An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition
Introduction
Let I = {1, 2, … , m}, J = {1, 2, … , n} and X = {x ∈ Rm : 0 ⩽ xi ⩽ 1 for i ∈ I}.
Suppose an m × n fuzzy matrix A = (aij) and an n-dimensional vector b = (bj) such that 0 ⩽ aij ⩽ 1, 0 ⩽ bj ⩽ 1, for each i ∈ I and j ∈ J. We are interesting in finding a solution vector x ∈ X such thatwhere “” denotes max-prod composition [1] i.e.The fuzzy relation equations topic is investigated by few numbers of researchers [2], [3], [4], [5], [8], [9], [10], [11], [12], [13], [14], [15], [16]; in this paper what we are interesting is learning specific form of these sort problems. Assume an m-dimensional cost coefficient vector c = (cj) where ci is associated with variable xi, for each i ∈ I. We would like to solve the problem below
First of all non-empty solution set of the fuzzy relation equations is generally non-convex set determined in term of the maximum solution and the finite number of minimum solutions [3], [8]. Next, we will show that these properties are also true for such equations with max-prod composition. On the other hand, these properties results in some structural differences between such problems and the traditional linear programming [6] both in their forms and their solution method. For example, the simplex method and also the interior point method and the other classical method can not be applied in solving the problems such as (3).
In Section 2, the feasible region of the fuzzy relation equations with max-prod composition is precisely investigated. Furthermore, the first method is introduced to find the maximum solution and minimum solutions in it. In Section 3, we convert the matrix A into another modified matrix and derive some structural properties which are very useful to obtain the optimum points. The matrix modification process improves the first method and accelerates it to find minimum solutions faster.
In Section 4, the second method is introduced. This method and its corollaries results in using 0–1 integer programming and a type of branch and bound techniques for linear optimization with the fuzzy relation equations regarding max-min composition [7]. Although, we can apply these methods again for max-prod composition, but it would be quite better to choose a tabular method [17] which is more effective. At the end of this section, it is proved the first method with the matrix modification is the same as second method. Furthermore, we give the necessary and sufficient condition for solution set. In Section 5, we present an algorithm in base of Sections 3 Modification of the matrix, 4 Second method in order to optimizing linear objective functions. Finally, an example is given to illustrate what have been presented, and then the conclusion is derived.
Section snippets
Characterization of feasible solutions set
Definition 1 For each 1x, 2x ∈ X[A, b], 1x ⩽ 2x iff 1xi ⩽ 2xi for ∀i ∈ I. where X[A, b] denotes the feasible solutions set of problem (3). Definition 2 is the maximum solution if for ∀x ∈ X[A, b]. Similarly, is the minimum solution if implies for ∀x ∈ X[A, b].
To determine X[A, b], we take apart (1) into the following equation:where aj is the j the column of matrix A.
If x is a feasible solution in (4), for a fixed j ∈ J, surely, we have got to haveLet 1Ij = {i ∈ I
Modification of the matrix A
In this section, we accelerate the finding process of the minimum solutions proposed in the first method by modifying the matrix A. Lemma 2 If for j1, j2 ∈ J we have , and then can be zero value. Proof See Lemma 2 in [16]. □
Matrix attained by Lemma 2 is called the modified one. It is obvious, where denotes the modified matrix A. The first method gives many points as candidate for being minimum solution. Actually, the final set generated by this method
Second method
Let Ij(x) = {i ∈ I : xi · aij = bj}, ∀j ∈ J, and I(x) = I1(x) × I2(x) × ⋯ × In(x).
If x ∈ X[A, b] then from (6), Ij(x) ≠ ∅ for ∀j ∈ J and hence, I(x) ≠ ∅. Let f = (f(1), f(2), … , f(n)) such that f(j) = i ∈ Ij(x) for ∀j ∈ J and Jf(i) = {j ∈ J : f(j) = i}. Definition 4 For each f ∈ I(x) let f[x] = (f[x]1, f[x]2, … , f[x]m) such thatand let Lemma 5 For each f ∈ I(x) and j, j′ ∈ Je(i), . Proof Since j, j′ ∈ Je(i), xi · aij = bj and . Hence, for any j, j′ ∈ Je(i). □ Lemma 6 Suppose x ∈ X[A, b] and f ∈ I(x)
An algorithm for optimizing the linear functions
Consider problem (3). We can split it into two sub-problems as follows:where
From Lemmas 4, 5 and Theorem 2 in [7], is the optimal solution for SP2 and one of the minimum solutions is the optimal solution of SP1. Also, the optimal solution of problem (3) is achieved aswhere are optimal in SP2 and SP1, respectively.
The optimal
Conclusion
In this paper, firstly, we defined the max-prod fuzzy relation composition and the feasible region, then with applying the first method a number of candidate’s points are obtained that none of them are optimum. In order to put away the additional points, we try to simplify the matrix A. The simplification process makes easier access to optimal points. Secondly, we introduced the second approach which represent a new effective method to obtain the optimal points, and finally, it is proved by
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