Decision Support Soft set theory and uni–int decision making Naim Çag ˘man * , Serdar Enginog ˘lu Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpas ßa University, 60250 Tokat, Turkey article info Article history: Received 27 December 2008 Accepted 3 May 2010 Available online 12 May 2010 Keywords: Soft sets Soft operations Soft products uni–int Decision function uni–int Decision making abstract We firstly redefine the operations of Molodtsov’s soft sets to make them more functional for improving several new results. We also define products of soft sets and uni–int decision function. By using these new definitions we then construct an uni–int decision making method which selects a set of optimum ele- ments from the alternatives. We finally present an example which shows that the method can be success- fully applied to many problems that contain uncertainties. Ó 2010 Elsevier B.V. All rights reserved. 1. 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, oper- ations research, Riemann-integration, Perron integration, probability, theory of measurement and so on. At present, works on soft set the- ory 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 infor- mation 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. Aktas ß and Çag ˘man [2] gave a definition of soft groups. They also compared soft sets to the related con- cepts 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 re- lated 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 formu- lated. This technique is applied to soft optimization problems by Kovkov et al. [12]. Xiao et al. [25] introduced the soft set theory for fore- casting 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. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.05.004 * Corresponding author. E-mail addresses: ncagman@gop.edu.tr (N. Çag ˘man), serdarenginoglu@gop.edu.tr (S. Enginog ˘lu). European Journal of Operational Research 207 (2010) 848–855 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor