end{matrix}$ where $R^{+} = {j | α_{j} ⩾ 0, j \in J}$ and $R^{-} = {j | α_{j} < 0, j \in J}$ .

Lemma 6

The optimal solution of Problem (4b) is $\bar{x}$ .

Proof

In objective function (4b) $α_{j} < 0$ therefore, $x_{j}^{α_{j}}$ is a monotone decreasing function of $x_{j}$ in the interval $0 ⩽ x_{j} ⩽ 1$ for each $j \in R^{-}$ . As a result $\prod_{j \in R^{-}} x_{j}^{α_{j}}$ is too. Hence, $\bar{x}$ is the optimal solution because $\bar{x}$ is the greatest element in set $S (A, B, d^{1}, d^{2})$ . □

Lemma 7

The optimal solution of

Algorithm for finding an optimal solution and examples

Definition 5

Consider Problem (1). We call $\bar{A} = ({\bar{a}}_{ij})_{m \times n}$ and $\bar{B} = ({\bar{b}}_{ij})_{l \times n}$ the characteristic matrices of matrix A and matrix B, respectively, where ${\bar{a}}_{ij} = \frac{d_{i}^{1}}{a_{ij}}$ for each $i \in I^{1}$ and $j \in J$ , also ${\bar{b}}_{ij} = \frac{d_{i}^{2}}{b_{ij}}$ for each $i \in I^{2}$ and $j \in J$ . (Set $\frac{0}{0} = 1$ and $\frac{k}{0} = \infty) .$

Algorithm

Given Problem (2),

(1)
Find matrices $\bar{A}$ and $\bar{B}$ by Definition (5).
(2)
If there exists $i \in I^{1}$ such that ${\bar{a}}_{ij} > 1$ , $\forall j \in J$ , then stop. Problem (2) is infeasible (see Theorem (1)).
(3)
Calculate $\bar{x}$ from $\bar{B}$ by Definition (1).
(4)
If there exists $i \in I^{1}$ such that $d_{i}^{1} = 0$ , then remove the i’th row of

Conclusion

In this paper, we studied the monomial geometric programming problem with fuzzy relational inequality constraints defined by the max-product operator. Since the difficulty of this problem is finding the minimal solutions optimizing the same problem with the objective function $\prod_{j \in R^{+}} x_{j}^{α_{j}}$ , we presented an algorithm together with some simplification operations to accelerate the problem resolution. At last, we gave three numerical examples to illustrate the proposed algorithm. Example (2) shows

Acknowledgements

The author is very grateful to the anonymous referees for their comments and suggestions which have been very helpful in improving the paper.

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