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ARTICLE
A method based on distance measure in linguistic space and its
application for CMADM problem
Rong Zhaoa,b, Peng Gea, Maozhu Jinaand Peiyu Rena
aBusiness School, Sichuan University, No. 24 Southern Yihuan Road, Chengdu, 610064,
China; bThe Department of Civil & Environmental Engineering, University of
Washington(Seattle), Seattle, WA 98195, USA
ARTICLE HISTORY
Compiled October 2, 2018
ABSTRACT
Owing to the complexity of decision environment, not all the attributes in mul-
tiple attribute decision making are quantitative. There are also some qualitative
attributes, which are related to the integration of multiple attribute decision mak-
ing(MADM) and linguistic multiple attribute decision making (LMADM). The spe-
cific method for composite multiple attribute decision making(CMADM) problems
is crucial for decision maker(DM)to make scientific decision.In this paper, the Tech-
nique for Order Preference by Similarity to an Ideal Solution(TOPSIS) method is
extended to a Composite Technique for Order Preference by Similarity to an Ideal
Solution(CTOPSIS) method to solve the CMADM problems. As the basis of the C-
TOPSIS method, the distance measure model in linguistic space and in n-dimension
linguistic space are generated based on the non-linear mapping. Based on the dis-
tance measure in linguistic space, a standard deviation method is taken to get the
attribute weight. At the same time, the distance measure models are proposed based
on the distance measure in n-dimension linguistic space, which are used to calcu-
late the distance between the alternatives and the positive and negative idea points
separately. Furthermore, a CTOPSIS method is generated to solve the CMADM
problems. Finally, a numerical example is illustrated to explain the process. And
the result shows that the CTOPSIS method is quite practical and more approxi-
mate to the real decision making situation.
KEYWORDS
Composite multiple attribute decision making; Distance measure; Linguistic
variable; CTOPSIS; Attribute weight
1. Introduction
Multiple attribute decision making(MADM) is an important part of modern decision
science, which has been widely studied by researchers and practitioners [33, 42, 55, 62].
It concerns with evaluating, assessing and selecting alternatives from the best to the
worst under conflicting attributes with respect to decision maker(s) preferences [5, 41].
In details, MADM generally involves three aspects [48]: (1) collecting the information
about attribute weights and attribute values; (2) obtaining an overall value by weighted
aggregation of the attribute values for each alternative; (3) ordering the overall values
and obtaining the optimal alternative(s).
CONTACT Maozhu Jin. Email: jinmaozhu@scu.edu.cn
Specifically, MADM emphasizes the problem of choosing an optimum in two or more
feasible solutions with multiple attributes so as to achieve a certain target, in which
attributes are quantitative and the value of the attribute can be expressed by a num-
ber. Till to now, researchers have paid plenty of attention to MADM problems with
numerical data [3, 14, 27, 29]. While, not all the attributes in MADM are quantitative.
There are also some qualitative attributes. The value of these attributes are expressed
by a word in language rather than a number. With the in-depth research, researcher
realized that the human decision making process is usually not based on the absolute
quantities of attributes, but based on the subjective estimation of the level of perfor-
mance variables, namely, by using linguistic variables instead of numerical variables
[11]. Thus, a more rational and practical approach,utilizing linguistic evaluations on
the basis of linguistic terms, was introduced and widely studied [31, 32, 46, 48, 61].
Tong and Bonissone solved a single stage multi-attribute decision problem by using
ideas in linguistic approximation and truth qualification [39]. Liu et al originally de-
signed a multiple criteria linguistic decision model for the screening of research projects
based on fuzzy set theory [32]. Xu and Da defined the concept of deviation degree to
derive the attribute weights and developed a method based on the possibility degree
between uncertain linguistic variables to rank the alternatives [48]. Wei and Zhao de-
veloped some argument-dependent approach to determining the OWA weights with
2-tuples linguistic information when solving multiple attribute group decision making
problems [46]. Lan proposed a new uncertain linguistic aggregation operator based
on linguistic evaluation scales [31]. Garg and Kumar developed various aggregation
operators in linguistic intuitionistic fuzzy set(LIFS) environment [18].
However, when making decisions in real world, the decision information usually con-
tains both real numbers and linguistic variables. Generally, the value of some attributes
(objective attribute) are real numbers, such as length, height, width, weight et al.. And
the value of others (subjective attribute) are linguistic variables, such as the degree of
beauty, satisfaction etc.. In this case, the decision making problem can be taken as a
composite multiple attribute decision making (CMADM) problem, which can not be
simply treated as a MADM problem or a linguistic multiple attribute decision making
(LMADM) problem. Few literature puts emphases on the CMADM problems with
both numerical and linguistic information, although it occurs more often in real deci-
sion making situation. It is necessary to pay more attention to this issue. To do this,
the rest of the paper is organized as follows: Section 2 introduces the basic concept
of linguistic variable and linguistic term set. In Section 3, a distance measure model
between two linguistic variables and a distance measure model between two linguis-
tic vectors are proposed to measure the distance between two linguistic vectors in n
dimension linguistic space(Ln). In Section 4, we extend the Technique for Order Pref-
erence by Similarity to an Ideal Solution (TOPSIS) method to rank the alternatives
and the Composite Technique for Order Preference by Similarity to an Ideal Solution
(CTOPSIS) is proposed since an aggregation operator is very difficult to integrate the
attributes’ value for the complexity of the type of decision information in CMADM
problem when comparing alternatives. Furthermore, a numerical example is given in
Section 5 to illustrate the CTOPSIS method. Finally, conclusion is given in Section 6.
2
2. Literature review
2.1. Methods for linguistic information aggregation
Aggregation plays a vital role in decision making field since decision makers (DMs) are
faced with a variety of issues related to the fusion of information [58]. Many approach-
es have been proposed for aggregating linguistic information up to now [7, 12, 21].
Degani and Bortolan put forward an ordinal structure of the linguistic term sets [9].
Yager was primarily concerned with the problem of aggregating multi-criteria to form
an overall decision function and introduced an ordered weighted aggregation (OWA)
operator [57, 58]. Then, Wang and Liu extended weighted averaging operator and
ordered weighted averaging operator by using Einstein norm operations under IFS en-
vironment [43]. Meanwhile, Chang and Chen developed an approximate computations
over the linguistic variables [7]. In Ref. [7, 21], fuzzy linguistic representation model
was established, which presents the linguistic information with a pair of values called
2-tuple, composed by a linguistic term and a number. Xu defined some operational
laws of linguistic variables and developed a series of new aggregation operators for deci-
sion making with linguistic preference relations, such as linguistic geometric averaging
(LGA) operator, linguistic weighted geometric averaging (LWGA) operator, linguistic
ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid ge-
ometric averaging (LHGA) operator, etc. [50]. Furthermore, Xu and Yager proposed
some new geometric aggregation operators, such as the intuitionistic fuzzy weighted
geometric (IFWG) operator for the environment in which the given arguments are
intuitionistic fuzzy sets [56]. Moreover,Xu developed a weighted averaging operator
for aggregating the different intuitionistic fuzzy numbers [53]. With the in-depth re-
search, dependency among criteria was taken into consideration when developing new
linguistic aggregation operators in Ref. [1]. Garg presented a new generalized improved
score function in the interval-valued intuitionistic fuzzy sets environment[15]. Then,
Garg also proposed some series of new intuitionistic fuzzy averaging aggregation oper-
ators under the intuitionistic fuzzy sets environment [16]. What’s more, some picture
fuzzy aggregation operators were presented under the picture fuzzy information for
aggregating the different preference of the decision-makers in Ref. [17]. As reviewed
above, Many approaches for aggregation have been widely researched. In order to ex-
press DMs’ thoughts and preferences as truthfully as possible, we introduce a method
based on the new distance measurement between two linguistic variables and extend-
ed this method to a composition decision information environment with linguistic and
numerical information.
2.2. TOPSIS Methods for MADM problem
MADM is a discipline aimed at supporting decision makers who are faced with nu-
merous and conflicting alternatives to make an optimal decision [41]. When DM eval-
uates alternatives with multiple attributes, it seems full of complication and needs to
overcome various problems by pretty sophisticated methods. Technique for order per-
formance by similarity to ideal solution (TOPSIS), one of the known classical MADM
method, was first developed by Hwang and Yoon [25] for solving MADM. It based
upon the concept that the chosen alternative should have The basic idea of TOPSIS
is that the ideal alternative should have the shortest distance from the positive ideal
solution (PIS) and the farthest from the negative ideal solution(NIS), which is a sound
logic that represents the rationale of human choice. The TOPSIS is used in situations
3
where the attribute values and the attribute weights are expressed in exact and inexact
numerical values.
Various extensions of TOPSIS have been made to incorporate fuzzy numbers in
the process [4, 8, 24, 26, 30, 38, 40, 44, 45]. Chen proposed a vertex method into the
TOPSIS to calculate the distance between two triangular fuzzy numbers [8]. Tsaur et
al. transformed the fuzzy MCDM problem into crisp one and then solved the nonfuzzy
MCDM problem by TOPSIS method [40]. Shih et al internally aggregated the prefer-
ences of more than one decision maker into the TOPSIS procedure [38]. Wang and Fan
extended the TOPSIS to solve the group decision making problems with 2-tuple lin-
guistic assessment information in which both the attribute values and attribute weight
take the form of linguistic information [44]. Considering the situation of some attribute
values with incomplete linguistic information and attribute weight, Wei developed de-
velop a more practical method based on the traditional ideas of TOPSIS for 2-tuple
linguistic group decision making problem with incomplete weight information [45]. An
Integrated Fuzzy TOPSIS-Knapsack Problem Model was built for order selection in
a bakery in Ref. [26]. Kuo detected a flaw of traditional TOPSIS and introduced a
modified TOPSIS with a different ranking index to overcome the flaw [30]. Huang and
Jiang defined an optimism coefficient to expand the physical meaning of the standard
TOPSIS and break its restriction in dealing with practical problems [24].Afsordegan
et al. proposed a modified TOPSIS method with qualitative linguistic labels [2]. Kang
et al. introduced Z-number for supply selection to describe the knowledge of human
being with uncertain information considering both restraint and reliability [28]. As
a result of the efficiency to modeling uncertainty [22, 24], TOPSIS method is also
extended via D numbers [13, 22] and belief function [10].
The above abundant extensions of TOPSIS for different kinds of MADM problems
indicate that TOPSIS has a great extensibility for dealing with MADM issues. In
this study, considering that usual aggregation operator is too difficult to integrate the
attributes’ value for the complexity of the type of decision information in CMADM
problem when comparing multiple alternatives, we extend the TOPSIS method to a
composite decision information environment based on a new distance measure operator
with the non-linear mapping.
3. Linguistic variable and linguistic term set
Suppose that L={lj|j=−t, −(t−1), ..., 0, ..., (t−1), t}is a finite and totally ordered
discrete term set, where ljrepresents the linguistic variable, and Nis a set of natu-
ral numbers [37]. For example, when evaluating the quality of a product, a DM can
describe the quality by linguistic variable in a set of linguistic term: Extremely poor,
Very poor, Poor, Slightly poor, Fair, Slightly good, Good, Very good, Extremely good
et al.. Then, the linguistic term set Lcould be L={l−4=Extremely poor, l−3=Very
poor, l−2=Poor, l−1=Slightly poor, l0=Fair, l1=Slightly good, l2=Good, l3=Very
good, l4=Extremely good}, where li< lj, if i<j(i,j=-4,-3,-2,-1,0,1,2,3,4) [51]. At
the same time, the linguistic term set L={lj|j=−t, −(t−1), ..., 0, ..., (t−1), t}(t∈
Z∗andt≥1) should satisfy the following additional characteristics [19, 20]:
(1) the set is ordered: li< lj, if i<j;
(2) there is a negation operator: Neg(lj)=l−j, especially, Neg(l0)=l0;
(3) maximum operator: max(li, lj)=li, if i≥j;
(4) minimum operator: min(li, lj)=lj, if i≥j.
To aggregate the linguistic information and avoid losing linguistic information, the
4
discrete linguistic term set Lis extended to a continuous linguistic term set [52]
¯
L={lα| − (t+ 1) ≤α≤t+ 1, α ∈R}(t∈Z∗andt≥1) (1)
where l−tmeans the worst, l0means fair and ltmeans the best. Base on the mapping
g:¯
L→(−∞,+∞),
g(lα) = tan( πα
2t+t), lα∈¯
L(2)
and its inverse mapping g−1: (−∞,+∞)→¯
L,
g−1(x) = lα,where α=(2t+t) arctan(x)
π(3)
Hu et al defined the addition and scalar multiplication operator that ∀lα, lβ∈¯
L, λ ∈
R[23]:
(1) lα⊕lβ=g−1[g(lα) + g(lβ)] ;
(2) λlα=g−1[λg(lα)] .
To complete the operator in the linguistic term set, the distance measurement op-
erator will be proposed in next section.
4. The distance measurement model
4.1. The distance measurement model in ¯
L
Definition 1 Let lα, lβ∈¯
Lbe two linguistic variables.
dl(lα, lβ) = ||lα−lβ|| =q[g(lα)−g(lβ)]2(4)
is called the distance between lαand lβ, where g(lα) = tan( πα
2t+2 )∈R.
For example, let t= 2 and ¯
L={lα| − 3< α < 3, α ∈R}. Then,
dl(l−2, l1)
=q[g(l−2)−g(l1)]2
=qtan(−2π
6)−tan(π
6)2
=q[−1.7321 −0.5774]2
= 2.3095.
5
dl(l0, l1)
=q[g(l0)−g(l1)]2
=qtan(0) −tan(π
6)2
=q[0 −0.5774]2
= 0.5774.
The distance between each linguistic variable lα(−3< α < 3) and the linguistic
variable l1is shown in Fig.1.
Figure 1. The distance between lα(−3<α<3) and l1
Which is shown in Fig.1 is that
(1) the distance mapping is continues;
(2) the distance value is greater than 0;
(3) the distance mapping function is decreasing when linguistic variable lα< l1;
(4) the distance mapping function is increasing when linguistic variable lα> l1;
(5) the distance mapping function is non-linear;
(6) the distance value is 0 when the linguistic variable lα=l1;
(7) the distance between l2and l1is larger than that between l0and l1. It goes
along with the non-linear property of people’s thinking.
The distance between each pair of linguistic variable in continuous linguistic term
set dl(lα, lβ) is shown in Fig.2. It also shows that the distance value between each pair
of linguistic variable in linguistic term set is not negative, and the distance value is
zero when the two linguistic variables are equal. These properties will be proposed and
proved in Property 1.
Property 1 For all lα, lβ, lγ∈¯
L,dl(lα, lβ) is the distance of two linguistic variables
in a continuous linguistic term set ¯
L. Then,
(1) dl(lα, lβ)≥0;
(2) dl(lα, lβ) = 0, if and only if lα=lβ;
6
Figure 2. The distance between each pair of linguistic variable
(3) dl(lα, lβ)≤dl(lα, lγ) + dl(lβ, lγ).
Proof (1) For any two linguistic variables lα, lβ∈¯
L, [g(lα)−g(lβ)]2≥0,
and then
q[g(lα)−g(lβ)]2≥0.
So
dl(lα, lβ)≥0.
(2) a. If dl(lα, lβ) = 0, then lα=lβ).
For dl(lα, lβ) = 0, then
q[g(lα)−g(lβ)]2= 0.
So,
[g(lα)−g(lβ)]2= 0.
Thus,
g(lα) = g(lβ).
g(lα) is a strictly monotonous and continuous mapping in linguistic space, so
lα=lβ.
7
b. If lα=lβ, then dl(lα, lβ) = 0. lα=lβ, and g(lα) is a strictly monotonous and
continuous mapping, so
g(lα) = g(lβ).
Then
q(g(lα)−g(lβ))2= 0
So,
dl(lα, lβ)=0.
(3) For all lα, lβ, lγ∈¯
L,
dl(lα, lγ) + dl(lβ, lγ)
=q(g(lα)−g(lγ))2+q(g(lβ)−g(lγ))2
According to Minkowski inequality [34], then
dl(lα, lγ) + dl(lβ, lγ)
=q(g(lα)−g(lγ))2+q(g(lβ)−g(lγ))2
≥q(g(lα)−g(lβ))2
=dl(lα, lβ).
So,
dl(lα, lβ)≤dl(lα, lγ) + dl(lβ, lγ).
From Property 1, we can see that the distance measurement dl(lα, lβ) can satisfy
the axiom of distance measurement. And dl(lα, lβ) can be taken as a distance measure
model in linguistic term set. And the distance measurement model in another space
¯
Lnwill be proposed in the next part.
4.2. The distance measurement model in ¯
Ln
The distance measurement model is the key to solve linguistic multiple attribute deci-
sion making problem by TOPSIS method [35]. In this part, the distance measurement
model will be introduced, which can be used to measure the distance between each
alternative and the positive idea point, and the distance between each alternative and
the negative idea point.
Definition 2 Let Lα= (lα1, lα2, ..., lαn)Tand Lβ= (lβ1, lβ2, ..., lβn)Tbe the linguis-
tic vector in the linguistic space ¯
Ln, and then the linguistic distance between Lαand
8
Lβcan be defined by
dv(Lα, Lβ) = v
u
u
t
n
X
i=1
[g(lαi)−g(lβi)]2(5)
where g(lα) = tan πα
2t+2 , lα∈¯
L.
Property 2 For all Lα, Lβ, Lγ∈¯
Ln,dv(Lα, Lβ) is the distance of two linguistic
vectors in a continuous linguistic space ¯
Ln. Then,
(1) dv(Lα, Lβ)≥0;
(2) dv(Lα, Lβ) = 0, if and only if Lα=Lβ;
(3) dv(Lα, Lβ)≤dv(Lα, Lγ) + dv(Lβ, Lγ).
Proof. (1) For any two linguistic vectors Lα, Lβ∈¯
Ln,
n
X
i=1
[g(lαi)−g(lβi)]2≥0,
So
dv(Lα, Lβ)≥0.
(2) a. If dv(Lα, Lβ) = 0, then Lα=Lβ).
For dv(Lα, Lβ) = 0, then
v
u
u
t
n
X
i=1
[g(lαi)−g(lβi)]2= 0.
So,
n
X
i=1
[g(lαi)−g(lβi)]2= 0.
Then,
g(lαi) = g(lβi), i = 1,2, ..., n.
For g(lα) is a strictly monotonous and continuous mapping in linguistic space, so
lαi=lβi. Thus
Lα=Lβ.
b. If Lα=Lβ, then dv(Lα, Lβ)=0.For Lα=Lβ, and g(lα) is a strictly monotonous
9
and continuous mapping, so
g(lαi) = g(lβi)(i= 1,2, ..., n).
Then
v
u
u
t
n
X
i=1
(g(lαi)−g(lβi))2= 0
So,
dv(Lα, Lβ)=0.
(3) For all Lα, Lβ, Lγ∈¯
Ln,
dv(Lα, Lγ) + dv(Lβ, Lγ)
=qPn
i=1 (g(lαi)−g(lγi))2
+qPn
i=1 (g(lβi)−g(lγi))2
According to Minkowski inequality [34], then
dv(Lα, Lγ) + dv(Lβ, Lγ)
=qPn
i=1 (g(lαi)−g(lγi))2
+qPn
i=1 (g(lβi)−g(lγi))2
≥qPn
i=1 (g(lαi)−g(lβi))2
=dv(Lα, Lβ).
So,
dv(Lα, Lβ)≤dv(Lα, Lγ) + dv(Lβ, Lγ).
Property 2 shows that the distance measurement in the n-dimensional linguistic
term set Lncan satisfy the axiom of distance measurement. dv(Lα, Lβ) can be used
to measure the distance in Ln.
In linguistic multiple attribute decision making, the distance between each linguistic
vector and positive idea point, negative idea point can be calculated with the distance
measurement when using the TOPSIS method. Then, the TOPSIS method can be
extended to LTOPSIS to solve linguistic multiple attribute decision making prob-
lem(LMADM). However, in multiple attribute decision making, achieving the subjec-
tive decision information of some attributes is difficult, such as ’satisfaction degree’.
10
We can not get the decision information by measurement or calculation. The deci-
sion information of this kind of attribute can only be obtained by people’s subjective
evaluation. In this way, both subjective and objective decision information exist in
the decision information matrix. And these decision information compose a CMADM
problem. In next part, the TOPSIS method will be extended to solve the CMADM
problem.
5. CTOPSIS method
Suppose that a Composite Multiple Attribute Decision Making(CMADM) problem
with nalternatives x1, x2, ..., xnand mattributes u1, u2, ..., um. These attributes can
be divided into two types: the subjective attributes and the objective attributes. The
subjective attribute value of each alternative can not be obtained directly from cal-
culation or measurement. They are always obtained by the description of the de-
cision maker. The objective attribute value can be obtained directly by calcula-
tion or measurement. Suppose that u1, u2, ..., um1(0 ≤m1≤m) are the subjective
attributes and um1+1,um1+2, ..., umare the objective attributes. A decision mak-
er can choose a linguistic variable from Lto describe the value of subjective at-
tributes and calculate the objective attribute value of each alternative. All these at-
tribute values of each alternative can compose a composite decision information matrix
D= [(lαij )n×m1,(aij )n×m−m1](lαij ∈L, aij ∈R).
Table 1. Composite decision information matrix
u1u2... um1um1+1 um1+2 ... um
x1lα11 lα12 ... lα1m1a1m1+1 a1m1+2 ... a1m
x2lα21 lα22 ... lα2m1a2m1+1 a2m1+2 ... a2m
. . . . . . . . . . . . . . . . . . . . . . . . . . .
xnlαn1lαn2... lαnm1anm1+1 anm1+2 ... anm
To select the best alternative and solve this CMADM problem, the normally
solving methods (such as aggragation operator WAA, OWA, LWAA, LOWA, TOP-
SIS and some other methods) can’t be used directly. Thus, an extended TOPSIS
method(CTOPSIS) will be proposed to solve this kind of problem. For the different
importance of different attribute, the attribute weights are taken into account.
5.1. Generate the attribute weight by a standard deviation method
To generate the attribute weight, Wang and Zhang proposed a standard and mean
deviation method to deal with MADM problems with numerical information [44]. Xu
and Da extended this method to generate the attribute weight in a uncertain MADM
with interval decision information [49]. Furthermore, Xu and Da extended this method
to calculate the attribute weight in a linguistic decision information environment [47].
Chang and Cheng created a combination of fuzzy OWA and the DEMATEL method
to rank the risk of failure [6]. Ye put forward a multi-criteria decision-making method
based on a simplified neutrosophic weighted arithmetic average operator and a simpli-
fied neutrosophic weighted geometric average operator [59]. Furthermore, Ye developed
an interval neutrosophic number ordered weighted averaging operator and an interval
neutrosophic number ordered weighted geometric operator [60]. In this part, we adapt
11
this method based on the new distance measurement between two linguistic variable
and extended this method to a composition decision information environment with
linguistic and numerical information.
Suppose that W= (w1, w2, ..., wm)Tis the attribute weight vector, where
wj≥0(j= 1,2, ..., m),
m
X
j=1
wj= 1.(6)
For a subjective attribute uj(j= 1,2, ..., m1), the standard deviation between al-
ternatives is
SDj
=q1
nPn
i=1 lαij ·wj−1
nPn
k=1 lαkj ·wj2
=wjq1
nPn
i=1 dl(lαij , l¯αj)2
=wj·δj, j = 1,2, ..., m1,
(7)
where
δj=v
u
u
t
1
n
n
X
i=1
dl(lαij , l¯αj)2,(8)
,and
l¯αj=1
n
n
X
i=1
lαij (9)
is the mean value of the attribute uj.dl(lαij, l ¯αj) is the distance between the jth
attribute value lαij of alternative xiand the jth attribute mean value l¯αj.
For the objective attribute uj(j=m1+ 1, m1+ 2, ..., m), the standard deviation
between alternatives is
SDj
=q1
nPn
i=1 aij ·wj−1
nPn
k=1 akj ·wj2
=wjq1
nPn
i=1 d(aij ,¯aj)2
=wj·δ0
j, j =m1+ 1, m1+ 2, ..., m.
(10)
12
where
δ0
j=v
u
u
t
1
n
n
X
i=1
d(aij ,¯aj)2,(11)
,and
¯aj=1
n
n
X
i=1
aij (12)
is the mean value of the attribute uj. d(aij,¯aj) is the distance between the jth attribute
value aij of alternative xiand the jth attribute mean value ¯aj.
To get a group weight such that the sum of standard deviation is maximum, a
non-linear programming model is established.
max F(w)
=Pm1
j=1 wj·δj+Pm
j=m1+1 wj·δ0
j
s.t. Pm
i=1 w2
j= 1, wj≥0, j = 1,2, ..., m
(13)
To solve the model, we construct the Lagrange function:
L(w, λ)
=Pm1
j=1 wj·δj+Pm
j=m1+1 wj·δ0
j
+1
2λPm
i=1 w2
j−1
(14)
where λis the Lagrange multiplier. Setting the partial derivatives equal to zero, we
get that
∂L
∂wj=δj+λwj= 0, j = 1,2, ..., m1;
∂L
∂wj=δ0
j+λwj= 0, j =m1+ 1, ..., m;
∂L
∂λ =1
2Pm
i=1 w2
j−1= 0.
By solving this equation set, we obtain that
λ=−v
u
u
t
m1
X
j=1
δ2
j+
m
X
j=m1+1
δ02
j;
wj=−δj
λ=δj
qPm1
j=1 δ2
j+Pm
j=m1+1 δ02
j
13
(j= 1,2, ..., m1);
wj=−δ0
j
λ=δ0
j
qPm1
j=1 δ2
j+Pm
j=m1+1 δ02
j
(j=m1+ 1, m1+ 2, ..., m).
Thus, we get the extreme point of Model(13). Obviously, wjis positive and satisfied
the condition of (13). It is the optimal solution.
By normalizing wj, we have
w∗
j=δj
Pm1
k=1 δk+Pm
k=m1+1 δ0
k
,
(j= 1,2, ..., m1)
(15)
w∗
j=δ0
j
Pm1
k=1 δk+Pm
k=m1+1 δ0
k
,
(j=m1+ 1, m1+ 2, ..., m)
(16)
where δjand δ0
jcan be generated by Eq.(8) and Eq.(11), separately.
With Eq.(15), the weight of the subjective attribute w∗
j(j= 1,2, ..., m1) can be
determined. At the same time, the weight of the objective attribute w∗
j(j=m1+
1, m1+ 2, ..., m) can be determined by Eq.(16).
5.2. CTOPSIS
As an effective method solving the MADM problem, the TOPSIS method has been
extended to various different environments. In this part, the TOPSIS method is ex-
tended to a composite decision information environment and a CTOPSIS method is
introduced.
For a MADM problem with composite decision information
D= [(lαij )n×m1,(aij )n×(m−m1)], the CTOPSIS can be divided into five steps as follows:
Step 1: Unify the range of the attribute value for the composite decision information
matrix D= [(lαij )n×m1,
(aij )n×(m−m1)].
Let lα= (lαij )n×m1and A= (aij )n×(m−m1), then D= [lα, A]. lαis the linguistic
fuzzy decision information matrix and Ais the accurate decision information matrix.
For the linguistic fuzzy decision information matrix lα= (lαij )n×m1, the range of
these attributes’ values are the same, only the accurate decision information matrix A
should be unified by the following formula so that they can go along with the linguistic
fuzzy decision information matrix.
14
bij =g(lt)·2aij −max1≤i≤naij −min1≤i≤naij
max1≤i≤naij −min1≤i≤naij
(aij ∈A;i= 1,2, ..., n;j=m1+ 1, m1+ 2, ..., m.)
(17)
Then, the accurate decision information matrix Achanges to B= (bij)n×(m−m1),
where −g(lt)≤bij ≤g(lt). And the composite decision information matrix D= [lα, A]
changes to D0= [lα, B], where the range of all the attribute value keep in the same
from −g(lt) to g(lt).
Step 2: Calculate the weighted composite decision information matrix:
D00 = [lβ, C] = [(lβij )n×m1,(cij )n×(m−m1)] by
lβij =w∗
jlαij =g−1[w∗
jg(lαij )],
i= 1,2, ..., n;j= 1,2, ..., m1.
(18)
cij =w∗
jbij , i = 1,2, ..., n;
j=m1+ 1, m1+ 2, ..., m.
(19)
Where w∗
jis generated by Eq.(15) and (16).
Step 3: Set the positive idea point
(C+={c+
1, c+
2,· · · , c+
n}) and the negative idea point
(C−={c−
1, c−
2,· · · , c−
n}).
The positive idea point(C+={c+
1, c+
2,· · · , c+
n}) can be set as follows:
c+
j=
max lβij ,if ujis the achive attribute;
min lβij ,if ujis the cost attribute;
i= 1,2, ..., m, j = 1,2, ..., m1;
max cij ,if ujis the achive attribute;
min cij ,if ujis the cost attribute;
i= 1,2, ..., m, j =m1+ 1, m1+ 2, ..., m.
(20)
The negative idea point(C−={c−
1, c−
2,· · · , c−
n}) can be set as follows:
c−
j=
min lβij ,if ujis the achive attribute;
max lβij ,if ujis the cost attribute;
i= 1,2, ..., m, j = 1,2, ..., m1;
min cij ,if ujis the achive attribute;
max cij ,if ujis the cost attribute;
i= 1,2, ..., m, j =m1+ 1, m1+ 2, ..., m.
(21)
Step 4: Calculate the distance d+
ibetween the composite decision information
[(lβij )1×m1,(cij )1×(m−m1)] of xiand the positive idea point C+={c+
1, c+
2,· · · , c+
n}
by
15
d+
i=
rPm1
j=1 hg(lβij )−g(c+
j)i2+Pm
j=m1+1 hcij −c+
ji2,
(i= 1,2, ..., n).
(22)
And calculate the distance d−
ibetween the composite decision information
[(lβij )1×m1,(cij )1×(m−m1)] of xiand the negative idea point
C−={c−
1, c−
2,· · · , c−
n}by
d−
i=
rPm1
j=1 hg(lβij )−g(c−
j)i2+Pm
j=m1+1 hcij −c−
ji2,
(i= 1,2, ..., n).
(23)
Step 5: Estimate the relative closeness degree vito the idea point by
vi=d−
i
d+
i+d−
i
, i = 1,2, ...n. (24)
And rank the alternatives x1, x2, ..., xnaccording to the value of v1, v2, ..., vn. The
bigger viis, the higher performance of the alternative is.
6. Case study
To illustrate the new extended CTOPSIS method, a case is introduced [36].
In order to reduce the traveling time on Istanbul’s Bosphorus, a highly congested sea
lanes, this part illustrates the choice about the Bosphorus’s double-ended passenger
ferries operating system [36]. This MADM problem involves 3 possible alternatives
xi(i= 1,2,3), where x1denotes the conventional propeller and the high lift rudder
(1×2); x2denotes the Z drive (2×2); x3denotes the cycloidal propeller (1 ×2). (1×2)
means that one drive is installed at both ends of passenger ferries, and (2 ×2) means
that two drives are installed at both ends of passenger ferries. The best one should
be selected out from x1, x2, x3. There are 7 main evaluation indexes (attributes) as
follows:
u1: Investment Cost Analysis;
u2: Operating Cost Analysis, including operating, repairing and maintaining cost
analysis;
u3: Operability Analysis;
u4: Shaking and noise analysis;
u5: Reliability Analysis, including safety of machinery, redundancy and service ex-
perience analysis;
16
u6: Propulsion needs analysis, including ships geometry, ship resistance, power re-
quirements and propulsive efficiency;
u7: Propulsion needs analysis.
The decision maker evaluates the possible alternatives above-mentioned xi(i=
1,2,3) by way of using linguistic assessment set (t= 4): L={l−4=Extremely
poor, l−3=Very poor, l−2=Poor, l−1=Slightly poor, l0=Fair, l1=Slightly
good, l2=Good, l3=Very good, l4=Extremely good}. And the decision information
shows in Table 2.
Table 2. Decision information matrix
u1u2u3u4u5u6u7
x1l3l2l−1l0l2l1l3
x2l2l1l4l2l3l1l2
x3l−2l2l−1l3l0l3l1
The CTOPSIS method is used to select the best one from the alternatives.
Step 1: Unify the range of the attribute value for the decision information matrix.
For all of these attribute value selected from L, the range of these attributes’ value
are the same. This step can be omitted and go to Step 2.
Step 2: Calculate the weighted composite decision information matrix.
According to Eq.(8) and (9), l¯αjand δjcan be calculated. Furthermore, w∗
jcan
generated by Eq.(15). Take l¯α1,δ1and w∗
1for example.
l¯α1
=1
3(l3⊕l2⊕l−2) = g−1[g(l3) + g(l2) + g(l−2)]
=1
3g−1htan( 3π
2×4+2 ) + tan( 2π
2×4+2 ) + tan( −2π
2×4+2 )i
=1
3g−1tan(3π
10 ) + tan(π
5) + tan(−π
5)
=g−1(1.3764+0.7265−0.7265
3)
=g−1(0.4588)
=l1.3692
δ1=
q1
3[dl(l3, l1.369)2+dl(l2, l1.369)2+dl(l−2, l1.369)2]
=q1
3(0.91762+ 0.26772+ 1.18532)
= 0.8791
w∗
j
=0.8791
0.8791+0.1893+1.6040+0.5622+0.5622+0.4957+0.4332
= 0.1860
17
All the results are shown in Table 3. Then calculate the weighted composite decision
information by Eq.(18), which is shown in Table 4.
Table 3. The weight of each attribute
u1u2u3u4u5u6u7
l¯αjl1.3692 l1.7030 l2.1657 l1.9461 l1.9461 l1.8909 l2.1657
δj0.8791 0.1893 1.6040 0.5622 0.5622 0.4957 0.4332
w∗
j0.1860 0.0401 0.3394 0.1190 0.1190 0.1049 0.0917
Table 4. Decision information matrix
u1u2u3u4u5u6u7
x1l0.7978 l0.0927 l−0.3496 l0l0.2745 l0.1085 l0.3996
x2l0.4276 l0.0415 l2.5694 l0.2745 l0.5168 l0.1085 l0.2118
x3l−0.4276 l0.0927 l−0.3496 l0.5168 l0l0.4564 l0.0948
Step 3: Set the positive idea point(C+={l0.7978,
l0.0927, l2.5694, l0.2745, l0.5168 , l0.4564, l0.3996 }) and the negative idea point(C−=
{l−0.4276, l0.0415, l−0.3496, l0,
l0, l0.1085, l0.0948})by Eq.(20) and (21).
Step 4: Calculate the distance d+
iand d−
iby Eq.(22) and Eq.(23), separately. The
results are shown in Table 5.
Table 5. d+
iand d−
i
d+
id−
i
x11.1742 0.4123
x20.1912 1.2010
x31.2340 0.1981
Step 5: Estimate the relative closeness degree viby Eq.(24) thus we have v1= 0.2599,
v2= 0.8626, v3= 0.1383. And rank the alternatives that
x2x1x3.
The best alternative is x2. The most desirable propulsion/manoeuvring system
should be the Z drive(2 ×2).
Xu proposed an Ideal Point-Based Model, which solved the linguistic multiple
attribute decision making problem [54]. With that model, the rank result is also
x2x1x3. It means that the new CTOPSIS method can get the same rank
result as what is generated by the Ideal Point-Based Model. Besides, the advantage
of the new CTOPSIS method is that it can be used in composite decision information
environment.
18
7. Conclusion
This paper concentrated on solving the CMADM problems with a new CTOPSIS
method. To begin with, some basic concepts about linguistic variable, linguistic term
set and non-linear operators of linguistic variables were introduced. Based on the
basic concepts and operators, the distance measurement in linguistic term set was
considered and a distance measure model dlwas proposed based on the non-linear
mapping. To compare two linguistic vectors in n-dimension linguistic space (Ln), the
distance measure model (dl) was extended and we proposed a distance measure model
(dv) in Ln. With these distance measure models, a TOPSIS method was extended to
a CTOPSIS method, which is used to solve a CMADM problem. With the composite
decision information, including linguistic variable decision information and real number
decision information, CTOPSIS uses a standard deviation method to get the attribute
weight. Furthermore, an example adapted from Ref [54] was illustrated to explain the
process of CTOPSIS. It showed that the result is the same as Ref [54]. The advantage
of CTOPSIS method is that it can be used to solve not only a CMADM problem, but
also a MADM problem or a LMADM problem.
Acknowledgement(s)
The authors acknowledge the support of research funds from the support of research
funds from National Natural Science Foundation of China (Nos. 71371130, 7150010715
and 7176010004), Key Research Base of Social Sciences in Sichuan Province – Sichuan
Tourism Development and Research Center project (No. LYB14-03), Next genera-
tion Internet technology innovation project(No. NGII20151207), Chinese Scholarship
Council of the Ministry of Education(CSC) (NO. 201606240014).
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