A New Method for Decision Making Based on Soft Matrix Theory

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INTRODUCTION
The soft set theory, originally proposed by Molodtsov [1], is a general mathematical tool for dealing with uncertainty. Since its appearance, soft set theory has a wide application in many practical problems, especially the use of soft sets in decision making. Maji and Roy [2] first introduced the soft set into the decision making problems with the help of rough sets [3]. By using a new definition of soft set parameterization reduction, Chen et al. [4] improved the soft sets based decision making in [2]. Çaðman and Enginoðlu [5] defined soft matrices and constructed a soft max-min decision making method which selected optimum alternatives from the set of the alternatives. It should be noted that the Çaðman and Enginoðlu's method has its inherent limitation. There exist some soft set based decision problems in which Çaðman and Enginoðlu's method is very likely to get an empty optimum set. The purpose of this paper is to point out the limitation of Çaðman and Enginoðlu's method by using an example. Moreover, to overcome this limitation, we present a new approach to soft set based decision making problems and give an illustrative example.

PRELIMINARIES
In the current section, we will briefly recall the notions of soft sets [1] and soft matrices [5]. Throughout this paper, let U be an initial universe of objects and E the set of parameters in relation to objects in U . Parameters are often attributes, characteristics, or properties of objects. Let ( ) P U denote the power set of U and A E ⊆ .
According to this definition, a soft set ( )

ÇAÐMAN AND ENGINOÐLU'S METHOD AND ITS LIMITATION
In [5], Çaðman and Enginoðlu constructed a soft max-min decision making (SMmDM) method by using soft max-min decision function. The method selected optimum alternatives from the set of the alternatives. In the current section, we introduce the Çaðman and Enginoðlu's method and show its limitation by an example.
which is called an optimum set of U .
By using above definitions, Çaðman and Enginoðlu constructed a SMmDM method by the following algorithm.

Algorithm 3.1 [5].
Step 1: Choose feasible subsets of the set of parameters, Step 2: construct the soft matrix for each set of parameters, Step 3: find a convenient product of the soft matrices, Step 4: find a max-min decision soft matrix, Step 5: find an optimum set of U .
It is worth noting that Çaðman and Enginoðlu's method has its inherent limitation. There exist some soft set based decision problems in which Algorithm 3.1 is very likely to get an empty optimum set. To illustrate this limitation, we consider the following example. , , B e e e = , respectively, to evaluate the candidates. After a careful evaluation, Mr. X and Mrs. X construct the following two soft matrices over U according to their own parameters, respectively, Following we shall select a house by using the SMmDM method. Here, we use And-product since both Mr. X and Mrs. X's choices have to be considered. We can obtain a product of the soft matrices ij a   We can find a max-min decision soft matrix as is a nonempty set, is so restrictive that it may limit the application of algorithm 3.1 in some practical problems. In other words, Çagman and Enginoglu's method is very likely to get an empty optimum set in some decision making problems.