INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, e-ISSN 1841-9844, 14(2), 220-232, April 2019.
Extended TODIM Method for MADM Problem under
Trapezoidal Intuitionistic Fuzzy Environment
H.P. Ren, M.F. Liu, H. Zhou
Haiping Ren*
Teaching Department of Basic Subjects
Jiangxi University of Science and Technology,330013 Nanchang, China
*Corresponding author: 9520060004@jxust.edu.cn
Manfeng Liu
The Collaborative Innovation Center
Jiangxi University of Finance and Economics,330013 Nanchang, China
liumanfeng@sina.com
Hui Zhou
School of Mathematics and Computer Science
Yichun University,336000 Yichun, China
huihui7978@126.com
Abstract: In actual decision making process, the final decision result is often af-
fected by decision maker’s psychological behavior, however, for the multiple attribute
decision making (MADM) problem in which attributes values are expressed with
trapezoidal intuitionistic fuzzy numbers, there are few literatures considering the de-
cision maker’s behavior factors in decision making process. For this case, this paper
first proposes a new distance measure of TIFNs and a new ranking method which
considers decision maker’s attitude behavior, and then develops an extended TODIM
decision making method. Finally an example is given to illustrate the validity and
practicability of the proposed method.
Keywords: TODIM method, trapezoidal intuitionistic fuzzy number, multiple at-
tribute decision making, ranking method.
1 Introduction
In recent years, with the increasing complexity of the managerial decision making envi-
ronment, many managerial decision-making problems contain qualitative properties which are
difficult to quantify. Zadeh’s fuzzy sets have been greatly successful in dealing with fuzzy manage-
ment decision making problems [3,4,18,20,22,35]. Zadeh’s fuzzy set is characterized by a single
scale (membership), which can only characterize the support and opposition of the two aspects
of the evidence. But some decision making problems have ambiguous hesitant phenomenon with
respect to evaluation of information, and Zadeh’s fuzzy set is hard or difficult to depict these sit-
uations. Therefore, many scholars developed Zadeh’s fuzzy set, and intuitionistic fuzzy (IF) set
is one of the most famous fuzzy sets among them. Originally proposed by Atanassov in 1986 [1],
IF sets can well describe the hesitation and uncertainty of judgment through the addition of a
non-membership parameters, which can describe the vague characters of things comprehensively.
Then IF sets have become a powerful and effective tool in dealing with uncertain or vague in-
formation in actual applications. In dealing with ambiguity and uncertainty, IF sets are more
flexible and practical than fuzzy sets, and thus they have been applied widely in decision making.
Because of the complexity and uncertainty of objective things and the limitation of decision
maker’s knowledge, membership and non-membership functions are sometimes difficult to repre-
sent by using the precise numbers. But interval number can be very useful to describe this kind
Copyright ©2019 CC BY-NC
Extended TODIM Method for MADM Problem under
Trapezoidal Intuitionistic Fuzzy Environment 221
of case, so Atanassov and Gargov [2] extended IF sets to interval-valued IF sets. Some scholars
put forward the concept of continuous IF numbers to describe an uncertain quantity or a difficult
quantification number on the basis of the concept of IF set. Grzegrorzewski [10] extended IF sets
to the continuous case of IF numbers . Nehi and Maleki [21] put forward trapezoidal intuition-
istic fuzzy number (TIFN), and defined the corresponding operation rules, which caused great
concern in the academic community. Shu [27] proposed the definition of triangular intuitionistic
fuzzy number, which is a special example of TIFN, and they put it to the application of fault
tree analysis, base on these research, Wang and Zhang [31] further expanded it and gave the
definition of a generalized TIFN. Different from the definitions of IF sets, TIFN is added to a
trapezoidal fuzzy number, which makes the membership degree and the non-membership degree
no longer be only a fuzzy concept Good or Excellent; then, the assessment information given by
the decision makers can be expressed exactly. Comparing with IF sets, they have more attractive
explanation, and are easy to be quantified and executed by the decision maker, and thus they
have more theoretical value in the field of decision science [8,14,16,29].
At present, the theory and application of fuzzy numbers, such as triangular intuitionistic
fuzzy number, trapezoidal intuitionistic fuzzy number, have been received great attention. But
most of the existing decision-making methods do not consider the influence of the behavior of
the decision-makers in the decision process, because they are assumed that the decision maker
is completely rational. However, the actual decision-making process is often accompanied by
the different psychological behavior of the decision-makers and the attitude of the risk and
other factors of behavior. Kahneman and Tversky [13] proposed the prospect theory, which can
describe the decision maker’s psychological behavior. Based on prospect theory, Gomes and
Lima ( [6,7]) developed a new MADM method named TODIM method, which has made many
successful applications, such as material evaluation [34], green supplier selection [26], logistics
outsourcing [30] etc. Fan et al. [5] proposed an extension of TODIM (H-TODIM) to solve the
hybrid MADM problems in which attribute values have three forms: crisp numbers, interval
numbers and fuzzy numbers. Qin [23] proposed a generalization of the TODIM method under
triangular intuitionistic fuzzy environment. Ren et al. [25] extended the TODIM method to
deal with the MADM problem in which attribute values are expressed with Pythagorean fuzzy
numbers. Zhang et al. [33] developed the TODIM method to solve the MADM problem in
which the attribute values are expressed with neutrosophic numbers. In this paper, we will
develop a new extension of TODIM method to solve the MADM problem in which attribute
values are expressed with TIFNs, and an application example is used to illustrate the validity
and practicability of the proposed method.
2 Preliminary knowledge
2.1 Definitions of TIFNs
Firstly, we recall the definition of the TIFN and the related theory. In order to use the
concept of IF sets to define an uncertain number or difficult to quantify the amount, Grze-
grorzewski [10] extended the IF sets to the continuous case of the IF sets, and gave the following
definition:
Definition 8. Let R be the set of real numbers, A is called an IF number in R , if its membership
function µA(x) and non-membership function νA(x) are respectively defined as follows ( [10]) :
222 H.P. Ren, M.F. Liu, H. Zhou
µA(x) =
0, x < a1
fA(x), a1 ≤ x ≤ a2
1, a2 ≤ x ≤ a3
gA(x), a3 ≤ x ≤ a4
0, a4 < x
and
νA(x) =
1, x < b1
hA(x), b1 ≤ x ≤ b2
0, b2 ≤ x ≤ b3
kA(x), b3 ≤ x ≤ b4
1, ab < x
where 0 ≤ µA(x) + νA(x) ≤ b2,ai,bi ∈ R,i = 1,2,3,4, and they satisfy b1 ≤ a1 ≤
b2 ≤ a2 ≤ b3 ≤ a3 ≤ b4 ≤ a4 . The four functions fA(x),gA(x),hA(x) and kA(x) are real
value aunctions defined in interval [0,1]. The functions fA(x) , kA(x) are non-decreasing
coctinuous functions and gA(x) , hA(x) are non-increasing continuous functions.
On the basis of Grzegrorzewski’s IF numbers, Nehi [21] developed the TIFN in 2005, which
is given in Definition 2.
Definition 9. Let b1 ≤ a1 ≤ b2 ≤ a2 ≤ b3 ≤ a3 ≤ b4 ≤ a4 ∈ R. A fuzzy number A is called
a TIFN, if its membership function µA(x) and non-membership function νA(x) are respectively
defined as follows( [21]):
µA(x) =
0, x < a1
x−a1
a2−a1 , a1 ≤ x ≤ a2
1, a2 ≤ x ≤ a3
a4−x
a4−a3 , a3 ≤ x ≤ a4
0, a4 < x
and
νA(x) =
1, x < b1
x−b1
b2−b1 , b1 ≤ x ≤ b2
0, b2 ≤ x ≤ b3
b4−x
b4−b3 , b3 ≤ x ≤ b4
1, ab < x
We denote A by A =< (a1,a2,a3,a4), (b1,b2,b3,b4) >.
Definition 10. For two TIFNs A1 =< (a11,a12,a13,a14), (b11,b12,b13,b14) > and A2 =<
(a21,a22,a23,a24), (b21,b22,b23,b24) > , the operational laws are defined as follows [2]:
(1) A1+A2 =< (a11+a21,a12+a22,a13+a23,a14+a24), (b11+b21,b12+b22,b13+b23,b14+b24) >
(2) kA1 =< (ka11,ka12,ka13,ka14), (kb11,kb12,kb13,kb14) >, for k > 0
Usually, α−cut set is a very effective tool for describing the number of fuzzy numbers [11],
and TIFNs have two classes α−cut sets: (A+)α and (A−)α.
Definition 11. Let A =< (a1,a2,a3,a4), (b1,b2,b3,b4) > be a TIFN, then the two classes α−cut
sets (A+)α and (A−)α are defined as follows, respectively:
(A+)α = {x ∈ R|µA(x) ≥ α}
(A−)α = {x ∈ R|1 −νA(x) ≥ α}
Extended TODIM Method for MADM Problem under
Trapezoidal Intuitionistic Fuzzy Environment 223
According to Definition 4, each α−cut set is a closed interval,thus they can be denoted as
(A+)α = [(A
+
L )α, (A
+
U )α] and (A
−)α = [(A
−
L )α, (A
−
U )α], where
(A+L )α = inf{x ∈ R|µA(x) ≥ α},
(A+U )α = sup{x ∈ R|µA(x) ≥ α},
(A−L )α = inf{x ∈ R|1 −νA(x) ≥ α},
(A−U )α = sup{x ∈ R|1 −νA(x) ≥ α}.
Then, for a TIFN A =< (a1,a2,a3,a4), (b1,b2,b3,b4) > , we can easily derive the following
results:
(A+)α = [(A
+
L )α, (A
+
U )α] = [a1 + (a2 −a1)α,a4 − (a4 −a3)α],
(A−)α = [(A
−
L )α, (A
−
U )α] = [b1 + (b2 − b1)(1 −α),b4 − (b4 − b3)(1 −α)],
2.2 A novel distance measure of TIFNs
In the follow we will develop a novel distance of TIFNs based on the following distance
measure between two fuzzy numbers proposed by Grzegorzewski in 1998.
Lemma 2.1. For two any fuzzy numbers A and B, and the corresponding α−cut sets are re-
spectively [(A+L )α, (A
+
U )α] and [(A
−
L )α, (A
−
U )α], then the distance measure between them is defined
as [9].
d(A,B) = (
∫ 1
0
((AL)α − (BL)α)
2
dα +
∫ 1
0
((AU )α − (BU )α)
2
dα)1/2.
Inspired by the Lemma 1, we define the following distance measure for two arbitrary TIFNs
and as follows:
d(A,B) = 1
2
(
∫ 1
0
((A+L )α − (B
+
L )α)
2
+ ((A+U )α − (B
+
U )α)
2
dα)1/2
+ 1
2
(
∫ 1
0
((A−L )α − (B
−
L )α)
2
+ ((A−U )α − (B
−
U )α)
2
dα)1/2
It is easy to prove that the new distance measure can satisfy the non negativity, symmetry and
triangle inequality. By straightforward calculation, we can get Theorem 1.
Theorem 1. Let A1 =< (a11,a12,a13,a14), (b11,b12,b13,b14) > and A2 =< (a21,a22,a23,a24),
(b21,b22,b23,b24) > be two TIFNs,then the distance measure between A1 and A2 is defined as
follows:
d(A1,A2) =
1
2
[
(I1 + I2)
1/2
+ (I3 + I4)
1/2
]
(1)
Proof: Here
I1 =
∫ 1
0
((A+1L)α − (A
+
2L)α)
2
dα
=
∫ 1
0
[(a21 −a11) + (a22 −a21 −a12 + a11)α]2dα
=
∫ 1
0
[x + (y −x)α]2dα
= x2 + x(y −x) + 1
3
(y −x)2
= 1
3
(x2 + xy + y2)
= 1
3
((a21 −a11)2 + (a21 −a11)(a22 −a12) + (a22 −a12)2)
where x = a21 −a11,y = a22 −a12. Similarly, we have
I2 =
∫ 1
0
((A+1U )α − (A
+
2U )α)
2
dα
= 1
3
((a23 −a13)2 + (a23 −a13)(a24 −a14) + (a24 −a14)2)
224 H.P. Ren, M.F. Liu, H. Zhou
I3 =
∫ 1
0
((A−1L)α − (A
−
2L)α)
2
dα
= 1
3
((b21 − b11)2 + (b21 − b11)(b22 − b12) + (b22 − b12)2)
I4 =
∫ 1
0
((A−1U )α − (A
−
2U )α)
2
dα
= 1
3
((b23 − b13)2 + (b23 − b13)(b24 − b14) + (b24 − b14)2)
Then by lemma 1, we can derive the conclusion (1). 2
3 A new ranking function of TIFNs
In actual decision making process, the final decision result is often affected by the different
attitudes of decision-makers, although many scholars have already considered the influence of
different attitude index for the MADM problems in which attributes values are expressed with
interval number and triangular fuzzy number [17, 24, 28]. However, for the MADM problem
in which attributes values are expressed with TIFNs, and there is no literature considering
the decision maker’s attitude in decision making process. Thus, we take the decision maker’s
mentality into the decision-making process, and put forward a new TIFN ranking method.
Definition 12. Let ã = [aL,aU ] be an interval fuzzy number, Mã = 0.5(aL + aU ) and Dã =
0.5(aU −aL). Fã(λ) : [0, 1] → ã is a function of parameter with the following form:
Fã(λ) = Mã + (2λ− 1)Dã = (1 −λ)aL + λaU.
Here, the parameter λ is called attitude index of interval number ã.
Remark 3.1. Apparently, Fã(λ) is a monotonic increasing function on interval[0, 1]. When an
attribute is a benefit type attribute, i.e. the value of it is the-larger-the-better. When λ =
0,then Fã(0) = aL = [aL,aL] is smaller than the fuzzy number ã = [aL,aU ],thus for benefit
type attribute, the parameter λ = 0 demonstrates a pessimistic attitude. Similarly, λ = 1
demonstrates an optimistic attitude and λ = 0.5 demonstrates a moderate attitude.
Lemma 3.1. Let ã = (aL,aM,aU ) be a triangular fuzzy number, and for any real number
α ∈ [0, 1], α− cut set of ã can be easily derived as follows [15]:
ãα = [a
L(α),aU (α)] = [aL + (aM −aL)α,aU − (aU −aM )α]
Remark 3.2. For two arbitrary triangular fuzzy number ã and b̃, α − cut sets are often used
to compare them. Considering the α − cut sets of triangular fuzzy numbers are still interval
numbers, and α is an arbitrary value in interval [0, 1],to eliminate the arbitrariness and reflect
the decision maker’s attitude behavior, Ren and Liu [24]developed a new ranking function of
triangular fuzzy number considering with attitude of decision maker(s) motivated by Definition
5.
Definition 13. Let ã = (aL,aM,aU ) be a triangular fuzzy number, then for any parameter
λ ∈ [0, 1], the function F(ã,λ) is a new ranking function of triangular fuzzy number considering
attitude of decision maker(s) with the following formula [24]:
F(ã,λ) =
∫ 1
0
(1 −λ)aL(α) + λaU (α)dα
Obviously, F(ã,λ) can be rewritten as the following form:
F(ã,λ) =
∫ 1
0
(1 −λ)(aL + (aM −aL)α) + λ(aU − (aU −aM )α)dα
= [(1 −λ)aL + aM + λaU ]/2
Extended TODIM Method for MADM Problem under
Trapezoidal Intuitionistic Fuzzy Environment 225
Let r ∈ [0, 1], then according to Definition 6, Ren and Liu [28] gave the following rule for
comparing two triangular fuzzy numbers ã = (aL,aM,aU ) and b̃ = (bL,bM,bU ):
(i) For arbitrary λ ∈ [0, 1], if F(ã,λ) ≤ F(b̃,λ),then ã is smaller than b̃, and noted ã ≤ b̃;
(ii)For arbitrary λ ∈ [0, 1], if F(ã,λ) = F(b̃,λ),then ã is equal to b̃, and noted ã = b̃;
(iii)For arbitrary λ ∈ [0,r],if F(ã,λ) ≤ F(b̃,λ),while when λ ∈ [r, 1], F(ã,λ) ≥ F(b̃,λ); then
for the decision maker whose attitude is pessimistic, the ranking result is ã ≤ b̃, which for the
decision maker whose attitude is optimistic, the ranking result is ã ≥ b̃.
Motivated by Definition 6, we will develop a new ranking function of trapezoidal intuitionistic
fuzzy number defined in Definition 7.
Definition 14. Let A =< (a1,a2,a3,a4), (b1,b2,b3,b4) > be a TIFN, and p(α) is a real function
defined on [0, 1] , then a new ranking function F(A,λ) including the attitude behavior of decision
maker is defined as follows:
F(A,λ) = 1
2
∫ 1
0
(1 −λ)(A+L )α + λ(A
+
U )αdP(α)
+ 1
2
∫ 1
0
(1 −λ)(A−L )α + λ(A
−
U )αdP(α)
Particularly, if P(α) = αr+1, then we can get
F(A,λ) = 1
2(r+2)
[(1 −λ)(a1 + b1) + (r + 1)(1 −λ)(a2 + b2)
+(r + 1)λ(a3 + b3) + λ(a4 + b4)]
Remark 3.3. If r = 0,λ = 1/2, then F(A,λ) is as same as that ranking function of Ye [32].
Similar discussion with Remark 1, the parameter λ is the attitude index. Then the ranking
function F(A,λ) can reflect the attitude behavior, and thus it can better depict the actual
decision process with the help of different values of λ than that ranking function of Ye [32].
Definition 15. For two given TIFNs A1 and A2 , and r ∈ [0, 1], the relationship of A1 and A2
can be defined as follows:
(i) For any λ ∈ [0, 1], if F(A1,λ) ≤ F(A2,λ),then A1 is smaller than A2, and noted A1 ≤ A2;
(ii)For any λ ∈ [0, 1], if F(A1,λ) = F(A2,λ),then A1 is equal to A2, and noted A1 = A2;
(iii)For any λ ∈ [0,r],if F(A1,λ) ≤ F(A2,λ),while when λ ∈ [r, 1], F(A1,λ) ≥ F(A2,λ); then
for the decision maker whose attitude is pessimistic, the ranking result is A1 ≤ A2, while for the
decision maker whose attitude is optimistic, the ranking result is A1 ≥ A2.
4 Extended TODIM method for MADM under TIFN environ-
ment
For a given MADM problem, let X = {x1,x2, · · · ,xm} be a possible alternatives set, and
O = {o1,o2, · · · ,on} be the evaluation attribute set.D = {D1,D2, · · · ,Ds} is the expert set.
Suppose the rating of xi (i = 1, 2, · · · ,m)with respect to oj(j = 1,2, · · · ,n)given by expert
Dk (k = 1,2, · · · ,s)is a linguistic term noted by s̃kij , which belongs to the linguistic terms
set { Absolutely low, Low, Fairly low, Fairly high, High, Absolutely high }. Then the
MADM problem can be expressed with matrices S̃k = (skij)m×n,k = 1,2, ...,s.
Let w = (w1,w2, ...,wn)T be the attribute weight vector, and each element wj represents
the degree of importance of attribute, which can be given by decision maker or deter-
mined by some weighting methods, such as AHP method or entropy weighting method.
226 H.P. Ren, M.F. Liu, H. Zhou
The calculation steps of the extended TODIM method considering the decision maker’s
attitude are given as follows:
Step 1. According to Table 1 [32],s̃kij can be transformed with TIFNs ãij(k),i =
1,2, ...,m,j = 1,2, ...,n and k = 1,2, ...,s.
Table 1: Linguistic terms and corresponding TIFNs
Linguistic terms TIFNs
Absolutely low (AL) < (0.001, 0.001, 0.001, 0.001), (0.001, 0.001, 0.001, 0.001) >
Low (L) < (0.0, 0.1, 0.2, 0.3), (0.0, 0.1, 0.2, 0.3) >
Fairly low (FL) < (0.1, 0.2, 0.3, 0.4), (0.0, 0.2, 0.3, 0.5) >
Medium (M) < (0.3, 0.4, 0.5, 0.6), (0.2, 0.4, 0.5, 0.7) >
Fairly high (FH) < (0.5, 0.6, 0.7, 0.8), (0.4, 0.6, 0.7, 0.9) >
High (H) < (0.7, 0.8, 0.9, 1.0), (0.7, 0.8, 0.9, 1.0) >
Absolutely high (AH) < (1.0, 1.0, 1.0, 1.0), (1.0, 1.0, 1.0, 1.0) >
The linguistic terms decision matrices S̃k = (skij)m×n are transformed into trapezoidal
intuitionistic fuzzy decision matrices Ãk = (ãkij)m×n (k = 1,2, ...,s).
Step 2. Let s̃ij be the total score of alternative xi with respect to attribute oj given
by all decision makers, and it is defined as
ãij =
1
s
s∑
k=1
ãkij. (2)
Step 3. Determine the weights of evaluation attributes. Using the Definition 7
and P(α) = αr+1, we can get the intuitionistic fuzzy sorting function matrix F(λ) =
(F(ãij,λ))m×n,where F(ãij,λ) is the ranking function of fuzzy number ãij considering with
the attitude of decision maker. For the maximum TIFN ã∗ =< (1,1,1,1) ,(1,1,1,1) >,
F(ã∗,λ) = 1.
Now, we will propose a new weighting method by means of the proposed ranking
function. The reasonable weight should be the minimum of the total deviation of the
alternative xi(i = 1,2, · · · ,m) and the positive ideal solution ã∗. Therefore, we can
establish the following optimization model:
minG(w) =
n∑
j=1
m∑
i=1
wj(1−F(ãij))
s.t.
w ∈ H
n∑
j=1
wj = 1
wj ≥ 0, j = 1,2, ...,n
(3)
By solving the Eq. (3), the optimal solution w∗ = arg maxS is chosen as the optimal
attribute weights.
Step 4. Calculate TODIM score as follows:
(i) Calculate wrc = wcwr ,where the value wrc represents the weight value of criteria r divided
by the weight of the reference point c,and wr = max
1≤c≤n
{wc}.
Extended TODIM Method for MADM Problem under
Trapezoidal Intuitionistic Fuzzy Environment 227
(ii) For given value of attitude index λ ,calculate
φc(xi,xj) =
√
d(xi,xj )
wrc
, F(xic,λ)−F(xjc,λ) > 0
0 F(xic,λ)−F(xjc,λ) = 0
−1
θ
√
d(xi,xj )
wrc
,F(xic,λ)−F(xjc,λ) < 0
Here d(xi,xj) =
n∑
c=1
wrcd(ãic, ãjc). Here the parameter θ is an important parameter in
prospect theory, and θ > 1 shows that the individual is losses aversion, and θ < 1 shows
the individuals are attenuated when facing the losses [19]. Here we set θ = 2.25 , which
is the most often used value of θ in prospect theory.
(iii) Let δ(Ai,Aj) =
n∑
c=1
φc(Ai,Aj), i,j = 1,2, ...,m, calculate the comprehensive evalua-
tion index value:
ξi =
m∑
j=1
δ(Ai,Aj)− min
1≤i≤m
m∑
j=1
δ(Ai,Aj)
max
1≤i≤m
m∑
j=1
δ(Ai,Aj)− min
1≤i≤m
m∑
j=1
δ(Ai,Aj)
, i = 1,2, · · · ,m.
Step 5. Rank the alternatives according to ξi(i = 1,2, · · · ,m) in decreasing order.
5 Applied example
Suppose that a company wants to invest a large amount of money in the best op-
tions(Herrera and Herrera-Viedma [12]; [32]). There are four parallel alternatives: x1 (a
car company), x2 (a food company), x3 (a computer company), x4 (an arms company)
and three evaluation attributes o1(the risk analysis), o2 (the growth analysis), and o3 (the
environmental impact analysis). The risk investment company now employs four experts
to evaluate these four alternative enterprises. The evaluation values are expressed with
linguistic terms, and the corresponding trapezoidal intuitionistic fuzzy evaluation decision
matrices are listed in Table 2 to Table 4. Our task is to choose the best investment plan
by the method presented in this paper.
Table 2: Linguistic evaluation values given by expert 1
Alternatives o1 o2 o3
x1 M M FL
x2 FH FH M
x3 M FH M
x4 H M FL
The specific calculation steps of the proposed decision making method considering
with the psychological behavior of the decision makers are given below:
Step 1. The linguistic terms decision matrices S̃k = (skij)m×n are transformed into
Ãk = (ãkij)m×n (k = 1,2, ...,s) and given in Table 5 to Table 7.
Step 2. The evaluation information of the expert group is gathered and expressed
with decision matrix à = (ãij)m×n , which is shown in Table 8.
228 H.P. Ren, M.F. Liu, H. Zhou
Table 3: Linguistic evaluation values given by expert 2
Alternatives o1 o2 o3
x1 FL M L
x2 FH H M
x3 M FH FL
x4 H FH FL
Table 4: Linguistic evaluation values given by expert 3
Alternatives o1 o2 o3
x1 M FH FL
x2 M FH M
x3 FH FH M
x4 H H M
Table 5: Trapezoidal intuitionistic fuzzy decision matrix given by expert 1
Alternatives o1 o2 o3
x1 <(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.1,0.2,0.3,0.4),
(0.0,0.2,0.3,0.5)>
x2 <(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
x3 <(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
x4 <(0.7,0.8,0.9,1.0),
(0.7,0.8,0.9,1.0)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.1,0.2,0.3,0.4),
(0.0,0.2,0.3,0.5)>
Table 6: Trapezoidal intuitionistic fuzzy decision matrix given by expert 2
Alternatives o1 o2 o3
x1 <(0.1,0.2,0.3,0.4),
(0.0,0.2,0.3,0.5)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.0,0.1,0.2,0.3),
(0.0,0.1,0.2,0.3)>
x2 <(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.7,0.8,0.9,1.0),
(0.7,0.8,0.9,1.0)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
x3 <(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.1,0.2,0.3,0.4),
(0.0,0.2,0.3,0.5)>
x4 <(0.7,0.8,0.9,1.0),
(0.7,0.8,0.9,1.0)>
<(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.1,0.2,0.3,0.4),
(0.0,0.2,0.3,0.5)>
Step 3. In order to facilitate the comparison with the results of Ye [32], here we
also assume that the attribute weights are known with w1 = 0.3490, w2 = 0.3020 and
w3 = 0.3490.
Step 4. For given attitude index value λ = 1/2 and r = 0 , the comprehensive evaluation
values of the extended TODIM method are calculated as
ξ1 = 0,ξ2 = 1.0000,ξ3 = 0.7421,ξ4 = 0.4370,
Extended TODIM Method for MADM Problem under
Trapezoidal Intuitionistic Fuzzy Environment 229
Table 7: Trapezoidal intuitionistic fuzzy decision matrix given by expert 3
Alternatives o1 o2 o3
x1 <(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.1,0.2,0.3,0.4),
(0.0,0.2,0.3,0.5)>
x2 <(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
<(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
x3 <(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.5,0.6,0.7,0.8),
(0.4,0.6,0.7,0.9)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
x4 <(0.7,0.8,0.9,1.0),
(0.7,0.8,0.9,1.0)>
<(0.7,0.8,0.9,1.0),
(0.7,0.8,0.9,1.0)>
<(0.3,0.4,0.5,0.6),
(0.2,0.4,0.5,0.7)>
Table 8: Evaluation information of the expert group
xi o1 o2 o3
x1 <(0.2333,0.3333,0.4333,
0.5333),(0.1333,0.3333,
0.4333,0.6333) >
<(0.3667,0.4667,0.5667,
0.6667),(0.2667,0.4667,
0.5667,0.7667) >
<(0.0667,0.1667,0.2667,
0.3667),(0.0000,0.1667,
0.2667,0.4333) >
x2 <(0.4333,0.5333,0.6333,
0.7333),(0.3333,0.5333,
0.6333, 0.8333)>
<(0.5667,0.6667,0.7667,
0.5667),(0.5000,0.6667,
0.7667, 0.9333)>
<(0.3000,0.4000,0.5000,
0.6000),(0.2000,0.4000,
0.5000,0.7000)>
x3 <(0.3667,0.4667,0.5667,
0.6667),(0.2667,0.4667,
0.5667,0.7667) >
<(0.5000,0.6000,0.7000,
0.8000),(0.4000,0.6000,
0.7000,0.9000)>
<(0.2333,0.3333,0.4333,
0.5333),(0.1333,0.3333,
0.4333,0.6333)>
x4 <(0.7000,0.8000,0.9000,
1.0000),(0.7000,0.8000,
0.9000,1.0000)>
<(0.5000,0.6000,0.7000,
0.8000),(0.4333,0.6000,
0.7000,0.8667)>
<(0.1667,0.2667,0.3667,
0.4667),(0.0667,0.2667,
0.3667,0.5667)>
Step 5. Based on the values of ξi(i = 1, 2, 3, 4), the ranking order of the alternatives is
obtained as
x2>x3>x4>x1,
and x2 is the best alternative.This result is in agreement with the one obtained in (Ye [32]).
6 Conclusion
Interactive multiple criteria (TODIM) decision method is developed on the basis of the
prospect theory, which can describe the psychological behavior of human under uncertain envi-
ronment, and has been successfully applied to many MADM problems. TODIM method is easier
than prospect theory in processing fuzzy numbers, and some authors have already developed it to
solve MADM problems in which the attributes values are expressed with crisp numbers, triangu-
lar fuzzy numbers, intuitionistic fuzzy numbers, Pythagorean fuzzy and neutrosophic numbers.
However, there is no research on the trapezoidal intuitionistic fuzzy environment, and the main
work of this paper is to extend TODIM method to solve MADM problems under TIFN environ-
ment. First, the article proposes a new class of distance measure of TIFNs, the distance measure
can better measure the difference between two TIFNs. Then a new ranking function of TIFNs
is introduced, which can take into account the decision-makers’ attitude with an attitude index.
Finally, the extended TODIM method is put forward to solve the MADM problem in which
the attribute evaluation values are expressed with TIFNs. The advantage of this method lies
230 H.P. Ren, M.F. Liu, H. Zhou
in the decision making process which can take into account the decision maker’s mentality and
the decision maker’s perceived value of the gain and loss, so that the decision-making process is
more consistent with the objective reality. The proposed distance measure and ranking function
can also be used to other MADM methods when the attribute values are expressed with TIFNs.
Funding
The authors would like to thank the support of the National Natural Science Foundation
of China (No.71661012) and Science & Technology Research Project of Jiangxi Educational
Committee (No. GJJ170496 and No. GJJ180829).
Author contributions. Conflict of interest
The authors contributed equally to this work. The authors declare no conflict of interest.
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