DECISION MAKING UNDER LINGUISTIC UNCERTAINTY CONDITIONS ON BASE OF GENERALIZED FUZZY NUMBERS

This article is devoted to the problem of decision making under linguistic uncertainty. The effective method for modelling linguistic uncertainty is the fuzzy set theory. There are several types of fuzzy number types proposed by L. Zadeh: fuzzy type-1, fuzzy type-2, Z-numbers. Chen proposed concept of generalized fuzzy numbers. Generalized trapezoidal fuzzy numbers (GTFN) one of effective approach which can be used for modeling linguistic uncertainty. GFTN very convenient model which allow take in account second order uncertainty. GFTN are formalized and major operations are described as practical problem is considered group decision making for supplier selection. In this case the criteria assessments are expressed by experts in linguistic form. Group decision making model is presented as 2 step aggregation procedure, in first step is aggregated value of alternative by expert, in second step by criteria. Numerical example with four criteria and three alternatives are presented and solved.


Introduction.
Decision making problem with imperfect information is very actual problem. As known in many practical cases we need to be satisfied of expert information and the linguistic assessments. One is effective method of modelling linguistic information is fuzzy set approach. The are many scientific works dedicated to applications of classical fuzzy approach which is named fuzzy type-1 proposed by L. Zadeh (1965) [1]. In 1975 L. Zadeh [2] proposed more general approach fuzzy type-2, which expands the features of classical fuzzy type-1 model. Chen in 1985 proposed generalized fuzzy set concept [3]. L. Zadeh in 2011 proposed fuzzy Z-numbers approach [4]. All these approaches allow not only modelling our imprecise knowledge about factors and also take in account our imprecision about membership function. All these models have more powerful features for modelling uncertainty [6-16]. 2. Preliminaries. In this article we discuss about application of generalized trapezoidal fuzzy numbers (GTFN) for modelling MADM problem [6].
Definition: General fuzzy number. A fuzzy set ̃, defined on the universal set of the real numbers R, is said to be generalized fuzzy number if it is membership function has the following characteristics: (i)

RS Global
Generalized trapezoidal fuzzy number ̃= ( , , , , ) is said to be generalized fuzzy number if its membership function is given

Fig 1. Comparison between membership function of TFN and GTFN
Here plays role of confidence level. Consider arithmetical operations on two trapezoidal GTFN numbers: ̃1 and ̃2 numbers are given:

Ranking function
For ranking alternatives we have used following centroid method /6/ Ranking function (̃) = √̃2 +̃2 (1) Let ̃ and ̃ two fuzzy numbers, With GTFN we can represent the crisp interval and also imprecise interval. If а=b and с=d and W  1 we have imprecise interval with confidence level .
If a=b, c=d and w=1 then we have crisp interval.

Fig. 2. Linguistic terms for alternatives evaluation
Experts using these terms have evaluated any potential suppliers and results are presented in following tables 3-5  ) On next step we carry out aggregation by attributes using formula As result we have global evaluation of all alternatives (Table 6) For comparison alternative decisions we will use Rank function (1) Rank( 1 )=3.52> Rank( 3 )=3.49> Rank( 2 )=3.45 It means that best is supplier 1 Conclusions. In this article have been considered problem of MADM under linguistic uncertainty. As model of decision making used group decision making approach and as model for modeling uncertainty have been used GTFN model. As test problem for proposed model have been used the supplier selection problem.