A DIFFERENCE-INDEX BASED RANKING METHOD OF TRAPEZOIDAL INTUITIONISTIC FUZZY NUMBERS AND APPLICATION TO MULTIATTRIBUTE DECISION MAKING

-The order relation of fuzzy number is important in decision making and optimization modeling, and ranking fuzzy numbers is difficult in nature. Ranking trapezoidal intuitionistic fuzzy numbers (TrIFNs) is more difficult due to the fact that the TrIFNs are a generalization of the fuzzy numbers. The aim of this paper is to develop a new methodology for ranking TrIFNs. We define the value-index and ambiguity-index based on the value and ambiguity of the membership and non-membership functions, and then propose a difference-index based ranking method, which is applied to multiattribute decision making (MADM) problems. The proposed method is compared to show its advantages and applicability.


INTRODUCTION
There is always existing uncertainty and imprecision in real-life decision making, the concept of the intuitionistic fuzzy (IF) set (IFS), introduced by Atanassov [1], is considered as a representation for these uncertain factors in real-life decision situations.Trapezoidal intuitionistic fuzzy numbers (TrIFNs) are special cases of IFSs defined on the set of real numbers, which may deal with more ill-known quantities, knowledge or experience.So TrIFNs play an important role in decision making and optimization modeling [2][3][4].
Different ranking methods of fuzzy numbers maybe produce different ranking results, which can bring some difficulties for decision makers.In addition, ranking fuzzy numbers is difficult in nature, especially the ranking methods of IF and IFS.Nowadays, there are some researches on the field.Nayagam et al. [5] described a type of special IFNs and introduced a scoring method of the special IFNs, which is a generalization of the scoring method for ranking fuzzy numbers.Zhang and Xu [6]propose a new method for ranking intuitionistic fuzzy values (IFVs) by using the similarity measure and the accuracy degree.Dymova et al. [7]proposed a new approach to estimate the strength of relations between real-valued and interval-valued IF values by the score and accuracy functions.Shu et al. (2006) developed an algorithm of the IF fault tree analysis for triangular IF numbers (TIFNs).Li [9] proposed a ratio ranking method of TIFNs based on the concept of value-index and ambiguity-index.Li et al. [10] proposed a value and ambiguity based ranking method through defining the values and ambiguities of the membership and non-membership degrees for TIFNs.Wang and Zhang [2] defined the TrIFNs and gave a ranking method, which transformed the ranking of TrIFNs into the ranking of interval numbers.
From the existing research results, we can see that there exists little investigation on the ranking of TrIFNs.In addition, the TrIFNs are a generalization of IF numbers, and which are commonly used in real decision problems with the lack of information or imprecision of the available information in real situations is more serious.So the research of ranking TrIFNs is very necessary.However, the ranking problem is more difficult than ranking fuzzy numbers due to additional non-membership functions [7][8][9][10][11][12][13][14].The possibility value and possibility ambiguity are the important mathematical characteristics of fuzzy numbers.Therefore, introducing the value-index and ambiguity-index based ranking method is developed for TrIFNs and used in MADM problems.Compared with the existing research, the proposed method has a natural appealing interpretation and possesses some good properties such as the linearity , as well as it is more easily to be handled and calculated.And the method can be extended to more general IFNs.So the proposed method is of a great importance for scientific researches and real applications.This paper is organized as follows.In Section 2, the concepts of TrIFNs and arithmetical operations as well as cut sets are introduced.Section 3 defines the concepts of value-index and ambiguity-index based on the value and ambiguity of the membership and non-membership functions.Hereby a difference-index based ranking method is developed.Section 4 formulates MADM problems with TrIFNs, which is solved by using the extended simple weighted average method according to the proposed ranking method.A numerical example and comparison analysis are given in Section 5. Section 6 contains the conclusion.

The definition and operations of TrIFNs
A TrIFN 12 ( , , , ); , aa a a a a a w u   is a special IFS on a real number set R , whose membership function and non-membership function are given as follows: x a w a a a x a w a x a x a x w a a a x a x a x a (1) and respectively, depicted as in Fig. 1. a w and a u respectively represent the maximum membership degree and minimum non-membership degree so that they satisfy the conditions: 01 where the symbols "  " and "  " are the min and max operators, respectively.

Cut sets of TrIFNs
A  -cut set of a TrIFN a is a crisp subset of R , which can be expressed as

Value and ambiguity of a TrIFN
The values of the membership and non-membership functions for a TrIFN 12 ( , , , ); , aa a a a a a w u   are defined as follows: respectively, where g  can be considered as weighting functions, and have various specific forms in actual applications, which can be chosen according to the real-life situations.Here, ( ) ( 1) /(1 ) ). ( 14) The function ()  f  gives different weights to elements at different  -cuts so that it can lessen the contribution of the lower  -cuts, since these cuts arising from values of () a x  have a considerable amount of uncertainty.Therefore, () Va  and () Va  synthetically reflects the information on membership and non-membership degrees.
According to Eqs. ( 9), ( 11) and ( 13), the value of the membership function of a TrIFN a is calculated as follows: In a similar way, according to Eqs. ( 10), ( 12) and ( 14), the value of the non-membership function can be obtained as follows: It is directly derived from the condition 01 which may be concisely expressed as an interval , we have  .Using Eq. ( 15), we obtain ) ( ) ( ) 12 12 Thus, Theorem 1 has been proven.In the same way to Theorem 1, we can prove Theorem 2 as follows. .
. The ambiguities of the membership and non-membership functions for a TrIFN a are defined as follows: Likewise, according to Eqs. ( 10), ( 14) and ( 18), the ambiguity of the non-membership function of a TrIFN a is calculated as follows: . Thus, the ambiguities of the membership and non-membership functions of a TrIFN a can be expressed as an interval , we have

The difference-index based ranking method
In this subsection, we propose a new ranking method based on the difference-index of the value-index to the ambiguity-index for a TrIFN.
A value-index and an ambiguity-index for a TrIFN a are defined as follows: respectively, where   is a weight which represents the decision maker's preference information.

 
shows that decision maker prefers to uncertainty or negative feeling; shows that the decision maker prefers to certainty or positive feeling; shows that the decision maker is indifferent between positive feeling and negative feeling.Therefore, the value-index and the ambiguity-index may reflect the decision maker's subjectivity attitude to the TrIFN.Remark 1.It is easily seen that the value-index ( , )  Va should be maximized whereas the ambiguity-index ( , )  Aa should be minimized.Furthermore, ( , )  Va and ( , )   Aa have some useful properties, which are summarized as in Theorems 5-7, respectively.

Theorem 5. ( , )
Va and ( , ) Aa are continuous non-decreasing and non-increasing functions of the parameter    can be calculated as follows: . Therefore, , where Proof.Using Eq. ( 21), we have . Combining with Theorems 1 and 2, we , where   .In the same way to Theorem 6, combining with Theorems 3 and 4, Theorem 7 can be easily proved.
A difference-index of a TrIFN a is defined as follows: , where   .Proof.According to Theorems 7 and 8, it is derived from Eq. ( 23) that . if and only if a is bigger than b , denoted by ab  ; (2) if and only if a is equal to b , denoted by ab  ; (3) if and only if ab  or ab  .The proposed ranking method satisfies five properties proposed by Wang and Kerre [16], which serve as the reasonable properties for the ordering of fuzzy quantities.In addition, it is a kind of two-index approaches, which aggregates both the value-index and ambiguity-index.Especially, this method satisfies the linearity.

AN EXTENDED MADM METHOD USING THE DIFFERENCE-INDEX BASED RANKING METHOD
In this section, we will extend the simple weighted average method to solve the MADM problems with TrIFNs.Suppose that there exists an alternative set  ) are calculated as follows:

APPLICATION AND COMPARISON ANALYSIS
In this section, an example for a multiattribute decision making problem of alternatives is used with the proposed method, and compared to show its advantages and applicability.Due to information of compared examples are expressed with TIFN, and TIFN is a special form of TrIFN, so we use TIFN in the numerical example.

A personnel selection problem and analysis process
The proposed decision method is illustrated with a personnel selection problem, which is adapted from [9] and [10].Suppose that a software company desires to hire a system analyst.After preliminary screening, three candidates (i.e., alternatives) stipulate the arithmetical operations as follows: b a b a b w w u b a b a b a b a b w w u u a b a b a b a b a b w w u u a b

g
()f  is a non-negative and non-decreasing function on the interval [ is a non-negative and non-increasing function on the interval[ ,1]

12 (
some useful properties, which are summarized as in Theorems 1 and 2, respectively.Theorem 1. Assume that properties, which are summarized as in Theorems 3 and 4, respectively.
3 has been proven.In the same way to Theorem 3, we can prove Theorem 4 as follows.
function of any TrIFNs.Furthermore, it is can be seen that the larger the difference-index the bigger the TrIFN.Thus, we propose the difference-index based ranking method of TrIFNs as follows.

Theorem 9 . 1 F and 2 F, then ab  on 1 F 2 F
The difference-index based ranking method of TrIFNs has the following properties.(P1) For a TrIFN a , then aa  ; (P2) For any TrIFNs a and b , if ab  and ba  , then ab  ; (P3) For any TrIFNs a , b and c , if ab  and bc  , then ac  ; (P4) Assume that are two arbitrary finite subsets of TrFNs.For any TrIFNs 12 if and only if ba  on Using Definition 1 and Eq.(23), Theorem 9 can be proven in a similar way to that Wang and Kerre (2001) (omitted).

12 { 12 {.
of m non-inferior alternatives from which the most preferred alternative has to be selected.Each alternative is assessed on n attributes.Denote the set of all attributes by Assume that ratings of alternatives on attributes are expressed with TrIFNs.Namely, the rating of each alternative , an MADM problem with TrIFNs can be expressed concisely in the matrix format as()   ij m n a  .Due to the fact that attributes may have different importance degree.Assume that the relative weight of the attribute weight vector of all attributes.The extended simple weighted average method for the MADM problems with TrIFNs can be summarized as follows:(a) Normalize the TrIFN decision matrix.In order to eliminate the effect of different physical dimensions on the final decision making results, the normalized TrIFN decision matrix can be calculated using the following formulae: B and C are the sets of benefit attributes and cost attributes, and max{ Construct the weighted normalized TrIFN decision matrix.Using Eq. (7), the weighted normalized TrIFN decision matrix can be calculated as () Calculate the weighted comprehensive values of alternatives.Using Eq. (3), the weighted comprehensive values of alternatives alternative is the one with the largest difference-index, i.e., max{ ( , ) | 1,2, , }

1 A , 2 A and 3 A 1 X 2 X 3 X 4 X
remain for further evaluation.The decision making committee assesses the three candidates based on five attributes, including emotional steadiness ( ), oral communication skill ( ), personality ( ), past experience ( ) and self-confidence . And it directly follows from membership and non-membership functions of a TrIFN that a  and 