INTUITIONISTIC FUZZY PRIORITIZED OPERATORS AND THEIR APPLICATION IN MULTI-CRITERIA GROUP DECISION MAKING

. Intuitionistic fuzzy set is a very useful tool to depict uncertainty. Lots of multi-criteria group decision making methods under intuitionistic fuzzy environment have been developed. Current methods are under the assumption that the criteria and the decision makers are at the same priority level. However, in real group decision making problems, criteria and decision makers have different priority level commonly. In this paper, multi-criteria group decision making problems where there exists a prioritization relationship over the criteria and decision makers are studied. First, the intuitionistic fuzzy prioritized weighted average (IFPWA) and the intuitionistic fuzzy prioritized weighted geometric (IFPWG) operators are proposed. Then, some of their desirable properties are investigated in detail. Furthermore, the procedure of multi-criteria group decision making based on the proposed operators is given under intuitionistic fuzzy environment. Finally, a practical example about talent introduction is provided to illustrate the developed method.


Introduction
extended the concept of fuzzy sets which was proposed by Zadeh (1965), and introduced the intuitionistic fuzzy set (IFS). IFS is characterized by a membership degree and a non-membership degree, so it is more effective to deal with uncertainty and vagueness in real applications than Zadeh's fuzzy sets. Since its appearance, IFS has been investigated by many researchers and applied to many fields, such as pattern recognition (Hong, Choi 2000;Hung, Yang 2004, 2007, market prediction (Liang, Shi 2003) and decision making (Liu 2009;Liu, Wang 2007;Chen 2011;Li 2010;Wei 2010a, b;Wei 2011;Wei et al. 2011a, b;Yu 2012;Yu 2013;Yu et al. 2012;Xia et al. 2012).
As a hot topic in the theory of IFS, intuitionistic fuzzy information aggregation has been investigated widely by many researchers from different points of view. According to the relationships between the aggregated arguments, the aggregation operators can be divided into two categories. The first category is based on the assumption that the aggregated arguments are independent. Based on the OWA operator (Yager 1988), Xu (2007) developed some aggregation operators, such as the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, the intuitionistic hybrid aggregation (IFHA) operator. Xu and Yager (2006) developed some geometric aggregation operators for intuitionistic fuzzy values (IFVs), such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator. Based on the GOWA operator (Yager 2004a), Zhao et al. (2010) developed some generalized aggregation operators, such as the generalized intuitionistic fuzzy weighted averaging (GIFWA) operator, the generalized intuitionistic fuzzy ordered weighted averaging (GIFOWA) operator, and the generalized intuitionistic fuzzy hybrid averaging (GIFHA) operator. In the second category it is assumed that there are interactions phenomena among the decision making criteria. To deal with this issue, Xu (2011a) extended the power average (Yager 2001) to intuitionistic fuzzy environment and introduced a series of operators for aggregating intuitionistic fuzzy values (IFVs) whose weighting vectors depend on the input arguments. Xu (2011b) applied the Bonferroni mean (BM) (Yager 2009a) to intuitionistic fuzzy environment and introduced the intuitionistic fuzzy Bonferroni mean (IFBM) and the weighted Bonferroni mean (WIFBM) whose characteristic is that they can not only consider the importance of each criterion but also reflect the interrelationship of the individual criteria. Motivated by the induced Choquet ordered averaging operator (Yager 2004b), Xu (2010), Tan and Chen (2010) and Tan (2011) developed some intuitionistic fuzzy correlative operators, such as the intuitionistic fuzzy Choquet average (IFCA) operator, and the intuitionistic fuzzy Choquet geometric (IFCG) operator.
The above aggregation operators for IFVs is assuming that the criteria are at the same priority level although some of them consider the correlation phenomena between criteria. They are characterized by the ability to trade-off between criteria. For example, if i C and j C are two criteria with the weight i ω and j ω respectively. By the above aggregation operator, we can compensate for a decrease of θ in satisfaction to criteria i C by gain j i ω θ ω in satisfaction to criteria j C . However, in many real decision making problems, this kind of compensation between criteria is not feasible. Consider the situation in which a woman is making a decision based on consideration of powdered milk cost and safety for her child.
She should not allow a benefit with respect to cost of powdered milk compensate for a loss in safety. This is a typical kind of prioritization of the criteria, i.e. Safety has a higher priority than cost. When making decisions in Chinese universities, criteria desired by president usually have a higher priority than professor. Yager (2008) first investigated this kind of problem by introducing the prioritized "and" and "or" operators. Then, Yager (2009b) and Yager et al. (2011) have paid attention on this issue. In this paper, we research the aggregation method for IFVs which has prioritization relationships between the criteria. To do this, the remainder of this paper is organized as follows. In Section 1, we briefly review some basic concepts. Section 2 proposes the intuitionistic fuzzy prioritized weighted average (IFPWA) operator and intuitionistic fuzzy prioritized geometric average (IFPWG) operator to aggregate the IFVs, whose desirable properties are also studied in this Section. In Section 3, we develop a method for multi-criteria group decision making based on the proposed operators under intuitionistic fuzzy environment. A practical example about the introduction of talents is provided in Section 4.

Basic concepts and operations
Definition 1 (Atanassov 1986). Let fixed. An inuitionistic fuzzy set (IFS) A on X can be defined as: where the functions ( ) denote the membership and non-membership of i x to X in A with the condition that 0 ( ) 1, and Xu (2007) named the pair ( , ) α α µ ν intuitionistic fuzzy value (IFV) denoted as α with the condition 0 , 1, α α ≤ µ ν ≤ 1 α α µ + ν ≤ . Chen and Tan (1994) introduced the score func- to get the score of α , Then, Hong and Choi (2000) defined an accuracy function ( ) H v α α α =µ + to evaluate the accuracy degree of α . Xu and Yager (2006) gave an total order relation between two IFVs α and β . If It should be noted that the score function ( ) s α is between -1 and 1. In order to facilitate the following study, we introduce another score function proposed by Liu (2005 on the other hand, if ( ) ( ) ( ( ) ( )) S S S S α < β α ≥ β , then ( ) ( ) (( ( ) ( )) s s s s α < β α ≥ β . Therefore, if we replace the Chen's score function by Liu's score function, the order relation between two IFVs α and β introduced by Xu and Yager (2006) are also valid.
For three IFVs α, 1 α , 2 α ∈ V, some operational laws were given as follows (Xu, Yager 2006). 1) The Prioritized Average (PA) operator was originally introduced by Yager (2008), which was defined as follows: Definition 2 (Yager 2008 , ,..., n C C C C = be a collection of criteria and there is a prioritization between the criteria expressed by the linear ordering 1 2 3 ... x is the performance of any alternative x under criteria j C , and satisfies ( ) 0,1

Intuitionistic fuzzy prioritized average operators
In this Section, we shall investigate the PA operator under intuitionistic fuzzy environments. Based on Definition 2, we give the definition of the IFPWA as follows:  Based on operations of the IFVs described in Section 1, we can drive the Theorem 1.
Theorem 1. Let ( , ) j j j α α α = µ ν ( 1,2,..., ) j n = be a collection of IFVs, then their aggregated value by using the IFPWA operator is also an IFV, and where 1 1 ( ) Proof. The first result follows quickly from Definition 1 and Theorem 1. In the following, we prove By using mathematical induction on n : 1) For 2 n = : Since We have 2) If Eq. (6) holds for n k = , that is then, when 1 n k = + , by the operational laws described in Section 1, we have 1 2 1 IFPWA( , ,..., ) i.e. Eq. (6) holds for 1 n k = + . Thus, Eq. (6) holds for all n . Then Now, we look at some desirable properties of the IFPWA operator.
Proof. By Definition 3, we have which is also the largest IFV.
Proof. Similar to the proof of Theorem 2, we can get Corollary easily. Corollary 2. (Non-compensatory) If 1 1 1 ( , ) α α α = µ ν is the smallest IFV, i.e., 1 * (0,1) α =α = , then which is also the smallest IFV. Proof. Since 1 (0,1) α = , then by the definition of the score function defined in Section 1, we have, We have By Definition 3, we have Corollary 2 indicated that when the satisfaction to the criteria which owns the highest priority is the smallest IFV, we can't get any compensation from other criteria even if they are satisfied.
Proof. Since ( ) According to Theorem 1, we have Proof. According to operation laws described in Section 1, we have According to Theorem 1, we have According to Theorems 5 and 6, we can get Theorem 7 easily.
Based on the IFPWA operator and the geometric mean, here we define an intuitionistic fuzzy prioritized weighted geometric (IFPWG) operator.
Proof: The proof of Theorem 16 is similar to Theorem 8.

An approach to multi-criteria group decision making under intuitionistic fuzzy environment
In this Section, we utilize the proposed aggregation operators to group decision making under intuitionistic fuzzy environment. In a group decision making problem, suppose If all the attributes ( 1,2,..., ) j C j n = are of the same type, then the attribute values do not need normalization. Whereas, there are generally benefit attributes (the bigger the attribute values the better) and cost attributes (the smaller the attribute values the better) in multi-attribute decision making (Xu, Hu 2010). We may transform the attribute values of cost type into the attribute values of benefit type, in such a case, can be transformed into the intuitionistic fuzzy decision matrices Then, we utilize the IFPWA (or IFPWG) operator to develop an approach to multi-criteria decision making under intuitionistic fuzzy environment, the main steps are as follows: Step 1. Calculate the values of ( ) q ij T , ( 1,2,..., ) q p = based on the following equations.
Step 4. Aggregate the intuitionistic fuzzy values ij r for each alternative i x by the IFPWA (or IFPWG) operator: Step 5. Rank all the alternatives by the score function described in Section 1 then the bigger the value of ( ) i s r , the larger the overall IFV i r and thus the alternative ( 1,2, , )

Practical example
Work to strengthen academic education, promote the building of teaching body, the school of management in a Chinese university wants to introduce oversea outstanding teachers. This introduction has raised great attention from the school; university president 1 e , dean of management school 2 e , and human resource officer 3 e sets up the panel of decision makers which will take the whole responsibility for this introduction. They made strict evaluation for 5 candidates ( 1,2, ,5) i x i =  from four aspects, namely morality 1 C research capability 2 C teaching skill 3 C , educational background 4 C . university president have the absolute priority for decision making, dean of the management school comes next. Besides, this introduction will be in strict accordance with the principle of combined ability with political integrity. The prioritization relationship for the criteria is as below, 1 2 3 4 C C C C    . Three decision makers evaluated the candidates ( 1,2,3,4,5) i x i = with respect to the attributes ( 1,2, ,4) j C j =  and construct the following three intuitionistic fuzzy decision Tables 1-3). Since all the attributes ( 1,2, ,5) j C j =  are of the benefit type, then the attribute values do not need normalization, therefore, Based on the IFPWA operator, the main steps are as follows.
Step 5 Based on the IFPWG operator, the main steps are as follows: Step 1′. See step 1.
Step 2′. utilize the IFPWG operator (Eq. (69)  Step 3′: Calculate the values of ij T′ , ( 1,2,..., , 1,2,..., ) i m j n = = based on Eqs. (70) and (71) Step 4′: utilize the IFPWG operator (Eq. (73)) to aggregate all the preference values ij r′ ( 1,2,3,4,5) i = in the i th line of R′ , and get the overall preference values i r′: Step 5′: Calculate the scores of i r′ ( 1,2,3,4,5) i = x x x x x     . The optimal decision have changed the sort result by the IFPWG operator is different from that by the IFPWA operator. The IFPWA operator focuses on the impact of overall data while the IFPWG operator highlights the role of individual data. For example when we sort by IFPWG operator several smaller attribute values correlated with candidate 5 x , such as (1)  34 (0.45,0.45) r = have bigger impact on the variation of its position in the sort, which leads to candidate 3 x being in the second place. If the criteria or the decision makers are at the same priority level, then the proposed operators are reduced to the traditional intuitionistic fuzzy aggregation operators. However, there are different priority levels among these four criteria and three decision makers. For example, the candidate is very hard to be selected when he received poor evaluation from the president of the university. From another point of view, if a candidate owns bad moral character, then he is impossible to be selected no matter how good performance he has received on research capabilities, teaching skill and education background. Therefore, we must consider the prioritization among the criteria or the decision makers. To deal with such situations, the IFPWA and the IFPWG operators are useful tools. From the above analysis, the main advantages over the traditional intuitionistic fuzzy operators are not only due to the fact that our operators accommodate the intuitionistic fuzzy environment but also due to the consideration of the prioritization among the criteria and the decision makers, which makes it more feasible and practical.

Conclusions
In this paper, we have studied the intuitionistic fuzzy information aggregation problems where there is a prioritization relationship over the criteria, and have proposed the intuitionistic fuzzy prioritized weighted average operator and the intuitionistic fuzzy prioritized weighted geometric operator on the basis of the idea of prioritized average. The significant feature of the proposed operators is that they consider prioritization among the criteria. Some of their desirable properties are investigated in detail. Then, we have applied our operators to develop a method of multi-criteria group decision making under intuitionistic fuzzy environment. Finally, an example is given to illustrate the given method. The proposed multi-criteria group decision making method considers prioritization relationship among these criteria and decision makers, which allows our method to have wider practical application potentials. It is worth noting that the results of this paper can be extended to the interval-valued intuitionistic fuzzy environment.