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International Journal for Uncertainty Quantification, 7(5):463–474 (2017)
A NEW IMPROVED SCORE FUNCTION OF AN
INTERVAL-VALUED PYTHAGOREAN FUZZY SET
BASED TOPSIS METHOD
Harish Garg
School of Mathematics, Thapar University, Patiala 147004, Punjab, India,
E-mail: harishg58iitr@gmail.com
Original Manuscript Submitted: 3/25/2017; Final Draft Received: 6/15/2017
In the present study, an improved score function for the ranking order of interval-valued Pythagorean fuzzy sets
(IVPFSs) has been proposed. Based on it, a Pythagorean fuzzy technique for order of preference by similarity to
ideal solution (TOPSIS) method by taking the preferences of the experts in the form of interval-valued intuitionis-
tic Pythagorean fuzzy decision matrices has been presented. A positive and negative ideal separation measures solution
has been computed based on the proposed score function to determine the relative closeness coefficient and hence the
most desirable one/s is/are selected. Finally, an illustrative example for a multicriteria decision-making (MCDM) prob-
lem has been taken to demonstrate the proposed approach.
KEY WORDS: Pythagorean fuzzy set, multicriteria decision-making, TOPSIS, score function, interval-
valued numbers
1. INTRODUCTION
In today’s world, uncertainties play a dominant role during analysis and without proper handling of it, the decision-
makers (DMs) cannot give their preferences to an accurate level. Therefore, the main objective of an analysis is to
handle the data so as to minimize the uncertainties level. For handling this, the fuzzy set (FS) [1] and its exten-
sions such as intuitionistic fuzzy set (IFS) [2], interval-valued intuitionistic fuzzy set (IVIFS) [3], etc., have been
proposed by the researchers during the last decades in the fields of the decision-making process. MCDM is one of
the important branches of the decision-making process for finding the best alternative among the set of the feasi-
ble ones. However, apart from that, the technique for order of preference by similarity to ideal solution (TOPSIS),
developed by Hwang and Yoo [4], is another well known MCDM approach for finding the best alternative based
on its ideal values. Chen [5] extended the TOPSIS approach to fuzzy decision-making by defining the crisp Eu-
clidean distance between the two fuzzy numbers. Park et al. [6] extended the TOPSIS method from the intuitionistic
fuzzy set to the IVIFSs in which all the preference information provided by the decision-makers is in the form
of interval-valued intuitionistic fuzzy numbers (IVIFNs). Hung and Chen [7] presented a fuzzy TOPSIS method
with the entropy weight to solve the decision-making problems under the intuitionistic fuzzy environment. Kumar
and Garg [8] presented a TOPSIS method in the IVIFS environment based on the connection number of the set
pair analysis theory for solving MCDM problems. In the past few decades, the IFS and/or IVIFS has gained great
attention and has been successfully applied to many practical areas such as decision-making, pattern recognition,
medical diagnosis, and clustering analysis [9–19]. From these studies, it has been concluded that they are valid un-
der the restrictions that the sum of the grades of memberships is not greater than 1. However, in day-to-day life, it
is not always possible for the DMs to give their preferences under this restriction. For instance, a person may ex-
press that his preference towards any object is 0.8 while dissatisfaction is 0.6; then clearly 0.8+0.61. Thus,
such types of preferences are not handled with IFS theory. To overcome this, Yager [20,21] relaxes the IFS con-
dition to its square sum not being greater than 1 and called the corresponding set to be the Pythagorean fuzzy set
2152–5080/17/$35.00 © 2017 by Begell House, Inc. www.begellhouse.com 463
464 Garg
(PFS). Thus, PFS is an extension of the existing IFS and easily able to handle the situations where IFS theory
fails. For instance, corresponding to the above-considered example, we see that (0.8)2+ (0.6)2≤1 and hence
analysis conducted in the PFS environment will be better able to get the correct decision to the DM than the IFS.
After their pioneering work, Yager and Abbasov [22] studied the relationship between Pythagorean fuzzy numbers
(PFNs) and complex numbers. Yager [21], further, presented some aggregation operators in the PFS environment
while Zhang and Xu [23] extended the TOPSIS approach in terms of the Pythagorean fuzzy environment. Garg
[24,25] presented some generalized averaging and geometric aggregation operators based on the Einstein t-norm op-
erations. Garg [26], further, presented confidence level based aggregation operators, by incorporating the confidence
level of the decision-makers during the analysis, and called them confidence based Pythagorean fuzzy weighted av-
eraging and geometric aggregation operators. Peng and Yang [27] defined the interval-valued PFSs (IVPFSs) and
developed an elimination and choice translating reality (ELECTRE) method to solve the decision-making prob-
lem with interval-valued Pythagorean fuzzy information. Later on, Gar [28] presented interval-valued Pythagorean
fuzzy weighted average (IPFWA) and interval-valued Pythagorean fuzzy weighted geometric (IPFWG) operators
for solving the decision-making problem in the IVPFS environment. Also, a novel accuracy function has been de-
fined in it for ranking the different interval-valued Pythagorean fuzzy numbers (IVPFNs). Now, in order to compare
the interval numbers, some scores, as well as accuracy functions, have been taken for measurement and then ap-
plied to solve MCDM problems. Garg [29] defined the concepts of correlation and correlation coefficients of PFSs.
Garg [30], further, presented improved accuracy functions under the IVPFS for solving the decision-making prob-
lems.
In the decision context of IVPFSs, substantial research used the TOPSIS technique as the main structure to
deal with multicriteria evaluation information and to construct a priority ranking for the best alternative. The main
advantages of the TOPSIS are to consider positive- and negative-ideal solutions as anchor points to reflect the contrast
of the currently achievable criterion performances. The basic concept of the TOPSIS is that the chosen alternative
should have the shortest distance from the positive-ideal solution and the farthest distance from the negative-ideal
solution. The difference between the TOPSIS and the Pythagorean TOPSIS is in their rating approaches. The merit
of Pythagorean TOPSIS is to assign the importance of attributes and the performance of alternatives with respect
to various attributes by using Pythagorean numbers instead of precise numbers. Furthermore, it has been observed
from the above studies that the measures defined for ranking the PFSs or IVPFSs have a vital shortcoming in some
cases. Therefore, in order to remove these features, an improved score function for IVPFSs has been proposed in this
manuscript and the subsequent TOPSIS method to solve MCDM problems.
Thus, the objective of the work is to present an improved score function of IVPFSs and hence the Pythagorean
fuzzy TOPSIS method based on it. In this methodology, we assume that the membership and nonmembership de-
grees of alternatives on attributes or criteria are variables, whose values belong to the intervals determined by the
given IVPFSs. A score matrix based on the proposed improved score function has been constructed for the given
preferences. The interval-valued Pythagorean (IVP)-positive ideal solution (IVPPIS) and the IVP-negative ideal so-
lution (IVPNIS) are chosen as reference points. The differences between the alternatives and the IVPPIS, as well
as the IVPNIS are measured by the weighted Euclidean distances [31]. These distances are used to define relative-
closeness coefficients of alternatives to the IVPPIS, which are continuous and monotonic functions with respect to
the membership and nonmembership degrees of alternatives on attributes. As a result, the relative-closeness coeffi-
cients of alternatives to the IVPPIS are determined to rank the given alternatives. Finally, a decision-making has been
presented and illustrated with a numerical example to show the superiority of the approach.
The rest of the manuscript is summarized as follows. Section 2 presents the basic definitions related to PFS,
IVPFS, score and accuracy functions. In Section 3, an improved score function of IVPFS is proposed for ranking
the IVPFNs by taking the degree of hesitation of IVPFSs. In Section 4, a TOPSIS method is presented based on the
proposed function. The proposed approach is illustrated with a numerical example in Section 5. Finally, in Section 6
the paper ends with a conclusion.
2. BASIC CONCEPTS
In this section, some basic concepts on PFS, IVPFS, and their relevant terms have been defined.
International Journal for Uncertainty Quantification
A New Improved Score Function of an Interval-Valued Pythagorean Fuzzy Set 465
2.1 Pythagorean Fuzzy Set
Definition 2.1. Pythagorean fuzzy set (PFS) Ais defined as a set of ordered pairs over a universal set Xgiven by
[20]
A={hx, µA(x),νA(x)i | x∈X},
where µA,νA:X−→ [0,1]are the degrees of membership and nonmembership of the element x∈X, respec-
tively, with the condition that (µA(x))2+ (νA(x))2≤1. Corresponding to its membership functions, the degree of
indeterminacy is given by πA(x) = p1−(µA(x))2−(νA(x))2.
Zhang and Xu [23] called the pairs of these membership functions a Pythagorean fuzzy number (PFN) denoted
by α=hµA,νAiand defined the score function as
S(α) = (µA)2−(νA)2,(1)
where S(α)∈[−1,1]and the accuracy function is
H(α) = (µA)2+ (νA)2,(2)
where H(α)∈[0,1].
2.2 Interval-Valued Pythagorean Fuzzy Set
Considering the fact that information provided by the DMs are always imprecise numbers due to variousconstraints,
it is not possible for the decision-maker to give their preference in terms of crisp numbers. In such case, it is always
preferable and convenient to give them in the form of interval numbers. So for this, Peng and Yang [27] and Gar [28]
extended the definition of PFS to IVPFS which is defined as follows:
Definition 2.2. An interval-valued Pythagorean fuzzy set (IVPFS) Adefined in Xis given as [27,28]
A={hx, [µL
A(x),µU
A(x)],[νL
A(x),νU
A(x)]i | x∈X},(3)
where 0 ≤µL
A(x)≤µU
A(x)≤1, 0 ≤νL
A(x)≤νU
A(x)≤1, and (µU
A(x))2+ (νU
A(x))2≤1 for all x∈X. Similar to
PFSs, corresponding to interval-valued membership values, its hesitation interval relative to Ais given as
πA(x) = [πL
A(x), πU
A(x)] (4)
=·q1−(µU
A(x))2−(νU
A(x))2,q1−(µL
A(x))2−(νL
A(x))2¸.
If for every x∈X,µA(x) = µL
A(x) = µU
A(x),νA(x) = νL
A(x) = νU
A(x), then IVPFS reduces to PFS. For an
IVPFS A, the pair h[µL
A(x),µU
A(x)],[νL
A(x),νU
A(x)]iis called an interval-valued Pythagorean fuzzy number (IVPFN)
[28]. For convenience, this pair is often denoted by α=h[a, b],[c, d]i, where
[a, b]⊆[0,1],[c, d]⊆[0,1],and b2+d2≤1.(5)
The score function of an IVPFN α=h[a, b],[c, d]iis defined as
S(α) = a2+b2−c2−d2
2,(6)
where S(α)∈[−1,1]. But, sometimes, it has been observed that this function is unable to rank the IVPFNs. For
instance, if we take α1=h[0.2,0.4],[0.2,0.4]iand α2=h[0.5,0.6],[0.5,0.6]ito be two IVPFNs, then by Eq. (6), we
get S(α1) = S(α2) = 0. Thus, it is unable to find the best among them. For resolving it, there is an another function
called the accuracy function for an IVPFN αwhich is defined as follows:
H(α) = a2+b2+c2+d2
2,(7)
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466 Garg
where H(α)∈[0,1].
In particular, if a=band c=dthen Eqs. (6) and (7) become Eqs. (1) and (2), respectively.
Now, based on these two functions, a comparison method for any two IVPFNs αand βis defined as follows:
Definition 2.3. Let αand βbe any two IVPFNs. Then
(i) If S(α)< S(β), then α≺β.
(ii) If S(α)> S(β), then αÂβ.
(iii) If S(α) = S(β),
(a) If H(α)< H(β), then α≺β.
(b) If H(α)> H(β), then αÂβ.
(c) If H(α) = H(β), then α∼β.
2.3 Shortcoming of the Existing Score Functions
The following examples show that the existing score functions are unable to give the correct justification for the set
of the alternatives in the decision-making process.
Example 2.1. Let α1=£0,0.5¤,£0.1,0.7¤® and α2=£0.3,0.4¤,£0.5,0.5¤®be two IVPFNs; then, by using
Eq. (6), we get
S(α1) = −0.1250 and S(α2) = −0.1250.
Hence, by Definition 2.3, we compute their accuracy function by using Eq. (7) and get
H(α1) = 0.3750 and H(α2) = 0.3750.
Therefore, based on comparison law, α∼β. But it is clearly seen that α6=β. Thus, the existing functions of the
IVPFNs are unable to give the exact position and hence it is necessary to modify it.
Example 2.2. Let α1=h[0.1,0.2],[0.4,0.5]iand α2=h[0.1,0.2],[1/√20,0.6]ibe two IVPFNs; then, by using
Eqs. (6) and (7), we get
S(α1) = −0.1800 and S(α2) = −0.1800,
H(α1) = 0.2300 and H(α2) = 0.2300.
Thus, by Definition 2.3, we get that α1and α2are equivalent. But this is not true. Hence, again, it is unable to
distinguish between the IVPFNs.
Thus, we can see that the functions are given in Eqs. (6) and (7) fail to rank correctly when the squares of their
sum of lower and upper bounds of membership and nonmembership degrees are equal, respectively. Furthermore, the
uncertainties degree has not been considered during the formulation, so the indeterminacy information has not been
completely extracted. Therefore, in order to handle this, an improved score function has been proposed in the next
section by sufficiently considering the indeterminacy information of an IVPFS.
3. PROPOSED IMPROVED SCORE FUNCTION
In this section, an improved score function is proposed by considering the indeterminacy information of an IVPFS.
It is known that the score functions for IVPFS mainly depend on two parameters, where one parameter is the
hesitation interval index π(·), and the other parameter is the score function S(·). Unfortunately, Eq. (6) has only
considered the parameter S(·)and it neglects the hesitation interval index π(·), which is illogical. For handling this,
we introduce the following concepts for an IVPFN α=h[µL
A(x),µU
A(x)],[νL
A(x),νU
A(x)]iwhich are denoted by
α=h[a, b],[c, d]i:
International Journal for Uncertainty Quantification
A New Improved Score Function of an Interval-Valued Pythagorean Fuzzy Set 467
(i) [√1−b2−d2,√1−a2−c2]is called the hesitation interval index relative to αfor x∈X.
(ii) a2+a2√1−a2−c2and b2+b2√1−b2−d2are called favorable degrees relative to αfor x∈X.
(iii) c2+c2√1−a2−c2and d2+d2√1−b2−d2are called unfavorable degrees relative to αfor x∈X.
Based on these concepts, the improved score function of an IVPFN α=h[a, b],[c, d]iis proposed by the following
formula:
M(α) = (a2−c2)(1+√1−a2−c2)+(b2−d2)(1+√1−b2−d2)
2,(8)
where M(α)∈[−1,1].
In particular, when a=b=µ(say) and c=d=ν(say), an IVPFN becomes a PFN and the improved score
function of an IVPFN becomes the score function of a PFN. That is, the improved score function of PFN α=hµ,νi
becomes the following formula:
M(α) = (µ2−ν2)(1+p1−µ2−ν2).
Based on the improved score function, we give the following comparison law for the decision-making process
by IVPFSs. Let αand βbe two IVPFNs; then
(i) If M(α)> M(β), then αÂβ.
(ii) If M(α)< M(β), then α≺β.
(iii) If M(α) = M(β), then α∼β.
To illustrate the effectiveness of the proposed score function for IVPFNs, consider the following examples.
Example 3.1. If we apply the proposed function M(·)as given in Eq. (8) to Example 2.1 then we get M(α1) =
−0.1912 and M(α2) = −0.2246. Hence, based on comparison laws, we get α1Âα2; i.e., the alternative α1is better
than the alternative α2.
Example 3.2. If we apply Eq. (8) to the above-considered Example 2.2, then we have M(α1) = −0.3368 and
M(α2) = −0.3233. Since M(α2)> M(α1)hence IVPFN α2is better than the IVPFN α1.
From the above examples, we can see that the proposed function is reasonable and provides an effective algorithm
for the decision analysis process. For an IVPFN α, the proposed score function M(α)has the following properties.
Property 3.1. The proposed improved score function M(α), where α=h[a, b],[c, d]i, lies between [−1,1].
Proof. Since α=h[a, b],[c, d]iis an IVPFN, hence a, b ⊆[0,1]and [c, d]⊆[0,1]such that b2+d2≤1. Therefore
√1−b2−d2≥0. Similarly, √1−a2−c2≥0. Thus,
M(α) = (a2−c2)(1+√1−a2−c2)+(b2−d2)(1+√1−b2−d2)
2
≥a2−c2+b2−d2
2
≥a2−c2+a2−c2
2=a2−c2
≥ −1.
Further,
M(α) = (a2−c2)(1+√1−a2−c2)+(b2−d2)(1+√1−b2−d2)
2
≤(1−2c2) + (1−2d2)
2=1−(c2+d2)
≤1.
Thus, −1≤M(α)≤1, where α=h[a, b],[c, d]i.
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468 Garg
Property 3.2. For an IVPFN α, the proposed score function M(α)and existing score function S(α)satisfy the
inequality M(α)≥S(α).
Proof. Consider α=h[a, b],[c, d]ito be an IVPFN such that [a, b]⊆[0,1],[c, d]⊆[0,1],a2+c2≤1, and
b2+d2≤1. Therefore, √1−a2−c2≥0 and √1−b2−d2≥0. Hence, by definition of M(α)we have
M(α) = (a2−c2)(1+√1−a2−c2)+(b2−d2)(1+√1−b2−d2)
2
≥(a2−c2)+(b2−d2)
2
≥S(α).
Property 3.3. (Zero Property) If IVPFN α=h[0,0],[1,1]ithen M(α) = −1.
Proof. If α=h[0,0],[1,1]iis an IVPFN then
M(α) = (02−12)(1+√1−0−1) + (02−12)(1+√1−0−1)
2=−1.
Property 3.4. (One Property) If IVPFN α=h[1,1],[0,0]ithen M(α) = 1.
Proof. If α=h[1,1],[0,0]iis an IVPFN then
M(α) = (12−02)(1+√1−1−0) + (12−02)(1+√1−1−0)
2=1.
Property 3.5. For a subset α=h[a, b],[√1−a2,√1−b2]ithe proposed score function is M(α) = a2+b2−1. In
particular, if a=bthen M(α) = 2a2−1.
Proof. As α=h[a, b],[√1−a2,√1−b2]iis an IVPFN, then from Eq. (8), we have
M(α) = (a2−(1−a2))(1+√1−a2−1+a2)+(b2−1+b2)(1+√1−b2−1+b2)
2
=2a2−1+2b2−1
2
=a2+b2−1.
4. TOPSIS METHOD BASED ON THE PROPOSED SCORE FUNCTION
Hwang and Yoo [4] introduced the TOPSIS method for finding the best alternative based on the shortest distance
from an ideal solution. In this section, we propose a TOPSIS method for solving the decision-making problems of
IVPFNs with known attributes weight ωbased on the proposed improved score function defined in Eq. (8). Let
A={A1, A2, . . . , Am}be the set of malternatives and G={G1, G2, . . . , Gn}be a set of ncriteria such that the
information about their weights be known. Then, the procedure for finding the best alternative by the TOPSIS method
has been summarized as below.
Step 1. Construction of Pythagorean fuzzy decision matrix: Assume that each alternative Ai(i=1,2, . . . , m)is
evaluated by the decision-maker under the set of criteria Gj(j=1,2, . . . , n)and gives their preference
in terms of the IVPFNs denoted by αij =[aij , bij ],[cij , dij ]®, where [aij , bij ]represents the degree of
satisfaction of the alternative Aiwith respect to Gj, while [cij, dij ]represents the degree of unsatisfactoriness
International Journal for Uncertainty Quantification
A New Improved Score Function of an Interval-Valued Pythagorean Fuzzy Set 469
such that b2
ij +d2
ij ≤1 for all i, j. Therefore, the collection information of these has been represented as a
decision matrix, called the Pythagorean fuzzy decision matrix D, summarized as follows:
D=
α11 α12 ... α1n
α21 α22 ... α2n
.
.
..
.
.....
.
.
αm1αm2... αmn
.
Step 2. Normalization: Transform the different types of criteria into a single one by using the following transforma-
tion:
rij =(αij ;j∈B
αc
ij ;j∈C,(9)
where αc
ij =h[c, d],[a, b]iis the complement of αij =h[a, b],[c, d]iand B, C represent the benefit (B) and
cost (C) type criteria, respectively.
Step 3. Construct the score matrix: The score matrix R, corresponding to the Pythagorean fuzzy decision matrix,
has been constructed by using Eq. (8), and is given as
R=
M(r11)M(r12). . . M (r1n)
M(r21)M(r22). . . M (r2n)
.
.
..
.
.....
.
.
M(rm1)M(rm2). . . M (rmn)
.
Step 4. Determine the distance separation of each alternative from ideal and anti-ideal alternatives: As 0 ≤aij ≤
bij ≤1, 0 ≤cij ≤dij ≤1, hence all rating values of the alternative as given in the decision ma-
trix D= (αij )m×nare IVPFS. Therefore, accordingly the membership degrees of the interval-valued
Pythagorean positive-ideal solution (IVPPIS)(a+) may be chosen as 1 and 0 and thus their rating values
may be expressed as a+=h[1,1],[0,0]i1×n. Similarly, the membership degrees of the interval-valued
Pythagorean negative-ideal solution (IVPNIS) (a−) may be chosen as 0 and 1 and hence their rating values are
summarized as a−=h[0,0],[1,1]i1×n. From these, it has been seen that a+and a−are complements to each
other. Furthermore, instead of fixing the degree of a+to be 1 and 0, the decision-maker may vary it and hence
define an IVPPIS a+and IVPNIS a−as h[e+
j, f +
j],[g+
j, h+
j]i)1×n, and h[e−
j, f −
j],[g−
j, h−
j]i)1×n, where
e+
j= max{aij |i=1,2, . . . , m},f+
j= max{bij |i=1,2, . . . , m},g+
j= min{cij |i=1,2, . . . , m},
h+
j= min{dij |i=1,2, . . . , m},e−
j= min{aij |i=1,2, . . . , m},f−
j= min{bij |i=1,2, . . . , m},
g−
j= max{cij |i=1,2, . . . , m}, and h−
j= max{dij |i=1,2, . . . , m}. From this, it has been clearly
seen that (h[e−
j, f −
j],[g−
j, h−
j]i)⊆(h[aij , bij ],[cij , dij ]i)⊆(h[e+
j, f +
j],[g+
j, h+
j]).
In order to compare the different alternatives Ai(i=1,2, . . . , m), the weighted-Euclidean distance measures
[31] have been used to measure differences between an alternative Aiand the IVPPIS a+= (h[1,1],[0,0]i)
1×n,
as well as the IVPNIS a−= (h[0,0],[1,1]i)1×nwhich are defined as follows:
d(Ai, a+) = v
u
u
t
n
X
j=1½ωj¡M(a+)−M(rij )¢2¾2
,(10)
and
d(Ai, a−) = v
u
u
t
n
X
j=1½ωj¡M(rij )−M(a−)¢2¾2
.(11)
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470 Garg
On the other hand, if the decision-maker gave their preference of ideal solution as a+= (h[e+
j, f +
j],[g+
j, h+
j]i)
1×n,
and a−= (h[e−
j, f −
j],[g−
j, h−
j]i)1×nthen the weighted-Euclidean distance [31] between the alternative Ai
and ideal alternatives a+and a−are defined as
d(Ai, a+) = v
u
u
t
n
X
j=1½ωj¡M(a+)−M(rij )¢2¾2
,(12)
and
d(Ai, a−) = v
u
u
t
n
X
j=1½ωj¡M(rij )−M(a−)¢2¾2
.(13)
Step 5. Compute the closeness coefficient (CC): Based on the weighted-Euclidean distance as defined either in Eqs.
(10) and (11) or in Eqs. (12) and (13), the relative-closeness coefficient of the alternative Aiwith respect to
a+and a−is defined as follows:
CCi=di(Ai, a−)
di(Ai, a+) + di(Ai, a−), i =1,2, . . . , m. (14)
It has been seen that 0 ≤d(Ai, a−)≤d(Ai, a−) + d(Ai, a+)and hence 0 ≤C Ci≤1.
Step 6. Rank the alternative: Based on the descending order of the values of CCi, we rank the alternatives and hence
select the best alternative(s).
5. NUMERICAL EXAMPLE
The above-mentioned approach has been illustrated with a practical example of the DM which can be read as follows:
Owing to the field of investment in India, an investor has usually faced a stronger challenge in the market. Com-
panies have attracted the investor by reducing the price, even though they have analyzed that customer satisfaction
is one of the most important and fundamental elements to survive in the market. To make a decision more clear, an
investor wants to know which market is best for investments and then call an expert to learn from them. So after
their preliminary screening, a committee has been constituted to invest the money in four major markets, which are
Southern Asian markets (A1), Eastern Asian markets (A2), Northern Asian markets (A3), and Local markets (A4)
according to the following four major criteria: the growth analysis (G1), the risk analysis (G2), the social-political
impact analysis (G3), and the environmental impact analysis (G4) with weight vector ω= (0.15,0.25,0.35,0.25)T.
5.1 By Proposed Approach
We utilize the approach developed to get the most desirable alternative(s) as follows.
Step 1. Assume that these alternatives are assessed with respect to the each criterion and their corresponding assess-
ments are provided by an expert in the form of IVPFNs given as follows:
D=
G1G2G3G4
A1h[0.4,0.5],[0.3,0.4]i h[0.2,0.4],[0.4,0.6]i h[0.5,0.6],[0.1,0.3]i h[0.2,0.4],[0.4,0.5]i
A2h[0.6,0.7],[0.2,0.3]i h[0.2,0.3],[0.6,0.7]i h[0.1,0.2],[0.4,0.7]i h[0.4,0.6],[0.3,0.4]i
A3h[0.3,0.6],[0.3,0.4]i h[0.3,0.4],[0.5,0.6]i h[0.1,0.3],[0.5,0.6]i h[0.3,0.5],[0.4,0.5]i
A4h[0.7,0.8],[0.1,0.2]i h[0.1,0.3],[0.6,0.7]i h[0.1,0.2],[0.3,0.4]i h[0.1,0.3],[0.5,0.6]i
.
In this matrix, the number h[0.3,0.4],[0.4,0.5]i, corresponding to A1and G1, represents that the degree to
which alternative A1(Southern Asian markets) satisfies the criteria G1(growth analysis) lies in the interval
[0.3, 0.4], and the degree to which the alternative A1dissatisfies the criteria G1lies in [0.4, 0.5]. The other
values in the matrix have similar meanings.
International Journal for Uncertainty Quantification
A New Improved Score Function of an Interval-Valued Pythagorean Fuzzy Set 471
Step 2. As the criteria G1and G4are benefit while G2and G3are from the cost, in order to normalize the Pythagorean
fuzzy decision matrix, the cost-type criteria are converted into the benefit-type criteria and hence the normal-
ized Pythagorean fuzzy decision matrix Ris given by
R=
G1G2G3G4
A1h[0.4,0.5],[0.3,0.4]i h[0.4,0.6],[0.2,0.4]i h[0.1,0.3],[0.5,0.6]i h[0.2,0.4],[0.4,0.5]i
A2h[0.6,0.7],[0.2,0.3]i h[0.6,0.7],[0.2,0.3]i h[0.4,0.7],[0.1,0.2]i h[0.4,0.6],[0.3,0.4]i
A3h[0.3,0.6],[0.3,0.4]i h[0.5,0.6],[0.3,0.4]i h[0.5,0.6],[0.1,0.3]i h[0.3,0.5],[0.4,0.5]i
A4h[0.7,0.8],[0.1,0.2]i h[0.6,0.7],[0.1,0.3]i h[0.3,0.4],[0.1,0.2]i h[0.1,0.3],[0.5,0.6]i
.
Step 3. By utilizing the proposed improved score function given in Eq. (8), convert the matrix Rinto their collective
score matrix as
M=
G1G2G3G4
A10.1449 0.2829 −0.4583 −0.1932
A20.6136 0.6136 0.5226 0.2346
A30.1693 0.3143 0.4583 −0.0653
A40.8794 0.6435 0.1916 −0.4583
.
Step 4. By employing Eqs. (10) and (11) to calculate the distance between the alternative Aifrom the ideal alterna-
tives a+= (h[1,1],[0,0]i)1×4and a−= (h[0,0],[1,1]i)1×4, we get d1(A1, a+) = 0.8422, d2(A2, a+) =
0.1724, d3(A3, a+) = 0.3400, and d4(A4, a+) = 0.5797; d1(A1, a−) = 0.0768, d2(A2, a−) = 0.1462,
d3(A3, a−) = 0.0777, and d4(A4, a−) = 0.1646.
Step 5. The relative closeness coefficient (CCi)of the alternative Ai(i=1,2,3,4)is computed by using Eq. (14);
we get CC1=0.0836, C C2=0.4589, CC3=0.1860, and CC4=0.2212.
Step 6. Therefore, the optimal ranking of these four major alternatives are A2ÂA4ÂA3ÂA1, and thus, the best
alternative is A2, namely, Eastern Asian markets.
5.2 Comparative Study
In order to compare the performance of the proposed approach with some existing approaches, we conducted a com-
parison analysis based on the different approaches as given by the authors in [9,20,24,27,28,32,33]. The results corre-
sponding to these are summarized in Table 1. Thus, from this table, it has been concluded that the results computed by
the existing approaches coincide with the proposed one which validates the proposed approach. Therefore, the pro-
posed technique can more suitably be utilized to solve the decision-making problem than the other existing measures
and hence the proposed approach equivalently solved the decision-making problems in the PFS environment.
TABLE 1: Comparative analysis
Overall values of the alternatives Order of alternatives
A1A2A3A4
Ye [32] 0.0627 0.3879 0.3106 0.0691 A2ÂA3ÂA4ÂA1
Chen et al. [33] 0.1002 0.1734 0.1443 0.1636 A2ÂA3ÂA4ÂA1
Garg [9] 0.3795 0.6615 0.5895 0.5260 A2ÂA3ÂA4ÂA1
Peng and Yang [27] –0.0362 0.2992 0.1647 0.2232 A2ÂA4ÂA3ÂA1
Gar [28] –0.6763 –0.4115 –0.5350 –0.5398 A2ÂA3ÂA4ÂA1
Yager [20] –0.0993 0.2706 0.1266 0.0500 A2ÂA3ÂA4ÂA1
Garg [24] –0.0945 0.2731 0.1295 0.0644 A2ÂA3ÂA4ÂA1
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472 Garg
5.3 Superiority of the Proposed Approach
In this section, we have presented a counter-example which shows that the existing TOPSIS methods under the IFS
environment fail to rank the given alternatives while the proposed approach can overcome their shortcomings.
Example 5.1. Consider a decision-making problem in which there are two alternatives denoted by A1and A2which
are evaluated by an expert under the set of the three different attributes denoted by G1,G2, and G3, whose weight
vector is ω= (0.2,0.5,0.3)T. Then the objective of the problem is to find the best alternative under the given set.
In order to do so, an expert evaluated these alternatives and gave their preferences in terms of intuitionistic fuzzy
numbers which are summarized as follows:
D=
G1G2G3
· ¸
A1h[0.5,0.6],[0.2,0.4]i h[0.5,0.6],[0.3,0.4]i h[0.3,0.5],[0.2,0.4]i
A2h[0.4,0.6],[0.3,0.4]i h[0.5,0.6],[0.2,0.4]i h[0.4,0.5],[0.3,0.4]i.(15)
Based on these decision matrices, if we utilize the existing TOPSIS approach [6] to find the best alternative, then the
following steps have been executed as follows:
Step 1. The information related to the alternatives is represented in the form of the decision matrix Das given in
Eq. (15).
Step 2. The positive- and negative-ideal solutions of these two alternatives are found as a+={h[0.5,0.6],[0.2,0.4]i,
h[0.5,0.6],[0.2,0.4]i,h[0.4,0.5],[0.2,0.4]i}and a−={h[0.4,0.6],[0.3,0.4]i,h[0.5,0.6],[0.3,0.4]i,h[0.3,0.5],
[0.3,0.4]i}, respectively.
Step 3. Based on these values, the distance measure values between the alternatives Ai(i=1,2)from their ideals
values are computed as d(A1, a+) = 0.0667, d(A2, a+) = 0.0667, d(A1, a−) = 0.0667, and d(A2, a−) =
0.0667.
Step 4. The relative-closeness coefficient C(·)of each alternative is C(A1) = d(A1, a−)/[d(A1, a−) + d(A1, a+)] =
0.5 and C(A2) = d(A2, a−)/[d(A2, a−) + d(A2, a+)] = 0.5. Since C(A1) = C(A2)we conclude that the
existing TOPSIS approach is unable to rank the given alternatives.
On the other hand, if we utilize the proposed TOPSIS approach to the above-considered data, then we get that the
distance measures are d(A1, a+) = 0.0017, d(A2, a+) = 0.0034, d(A1, a−) = 0.0034, and d(A2, a−) = 0.0017.
Thus, the relative-closeness-coefficient degrees of each alternative are CC (A1) = 0.0034/(0.0034 +0.0017) =
0.6590 and CC (A2) = 0.0017/(0.0017 +0.0034) = 0.3409. Since CC(A1)> CC (A2)we hence conclude that
the alternative A1is better than A2. Therefore, the proposed approach is suitably working in those cases where the
existing TOPSIS method fails.
5.4 Advantages of the Proposed Approach
According to the above comparison analysis, the proposed method for addressing the decision-making problems has
the following merits with respect to the existing ones.
(i) As discussed above, the classical, fuzzy, and intuitionistic fuzzy sets all are special cases of the Pythagorean
fuzzy set. So far, some of the existing authors have used the IFS which is characterized by the degree of
membership and nonmembership of a particular element such that their sum is less than or equal to 1. How-
ever, in most of the day-to-day life problems, this condition may not be satisfied when an expert gives their
preferences towards the elements. For handling this, PFS is one of the generalized theories which can handle
not only incomplete information but also the indeterminate information and inconsistent information, which
exist commonly in real situations. Therefore, the existing studies are more suitable than the existing ones for
solving real-life and engineering design problems.
International Journal for Uncertainty Quantification
A New Improved Score Function of an Interval-Valued Pythagorean Fuzzy Set 473
(ii) It has been observed from the existing studies that the existing score and accuracy functions in the IVPFS
environment havebeen proposed by the researchers, but there are some situations that cannot be distinguished
by these existing measures, and hence their corresponding algorithm may give an irrelevant result. On the
other hand, the proposed measure has the ability to overcome those flaws, so it is the more suitable measure
to tackle problems.
(iii) Also, it has been observed from Table 1 that the decision results obtained by our proposed method coincide
with the existing approach results in the IVIFS and/or IVPFS environment. Thus, from the comparative study,
it has been concluded that the proposed measures are more suitable and practicable, and provide a better way
in the IVPFS environment where the existing measures are unable to rank the alternatives.
(iv) The existing measure functions for IFS are a special case of the proposed operators. Furthermore, some of
the existing score functions for IVPFS are also a special case of the proposed operators. Therefore, it has
been concluded that the proposed improved score function is more generalized and suitable to solve real-life
problems more accurately than the existing ones.
6. CONCLUSION
In this paper, an improved score function of IVPFS and an interval-valued Pythagorean fuzzy TOPSIS method based
on it have been presented where the characteristic of each alternative is taken in the form of IVPFNs. The shortcoming
of the existing score functions has been illustrated with some numerical examples. During the formulation, firstly the
Pythagorean fuzzy decision matrix has been converted into the score function by using an improved score function.
Based on the ideal measures values, a relative-closeness coefficient has been determined to find the most desirable
alternative among the feasible ones. The approach has been illustrated with a numerical example for the field of
decision-making to show its effectiveness as well as stability. A comparison between the proposed and existing
measures has been formulated in terms of the counterintuitive cases for showing the validity of it. From the computed
results, it has been observed that the proposed approach can be equivalently utilized to solve MCDM problems where
DMs give their preferences in terms of PFNs rather than IFNs. In the future, we may extend this technique to other
domains such as multiobjective programming, clustering, uncertain system, and pattern recognition.
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International Journal for Uncertainty Quantification
