DENSITY AGGREGATION OPERATORS BASED ON THE INTUITIONISTIC TRAPEZOIDAL FUZZY NUMBERS FOR MULTIPLE ATTRIBUTE DECISION MAKING

. With respect to the multiple attribute decision making problems in which the attribute values take the form of the intuitionistic trapezoidal fuzzy numbers, some methods based on density aggregation operators are proposed. Firstly, the definition, expected value and the ranking method of intuitionistic trapezoidal fuzzy numbers are introduced, and the method of calculating density weighted vector is proposed. Then some density aggregation operators based on interval numbers and intuitionistic trapezoidal fuzzy numbers are developed, and a multiple attribute decision making method is presented. Finally an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.


Introduction
Multi-attribute decision making (MADM) has a wide range of applications, such as personal assessment, product evaluation, employee performance evaluation, economic evaluation, investment decision making, risk assessment, etc. Due to the complexity and uncertainty of the decision-making environment, we need consider various aspects in the evaluation process so as to make a scientific and rational decision.
In recent years, Research on the MADM problems with the intuitionistic fuzzy information has made a lot of achievements (Atanassov 1989;Liu 2009Liu , 2013aXu, Yager 2006;Xu 2007;Yu 2013). Atanassov (1986Atanassov ( , 1989 proposed the intuitionistic fuzzy set (IFS) which is the generalization of the concept of fuzzy set. Xu and Yager (2006), Xu (2007) proposed some aggregation operators with intuitionistic fuzzy information. Yu (2013) proposed some intuitionistic fuzzy prioritized operators and applied them to multi-criteria group decision making. Razavi Hajiagha et al. (2013) proposed a complex proportional assessment method for group decision making in an interval-valued intuitionistic fuzzy environment. Zhang and Liu (2010) proposed the triangular intuitionistic fuzzy number which used the triangular fuzzy number to denote the membership degree and the non-membership degree, and then the weighted arithmetic average operator was defined. Further, Wang (2008), Wang and Zhang (2008) gave the definition of intuitionistic trapezoidal fuzzy number, and defined the expected values of intuitionistic trapezoidal fuzzy number and proposed a decision method based on intuitionistic trapezoidal fuzzy number. Wang and Zhang (2009a) proposed the Hamming distance between intuitionistic trapezoidal fuzzy numbers and intuitionistic trapezoidal fuzzy weighted arithmetic averaging (ITFWAA) operator. Wang and Zhang (2009b) proposed some aggregation operators, including intuitionistic trapezoidal fuzzy weighted arithmetic averaging operator and weighted geometric averaging operator. Du and Liu (2011) proposed the extended VIKOR method based on the intuitionistic trapezoidal fuzzy numbers. Zhang et al. (2013) proposed the grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy numbers. Wan (2013) proposed some power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. WAA (or WGA) operator can only weight the information by importance of all attributes, and OWA operator weights the ordered positions of all attributes. But the two weighted methods don't consider the distribution of density degree of the attribute values. However, density of decision making information is an important index for decision making. In a set of data, the high concentration of data reflects the high consistency of information; on the contrary, the high dispersion of the data reflects the low consistency of information. According to the preferences of decision makers, we could pay attention to the high consistency of information (emphasis on group opinion) or to the low consistency of information (emphasis on individual opinions). Based on the distribution of density degree of attributes, Yi et al. (2007) proposed the density-weighted averaging (DWA) operator to aggregate attribute values which are crisp numbers, then Hou and Guo (2008), Li et al. (2012) proposed the density-weighted averaging (DWA) operators to aggregate interval numbers.
The proposed density aggregation operators above are only used for crisp numbers or interval numbers. In this paper, we will extend them to the intuitionistic trapezoidal fuzzy numbers, and propose a decision making method.

The definition and operational laws of intuitionistic trapezoidal fuzzy numbers
Definition 1 (Wang, Zhang 2009a): let a  be an intuitionistic trapezoidal fuzzy number in the set R of real numbers, its membership function is given by: and its non-membership function is given by: is called the degree of indeterminacy of x to a  . The smaller value of ( ) a x π  is, the more certain of x to a  is. Some operational laws on intuitionistic trapezoidal fuzzy numbers are shown as follows: D ef i n iti on 2 ( Wa n g , Z h a n g 2 0 0 9 a ) : L e t

The expected value of an intuitionistic trapezoidal fuzzy number
For an intuitionistic trapezoidal fuzzy number a  , we can define ( ) . According to the characteristics of these two functions, we know that is a monotone decreasing function on its interval [ , ] c d , and then their inverse functions are respectively given as follows (Wang, Zhang 2008):  (Wang, Zhang 2008). Definition 3 (Wang, Zhang 2008 is called the minimum expected value for the intuitionistic trapezoidal fuzzy number a , where, 0,1 λ ∈     denotes the risk preferences of decision makers. If 0.5 λ > , decision makers tend to risk-pursuit; if 0.5 λ < , decision makers tend to risk-aversion, if 0.5 λ = , decision makers tend to risk-neutrality. Then we can get: Definition 4 (Wang, Zhang 2008): is called the maximum expected value for the intuitionistic trapezoidal fuzzy number of a  . Similarly, 0,1 λ ∈     denotes the risk preferences of decision makers. If 0.5 λ > , decision makers tend to risk-pursuit; if 0.5 λ < , decision makers tend to risk-aversion, if 0.5 λ = , decision makers tend to risk-neutrality. Then we can get: Therefore, we get the expected value interval of intuitionistic trapezoidal fuzzy number of a  is ( ), ( )  . For two intuitionistic trapezoidal fuzzy numbers, we could rank them by their expected value intervals according to the ranking method of the interval numbers. Yi et al. (2007) proposed a clustering method for point data element, which is called the ordered incremental segmentation method, but it is unable to handle multidimensional data elements, such as vector and matrix. Further, Yi and Guo (2010) proposed a new general method to cluster the data elements for point, vector and matrix.

The clustering method
Definition 5 (Yi, Guo 2010): Let i a and j a ( , , ) i j R i j ∈ ≠ be any two elements of A, , then the distance between i a and j a is given by: If i a and j a are two vectors, and If i a and j a are two matrixes, Definition 6 (Yi, Guo 2010): let be any data subset of 1 A , 2 A ,…., m A , and the number of its elements is t k , then the distance between i A and  ) denotes the distance between the hth element in i A and the gth element in j A . Based on (13)-(16), we can compute the distance between elements or between the data subsets respectively, and then complete the process of cluster grouping of data elements. For the detailed steps, please refer to Yu and Fan (2003).

Determining the density weighting vectors
Yi and Guo (2010) proposed a method for balancing the weight vector of "scale" and "function", and its form is given by: where: i ξ is the ith component of According to Yi and Guo (2010), if 1 2 , , , n ω ω ω  are the "function" weights of 1 2 , , , n a a a  , respectively, and 0 Based on the "scale" connotation of ( ) ), and we can specify an exponential function: where: ( ) f ± γ satisfies the following conditions: (1) Zero condition: (2) Normalization condition: (3) Ratio condition: where: η ( 0 η > ) is the ratio which is determined by the maker's judgment, and it is used to provide an entry point for the maker's preference judgment. If i k > j k , we can allow 1 η > in the function of ( ) f + γ and 1 η < in the function of ( ) f − γ . If this condition is not needed, we can allow 1 c = .
The above conditions are converted to the following formulas.
where: " ± " can get " + " or " − " which is determined by the characteristics of function. By solving Eq. (20), we can get parameters , a b and c , and get the precise form of function ( ) f ± γ . Then let: to calculate s i ω . Based on condition (2), we can get 1 After getting the values of f i ω and s i ω , for Eq. (17), if f s ω ≠ ω (this is a normal condition), the values of ρ and υ will be two parameters to be determined to calculate i ξ . The following, we will give some methods to determine their values.
Definition 8 (Yi, Guo 2010): The measuring degree of "group similarity" of the density weighting vector can be defined as follows: and the measuring degree of "group differences" can be given by: It is easily proved that ( ), ( ) [0,1] Ts Te ξ ξ ∈ . If ( ) 0.5 Ts ξ > (when the positive gain function of "scale" is chosen), it shows that decision makers pay more attention to "large group consensus" of groups; contrarily, if ( ) 0.5 Ts ξ < (when the negative gain function of scale is chosen), it shows that decision makers pay more attention to "small group consensus" of groups.
Definition 9 (Yi, Guo 2010): The measure of entropy of density weighting vector is given by: denotes the information amount of the density weighting vector ξ and reflects the balanced state between component values.
Based on the Definitions 8 and 9, two ways are given to calculate the values of ρ and υ .
(1) Preference coefficients method (also be called the subjective method). Suppose that decision maker gives a preference level value π on the degree of "group similarity" in advance, based on Eqs. (17) and (19), we can get: Theorem 1 (Yi, Guo 2010): If π denotes the measuring degree of "group similarity" ( ) Ts ξ , we can get: Combined Eq. (25)  The valid range of preference value on the degree of "group similarity" can be set as follows (Yi, Guo 2010) Eq. (25) can be seen as a linear function π with respect to ρ , and then we can calculate the range of the value of ( ) π ρ on [0,1] by considering the positive or negative value of d d π ρ .
(2) Maximum entropy method (also be called objective method). From the properties of entropy, the maximum entropy is the best balance of components' difference of density weighted vector ( ) 1 2 , , , m ξ = ξ ξ ξ  . So we can select the values of ρ and υ by maximizing the entropy value 1 En ( (1 ) ) ln ( (1 ) ) ln Then we can calculate the first and second derivatives of ( ) En ρ , and it is easy to know that ( ) En 0 ′′ ρ > for all ρ , i.e.

( )
En ρ is a strict concave function. So, the condition of maximizing ( ) En ρ is shown as follows: where: 0 ρ is the point which make that ( ) En 0 ′ ρ = .

Some density aggregation operators based on intuitionistic trapezoidal fuzzy numbers
Definition 10 where: , , , m ξ = ξ ξ ξ  is the density weighted vector; , , , m ξ = ξ ξ ξ  is the density weighted vector; is an associated weight vector with OWA, and satisfying ( ) Then IDOWA is called the interval density ordered weighted average operator, also called IDOWA operator.
Further, according to the operational rules of interval numbers, we can get: From Definitions 10 and 11, we know that the IDWA operator weights the interval numbers while the IDOWA operator weights the ordered positions of the interval numbers instead of weighting the arguments themselves. Therefore, weights in both the IDWA operator and the IDOWA operator represent different aspects. However, these two operators consider only one of them. To overcome this drawback, in the following we shall propose an interval density hybrid weighted average (IDHWA) operator.
Definition 12  n a n a n a , , , m ξ = ξ ξ ξ  is the density weighted vector; Similarly, we can get: The following, we can define some density aggregation operator based on intuitionistic trapezoidal fuzzy numbers.
where: ( ) is the density weighted vector; , then ITFDWA is called the intuitionistic trapezoidal fuzzy density weighted average operator. Further, according to the operational rules of intuitionistic trapezoidal fuzzy numbers, we can get: (1 (1 (1 ) )) , ( ( ) ) where: is an associated weight vector with OWA, and satisfying ( ) 1,2, , j n =  , then ITFDOWA is called the intuitionistic trapezoidal fuzzy density ordered weighted average operator.
Further, according to the operational rules of intuitionistic trapezoidal fuzzy numbers, we can get: (1 (1 (1 ) )) , ( ( From Definitions 13 and 14, we know that the ITFDWA operator weights the intuitionistic trapezoidal fuzzy numbers while the ITFDOWA operator weights the ordered positions of the intuitionistic trapezoidal fuzzy numbers instead of weighting the arguments themselves. Both the operators consider only one aspect. To overcome this drawback, in the following we shall propose an intuitionistic trapezoidal fuzzy density hybrid weighted average ( ITFDHWA ) operator.
Definition 15 where: j j a n a ′ = ω   ) for all 1,2, , j n =  , then ITFDHWA is called the intuitionistic trapezoidal fuzzy density hybrid weighted average operator.

Multi-attribute decision making methods based on intuitionistic trapezoidal fuzzy numbers
Consider a multiple attribute decision making problem with intuitionistic trapezoidal fuzzy numbers: let takes the form of the intuitionistic trapezoidal fuzzy number for alternative i A with respect to attribute j C . Then, the ranking of alternatives is required.
In the following, we will propose a multiple attribute decision making method based on the density aggregation operators. We firstly convert the decision matrix in intuitionistic trapezoidal fuzzy numbers to interval numbers, and then we use the interval density aggregation operators to derive the overall preference values in interval numbers. Finally, we can rank all alternatives by sorting the interval numbers. The method involves the following steps: Step 1: Calculate the expected value intervals for all attribute values by Eqs. (10) and (12), we can get: Step 2: Standardize the expected value intervals, we can obtain the normalized value , (1 ,1 ) Step 3: Cluster the expected value intervals by Eqs. (13)-(16).
Step 5: Apply the IDWA operator (or IDOWA, IDHWA operators) to derive the overall preference values in interval numbers.
Step 6: Rank all alternatives by sorting the interval numbers.

An illustrative example
In this section, we use the proposed methods to analysis an example which is used in Wang and Zhang (2008). An Engine part manufacturing company wants to selects the best suppliers according to their core competencies. Suppose that there are five suppliers ( 1 2 3 4 5 , , , , a a a a a ) whose core competencies are evaluated by the following five criterions (C1, C2, C3, C4, C5): the capability of supplying (C1) ,the capability of delivery (C2), the quality of service (C3), the capability of influence (C4), the strength of scientific research (C5). The criteria weight is * ω = (0.15, 0.30, 0.10, 0.15, 0.30). Decision makers give the evaluation information for all suppliers with respect to all criterions which are listed in Table 1.

The decision steps for this example
(1) Calculate the expected values for intuitionistic trapezoidal fuzzy numbers by Eqs. (10) and (12) which are listed in Table 2. (Supposed that the decision makers are indifferent to the risk, and then 0.5 λ = ) (2) Standardize the expected value intervals which are listed in Table 3. averaging operator and weighted geometric averaging operator proposed by Wang and Zhang (2009b) to the above example, and they can get the ranking result: 2 5 1 3 4 a a a a a     . Of course, they are the same as the result in Wang and Zhang (2008). Comparing with these methods, the proposed method in this paper can consider the distribution of density degree of the attribute values, and the attribute values can be weighted according to the density of decision making information. In addition, comparing with the methods proposed by Du and Liu (2011), Zhang et al. (2013), Wang and Zhang (2008), the proposed method not only provides a ranking of the alternatives, but also provides the overall preference values for all alternatives, However, these existing methods can only provide the ranking of the alternatives. Comparing with the methods proposed by Wang and Zhang (2009b), the proposed method is the extensions of the methods proposed by Wang and Zhang (2009b), and when the density weights are equal in each clustering class, the ITFDWA operator can reduce to ITFWA operator which was proposed by Wang and Zhang (2009b). So, the proposed method has more advantages than the existing methods.

Conclusions
In this paper, with respect to multiple attribute decision making (MADM) problems in which the attribute value takes the form of intuitionistic trapezoidal fuzzy number, some new decision making analysis methods are developed. Firstly, some operational laws and expected values of intuitionistic trapezoidal fuzzy numbers are introduced, and the comparison method for the intuitionistic trapezoidal fuzzy numbers is proposed. Then, the method of calculating density weighted vector has been discussed in detail, and some density aggregation operators based on interval numbers and intuitionistic trapezoidal fuzzy numbers are developed, and a multiple attribute decision making method is presented. The characteristics of these methods are that the density level of the information distribution is considered to weight the attribute values. Finally, an illustrative example is given to illustrate the decision-making steps, to verify the developed methods and to demonstrate its practicality and effectiveness. In the future, we shall continue working in the extension and application of the developed method to other domains, and extension of ITFDWA , ITFDOWA and ITFDHWA operators to the multiple attribute decision making.