The aim of this paper is to develop a linear programming technique for multidimensional analysis of preferences in multiattribute group decision making under fuzzy environments. Fuzziness is inherent in decision data and group decision making processes, and linguistic variables are well suited to assessing an alternative on qualitative attributes using fuzzy ratings. A crisp decision matrix can be converted into a fuzzy decision matrix once the decision makers’ fuzzy ratings have been extracted. In this paper, we first define group consistency and inconsistency indices based on preferences to alternatives given by decision makers and construct a linear programming decision model based on the distance of each alternative to a fuzzy positive ideal solution which is unknown. Then the fuzzy positive ideal solution and the weights of attributes are estimated using the new decision model based on the group consistency and inconsistency indices. Finally, the distance of each alternative to the fuzzy positive ideal solution is calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of the technique.
Multiple attribute decision making (MADM) problems are widespread in real life decision situations [3], [4], [8], [9], [10] and [11]. A MADM problem is to find a best compromise solution from all feasible alternatives assessed on multiple attributes, both quantitative and qualitative. Suppose the decision makers have to choose one of or rank n alternatives: A1,A2,…,An based on m attributes: C1,C2,…,Cm. Denote an alternative set by A={A1,A2,…,An} and an attribute set by C={C1,C2,…,Cm}. Let xij be the score of alternative Ai (i=1,2,…,n) on attribute Cj (j=1,2,…,m), and suppose ωj is the relative weight of attribute Cj, where ωj⩾0 (j=1,2,…,m) and ∑j=1mωj=1. Denote a weight vector by . A MADM problem can then be expressed as the following decision matrix:
The above MADM problem can be dealt with using several existing methods such as the technique for order preference by similarity to ideal solution (TOPSIS) developed by Hwang and Yoon [8], the linear programming technique for multidimensional analysis of preference (LINMAP) developed by Srinivasan and Shocker [13] and the nonmetric multidimensional scaling (MDS). The TOPSIS and LINMAP methods are two well-known MADM methods, though they require different types of information. In the TOPSIS method, the decision matrix D and the weight vector ω are given as crisp values a priori; a positive ideal solution (PIS) and a negative ideal solution (NIS) are generated from D directly; the best compromise alternative is then defined as the one that has the shortest distance to PIS and the farthest from NIS. However, in the LINMAP method, the weight vector ω and the positive ideal solution are unknown a priori. The LINMAP method is based on pairwise comparisons of alternatives given by decision makers and generates the best compromise alternative as the solution that has the shortest distance to the positive ideal solution.
Under many conditions, however, crisp data are inadequate or insufficient to model real-life decision problems [1], [2], [5], [6] and [12]. Indeed, human judgments are vague or fuzzy in nature and as such it may not be appropriate to represent them by accurate numerical values. A more realistic approach could be to use linguistic variables to model human judgments [3], [4], [5], [9] and [12]. In this paper, we further extend the LINMAP method to develop a new methodology for solving multiattribute group decision making problems in a fuzzy environment [6] and [7]. In this methodology, linguistic variables are used to capture fuzziness in decision information and group decision making processes by means of a fuzzy decision matrix. A new vertex method is proposed to calculate the distance between triangular fuzzy scores. Group consistency and inconsistency indices are defined on the basis of preferences between alternatives given by decision makers. Each alternative is assessed on the basis of its distance to a fuzzy positive ideal solution (FPIS) which is unknown. The fuzzy positive ideal solution and the weights of attributes are then estimated using a new linear programming model based upon the group consistency and inconsistency indices defined. Finally, the distance of each alternative to FPIS can be calculated to determine the ranking order of all alternatives. The lower value of the distance for an alternative indicates that the alternative is closer to FPIS.
The paper is organized as follows. In next section, the basic definitions and notations of fuzzy numbers and linguistic variables are defined as well as the fuzzy distance formula and the normalization method. Section 3 defines group consistency and inconsistency indices between preferences of alternatives given by decision makers and the results of the decision making model, and presents a new linear programming model to solve such multiattribute group decision making problems. The developed method is also illustrated with a real life example in Section 4. The paper is concluded in Section 5.
Most multiattribute decision making problems include both quantitative and qualitative attributes which are often assessed using imprecise data and human judgments. Fuzzy set theory is well suited to dealing with such decision problems. In this paper, the classical LINMAP method is further developed to solve multiattribute group decision making problems in fuzzy environments. Linguistic variables as well as crisp numerical values are used to assess qualitative and quantitative attributes. In particular, triangular fuzzy numbers are used in this paper to assess alternatives with respect to qualitative attributes.
A fuzzy linear programming (FLP) model was constructed to rank alternative decisions using the pairwise comparisons between alternatives, which can be used in both crisp and fuzzy environments. In the FLP model, the normalization constraints on weights are imposed, which ensures that the weights generated are not zero. The technique can be used to generate consistent and reliable ranking order of alternatives in question.
The developed method is illustrated using an air-fighter selection problem. It is expected to be applicable to decision problems in many areas, especially in situations where multiple decision makers are involved and the weights of attributes are not provided a priori.
