- 1 - FLHA Operator and Its Application to Group Decision Making with Triangular Fuzzy Linguistic Information Gui-wu WEI Department of Economics and Management, Chongqing University of Arts and Sciences, Yongchuan, Chongqing (402160) Abstract With respect to multiple attribute group decision making problem with triangular fuzzy linguistic information, in which the attribute weights and expert weights take the form of real numbers, and the preference values take the form of triangular fuzzy linguistic variables. A fuzzy linguistic aggregation operator called fuzzy linguistic hybrid aggregation (FLHA) operator is proposed. An approach to multiple attribute group decision making (MAGDM) with triangular fuzzy linguistic information is developed based on the FLWA and the FLHA operators. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness. Keywords: Multiple Attribute Group Decision Making; Triangular Fuzzy Linguistic Variables; Fuzzy Linguistic Ordered Weighted Averaging (FLOWA) Operator; Fuzzy Linguistic Hybrid Aggregation (FLHA) Operator 1. Introduction In the real world, human beings are constantly making decisions under linguistic environment 1-3. For example, when evaluating the “comfort” or “design” of a car, linguistic terms like “good”, “fair”, “poor” are usually be used 1. Sometimes, however, the decision makers are willing or able to provide only triangular fuzzy linguistic information because of time pressure, lack of knowledge, or data, and their limited expertise related to the problem domain 4. Thus, Xu4 developed some operators for aggregating triangular fuzzy linguistic variables, such as the fuzzy linguistic averaging (FLA) operator, fuzzy linguistic weighted averaging (FLWA) operator, fuzzy linguistic ordered weighted averaging (FLOWA) operator, and induced FLOWA (IFLOWA) operator, etc. In this paper, we shall develop a fuzzy linguistic aggregation operator called fuzzy linguistic hybrid aggregation (FLHA) operator. Based on the FLWA and the FLHA operators, we shall develop an approach to MAGDM with triangular fuzzy linguistic information. Finally, the method has been demonstrated by a practical example. 2. Triangular Fuzzy Linguistic Variables Let 0,1, i Ss it=L be a linguistic term set with odd cardinality. Any label, i s represents a possible value for a linguistic variable, and it should satisfy the following characteristics: The set is ordered: ij ss,if ij; There is the reciprocal operator: ( ) ij rec ss= such that itj= ; Max operator: ()max, iji s ss=, if ij ss; Min operator: ()min, iji s ss=, if ij ss. For example, S can be defined as 4 0123 456 , , Ssextremely poor svery poor spoor smedium sgood svery good sextremely good = = To preserve all the given information, we extend the discrete term set S to a continuous term set 0 ,0, aaq Ss sss aq=, where q is a sufficiently large positive integer. If a sS, - 2 - then we call a sthe original linguistic term, otherwise, we call a sthe virtual linguistic term. In general, the decision maker uses the original linguistic term to evaluate attributes and alternatives, and the virtual linguistic terms can only appear in calculation 4. Definition 1. Let,ssS , then we call () ,d ss = (1) the distance between s and s 4. In the following we introduce the concept of triangular fuzzy linguistic variable. Definition 2. Let,ssssS = % %, where ,sssS , ,ss and s are the lower, modal and upper values of s % , respectively, then we call s % a triangular fuzzy linguistic variable, which is characterized by the following member function 4 ( ) ()() ()() 0 0, , , 0, s q sss d ssd sssss d ssd sssss sss = % (2) Clearly, s gives the maximal grade of( )( )()1 ss = % , s and s are the lower and upper bounds with limit the field of the possible evaluation. Especially, if sss =, then s % is reduced to a linguistic variable. Let S %be the set of all triangular fuzzy linguistic variables. Consider any three triangular fuzzy linguistic variables(),ssss =%, () 111 1 ,ssss =%, () 222 2 ,ssssS = % %, and suppose that 0,1, then we define their operational laws as follows4: (1) ()() 111222 12 ,ssssssss =% () 121212 ,sss + =; (2) () () ,sssssss =%. In the following, we introduce a formula for comparing triangular fuzzy linguistic variables. Definition 3. Let () 111 1 ,ssss =%, () 222 2 ,ssssS =%, then the degree of possibility of 12 ss% is defined as4 ()()()()() 211122 12 max 1 max,0 ,0p ssd ssd ssd ss =+ % ()()()()() 211122 1max 1 max,0 ,0d ssd ssd ss + (3) where the value is an index of rating attitude. It reflects the decision makers risk-bearing attitude. If 0.5, the decision maker is risk avertor. From Definition 3, we can easily get the following results easily: (1) ()() 1221 01,01p ssp ss%; (2) ()() 1221 1p ssp