Fuzzy LINMAP method for multiattribute decision making under fuzzy environments

The Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) developed by Srinivasan and Shocker [V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference, Psychometrika 38 (1973) 337-342] is one of the existing well-known methods for multiattribute decision making (MADM) problems. However, the LINMAP only can deal with MADM problems in crisp environments. Fuzziness is inherent in decision data and decision making processes, and linguistic variables are well suited to assessing an alternative on qualitative attributes using fuzzy ratings. The aim of this paper is further extending the LINMAP method to develop a new methodology for solving MADM problems under fuzzy environments. In this methodology, linguistic variables are used to capture fuzziness in decision information and decision making processes by means of a fuzzy decision matrix. A new vertex method is proposed to calculate the distance between trapezium fuzzy number scores. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to a fuzzy positive ideal solution (FPIS) which is unknown. The FPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the FPIS can be calculated to determine the ranking order of all alternatives. A numerical example to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this paper can deal with MADM problems under not only fuzzy environments but also crisp environments. (C) 2005 Elsevier Inc. All rights reserved.