Multiattribute decision making (MADM) problems can be solved using several existing methods such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [22] and the Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) [33]. The TOPSIS and the LINMAP are two well-known MADM methods, though they require different types of information and decision conditions [11], [12], [13], [21], [30] and [32].
In the LINMAP, all the decision data are known precisely or given as numeric values. However, under many conditions, numeric values are inadequate or insufficient to model real-life decision problems [1], [2], [9], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [22], [23], [24], [25], [26], [28], [31], [34], [36] and [37]. Indeed, human judgments including preference information are vague or fuzzy in nature and as such it may not be appropriate to represent them by accurate numeric values. Since the concept of the fuzzy set was introduced by Zadeh [40] in 1965, the fuzzy set theory has been used to handle MADM problems and has achieved a great success [1], [2], [8], [10], [11], [12], [35], [36], [37], [38], [41] and [42]. In the fuzzy set theory, the degree of membership for an element x is μ(x)μ(x) and the degree of non-membership is 1-μ(x)1-μ(x) automatically, i.e., this membership degree combines the evidence for x and the evidence against x. The single number tells us nothing about the lack of knowledge. In real applications, however, information of an element belonging to a fuzzy concept may be incomplete, i.e., the sum of the membership degree and the non-membership degree may be less than one. There is no means to incorporate the lack of knowledge of the membership degree in the fuzzy set. A possible solution is to use the intuitionistic fuzzy (IF) set introduced by Atanassov [3], [4], [5] and [6], which is a generalization of the fuzzy set. The reason is that the IF set seems to be well suited for expressing hesitation of the decision makers (or experts) [7], [27], [29] and [39]. Therefore, the purpose of this paper is to further extend the LINMAP to develop a new methodology for solving multiattribute group decision making (MAGDM) problems using IF sets. In this methodology, IF sets are used to describe fuzziness in decision information and decision making process by means of IF decision matrices. An IF positive ideal solution (IFPIS) [43], [44] and [45] and weights of attributes are estimated using a new auxiliary linear programming model based upon the group consistency and inconsistency indices, which are defined on the basis of pairwise comparison preference relations on alternatives given by the decision makers. The distances of alternatives to the IFPIS are calculated to determine their ranking orders for the decision makers. The ranking order of alternatives for the group can be generated using the Borda’s function [22] and [24].
The rest of this paper is organized as follows. In Section 2, the concept of the IF set and the Euclidean distance between IF sets are introduced and the MAGDM problems using IF sets are formulated. In Section 3, the group consistency and inconsistency indices are defined. A new auxiliary linear programming model is constructed to estimate the IFPIS and attribute weights which are unknown a priori. Some properties of the auxiliary linear programming model and other generalizations or specializations are discussed in detail. The proposed methodology is illustrated with a numerical example of the extended air-fighter selection problem and compared with other similar methods in Section 4. A practical application of the proposed methodology is shown with the doctoral student selection problem in Section 5. Further discussions on the proposed methodology and conclusions are given in Sections 6 and 7, respectively.
