Decision SupportSoft set theory and uni–int decision making
Introduction
Problems in many fields involve data that contain uncertainties. Uncertainties may be dealt with using a wide range of existing theories such as theory of probability, fuzzy set theory [29], intuitionistic fuzzy sets [3], vague sets [7], theory of interval mathematics [8], rough set theory [20], etc. All of these theories have their own difficulties which are pointed out in [18]. To overcome these difficulties, Molodtsov [18] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties.
In [16], [18], Molodtsov pointed out several directions for the applications of soft sets, such as smoothness of functions, game theory, operations research, Riemann-integration, Perron integration, probability, theory of measurement and so on. At present, works on soft set theory and its applications are progressing rapidly. The help of rough mathematics of Pawlak [20], Maji et al. [14] defined a parameter reduction on soft sets, and presented an application of soft sets in a decision making problem. Chen et al. [5] and Kong et al. [11] presented a new definition of the parameter reduction. Xiao et al. [27], and Pei and Miao [22] discussed the relationship between soft sets and information systems. They showed that soft sets are a class of special information systems.
Maji et al. [13] published a detailed theoretical study on soft sets. By using this study, the algebraic structure of soft set theory has been studied increasingly in recent years. Aktaş and Çağman [2] gave a definition of soft groups. They also compared soft sets to the related concepts of fuzzy sets and rough sets. Jun [9] introduced the notion of soft BCK/BCI-algebras and soft subalgebras. Jun and Park [10] dealt with the algebraic structure of BCK/BCI-algebras by applying soft set theory. Park et al. [21] introduced the notion of soft WS-algebras and then derived their basic properties. Feng et al. [6] initiated the study of soft semirings by using the soft set theory and investigated several related properties. Sun et al. [24] introduced a basic version of soft module theory, which extends the notion of module by including some algebraic structures in soft sets.
A soft set is a parameterized family of subsets of the universe. In the soft set theory, the parameters are fuzzy concepts in real world from the viewpoint of fuzzy set theory. Some researchers have worked on fuzzy soft sets. Majumdar and Samanta [15] introduced several similarity measures of fuzzy soft sets. Roy and Maji [23] presented some results on an application of fuzzy soft sets in decision making problem. Yang et al. [28] defined the reduction of fuzzy soft sets and then analyzed a decision making problem by fuzzy soft sets. Based on the theory of soft sets, the analysis was developed in [17], and the notions of soft number, soft derivative, soft integral, etc. are formulated. This technique is applied to soft optimization problems by Kovkov et al. [12]. Xiao et al. [25] introduced the soft set theory for forecasting the export and import volume in international trade. They proposed a combined forecasting approach based on the fuzzy soft sets. Xiao et al. [26] introduced soft set theory into the research of business competitive capacity evaluation.
Furthermore, Zou and Xiao [30] presented data analysis approaches of soft sets under incomplete information. These approaches presented in [30] are preferable for reflecting actual states of incomplete data in soft sets. Mushrif et al. [19] proposed a new algorithm for classification of the natural textures. Ali et al. [1] introduced the analysis of several operations on soft sets.
Up to the present, the applications of the soft set theory generally solve problems with the help of the rough sets or fuzzy soft sets. In this paper, we first redefine the operations of soft sets which are more functional to make theoretical studies of soft set theory in more detail and improve several results. We then introduce soft decision making method which selects a set of optimum elements from the alternatives without using the rough sets and fuzzy soft sets. We finally give an example which shows that the method can be successfully applied to many fields.
The presentation of the rest of the paper is organized as follows. In the next section, the operations of soft sets are redefined. In Section 3, the products of soft sets are defined and their basic properties are studied. In Section 4, uni–int decision function is defined to construct uni–int decision making method, and the method is applied to a problem. In the final section, some concluding comments are presented.
Section snippets
Soft set theory
In this section, we give new definitions and various results on soft set theory. Throughout this work, U refers to an initial universe, E is a set of parameters, P(U) is the power set of U, and A ⊆ E. Definition 1 A soft set FA on the universe U is defined by the set of ordered pairswhere fA : E → P(U) such that fA(x) = ∅ if x ∉ A.
Here, fA is called approximate function of the soft set FA. The subscript A in the notation fA indicates that fA is the approximate function of FA. The value of
Products of soft sets
Until now, we defined the binary operations of soft sets which depend on an approximate function of one variable. Now, we define products of the soft sets which are binary operations of soft sets depending on an approximate function of two variables. We have four kinds of products in the soft set theory. They are And-product, Or-product, And–Not-product and Or–Not-product that are denoted by ∧-product, ∨-product, -product and ⊻-product, respectively. Definition 10 If FA, FB ∈ S(U), then ∧-product of two soft
uni–int Decision making
In this section, we define uni–int operators and uni–int decision function for the ∧-product to construct a uni–int decision making method. The method reduces a set to its subset according to the parameters of decision makers. Therefore, decision makers works on small number of alternatives instead of large numbers.
Throughout this section, assume that ∧(U) is a set of all ∧-products of the soft sets over U. Definition 14 Let FA ∧ FB ∈ ∧(U). Then uni–int operators for the ∧-products, denoted by unixinty and uniyint
Conclusion
Main aim of this paper is redefine the operations of soft sets and construct a uni–int decision making method. The method is constructed by using the union-intersection decision function for the ∧-products. It should be easy that, a group of a similar type uni–int decision making methods can be constructed for the ∨-product, -product and ⊻-product of the soft sets. In addition, by using intersection–union (int–uni), union-union (uni–uni) and intersection-intersection (int–int) decision
Acknowledgment
The authors are grateful for financial support from the Research Fund of Gaziosmanpasa University under Grant No.: 2007/11.
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