,*

1.Schoolof Transportation and Logistics,Southwest Jiaotong University,Chengdu 610031,China;

2.Nation and Region Combined Engineering Lab of Intelligentizing Integrated Transportation,Southwest Jiaotong University, Chengdu 610031,China;

3.Departmentof System Engineering and Engineering Management,City University of Hong Kong,Hong Kong 999077,China

Entropy measures oftype-2 intuitionistic fuzzy sets and type-2 triangular intuitionistic trapezodialfuzzy sets

Zhensong Chen1,2,Shenghua Xiong1,2,YanlaiLi1,2,*,and Kwai-Sang Chin3

1.Schoolof Transportation and Logistics,Southwest Jiaotong University,Chengdu 610031,China;

2.Nation and Region Combined Engineering Lab of Intelligentizing Integrated Transportation,Southwest Jiaotong University, Chengdu 610031,China;

3.Departmentof System Engineering and Engineering Management,City University of Hong Kong,Hong Kong 999077,China

In order to measure the uncertain information ofa type-2 intuitionistic fuzzy set(T2IFS),an entropy measure of T2IFS is presented by using the constructive principles.The proposed entropy measure is also proved to satisfy all of the constructive principles.Further,a novel concept of the type-2 triangular intuitionistic trapezoidal fuzzy set(T2TITrFS)is developed,and a geometric interpretation ofthe T2TITrFS is given to comprehend it completely or correctly in a more intuitive way.To dealwith a more general uncertain complex system,the constructive principles of an entropy measure of T2TITrFS are therefore proposed on the basis of the axiomatic de fi nition of the type-2 intuitionisic fuzzy entropy measure.This paper elicits a formula of type-2 triangular intuitionistic trapezoidalfuzzy entropy and veri fi es that it does satisfy the constructive principles.Two examples are given to show the ef fi ciency of the proposed entropy of T2TITrFS in describing the uncertainty of the type-2 intuitionistic fuzzy information and illustrate its application in type-2 triangular intuitionistic trapezodial fuzzy decision making problems.

type-2 intuitionistic fuzzy set,intuitionistic fuzzy entropy,type-2 triangular intuitionistic trapezoidalfuzzy entropy.

1.Introduction

Type-1 fuzzy sets(T1FSs),which represent uncertainties by numbers in the range[0,1],have been diffusely applied in fuzzy systems in different areas of applications[1-4]. However,the membership functions of T1FSs are often overly precise.They require that each element of the universalsetis assigned a precise real number[5,6].This has hampered the development of real applications on T1FSs and therefore no advantage has been taken of the ability of T1FSs to model higher levels of uncertainty.In this sense,an interesting alternative is to employ type-2 fuzzy sets(T2FSs).The concept of the T2FS,as de fi ned by Zadeh in[5],is a fuzzy set with fuzzy membership function thatenables itto dealwith the second-orderuncertainties of a complex system[7].Moreover,augment fuzzy models with expressive power to develop models and effi ciently capture the factor of uncertainty[8-12].During the past decades,it has been previously treated in the literature both from theoretical and applied points of view, approaches and methodologies,and by considering differentextensions of the T2FS(see[13,14]for recentsurveys on the topic).

In many real-world problems involving decision making,system identi fi cation and modeling,and forecasting of time-series,medicalapplication,available inputs are quite often of uncertain nature.The three-dimensionalmembership functions of T2FSs offer additional degrees of freedom that make it possible to directly and more effectively account for model's uncertainties[6].Walker et al.[13] generated the truth-value algebras of both type-1 and of interval-valued fuzzy sets by investigating the algebra of truth values for T2FS.Furthermore,the situation when the unit interval is replaced by two fi nite chains is also addressed.Inspired by the notion ofhorizontalslice representation and the type-reduction method for a T2FS proposed by Liu[15],Wu et al.[6]explored some useful properties of theα-plane representation and ofthe type reduction for interval T2FSs,and a fast method was developed for computing the centroid of a T2FS.The number of computations and comparisons involved is greatly reduced.To overcome the de fi ciency that the defuzzi fi cation process has historically been slow and inef fi cient,Green fi eld etal. [16]provided a novel approach for improving the speed of defuzzi fi cation for discretised generalised T2FSs.Aliev et al.[14]developed novel type-2 fuzzy neural networksthat take advantage of capabilities of fuzzy clustering realized with the aid of differential evolution by generating type-2 fuzzy rule base,resulting in a small number of rules and then optimizing membership functions of T2FSs presentin the antecedentand consequentparts of the rules. Zdenko[17]extended inclusions and subsethood measures forinterval-valued fuzzy sets to generalT2FSs by using an α-plane representation for T2FSs,and proposed new subsethood measures for continuous,generalT2FSs.Zhaiand Mendel[7]generalized the fi ve uncertainty measures for interval type-2 fuzzy sets(IT2FSs),namely centroid,cardinality,fuzziness,variance and skewness,to T2FSs and, more importantly,derived a uni fi ed strategy for computing all different uncertainty measures with low complexity.In addition,there have been many successful applications offuzzy systems employing T2FSs in the realworld, such as pattern recognition[18,19],automatic control[20-22],function approximation[23,24],and data classi fi cation[25-27].

However,most of the existing studies focus on arguments of IT2FSs.They regarded the IT2FSs as a simplifi ed version of general T2FSs,whose membership grades are crisp intervals rather than functions[28-32].Hu and Wang[33]introduced IT2FSs and interval-valued type-2 fuzzy relations,and discussed their properties.Ulu et al. [34]proposed a new centroid type reduction method for piecewise linear IT2FSs based on the geometrical approach.For the purpose of allowing for an improved handling of uncertain assumptions about the distributions of noisy and erroneous inputs which are essential for correct design of the fuzzy voting scheme,Linda and Manic[35] presented an extension of the type-1 fuzzy voting scheme to the fuzzy voting scheme via incorporating interval type-2 fuzzy logic.Visconti and Tahayori[36]discussed the design and engineering of a biologically-inspired intrusion detection system forprotecting computernetworks based on the interval type-2 fuzzy set paradigm.Chen et al.[37]proposed a weighted fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on IT2FSs.In orderto overcome the drawbacks of the method in[38]and the method in[39]due to the fact that it can deal with the ranking of interval type-2 fuzzy sets to distinguish the preference order of the alternatives,Chen and Wang[40]proposed a fuzzy multiple attributes decision making method,which is more fl exible and more intelligent than the method in[41]due to the fact that it not only uses IT2FSs,but also considers the decision-maker's attitude towards risks.In fact,compared with the general type-2 fuzzy systems,the computation complexity of interval type-2 fuzzy systems is far lower[42-44].Wu et al.[6]pointed outthat“the intervaltype-2 fuzzy sets offer developers a disadvantage ofhaving less expressive power than generalT2FSs.”They also stated that“generaltype-2 fuzzy sets have more design degrees of freedom than interval type-2 fuzzy sets,and therefore,one system using type-2 fuzzy sets has the potential of outperforming another system thatuses intervaltype-2 fuzzy sets.”

Recently,the concept of intuitionistic fuzzy set(IFS) introduced in[45],which is the generalization of the fuzzy set originally investigated in[46],provided a theoretical basis to manage the hesitation information when people judge practicalproblems of complex uncertainties. Inspired by the strengths of T2FSs and intuitionistic fuzzy sets(IFSs),Zhao and Xiao[47]introduced the notions of type-2 IFSs(T2IFSs),which are based on the development of T2FSs and IFSs.They also proved that T2IFSs are the generalized forms of T1FSs,IFSs,interval-valued fuzzy sets and interval-valued IFSs,thus applications of T2IFSs are more ef fi cient and extensive than that of the existing forms offuzzy sets.Therefore,we investigated uncertainty measures for T2IFSs in this paper,which are advantageous to calculate the importance degrees of T2IFSs.They are essentially the extensions of uncertainty measures of T1FSs and T2FSs.It is necessary to quantify the uncertainty associated with T2IFSs because if uncertainty(and information)measures become well justi fi ed,then they can very effectively be utilized for managing uncertainty and associated information.Klir et al.[48]pointed out that“they can be utilized for extrapolating evidence,assessing the strength ofrelationship between given groups of variables, assessing the in fl uences of given input variables on given outputvariables,measuring the loss of information when a system is simpli fi ed,and the like”.Based on the aforementioned analysis,entropy of T2IFSs can be de fi ned similar to thatof IFSs,which is one ofthe key conceptionsin measuring uncertainties of type-2 intuitionistic fuzzy systems. Zadeh[49]fi rstpresented fuzzy entropy to measure fuzziness by probability methods.Burillo and Bustince[50,51] de fi ned entropies of IFS and interval-valued IFS.Szmidt and Kacprzk[52]improved the constructive axioms of the fuzzy entropy discussed in[53],meanwhile,another intuitionistic fuzzy entropy based new axioms is also proposed.However,the entropy measures de fi ned in[50-52] cannotbe expressed in the equilibrium state ofsupportability and opposability when neutralevidences are indicated in the hesitancy degree[54].Furthermore,severalentropy formulas and their constructive principles have been improved[55,56],but they still ignored effects induced by changes of hesitancy degree when their membership and non-membership functions are of same degree,so they cannot distinguish the uncertain information in the IFS which is different from the FS.To overcome all of these drawbacks,Mao et al.[54]re fi ned the constructive principles of entropy,and a novel intuitionistic fuzzy entropyformula was also established.

From the perspective of measuring uncertain information in T2IFS,we extend the axiomatic de fi nitions of intuitionistic fuzzy entropy to T2IFS.The rest of this paper is organized as follows.Section 2 reviews some basic concepts about T2FSs,T2IFSs and entropy of IFS.Section 3 investigates the constructive principles of the type-2 intuitionistic fuzzy entropy,moreover,establishes and veri fi es a noveltype-2 intuitionistic fuzzy entropy formula.We put forward a novel concept of the type-2 triangular intuitionistic trapezoidalfuzzy set(T2TITrFS)in Section 4,which is also a special case of the T2FS,and present the constructive principles of entropy of T2TITrFS.Similarly,a noveltype-2 triangular intuitionistic trapezoidalfuzzy entropy formula is de fi ned and veri fi ed in this section.Two examples for illustration are given in Section 5.Finally,a conclusion is given in Section 6.

2.Preliminaries

Zhao etal.[47]introduced the notions of T2IFSs based on the combination of T2FSs and IFSs,which is a generalization of the concepts of T2FSs,IFSs,interval-valued fuzzy sets and interval-valued IFSs.The main strength of T2FSs is its ability to dealwith the second-orderuncertainties that arise from several sources[57].One of the most remarkable characteristics of a T2FS is that its membership and non-membership functions are characterized by T1FSs.

In this section,we brie fl y review the mathematicalde finitions of the T2FS and T2IFS,and the axiomatic de fi nition of the entropy measure of IFSs.

Definition 1[47]A general T2FS~A in the universe of discourse U can be described by its vertical-slice representation,as

where x∈U is the primary variable,Jxrepresents the primary membership of x,u∈Jx?[0,1]is the secondary variable,fx(u)is the secondary membership function at x,denotes the union over alladmissible x and u.

Definition 2[47]An IFS A in the universe ofdiscourse U is de fi ned as an objectin the following form:

where the functions:uA:U→[0,1]andυA:U→[0,1] de fi ne the degree of membership and the degree of nonmembership of the element x∈U,respectively,and for every x∈U,uA(x),υA(x)≥0,0?uA(x)+υA(x)?1. DenoteπA(x)=1?uA(x)?υA(x)for all x∈U,called the hesitancy degree of x to A,which is also a mapping from U to[0,1],and satisfyπA(x)∈[0,1].

In particular,ifπA(x)=0 for all x∈U,then an IFS is degenerated to a fuzzy set in the sense that every fuzzy set can be viewed as a special case of the IFS.For convenience,the set IFS(U)is used to stand for all IFSs in U.

Definition 3[47]be an IFS in the universe of discourse U,which is represented byis called a T2IFS.In(3),the functions:U→[0,1]de fi ne the degree ofmembership and the degree of non-membership of the element x∈U, respectively,and the degree of hesitation of the element x∈U is de fi ned as

For convenience,the set T2IFS(U)is used to stand for all T2IFSs in U.

People have to vote during the election season,and each voteroften needs to evaluate allcandidates and then decide who to vote for based on their comprehensive consideration.However,considering the voters'point of view and the circumstances of the multiple criteria decision-making process,we fi nd that subjective opinions and judgments are inherently imprecise and involve many uncertainties. Therefore,it is dif fi cult for a voter to either completely support or completely oppose a candidate.In fact,voters may be more willing to support or oppose a candidate by giving a membership degree,rather than completely agree ortotally againsta candidate.Forexample,there are a total of 100 who are eligible to vote during an election process, 80 among them prepare to vote for candidate X and 60 of these voters give their membership degree by 0.9 with the others give their membership degree by 0.8,besides,another 20 people are ready to vote againstit,including 12 of them give their membership degree by 0.8,and another 8 voters give their membership degree by 0.7.In such situation,we can utilize the T2IFSs to explain the voting results, and the support level for candidate X can be representedand opposition forcandidate X can be repre-.Apiraticalexample can be referred in[47]and therefore we omitthe further explanation here.

In addition,to facilitate the further analysis,the axiomatic de fi nition ofthe entropy measure of the IFS in[54] is introduced as follows.

Definition4[54]Letthe intuitionistic fuzzy entropy measure is a real-valued function,and IFS(U)→ [0,1],satisfying the following axiomatic principles:=1 for all x∈U; 0 for all x∈U;

3.Entropy of T2IFS

In this section,the concept of the entropy of the T2IFS is proposed based on the axiomatic de fi nition of the entropy measure of IFSs and then veri fi ed.

Definition5(x)〉|x∈U}be two T2IFSs in U, where the degree of membership and the degree of non-

membership ofthe element x∈U corresponding to

respectively,and for every x∈U, the degree of hesitation of the element x∈U correspon-

Suppose that U is a fi nite set,and U={x1, x2,...,xn},the type-2 intuitionistic fuzzy entropy measure is a real-valued functionT2IFS(U)→[0,1], satisfying the following axiomatic principles:

for all xi∈U;

(T2IFS5):Denote|fxi(ui)ui?gxi(υi)υi|=ζ,clearly, ζ∈[0,1].For every xi∈U,?ζis a collection of T2IFSs

with the deviationζ,then the entropy of the T2IFS

A novelentropy measure of the T2IFS is de fi ned in the following theorem.

Theorem 1For a T2IFS,

is a type-2 intuitionistic fuzzy entropy measure.

ProofLet

As 0?fxi(ui),gxi(υi),ui,υi?1,0?fxi(ui)ui+ gxi(υi)υi?1,and|fxi(ui)ui?gxi(υi)υi|?1,then we

Let?i=|fxi(ui)ui?gxi(υi)υi|,then?i∈[0,1],and (20)can be transformed into

Considering the following function:

by differentiating(22),we obtain

holds for all?i∈[0,1].

Following(23),we can know that f(?i)is strictly decreasing in[0,1].Thus,f(?i)gets the maximal value (1?0)exp{02}=1 when?i=0,and gets the minimalvalue(1?1)exp{12}=0 when?i=1.

In summary,we have

Therefore,we obtain 0?E(~A)?1 from(19)and(24).

In what follows,we verify that the type-2 intuitionistic fuzzy entropy measure de fi ned in Theorem 1 satis fi es the following axiomatic principles.

(T2IFS1):For all xi∈U,if

and therefore

Conversely,for all xi∈U,if E(~A)=0,then we have Ei(~A)=0 from(19),which means eitherholds.

As exp{π~A(xi)(fxi(ui)ui?gxi(υi)υi)2}≥1,hence 1?|fxi(ui)ui?gxi(υi)υi|=0 musthold,thatis to say,

Furthermore,we can calculate the degree of hesitation

Combining(25),(26),(29),(30),(31)and(32),T2IFS1 is proven to be satis fi ed.

(T2IFS2):For all xi∈U,if

following which,we can easily obtain(19).

With respectto the arguments fxi(ui)uiand gxi(υi)υi, the conditions(33)and(34)are equivalentto thateither is satis fi ed.

By solving(35)and(36),we have the solutions of them are fxi(ui)ui=gxi(υi)υi=0.5 and fxi(ui)ui=gxi(υi)υi,respectively.Therefore,ifand only if fxi(ui)ui=gxi(υi)υi,i.e.which implies

T2IFS2 is proven to be satis fi ed base on the above analysis.

we know T2IFS3 is satis fi ed from the symmetrical characteristic of the following function:

(T2IFS4):If|fxi(ui)ui?gxi(υi)υi|=1,0,thus E(~A)?E(~B).Thus consider|fxi(ui)ui?gxi(υi)υi|＜1,when u~B(xi)■υ~B(xi)?hxi(ωi)ωi≥kxi(ρi)ρi,we have

f xi(u i)u i≥h xi(ωi)ωi≥k xi(ρi)ρi≥g xi(υi)υi

subsequently,we obtain

Furthermore,we have

Considering the following function:

by differentiating(45),we obtain

holds for all?i∈[0,1].

Therefore,we obtain

Combining(43),(44)and(47)yields

(T2IFS5):Since

for all xi∈U,we have

Considering the following function:

by differentiating(49),we obtain holds for all x∈[0,1].

Following(19),it is easy to know that E(~A)is strictly increasing withπ~A(xi)in?ζ.?

In fact,the type-2 intuitionistic fuzzy entropy measure de fi ned in Theorem 1 is capable of depicting the uncertain information by both intuitionism and fuzziness in a T2IFS. In an attempt to propose the geometric interpretation of T2IFSs,we divide the type-2 intuitionistic fuzzy entropy measure into two parts,which are called fuzzy information of a T2IFS and intuitionistic information of a T2IFS, respectively.They can be represented by a fuzzy information vector:

respectively.

Forthe fuzzy information vector,the absolute deviations of the degrees of the membership and non-membership ofa T2IFS for all xi∈U(i=1,2,...,n)re fl ect the fuzzy information to some degree,and we de fi ne

as the index of fuzzy information of the T2IFS.

Similarly,following T2IFS5,the degrees of hesitation of a collection of the T2IFS in U have a monotony relationship withthus we de fi ne as the index of intuitionistic information of the T2IFS.

Similar to the discussion in[36,37],intuitive relations between|fxi(ui)ui?gxi(υi)υi|,andπ~A(xi)can be described in Figs.1,2,and 3 when U={x},and the colors of Figs.1 and 2 re fl ectthe amountof the fuzzy and intuitiontistic information,whereas the color of Fig.3 denote the type-2 intuitionistic fuzzy entropy values.

Fig.1 Fuzzy information Efuzzyof T2IFS

Fig.2 Intuitionistic information Eintuitionistiof T2IFS

Fig.3 Type-2 intuitionistic fuzzy entropy

4.Entropy of T2TITrFSs

Definition 6Letbe a T2IFS,its primary membership function is de fi ned as

Its primary non-membership function is de fi ned as

Its secondary membership function is de fi ned as

Its secondary non-membership function is de fi ned as

is called a T2TITrFS.The geometric interpretation of a T2TITrFSand its memebership and non-membership functions are shown in Figs.4,5 and 6.

Fig.4 Geometric interpretation of a T2TITrFS

Fig.5 Geometric interpretation of the membership function of a T2TITrFS

Fig.6 Geometric interpretation of the non-membership function of a T2TITrFS

To give the detailed practicalbackgroundsof T2TITrFS, we use the aforementioned example thatwe utilized to illustrate the practical backgrounds of T2IFS.There are a total of 100 who are eligible to vote during an election process,80 of them prepare to vote for candidate X and an uncertain number of these voters,which can be represented by a trapezoidal number[54,56,58,60],give theirmembership degree by[0.7,0.8,0.9]with the others,which can be represented by a trapezoidalnumber[15,17,18,20], give their membership degree by[0.6,0.7,0.8],besides,another[16,18,19,20]people are ready to vote againstit,including[8,10,11,12]of them give their membership degree by[0.5,0.6,0.7,0.8],and another[5,6,7,8]voters give their membership degree by[0.5,0.6,0.6,0.7].In such situation,we can utilize the T2TITrFS to explain the voting results,and the supportlevelfor candidate X can be represented by

and the opposition for candidate X can be represented by

In order to measure the uncertain information of T2TITrFS,the axiomatic de fi nition of the entropy measure of T2TITrFS is introduced.

Definition7Suppose that U is a fi nite set,and U={y1,y2,...,yn},where yi(i= 1,2,...,n)are common trapezoidal fuzzy num-

we have

(ii)when

we have

where

(T2TITrFS5):For every yi∈U,?η,ψ,τis a collection of T2TITrFS in the form of

iii

whose low,medium,high deviations are denoted as

respectively.Then,the entropy of T2TITrFS

A novelentropy measure of T2TITrFS is de fi ned in the following theorem.

Theorem 2For a T2TITrFST2TITrFS(U),then is a type-2 triangular intuitionistic trapezoidal fuzzy entropy measure.

Where

Following(61)and(63),we obtain

By calculating the partial derivative of fwith respectto,we have

Similarly,by calculating the partial derivatives ofand

Following(66),(67)and(68),it is obvious thatis strictly decreasing with re-respectively.Thus,takes the maximal value

Thatis to say,

and hence we have

In what follows,we veri fi ed that the type-2 triangular intuitionistic trapezoidalfuzzy entropy measure de fi ned in Theorem 2 satis fi es the following axiomatic principles.

(T2TITrFS1):For all yi∈U,if

Conversely,for all yi∈U,if E=0,then we haveSinceis continuous and strictly decreasing with respect torespectively,we obtain

Following De fi nition 2,

Combing(74),(75),(76)and(77),T2TITrFS1 is proven to be satis fi ed.

(T2TITrFS2):For all yi∈U,if

T2TITrFS2 is proven to be satis fi ed base on the above analysis.

(T2TITrFS3):Since is trivial,we have T2TITrFS3 is satis fi ed from the sym-

metricalcharacteristic of

(T2TITrFS4):If

then(79)is transformed into

According to the theorem of integral fi rstmedian value,

holds for

(T2TITrFS5):Asη,ψ,τ∈[0,1]for all yi∈U andConsidering the following function

by calculating the partial derivative ofwe obtain

Obviously,(83)≥0 holds.Similarly,we have

Following(83),(84)and(85),it is easy to know thatis strictly increasing withandrespectively. ?

5.Illustrative examples

5.1CalculationprincipleofentropyofT2ITrFS

An example is given here to illustrate how to calculate the entropy of T2ITrFS.Forsimplicity,suppose U={y},that is,there is only one object in the universe of discourse U, we considerthe following T2ITrFSs:

Following Theorem 2,their entropies can be calculated as

In whatfollows,we illustrate the calculating principle of entropy of T2ITrFS.First,with respectto the T2ITrFSyyyyyyU},assume that any possibility value of[α1,α2,α3,α4] is denoted as x,then the entropy of any x can be calculated,which is sketched as curves by grey color in Figs.7-16.Furthermore,the asymptotic distribution of entropy(ADOE)of x is also calculated and sketched as curves by yellow color in Figs.7-16.Finally,the median value of entropy of any x can be obtained by calculating the integralof function in Theorem 2,which is the entropy value of~A and sketched as curves by blue color in Figs. 7-16.

Fig.7 Type-2 triangular intuitionistic trapezoidal fuzzy entropy of

Fig.8 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.9 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.10 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.11 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.12 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.13 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.14 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.15 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

Fig.16 Type-2 triangular intuitionistic trapezoidal fuzzy entropy

5.2 Type-2 triangular intuitionistic trapezodialfuzzy decision making

Consider that there is an investment company(adopted from[54])which wants to invest a sum of money in the best fund,now fi ve possible funds(x1,x2,...,x5)satisfy requirements under four attributes(a1,a2,...,a4), in order to choose the best fund,the company make some evaluations for these funds and results have been represented by using T2TITrFSslisted in Table 1.

Table 1 Evaluate results of five funds

To compare any two T2TITrFSs,we de fi ne the score function for T2ITrFS.

Definition 8Suppose that U is a fi nite set,and U={y1,y2,...,yn},where yi(i=1,2,...,n)

are common trapezoidal fuzzy numbers.

= 1,2,...,n)be a set of type-2 triangular intuitionistic

trapezodial fuzzy numbers de fi ned onits score function is de fi ned by

where E(yi)

Here suppose every attribute is bene fi t-type,according to(61),calculate type-2 triangular intuitionistic trapezodial fuzzy entropy of every object under each attribute E(xi,aj),and attribute weights are given by

Then we calculate a score function:

and then we have

Therefore the bestfund is x2.

Compared with the existing representative tools forconducting the multiple attributes fuzzy decision making process,the three-dimensional membership functions of the T2TITrFS offer additional degrees of freedom that make it possible to directly and more effectively account for model's uncertainties.Moreover,the proposed entropy and score function for T2TITrFS enable the decision makers to make a reasonable decision by constructing a logical and relative simple model.Similar to the modelconstructed in [54],the proposed decision making modelin this study can ef fi ciently and reasonablely addresses multiple attributes fuzzy decision making problems owing to that the proposed tools are more accurate than thatof[54].

6.Concluding remarks

This paper focuses on an extension version of T2FSs: T2IFSs.As the generalized forms of T1FSs,IFSs,intervalvalued fuzzy sets and interval-valued IFSs,the T2IFSs are ofhigherability in describing the high-orderuncertainty of fuzzy systems.In orderto measure the amountofuncertain information of a T2IFS,we investigate the constructive principles,present entropy measure of T2IFS and prove that it satis fi es all of the proposed constructive principles. Furthermore,we develop a novelconceptof the T2TITrFS and give geometric interpretations of the T2TITrFS.The constructive principles ofan entropy measure of T2TITrFS are also proposed on the basis of the axiomatic de fi nition of the type-2 intuitionisic fuzzy entropy measure.Similar to the T2IFS,a formula of type-2 triangular intuitionistic trapezoidal fuzzy entropy is elicited and further veri fi ed to satisfy its constructive principles.Finally,two examples are given to demonstrate the ef fi ciency of the proposed entropy measures of T2IFS and T2TITrFS in describing the uncertainty of the type-2 intuitionistic fuzzy information.

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Biographies

ZhensongChenwas born in 1988.He received his B.S.degree in statistics from Southwest Jiaotong University and he is currently a Ph.D.student in Department of Traf fi c Engineering.His current research interests include fuzzy multi-criteria decision analysis,fuzzy random programming and data mining.

E-mail:czs7328026@126.com

ShenghuaXiongwas born in 1988.He is a Ph.D.studentin Departmentof Traf fi c Engineering, Southwest Jiaotong University,China.His current research interests include fuzzy multi-criteria decision analysis,production and operation management.

E-mail:xsh1841@163.com

YanlaiLi was born in 1971.He is a professorin Nation and Region Combined Engineering Lab of Intelligentizing Integrated Transportation,Southwest Jiaotong University,China.He obtained his Ph.D. degree of Management Science and Engineering in School of Economics and Management,Southwest Jiaotong University.He has published more than 80 papers in journals.His major research interests are fuzzy multi-criteria decision,quality function deployment,data mining. E-mail:yanlaili@home.swjtu.edu.cn

Kwai-SangChinwas born in 1958.He is an associate professor at Departmentof Systems Engineering and Engineering Management,City University of Hong Kong.He is a charter engineer in U.K., a registered professional engineer in Hong Kong, and a fellow member of the American Society for Quality,Hong Kong Society for Quality and Hong Kong Quality Management Association.His current research interests are quality systems and management,new productdesign and development,and decision supportsystems.

E-mail:mekschin@cityu.edu.hk

10.1109/JSEE.2015.00086

Manuscriptreceived May 27,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(71371156;70971017)and the Research Grants Council of the Hong Kong Special Administrative Region,China(CityU112111).