Another modification from two papers of Ghodousian and Khorram and Khorram et al.

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Abstract

In this paper, we focus on the proposed algorithms to solve a linear programming problem with the convex combination of the max–min and the max–average composition and the max–star composition, respectively. They have been proposed by Ghodousian and Khorram [A. Ghodousian, E. Khorram, Solving a linear programming problem with the convex combination of the max–min and the max–average fuzzy relation equations, Appl. Math. Comput. 180 (2006) 411–418] and Khorram et al. [E. Khorram, A. Ghodousian, A. Abbasi Molai, Solving linear optimization problems with max–star composition equation constraints, Appl. Math. Comput. 179 (2006) 654–661], respectively. Firstly, we show that the “Tabular method algorithm” in the first paper and the “First procedure” in the second paper may not lead to the optimal solutions of the two models in some cases. Secondly, we generalize the proposed algorithm by Abbasi Molai and Khorram [A. Abbasi Molai, E. Khorram, A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian, Appl. Math. Comput. 190 (2007) 1161–1167] to solve the two models. In fact, it modifies the presented algorithms in the two papers. Finally, some numerical examples are given to illustrate the purposes.

Introduction

The aim of this paper is to show on two numerical examples that the “Tabular method algorithm” and the “First procedure” proposed in [1, p. 417] and [2, p. 657] may not lead to the optimal solutions of the two presented models in [1], [2] in some cases. Furthermore, we generalize the proposed algorithm in [3] to solve the presented models in [1], [2]. In fact, this generalized algorithm modifies the two proposed algorithms in [1], [2]. Let us recall the formulation of the considered optimization problems in [1], [2] using the same notations as in [1], [2]. We will consider two optimization problems as follows:minz=i=1mcixi,s.t.((x1,,xm)A)j=maxiIλmin{xi,aij}+(1-λ)xi+aij2=bjjJ,λ(0,1),xi[0,1]iI.minz=i=1mcixi,s.t.((x1,,xm)A)j=maxiI{xi+aij-xiaij}=bjjJ,xi[0,1]iI,where I = {1, …, m}, J = {1, …, n}, aij  [0, 1], and bj  [0, 1] ∀ i  I, j  J.

Of course, these kinds of problems and problems similar to them have been studied by many researchers since the resolution of fuzzy relation equations was proposed by Sanchez [4], for example, see Refs. [1], [2], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. An interested reader can find a comprehensive survey of done works about FRE and its applications in Ref. [14]. Currently, two models of such problems with the convex combination of the max–min and the max–average composition and the max–star (i.e. max–algebraic sum) composition have been studied by Ghodousian and Khorram [1] and Khorram et al. [2], respectively. In this paper, in Section 2, we show that the proposed Tabular method algorithm in [1] and the First procedure in [2] for solving two models (1), (2) may not lead to the optimal solutions of these models in some cases. Then, in Section 3, we generalize the proposed algorithm in [3] to solve two models (1), (2). This generalized algorithm solves the problems, in a general case. Conclusion is in Section 4.

Section snippets

Two examples

In this section, firstly, we show by an example that the “Tabular method algorithm” in [1] may not lead to the optimal solution of problem (1). Let us consider the following:

Example 1

Consider the following problemminz=2x1+x2+5x3,s.t.(x1,x2,x3)1000.80.50.10.20.50.45=[0.7,0.5,0.45],0xi1i=1,2,3,where I = {1, 2, 3}, J = {1, 2, 3} and λ = 0.25. Also, the maximum element of the set of feasible domain in this case is as [1]: xˆ=(0.52,0.5,0.45). According to the “Tabular method algorithm” in [1, p. 417], Table 1 is

A modified algorithm for solving problems (1) and (2)

In this section, we present some modifications for the “Tabular method algorithm” and the “First procedure” in [1, p. 417] and [2, p. 657], respectively, and generalize the proposed algorithm in [3] to solve two models (1), (2). This generalized algorithm solves the problems, in a general case. According to [1], [2], problems (1), (2) can be divided into two parts; one with non-negative cost coefficients and the other with negative cost coefficients. It has been proved that these two

Conclusion

In this paper, we showed that the “Tabular method algorithm” and the “First procedure” proposed by Ghodousian and Khorram in [1, p. 417] and Khorram et al. in [2, p. 657] may not lead to the optimal solutions of the two presented models in [1], [2] in some cases. Furthermore, the proposed algorithm by Abbasi Molai and Khorram in [3] was generalized to solve the models in a general case.

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