# MADM Problems with Correlation Coefficient of Trapezoidal Fuzzy Intuitionistic Fuzzy Sets

Authors: John Robinson; Henry Amirtharaj
Journal: Advances in Decision Sciences
 (2014)

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## Abstract

This paper discusses on the notion of trapezoidal fuzzy intuitionistic fuzzy sets (TzFIFSs) and some of the arithmetic operations of the same.  Correlation coefficient of TzFIFS is proposed based on the membership, nonmembership, and hesitation degrees.  The weighted averaging (WA) operator and the weighted geometric (WG) operator are proposed for TzFIFSs.  Based on these operators and the correlation coefficient defined for the TzFIFS, new multiattribute decision making (MADM) models are proposed and numerical illustration is given.


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## Body

## 1. Introduction

Intuitionistic fuzzy sets (IFSs) proposed by Atanassov [1–3] are a generalization of the concept of fuzzy sets. Atanassov and Gargov [4] expanded the IFSs, using interval value to express membership and nonmembership function of IFSs. Liu and Yuan [5] introduced the concept of fuzzy number IFSs as a further generalization of IFSs. Among the works done in IFSs, Szmidt and Kacprzyk [6–8] can be mentioned. To the best of our knowledge, Burillo et al. [9] proposed the definition of intuitionistic fuzzy number (IFN) and studied the perturbations of IFN and the first properties of the correlation between these numbers. Many researchers have applied the IFS theory to the field of decision making. Recently some researches [5, 10–14] showed great interest in the fuzzy number IFSs and applied it to the field of decision making. Based on the arithmetic aggregation operators, Xu and Yager [15], Xu and Chen [16, 17], and Wang [13] developed some new geometric aggregation operators and intuitionistic fuzzy ordered weighted averaging (IFOWA) operator. Szmidt and Kacprzyk [7] proposed some solution concepts in group decision making with intuitionistic (individual and social) fuzzy preference relations. Szmidt and Kacprzyk [8] investigated the consensus-reaching process in group decision making based on individual intuitionistic fuzzy preference relations. Herrera et al. [18] developed an aggregation process for combining numerical, interval valued, and linguistic information and then proposed different extensions of this process to deal with contexts in which information such as IFSs or multigranular linguistic information can appear. Xu and Yager [15] developed some geometric aggregation operators for MADM problems. Li [19] investigated MADM problems with intuitionistic fuzzy information and constructed several linear programming models to generate optimal weights for attributes.Multiattribute decision making (MADM) problems are of importance in most kinds of fields such as engineering, economics, and management. It is obvious that much knowledge in the real world is fuzzy rather than precise. Imprecision comes from a variety of sources such as unquantifiable information [20]. In many situations decision makers have imprecise/vague information about alternatives with respect to attributes. It is well known that the conventional decision making analysis using different techniques and tools has been found to be inadequate to handle uncertainty of fuzzy data. To overcome this problem, the concept of fuzzy approach has been used in the evaluation of decision making systems. For a long period of time, efforts have been made in designing various decision making systems suitable for the arising day-to-day problems. MADM problems are widespread in real-life decision making situations and the problem is to find a desirable solution from a finite number of feasible alternatives assessed on multiple attributes, both quantitative and qualitative [21]. In order to choose a desirable solution, the decision maker often provides his/her preference information which takes the form of numerical values, such as exact values, interval number values, and fuzzy numbers. However, under many conditions, numerical values are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information may be stated in intuitionistic fuzzy information, especially in trapezoidal fuzzy intuitionistic fuzzy information. Hence, MADM problems under intuitionistic fuzzy or trapezoidal fuzzy intuitionistic fuzzy environment are an interesting area of study for researchers in the recent days.It is well known that the conventional correlation analysis using probabilities and statistics has been found to be inadequate to handle uncertainty of failure data and modeling. The method to measure the correlation between two variables involving fuzziness is a challenge to classical statistical theory. Park et al. [22] proposed the correlation coefficient of interval valued intuitionistic fuzzy sets. Robinson and Amirtharaj [23–26] proposed correlation coefficient for vague sets, interval vague sets, triangular and trapezoidal IFSs, and a revised correlation coefficient for triangular and trapezoidal IFSs using graded mean integration representation. In this paper, a novel method of correlation coefficient of trapezoidal fuzzy intuitionistic fuzzy sets (TzFIFSs) is proposed and developed by taking into account the membership, nonmembership, and the hesitation degrees of TzFIFSs. The weighted averaging (WA) and weighted geometric (WG) operators for TzFIFSs are proposed for MADM problems. Based on these operators, an approach is suggested to solve uncertain multiple attribute group decision making problems, where the attribute values are trapezoidal fuzzy intuitionistic fuzzy numbers. A new algorithm is developed to solve the MADM problems in which the correlation coefficient of TzFIFSs is used for ranking alternatives.


## 2. Arithmetic Operations for TzFIFS

Arithmetic operations of TzFIFNs are based on the arithmetic operations of TzFNs. In TzFIFN the membership and nonmembership degrees take the form of trapezoidal fuzzy number. The basic concepts related to TzFIFNs are presented in the following.Definition 1 (trapezoidal fuzzy number (TzFN)).
A
=
(
a
,
b
,
c
,
d
) is called a trapezoidal fuzzy number, if the membership function μA:R→[0,1] is expressed as
(1)μA(x)={x-ab-afora≤x<b1forb≤x<cc-xd-cforc≤x<d0otherwise,
where x∈R,  0≤a≤b≤c≤d≤1.Definition 2 (trapezoidal fuzzy intuitionistic fuzzy number (TzFIFN)).
LetX be a nonempty set. Then A={〈x,μA(x),γA(x)〉/x∈X} is called a trapezoidal fuzzy intuitionistic fuzzy number if μA(x)=(μA1(x),μA2(x),μA3(x),μA4(x)) and γA(x)=(γA1(x),γA2(x),γA3(x),γA4(x)) are trapezoidal fuzzy numbers, which can express the membership degree and the nonmembership degree of x in X and fulfill 0≤μA4(x)+γA4(x)≤1,  for  all  x∈X.
An intuitionistic fuzzy number expressed on the basis of trapezoidal fuzzy number is called trapezoidal fuzzy intuitionistic fuzzy number.
π
A
(
x
)
=
(
π
A
1
(
x
)
,
π
A
2
(
x
)
,
π
A
3
(
x
)
,
π
A
4
(
x
)
) is called the hesitation degree of the given trapezoidal fuzzy intuitionistic fuzzy set. Also we have 0≤πA1(x), πA2(x), πA3(x), πA4(x)≤1,  for  all  x∈X.
Suppose(2)A={〈x,[μA1(x),μA2(x),μA3(x),μA4(x)],[γA1(x),γA2(x),γA3(x),γA4(x)]〉×(x∈X)-1},B={〈x,[μB1(x),μB2(x),μB3(x),μB4(x)],[γB1(x),γB2(x),γB3(x),γB4(x)]〉×(x∈X)-1}
are two TzFIFNs; then according to the above operation rules of intuitionistic fuzzy numbers and the operation rules of trapezoidal fuzzy numbers, the operational rules of TzFIFNs are as follows:
(3)A+B={〈[μA1(x),μA2(x),μA3(x),μA4(x)],[γA1(x),γA2(x),γA3(x),γA4(x)]〉+〈[μB1(x),μB2(x),μB4(x)],[γB1(x),γB2(x),γB3(x),γB4(x)]〉}={〈[μA1(x)+μB1(x)-μA1(x)·μB1(x),μA2(x)+μB2(x)-μA2(x)·μB2(x),μA3(x)+μB3(x)-μA3(x)·μB3(x),μA4(x)+μB4(x)-μA4(x)·μB4(x)],[γA1(x)·γB1(x),γA2(x)·γB2(x),γA3(x)·γB3(x),γA4(x)·γB4(x)]〉},(4)A·B={〈[μA1(x),μA2(x),μA3(x),μA4(x)],[γA1(x),γA2(x),γA3(x),γA4(x)]〉·〈[μB1(x),μB2(x),μB4(x)],[γB1(x),γB2(x),γB3(x),γB4(x)]〉}={〈[μA1(x)·μB1(x),μA2(x)·μB2(x),μA3(x)·μB3(x),μA4(x)·μB4(x)],[γA1(x)+γB1(x)-γA1(x)·γB1(x),γA2(x)+γB2(x)-γA2(x)·γB2(x),γA3(x)+γB3(x)-γA3(x)·γB3(x),γA4(x)+γB4(x)-γA4(x)·γB4(x)]〉},(5)λA=λ([μA1(x),μA2(x),μA3(x),μA4(x)],[γA1(x),γA2(x),γA3(x),γA4(x)])={〈[1-(1-μA1(x))λ,1-(1-μA2(x))λ,1-(1-μA3(x))λ,1-(1-μA4(x))λ],[(γA1(x))n,(γA2(x))n,(γA3(x))n,(γA4(x))n][1-(1-μA1(x))λ,1-(1-μA2(x))λ,〉},(6)Aλ=([μA1(x),μA2(x),μA3(x),μA4(x)],[γA1(x),γA2(x),γA3(x),γA4(x)])λ={〈[(1-γA4(x))λ(μA1(x))n,(μA2(x))n,(μA3(x))n,(μA4(x))n(μA1(x))n],[1-(1-γA1(x))λ,1-(1-γA2(x))λ,1-(1-γA3(x))λ,1-(1-γA4(x))λ]〉},wwwwwwwwwwwwwwwwwwwwwwλ≥0.
For the above operation rules, the following are true:(i)
A+B=B+A,(ii)
A·B=B·A,(iii)
λ(A+B)=λA+λB, λ≥0,(iv)
λ1A+λ2A=(λ1+λ2)A,λ1, λ2≥0,(v)
Aλ1·Aλ2=(A)λ1+λ2,λ1, λ2≥0.


## 3. Correlation Coefficient of TzFIFS

LetX={x1,x2,…,xn} be the finite universal set and let A, B∈ TzFIFS(X) be given by
(7)A={〈x,[μA1(x),μA2(x),μA3(x),μA4(x)],[γA1(x),γA2(x),γA3(x),γA4(x)]〉×(x∈X)-1},B={〈x,[μB1(x),μB2(x),μB3(x),μB4(x)],[γB1(x),γB2(x),γB3(x),γB4(x)]〉×(x∈X)-1},
which are two trapezoidal fuzzy intuitionistic fuzzy numbers.Then the correlation of trapezoidal fuzzy intuitionistic fuzzy numbers (TzFIFNs) is defined as follows.Definition 3.
LetA={〈x,μA(x),γA(x)〉/x∈X} be a trapezoidal fuzzy intuitionistic fuzzy number, where μA(x)=(μA1(x),μA2(x),μA3(x),μA4(x)) is the membership degree, γA(x)=(γA1(x),γA2(x),γA3(x),γA4(x)) is the nonmembership degree, and πA(x)=(πA1(x),πA2(x),πA3(x),πA4(x)) is the hesitation degree. Then the trapezoidal fuzzy intuitionistic energy of the sets A and B is defined as
(8)ETzFIFS(A)=14∑i=1n[μA2(xi)+γA2(xi)+πA2(xi)]=14∑i=1n{[(μA12(xi)+μA22(xi)+μA32(xi)+μA42(xi))]+[(γA12(xi)+γA22(xi)+γA32(xi)+γA42(xi))]+[(πA12(xi)+πA22(xi)+πA32(xi)+πA42(xi))]},(9)ETzFIFS(B)=14∑i=1n[μB2(xi)+γB2(xi)+πB2(xi)]=14∑i=1n{[(μB12(xi)+μB22(xi)+μB32(xi)+μB42(xi))]+[(γB12(xi)+γB22(xi)+γB32(xi)+γB42(xi))]+[(πB12(xi)+πB22(xi)+πB32(xi)+πB42(xi))]}.Definition 4.
LetA={〈x,μA(x),γA(x)〉/x∈X} be a trapezoidal fuzzy intuitionistic fuzzy number, where μA(x)=(μA1(x),μA2(x),μA3(x),μA4(x)) is the membership degree, γA(x)=(γA1(x),γA2(x),γA3(x),γA4(x)) is the nonmembership degree, and πA(x)=(πA1(x),πA2(x),πA3(x),πA4(x)) is the hesitation degree. Then the correlation between the sets A and B is defined as
(10)CTzFIFS(A,B)=14∑i=1n[μA(xi)μB(xi)+γA(xi)γB(xi)+πA(xi)πB(xi)]=14∑i=1n{[(μA1(xi)μB1(xi)+μA2(xi)μB2(xi)+μA3(xi)μB3(xi)+μA4(xi)μB4(xi))]+[(γA1(xi)γB1(xi)+γA2(xi)γB2(xi)+γA3(xi)γB3(xi)+γA4(xi)γB4(xi))]+[(πA1(xi)πB1(xi)+πA2(xi)πB2(xi)+πA3(xi)πB3(xi)+πA4(xi)πB4(xi))]}.
The correlation coefficient is given as(11)KTzFIFS(A,B)=CTzFIFS(A,B)ETzFIFS(A)·ETzFIFS(B),wwwwwwwwwwi0≤KTzFIFS(A,B)≤1.Proposition 5.
ForA, B∈ TzFIFS(X) the following are true:(1)
0≤KTzFIFS(A,B)≤1,(2)
CTzFIFS(A,B)=CTzFIFS(B,A),(3)
KTzFIFS(A,B)=KTzFIFS(B,A).Theorem 6.
ForA, B∈ TzFIFS(X), then 0≤KTzFIFS(A,B)≤1.Proof.
SinceCTzFIFS(A,B)≥0, it can be proved that KTzFIFS(A,B)≤1.
For any arbitrary real numberξ, the following relation is true:
(12)0≤∑i=1n‍{(μA1(xi)-ξμB1(xi))2+(μA2(xi)-ξμB2(xi))2+(μA3(xi)-ξμB3(xi))2+(μA4(xi)-ξμB4(xi))2+(γA1(xi)-ξγB1(xi))2+(γA2(xi)-ξγB2(xi))2+(γA3(xi)-ξγB3(xi))2+(γA4(xi)-ξγB4(xi))2+(πA1(xi)-ξπB1(xi))2+(πA2(xi)-ξπB2(xi))2+(πA3(xi)-ξπB3(xi))2+(πA4(xi)-ξπB4(xi))2}=∑i=1n‍{(μA12(xi)+γA12(xi)+πA12(xi))+(μA22(xi)+γA22(xi)+πA22(xi))+(μA32(xi)+γA32(xi)+πA32(xi))+(μA42(xi)+γA42(xi)+πA42(xi))-2ξ(μA1(xi)μB1(xi)+γA1(xi)γB1(xi)+πA1(xi)πB1(xi))-2ξ(μA2(xi)μB2(xi)+γA2(xi)γB2(xi)+πA2(xi)πB2(xi))-2ξ(μA3(xi)μB3(xi)+γA3(xi)γB3(xi)+πA3(xi)πB3(xi))-2ξ(μA4(xi)μB4(xi)+γA4(xi)γB4(xi)+πA4(xi)πB4(xi))+ξ2(μB12(xi)+γB12(xi)+πB12(xi))+ξ2(μB22(xi)+γB22(xi)+πB22(xi))+ξ2(μB32(xi)+γB32(xi)+πB32(xi))+ξ2(μB42(xi)+γB42(xi)+πB42(xi))}.
Hence,
(13){∑i=1n‍[(μA1(xi)μB1(xi)+μA2(xi)μB2(xi)+μA3(xi)μB3(xi)+μA4(xi)μB4(xi)+γA1(xi)γB1(xi)+γA2(xi)γB2(xi)+γA3(xi)γB3(xi)+γA4(xi)γB4(xi)+πA1(xi)πB1(xi)+πA2(xi)πB2(xi)+πA3(xi)πB3(xi)+πA4(xi)πB4(xi))]∑i=1n‍}2≤(∑i=1n‍{(μA12(xi)+γA12(xi)+πA12(xi))+(μA22(xi)+γA22(xi)+πA22(xi))+(μA32(xi)+γA32(xi)+πA32(xi))+(μA42(xi)+γA42(xi)+πA42(xi))}×∑i=1n‍{(μB12(xi)+γB12(xi)+πB12(xi))+(μB22(xi)+γB22(xi)+πB22(xi))+(μB32(xi)+γB32(xi)+πB32(xi))}+(μB42(xi)+γB42(xi)+πB42(xi))∑i=1n‍).
The above inequality can be written as
(14)({∑i=1n‍[μA1(xi)μB1(xi)+μA2(xi)μB2(xi)+μA3(xi)μB3(xi)+μA4(xi)μB4(xi)+γA1(xi)γB1(xi)+γA2(xi)γB2(xi)+γA3(xi)γB3(xi)+γA4(xi)γB4(xi)+πA1(xi)πB1(xi)+πA2(xi)πB2(xi)+πA3(xi)πB3(xi)+πA4(xi)πB4(xi)]2∑i=1n‍}×((∑i=1n‍{(μA12(xi)+γA12(xi)+πA12(xi))+(μA22(xi)+γA22(xi)+πA22(xi))+(μA32(xi)+γA32(xi)+πA32(xi))+(μA42(xi)+γA42(xi)+πA42(xi))}×∑i=1n{(μB12(xi)+γB12(xi)+πB12(xi))+(μB22(xi)+γB22(xi)+πB22(xi))+(μB32(xi)+γB32(xi)+πB32(xi))+(μB42(xi)+γB42(xi)+πB42(xi))}∑i=1n‍))-1).wwwwwwwwwwwwwwwwwwwwwwwwwwwww≤1.
Therefore [CTzFIFS(A,B)]2/ETzFIFS(A)·ETzFIFS(B)≤1; hence KTzFIFS(A,B)=CTzFIFS(A,B)/ETzFIFS(A)·ETzFIFS(B)≤1.Theorem 7.
ConsiderKTzFIFS(A,B)=1⇔A=B.Proof.
Considering the inequality in the proof of Theorem6, then the equality holds if and only if the following are satisfied(i)
μA1(xi)=ξμB1(xi), μA2(xi)=ξμB2(xi), μA3(xi)=ξμB3(xi),μA4(xi)=ξμB4(xi),(ii)
γA1(xi)=ξγB1(xi), γA2(xi)=ξγB2(xi), γA3(xi)=ξγB3(xi),γA4(xi)=ξγB4(xi),(iii)
πA1(xi)=ξπB1(xi), πA2(xi)=ξπB2(xi), πA3(xi)=ξπB3(xi),πA4(xi)=ξπB4(xi),for some positive real ξ.
As(15)μA1(xi)+γA1(xi)+πA1(xi)=μB1(xi)+γB1(xi)+πB1(xi)=1,μA2(xi)+γA2(xi)+πA2(xi)=μB2(xi)+γB2(xi)+πB2(xi)=1,μA3(xi)+γA3(xi)+πA3(xi)=μB3(xi)+γB3(xi)+πB3(xi)=1,μA4(xi)+γA4(xi)+πA4(xi)=μB4(xi)+γB4(xi)+πB4(xi)=1,
then it means ξ=1, and therefore A=B.Theorem 8.
C
T
r
F
I
F
S
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A
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B
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=
0
⇔
A and B are nonfuzzy sets and satisfy the condition μA(xi)+μB(xi)=1 or γA(xi)+γB(xi)=1 or πA(xi)+πB(xi)=1,  for  all  xi∈X.Proof.
For allxi∈X, the following are true:
(16)(μA1(xi)μB1(xi)+γA1(xi)γB1(xi)+πA1(xi)πB1(xi))≥0,(μA2(xi)μB2(xi)+γA2(xi)γB2(xi)+πA2(xi)πB2(xi))≥0,(μA3(xi)μB3(xi)+γA3(xi)γB3(xi)+πA3(xi)πB3(xi))≥0,(μA4(xi)μB4(xi)+γA4(xi)γB4(xi)+πA4(xi)πB4(xi))≥0.
Hence,
(17){(μA1(xi)μB1(xi)+γA1(xi)γB1(xi)+πA1(xi)πB1(xi))+(μA2(xi)μB2(xi)+γA2(xi)γB2(xi)+πA2(xi)πB2(xi))+(μA3(xi)μB3(xi)+γA3(xi)γB3(xi)+πA3(xi)πB3(xi))+(μA4(xi)μB4(xi)+γA4(xi)γB4(xi)+πA4(xi)πB4(xi))}≥0,{μA1(xi)μB1(xi)+μA2(xi)μB2(xi)+μA3(xi)μB3(xi)+μA4(xi)μB4(xi)+γA1(xi)γB1(xi)+γA2(xi)γB2(xi)+γA3(xi)γB3(xi)+γA4(xi)γB4(xi)+πA1(xi)πB1(xi)+πA2(xi)πB2(xi)+πA3(xi)πB3(xi)+πA4(xi)πB4(xi)}≥0.
If CTzFIFS(A,B)=0 for all xi∈X, then the following should be true:
(18)μA1(xi)μB1(xi)+μA2(xi)μB2(xi)+μA3(xi)μB3(xi)+μA4(xi)μB4(xi)=0,γA1(xi)γB1(xi)+γA2(xi)γB2(xi)+γA3(xi)γB3(xi)+γA4(xi)γB4(xi)=0,πA1(xi)πB1(xi)+πA2(xi)πB2(xi)+πA3(xi)πB3(xi)+πA4(xi)πB4(xi)=0.(i)
IfμA1(xi)=1 then μB1(xi)=0 and γA1(xi)=πA1(xi)=0,(ii)
ifμA2(xi)=1 then μB2(xi)=0 and γA2(xi)=πA2(xi)=0,(iii)
ifμA3(xi)=1 then μB3(xi)=0 and γA3(xi)=πA3(xi)=0,(iv)
ifμA4(xi)=1 then μB4(xi)=0 and γA4(xi)=πA4(xi)=0.
Also,(i)
ifμB1(xi)=1 then μA1(xi)=0 and γB1(xi)=πB1(xi)=0,(ii)
ifμB2(xi)=1 then μA2(xi)=0 and γB2(xi)=πB2(xi)=0,(iii)
ifμB3(xi)=1 then μA3(xi)=0 and γB3(xi)=πB3(xi)=0,(iv)
ifμB4(xi)=1 then μA4(xi)=0 and γB4(xi)=πB4(xi)=0.
HenceμA1(xi)+μB1(xi)=1, μA2(xi)+μB2(xi)=1, μA3(xi)+μB3(xi)=1, and μA4(xi)+μB4(xi)=1.
Conversely, whenA and B are nonfuzzy sets and μA1(xi)+μB1(xi)=1, μA2(xi)+μB2(xi)=1, μA3(xi)+μB3(xi)=1,  and  μA4(xi)+μB4(xi)=1.
IfμA1(xi)=1 then μB1(xi)=0 and γA1(xi)=πA1(xi)=0.
This property can be observed similarly for all other entries. ThereforeCTrFIFS(A,B)=0.
The casesγA(xi)+γB(xi)=1 and πA(xi)+πB(xi)=1 can be proved similarly.Theorem 9.
C
T
z
F
I
F
S
(
A
,
A
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=
1
⇔
A is a nonfuzzy set.Proof.
IfA is a nonfuzzy set, then CTzFIFS(A,A)=1 is obvious.
Conversely, it can be proved by the method of contradiction.
AssumeA is not a nonfuzzy set.
Then0≤μA(xi)<1,0≤γA(xi)<1, and 0≤πA(xi)<1, for some xi.
HenceμA2(xi)+γA2(xi)+πA2(xi)<1.
That is,(19)μA12(xi)+γA12(xi)+πA12(xi)<1,μA22(xi)+γA22(xi)+πA22(xi)<1,μA32(xi)+γA32(xi)+πA32(xi)<1,μA42(xi)+γA42(xi)+πA42(xi)<1.
Also,(20){μA12(xi)+μA22(xi)+μA32(xi)+μA42(xi)+γA12(xi)+γA22(xi)+γA32(xi)+γA42(xi)+πA12(xi)+πA22(xi)+πA32(xi)+πA42(xi)}<1.
Then(21)CTzFIFS(A,A)=13∑i=1n‍{μA12(xi)+μA22(xi)+μA32(xi)+μA42(xi)+γA12(xi)+γA22(xi)+γA32(xi)+γA42(xi)+πA12(xi)+πA22(xi)+πA32(xi)+πA42(xi)}<1.
This is contradictory, and so A is a nonfuzzy set.


## 4. WA and WG Operators for TzFIFNs

On the foundation of the definitions discussed by Wang [13], the weighted arithmetic operators for TzFIFNs are given as follows.Definition 10.
Leta~j=〈[μ1j,μ2j,μ3j,μ4j],[γ1j,γ2j,γ3j,γ4j]〉, j=1,2,…,n, be a collection of TzFIFN values. The trapezoidal fuzzy intuitionistic fuzzy weighted averaging (TzFIFWA) operator, TzFIFWA:Qn→Q, is defined as
(22)TzFIFWAω(a~1,a~2,…,a~n)=∑j=1nωja~j=(1-∏j=1n(1-μj)ωj,∏j=1n(γj)ωj)=([1-∏j=1n(1-μ1j)ωj,1-∏j=1n(1-μ2j)ωj,1-∏j=1n(1-μ3j)ωj,1-∏j=1n(1-μ4j)ωj],[∏j=1n(γ1j)ωj,∏j=1n(γ2j)ωj,∏j=1n(γ3j)ωj,∏j=1n(γ4j)ωj]),
where ω=(ω1,ω2,…,ωt)T is the weight vector of a~j=〈[μ1j,μ2j,μ3j,μ4j],[γ1j,γ2j,γ3j,γ4j]〉, such that ωj>0 and ∑j=1nωj=1.Definition 11.
Leta~j=〈[μ1j,μ2j,μ3j,μ4j],[γ1j,γ2j,γ3j,γ4j]〉, j=1,2,…,n, be a collection of TzFIFN values. The trapezoidal fuzzy intuitionistic fuzzy weighted geometric (TzFIFWG) operator, TzFIFWG:Qn→Q, is defined as
(23)TzFIFWGω(a~1,a~2,…,a~n)=∑j=1nωja~j=(∏j=1n(μj)ωj,1-∏j=1n(1-γj)ωj)=([∏j=1n(μ1j)ωj,∏j=1n(μ2j)ωj,∏j=1n(μ3j)ωj,∏j=1n(μ4j)ωj],[1-∏j=1n(1-γ1j)ωj,1-∏j=1n(1-γ2j)ωj,1-∏j=1n(1-γ3j)ωj,1-∏j=1n(1-γ4j)ωj]),
where ω=(ω1,ω2,…,ωt)T is the weight vector of a~j=〈[μ1j,μ2j,μ3j,μ4j],[γ1j,γ2j,γ3j,γ4j]〉, such that ωj>0 and ∑j=1nωj=1.


## 5. MADM Algorithm for Trapezoidal Fuzzy IFS

LetA={A1,A2,…,Am} be a set of alternatives, and let G={G1,G2,…,Gn} be the set of attributes; ω=(ω1,ω2,…,ωn) is the weighting vector of the attribute Gj, j=1,2,…,n, where ωj∈[0,1], such that ∑j=1nωj=1. Suppose that
(24)R~k=(r~hij(k))m×n=(μhij(k),γhij(k))m×n=([μ1ij(k),μ2ij(k),μ3ij(k),μ4ij(k)],[γ1ij(k),γ2ij(k),γ3ij(k),γ4ij(k)])m×n
be the trapezoidal fuzzy intuitionistic fuzzy number decision matrix, where [μ1ij(k),μ2ij(k),μ3ij(k),μ4ij(k)] is the degree of the membership value that the alternative Ai satisfies the attribute Gj given by the decision maker Dk and [γ1ij(k),γ2ij(k),γ3ij(k),γ4ij(k)] is the degree of nonmembership value for the alternative Ai, where μhij(k),γhij(k)⊂[0,1] and μ4ij(k),γ4ij(k)≤1,i=1,2,…,m, j=1,2,…,n, and k=1,2,…,t.The developed model of MADM is given as follows.Step 1.
Utilize the decision information given in the matrixR~ and the TzFIFWA or TzFIFWG operator which has the associated weighting vector ω=(ω1,ω2,…,ωn)T and aggregate the given decision matrix R~=(r~ij)m×n into group integrated attribute values r~i:
(25)r~i=TzFIFWAω(a~1,a~2,…,a~n)=∑j=1nωja~j=(1-∏j=1n(1-μj)ωj,∏j=1n(γj)ωj)=([1-∏j=1n(1-μ1j)ωj,1-∏j=1n(1-μ2j)ωj,1-∏j=1n(1-μ3j)ωj,1-∏j=1n(1-μ4j)ωj],[∏j=1n(γ1j)ωj,∏j=1n(γ2j)ωj,∏j=1n(γ3j)ωj,∏j=1n(γ4j)ωj])
or
(26)r~i=TzFIFWGω(a~1,a~2,…,a~n)=∑j=1nωja~j=(∏j=1n(μj)ωj,1-∏j=1n(1-γj)ωj)=([∏j=1n(μ1j)ωj,∏j=1n(μ2j)ωj,∏j=1n(μ3j)ωj,∏j=1n(μ4j)ωj],[1-∏j=1n(1-γ1j)ωj,1-∏j=1n(1-γ2j)ωj,1-∏j=1n(1-γ3j)ωj,1-∏j=1n(1-γ4j)ωj]).Step 2.
Utilize the correlation coefficient (8) to (11) of TzFIFNs to derive the closeness between the overall group integrated attribute values r~i and the TzFIFN positive ideal value r~+, where r~+=〈[1,1,1,1],[0,0,0,0]〉.Step 3.
Rank alternativesAi,i=1,2,…,m, and select the best in accordance with the highest closeness obtained from Step 2.


## 6. Numerical Illustration

A company intends to select one person to take the position of Assistant Manager from four candidates. Five indicators (attributes) must be evaluated. They are shown as follows:(i)
technical skill (G1),(ii)
professional ability (G2),(iii)
creative ability (G3),(iv)
analytical skill (G4),(v)
leadership ability (G5).The weights of the indicators are=(ω1,ω2,…,ωn)T=(0.18,0.26,0.15,0.19,0.22)T. The individual attributes of each candidate are to be evaluated in order to come to a good decision. The decision matrix R~=(r~ij)m×n is given as follows:
(27)R~=G1(〈[0.15,0.21,0.32,0.42],[0.19,0.23,0.42,0.51]〉〈[0.21,0.33,0.45,0.49],[0.12,0.19,0.23,0.35]〉〈[0.13,0.19,0.31,0.39],[0.11,0.23,0.27,0.31]〉〈[0.18,0.25,0.32,0.43],[0.17,0.23,0.34,0.41]〉G2〈[0.11,0.19,0.23,0.31],[0.13,0.24,0.29,0.35]〉〈[0.13,0.15,0.25,0.34],[0.19,0.21,0.26,0.32]〉〈[0.21,0.23,0.29,0.42],[0.28,0.34,0.41,0.46]〉〈[0.22,0.25,0.39,0.43],[0.33,0.35,0.41,0.45]〉G3〈[0.17,0.27,0.37,0.43],[0.11,0.19,0.23,0.30]〉〈[0.21,0.33,0.42,0.51],[0.25,0.32,0.41,0.45]〉〈[0.15,0.20,0.35,0.49],[0.22,0.26,0.35,0.42]〉〈[0.16,0.23,0.36,0.39],[0.27,0.35,0.41,0.46]〉G4〈[0.25,0.35,0.45,0.55],[0.15,0.20,0.32,0.41]〉〈[0.21,0.32,0.43,0.56],[0.12,0.23,0.33,0.39]〉〈[0.13,0.15,0.25,0.29],[0.11,0.25,0.34,0.41]〉〈[0.15,0.23,0.41,0.44],[0.21,0.32,0.39,0.46]〉G5〈[0.15,0.23,0.34,0.47],[0.25,0.28,0.34,0.45]〉〈[0.18,0.25,0.36,0.43],[0.26,0.37,0.47,0.51]〉〈[0.19,0.23,0.35,0.43],[0.27,0.31,0.41,0.52]〉〈[0.24,0.34,0.45,0.61],[0.15,0.25,0.28,0.30]〉).Step 1.
Utilize the decision information given in the matrixR~, the TzFIFWA operator, and the weighting vector of the attributes ω=(ω1,ω2,…,ωn)T=(0.18,0.26,0.15,0.19,0.22)T and aggregate the decision matrix R~. The collective values are as follows:
(28)r~1=〈[0.1629,0.2470,0.3375,0.4346],[0.1610,0.2297,0.3159,0.3986]〉,r~2=〈[0.1832,0.2673,0.3744,0.4598],[0.1789,0.2532,0.3245,0.3937]〉,r~3=〈[0.1677,0.2036,0.3092,0.4057],[0.1896,0.2813,0.3584,0.4248]〉,r~4=〈[0.1956,0.2642,0.3914,0.4720],[0.2192,0.2963,0.3729,0.4078]〉.Step 2.
Utilize the correlation coefficient of TzFIFNs to derive the closeness between the overall group integrated attribute valuesr~i and the TzFIFN positive ideal value r~+, where r~+=〈[1,1,1,1],[0,0,0,0]〉. Using (8) to (11), the correlation coefficients are calculated using the following formula:
(29)KTzFIFS(A,B)=CTzFIFS(A,B)ETzFIFS(A)·ETzFIFS(B),wwwwwwwwww0≤KTzFIFS(A,B)≤1.
Hence the calculated values are given as follows:
(30)CTzFIFS(r~1,r~+)=0.2955,KTzFIFS(r~1,r~+)=0.4663,CTzFIFS(r~2,r~+)=0.3212,KTzFIFS(r~2,r~+)=0.5140,CTzFIFS(r~3,r~+)=0.2715,KTzFIFS(r~3,r~+)=0.4332,CTzFIFS(r~4,r~+)=0.3308,KTzFIFS(r~4,r~+)=0.5347.Step 3.
Rank alternativesAi, i=1,2,…,m, from the highest closeness value (correlation coefficient) obtained from Step 2; the result is obtained as follows:
(31)A4>A2>A1>A3.
Hence the best alternative is A4.
Following the MADM model using the TzFIFWA operator, we have the following numerical results for the MADM model using TzFIFWG operator.Step 1.
Utilize the decision information given in the matrixR~k, the TzFIFWG operator, and the weighting vector of the attributes ω=(ω1,ω2,…,ωn)T=(0.18,0.26,0.15,0.19,0.22)T and aggregate the same decision matrix R~. The collective values are as follows:
(32)r~1=〈[0.1553,0.2388,0.3245,0.4202],[0.1695,0.2325,0.3236,0.4088]〉r~2=〈[0.1792,0.2514,0.3608,0.4467],[0.1906,0.2654,0.3431,0.4045]〉r~3=〈[0.1635,0.2006,0.3059,0.3973],[0.2094,0.2857,0.3385,0.4347]〉r~4=〈[0.1917,0.2600,0.3874,0.4596],[0.2331,0.3024,0.3670,0.4163]〉.Step 2.
Utilize the correlation coefficient of TzFIFNs to derive the closeness between the overall group integrated attribute valuesr~i and the TzFIFN positive ideal value r~+, where r~+=〈[1,1,1,1],[0,0,0,0]〉. Using (8) to (11), the correlation coefficients are calculated as follows:
(33)CTzFIFS(r~1,r~+)=0.2847,KTzFIFS(r~1,r~+)=0.4493,CTzFIFS(r~2,r~+)=0.3095,KTzFIFS(r~2,r~+)=0.4966,CTzFIFS(r~3,r~+)=0.2673,KTzFIFS(r~3,r~+)=0.4281,CTzFIFS(r~4,r~+)=0.3246,KTzFIFS(r~4,r~+)=0.5270.Step 3.
Rank alternativesAi, i=1,2,…,m, from the highest closeness (correlation coefficient) obtained from Step 2; the result obtained is
(34)A4>A2>A1>A3.
Hence the best alternative is A4.


## 7. Conclusion

In this paper an MADM model was proposed based on the correlation coefficient of TzFIFS for ranking the alternatives together with WA and WG operators. The trapezoidal fuzzy intuitionistic fuzzy weighted averaging (TzFIFWA) operator and the trapezoidal fuzzy intuitionistic fuzzy weighted geometric (TzFIFWG) operator were used to aggregate the trapezoidal fuzzy intuitionistic fuzzy information given in a decision matrix. A numerical illustration was given to show the effectiveness of the proposed approach in using correlation coefficient of TzFIFSs because it preserved the linear relationship between the variables.



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*Source: 159126-2014-08-04.xml*