Spherical fuzzy cross entropy for multiple attribute decision making problems

Corresponding Author: Princy.pjs@gmai.com
http://dx.doi.org/10.22105/jfea.2021.281447.1120
E-ISSN: 2717-3453 | P-ISSN: 2783-1442
|
Abstract
1 | Introduction
Entropy is very important and effective tool for measuring uncertain information. Firstly, Zadeh [15],
[16] introduced the fuzzy entropy. The starting point for the cross-entropy is information theory as
developed by Shannon [8]. Kullback and Leibler [5] proposed the “cross-entropy distance” measure
between two probability distributions. Later, Lin [6] proposed a modified cross-entropy measure.
Shang and Jiang [7] proposed a fuzzy cross-entropy measure and a symmetric discrimination
information measure between fuzzy sets. Vlachos and Sergiadis [9] developed the intuitionistic fuzzy-
cross entropy based on the De Luca-Termini nonprobabilistic entropy [2]. Zhang and Jiang [17]
defined the cross-entropy between vague sets. According to the cross-entropy of vague sets, Ye [12]
has investigated the fault diagnosis problem of turbine. Ye [13] has applied the intuitionistic fuzzy
cross-entropy to multicriteria fuzzy decision-making problems.
Journal of Fuzzy Extension and Applications
www.journal-fea.com
J. Fuzzy. Ext. Appl. Vol. 2, No. 4 (2021) 355–363.
Paper Type: Research Paper
Spherical Fuzzy Cross Entropy for Multiple
Attribute Decision Making Problems
Princy Rayappan 1,*, Krishnaswamy Mohana2
1 Department of Mathematics, Research Scholar, Nirmala College for Women, Coimbatore, India; Princy.pjs@gmai.com;
riyaraju1116@gmail.com.
Citation:
Rayappan, P., & Mohana, K. (2021). Spherical fuzzy cross entropy for multiple attribute decision
making problems. Journal of fuzzy extension and application, 2 (4), 355-363.
Accept: 22/07/2021
Revised: 21/06/2021
Reviewed: 07/05/2021
Received: 15/04/2021
In this paper, we investigate the multiple attribute decision making problems with spherical fuzzy information. The
advantage of spherical fuzzy set is easily reflecting the ambiguous nature of subjective judgments because the spherical
fuzzy sets are suitable for capturing imprecise, uncertain and inconsistent information in the multiple attribute decision
making analysis. Thus, the cross- entropy of spherical fuzzy sets called, spherical fuzzy cross-entropy, is proposed as an
extension of the cross-entropy of fuzzy sets. Then, a multiple attribute decision making method based on the proposed
spherical fuzzy cross entropy is established in which attribute values for alternatives are spherical fuzzy numbers. In
decision making process, we utilize the spherical fuzzy weighted cross entropy between the ideal alternative and an
alternative to rank the alternatives corresponding to the cross entropy values and to select the most desirable one(s).
Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach
and to demonstrate its practicality and effectiveness.
Keywords: Spherical fuzzy set, Spherical fuzzy cross entropy, Spherical fuzzy weighted cross entropy, Enterprise resource
planning system selection.
Licensee Journal
of Fuzzy Extension and
Applications. This article
is an open access article
distributed under the
terms and conditions of
the Creative Commons
Attribution (CC BY)
license
(http://creativecommons.
org/licenses/by/4.0).

356
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Ye [14] proposed an interval-valued intuitionistic fuzzy cross-entropy for multiple attribute decision
making problems on the basis of the vague-cross entrop Xia and Xu [10] proposed some cross-entropy
and entropy formulas for intuitionistic fuzzy sets and applied them to group decision-making.
For interval-valued intuitionistic fuzzy sets, Zhang and Jiang [18] proposed the entropy and cross-
entropy concepts and discussed the connections among some important measures. Xu and Xia [10]
introduced the concepts of entropy and cross-entropy for hesitant fuzzy information, and discussed
their desirable properties.
Antonov [1] introduced the new stereo metrical interpretation of the IFS elements and gave the
geometrical interpretation of the distance between two intuitionistic fuzzy points. Yang and Chiclana
[11] proposed a spherical representation, which allowed us to define a distance function between
intuitionistic fuzzy sets. They have showed that the spherical distance is different from the existing
distances in that it is non- linear with respect to the change of the corresponding fuzzy membership
degrees, and thus it seems more appropriate than usual linear distances for non- linear contexts in 3D
spaces. Their work is just on the usage of IFS on a sphere.
In the spherical representation, hesitancy can be calculated based on the given membership and non-
membership values since they only consider the surface of the sphere. Besides, they measure the
spherical arc distance between two IFSs. Furthermore, Gong et al. [4] introduced an approach
generalizing Yang and Chiclana’s work. They applied the spherical distance measure to obtain the
difference between two IFSs. They first introduced an ideal opinion and each individual’s opinion in
group decision, they briefly constructed a non- linear optimization model.
The Spherical Fuzzy Sets (SFSs) are based on the fact that the hesitancy of a decision maker can be
defined independently from membership and non-membership degrees, satisfying the following
condition:
0 ≤ μA
̃
2
(u) + ϑA
̃
2
(u) + πA
̃
2
(u) ≤ 1. ∀ u ∈ U
On the surface of the sphere, above Equation becomes
μA
̃
2
(u) + ϑA
̃
2
(u) + πA
̃
2
(u) = 1. ∀ u ∈ U
Since Yang and Chiclana [11] and Gong et al. [4] measure the arc distance on the surface of the sphere,
Euclidean distance is not measured in these works. In our spherical fuzzy sets approach, the sphere is
not solid but a spherical volume. Based on this fact, Euclidean distance measurement is meaningful.
This also means that any two points within the spherical volume are also on the surface of another
sphere. Euclidean distance gives the shortest distance between two points in the sphere.
The rest of the paper is organized as follows: Section 2 introduces the concept of SFSs. Section 3
introduces the concept of cross entropy between spherical fuzzy sets and spherical fuzzy weighted cross
entropy measure. Also a modal for spherical fuzzy cross entropy decision making environment is
studied. Section 4 illustrates the spherical fuzzy cross entropy measures through a numerical example.
2| Preliminaries
Definition 1. [1]. An IFS 𝐴 in 𝑋 is given by,
A = {< x, μA(x), ϑA(x) > |x ∈ X},

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where 𝜇𝐴
: 𝑋 → [0,1] and 𝜗𝐴: 𝑋 → [0,1], where 0 ≤ 𝜇𝐴(𝑥) + 𝜗𝐴(𝑥) ≤ 1, ∀ 𝑥 ∈ 𝑋. The number 𝜇𝐴(𝑥) and
𝜗𝐴(𝑥) represents, respectively the membership degree and non-membership degree of the element 𝑥 to the
set 𝐴.
Definition 2. [3]. A Spherical Fuzzy Set (SFS) 𝐴̃𝑠 of the universe of discourse U is given by,
A
̃ s = {〈μA
̃s
(u),ϑA
̃s
(u), πA
̃s
(u)|u ∈ U 〉} ,
where
μA
̃s
: U → [0,1], ϑA
̃s
: U → [0,1], πA
̃s
: U → [0,1] and
0 ≤ μA
̃s
2
(u) + ϑA
̃s
2
(u) + πA
̃s
2
(u) ≤ 1. ∀ u ∈ U
For each 𝑢, the numbers 𝜇𝐴̃𝑠
(𝑢),𝜗𝐴̃𝑠
(𝑢) 𝑎𝑛𝑑 𝜋𝐴̃𝑠
(𝑢) are the degree of membership, non-membership and
hesitancy of 𝑢 to 𝐴̃𝑠, respectively.
Definition 3. [3]. Basic operators of Spherical fuzzy sets:
Union:
A
̃ s ∪ B̃s = {max{μA
̃s
, μB̃s
},min{ϑA
̃s
,ϑB̃s
},min{πA
̃s
, πB̃s
} }.
Intersection:
A
̃ s ∩ B̃s = {min{μA
̃s
, μB̃s
}, max{ϑA
̃s
, ϑB̃s
},max{πA
̃s
, πB̃s
}}.
Addition:
A
̃ s⨁ B̃s = {(μA
̃s
2
+ μB̃s
2
− μA
̃s
2
μB̃s
2)
1 2
⁄
, ϑA
̃s
ϑB̃s
, πA
̃s
πB̃s
}.
Multiplication:
A
̃ s⨂ B̃s = {μA
̃s
μB̃s
, (ϑA
̃s
2
+ ϑB̃s
2
− ϑA
̃s
2
ϑB̃s
2
)
1 2
⁄
, πA
̃s
πB̃s
}.
Multiplication by a scalar; 𝝀 > 𝟎:
λ. A
̃ s = {(1 − (1 − μA
̃s
2)
λ
)
1 2
⁄
, ϑA
̃s
λ
, πA
̃s
λ
}.
Power of 𝑨̃𝒔 ; 𝝀 > 𝟎:
A
̃ s
λ
= { μA
̃s
λ
, (1 − (1 − ϑA
̃s
2
)
λ
)
1 2
⁄
, πA
̃s
λ
}.

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3| Cross-Entropy between Spherical Fuzzy Sets
In this section, we shall develop the cross-entropy and discrimination information measures between
two SFSs based on the extension of the concept of cross-entropy between two fuzzy sets.
To do this, we firstly introduce the concepts of cross-entropy and symmetric discrimination information
measures between two fuzzy sets which were proposed by Shang and Jiang [7].
Definition 4. [7]. Assume that 𝛼 = (𝛼(𝑥1), 𝛼(𝑥2),… , 𝛼(𝑥𝑛)) 𝑎𝑛𝑑 𝛽 = (𝛽(𝑥1), 𝛽(𝑥2), … , 𝛽(𝑥𝑛)) are two fuzzy
sets in the universe of discourse (𝑥1, 𝑥2, … , 𝑥𝑛). The fuzzy cross-entropy of 𝛼and 𝛽 is defined as follows:
which indicates the degree of discrimination of 𝛼 and 𝛽.
However, ℎ(𝛼,𝛽) is not symmetric with respect to its arguments. Shang and Jiang [7] proposed a
symmetric discrimination information measure:
Moreover, there are 𝐼(𝛼, 𝛽) ≥ 0 and 𝐼(𝛼, 𝛽) = 0 if and only if 𝛼 = 𝛽.
Then, the cross-entropy and symmetric discrimination information measures between two fuzzy sets
are extended to these measures between SFSs. In order to do so, let us consider two groups of Spherical
Fuzzy numbers 𝛼 = (𝜇𝛼𝑗
, 𝜗𝛼𝑗
, 𝜋𝛼𝑗
) and 𝛽 = (𝜇𝛽𝑗
,𝜗𝛽𝑗
, 𝜋𝛽𝑗
), 𝑗 = 1,2, . . , 𝑛.
Thus, based on Eq. (3), the amount of information for discrimination of 𝜇𝛼𝑗
from 𝜇𝛽𝑗
(𝑗 = 1,2, … , 𝑛) can
be given by:
Therefore, the expected information based on the single degree of membership for discrimination of 𝛼
against 𝛽 is expressed by:
Similarly, considering the degree of non-membership and the degree of hesitancy, we have the following
amounts of information:
h(α, β) = ∑
(
α(xj)ln
α(xj)
1
2
(α(xj) + β(xj))
+ (1 − α(xj)) ln
1 − α(xj)
1 −
1
2
(α(xj) + β(xj)))
n
j=1
, (1)
I(α, β) = h(α, β) + h(β, α). (2)
Cμ
(α, β,μ) = μαj
2
ln
μαj
2
1
2
(μαj
2 + μβj
2)
+ (1 − μαj
2)ln
1 − μαj
2
1 −
1
2
(μαj
2 + μβj
2)
. (3)
Cμ
(α, β) = ∑
(
μαj
2
ln
μαj
2
1
2
(μαj
2 + μβj
2)
+ (1 − μαj
2)ln
1 − μαj
2
1 −
1
2
(μαj
2 + μβj
2))
n
j=1
. (4)
Cϑ
(α, β) = ∑
(
ϑαj
2
ln
ϑαj
2
1
2
(ϑαj
2
+ ϑβj
2
)
+ (1 − ϑαj
2
)ln
1 − ϑαj
2
1 −
1
2
(ϑαj
2
+ ϑβj
2
))
.
n
j=1
(5)
Cπ
(α,β) = ∑
(
παj
2
ln
παj
2
1
2
(παj
2 + πβj
2)
+ (1 − παj
2)ln
1 − παj
2
1 −
1
2
(παj
2 + πβj
2))
n
j=1
. (6)

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Hence, a novel spherical fuzzy cross- entropy measure between 𝛼 and 𝛽 is obtained as the sum of the three
amounts:
which also indicates discrimination degree of 𝛼 from 𝛽. According to Shannon’s [8] inequality, one can
easily prove that 𝐶(𝛼,𝛽) ≥ 0 and 𝐶(𝛼, 𝛽) = 0 if and only if 𝛼 = 𝛽. Then, 𝐶(𝛼,𝛽) is not symmetric. So it should
be modified to a symmetric discrimination information measure for SFSs as:
The larger the difference between 𝛼 and 𝛽 is, the larger 𝐷(𝛼,𝛽) is.
If we consider the weights of 𝛼 and 𝛽, a spherical fuzzy weighted cross-entropy measure between 𝛼 and 𝛽
is proposed as follows:
where 𝜔 = (𝜔1, 𝜔2, … , 𝜔𝑛)𝑇
is the weight vector of 𝛼,𝛽(𝑗 = 1,2, … , 𝑛), with 𝜔𝑗 ∈ [0,1],𝑗 = 1,2, … , 𝑛, ∑ 𝜔𝑗 =
𝑛
𝑗=1
1,which also indicated discrimination degree of 𝛼 from 𝛽. According to Shannon’s inequality (3), one can
easily prove that 𝐶𝜔(𝛼, 𝛽) ≥ 0 and 𝐶𝜔(𝛼,𝛽) = 0 if and only if 𝛼 = 𝛽.
Then, 𝐶𝜔(𝛼,𝛽) is not symmetric. So it should be modified to a symmetric discrimination information
measure for SFSs as
The larger the difference between 𝛼 and 𝛽 is, the larger 𝐷𝜔(𝛼,𝛽) is.
C(α, β) = ∑
(
μαj
2
ln
μαj
2
1
2
(μαj
2 + μβj
2)
+ (1 − μαj
2)ln
1 − μαj
2
1 −
1
2
(μαj
2 + μβj
2))
n
j=1
+
∑
(
ϑαj
2
ln
ϑαj
2
1
2
(ϑαj
2
+ ϑβj
2
)
+ (1 − ϑαj
2
)ln
1 − ϑαj
2
1 −
1
2
(ϑαj
2
+ ϑβj
2
))
n
j=1
+
∑
(
παj
2
ln
παj
2
1
2
(παj
2 + πβj
2)
+ (1 − παj
2)ln
1 − παj
2
1 −
1
2
(παj
2 + πβj
2))
,
n
j=1
(7)
D(α, β) = C(α,β) + C(β,α). (8)
Cω
(α,β) = ∑ ωj
(
μαj
2
ln
μαj
2
1
2
(μαj
2 + μβj
2)
+ (1 − μαj
2)ln
1 − μαj
2
1 −
1
2
(μαj
2 + μβj
2))
n
j=1
+
∑ ωj
(
ϑαj
2
ln
ϑαj
2
1
2
(ϑαj
2
+ ϑβj
2
)
+ (1 − ϑαj
2
)ln
1 − ϑαj
2
1 −
1
2
(ϑαj
2
+ ϑβj
2
))
n
j=1
+
∑ ωj
(
παj
2
ln
παj
2
1
2
(παj
2 + πβj
2)
+ (1 − παj
2)ln
1 − παj
2
1 −
1
2
(παj
2 + πβj
2))
,
n
j=1
( )
Dω(α, β) = Cω(α, β) + Cω(β,α). (10)

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3.1| Models for Multiple Attribute Decision Making based on Cross-Entropy with
Spherical Fuzzy Information
Based on cross- entropy with spherical fuzzy information, in this section, we shall propose the
corresponding model for multiple attribute decision making with spherical fuzzy information. Let 𝐴 =
{𝐴1, 𝐴2, … , 𝐴𝑚} be a discrete set of alternatives, and 𝐺 = {𝐺1, 𝐺2, … , 𝐺𝑛} be the set of attributes, 𝜔 =
(𝜔1, 𝜔2, … , 𝜔𝑛) is the weighting vector of the attribute 𝐺𝑗(𝑗 = 1,2,… , 𝑛), where 𝜔𝑗 ∈ [0,1], ∑ 𝜔𝑗 = 1
𝑛
𝑗=1 .
Suppose that 𝑅̃ = (𝑟̃𝑖𝑗)
𝑚×𝑛
= (𝜇𝑖𝑗, 𝜗𝑖𝑗, 𝜋𝑖𝑗)
𝑚×𝑛
is the spherical fuzzy decision matrix, where 𝜇𝑖𝑗 indicates the
degree of membership that the alternative 𝐴𝑖 satisfies the attribute 𝐺𝑗 given by the decision maker, 𝜗𝑖𝑗
indicates the degree of non-membership that the alternative 𝐴𝑖 doesn’t satisfy the attribute 𝐺𝑗 given by
the decision maker and 𝜋𝑖𝑗 indicates the hesitancy degree that the alternative 𝐴𝑖 doesn’t satisfy the
attribute 𝐺𝑗, 𝜇𝑖𝑗 ∈ [0,1],𝜗𝑖𝑗 ∈ [0,1],𝜋𝑖𝑗 ∈ [0,1],
0 ≤ μij
2
+ ϑij
2
+ πij
2
≤ 1, i = 1,2, … , m, j = 1,2, … , n.
In multiple attribute decision-making environments, the concept of ideal point has been used to help
identify the best alternative in the decision set. Although the ideal alternative does not exist in real world,
it does provide a useful theoretical construct against which to evaluate alternatives. Hence, we can define
an ideal alternative value in the ideal alternative 𝐴+
.
Let r̃j
+
= (μj
+
, ϑj
+
,πj
+)(j = 1,2, … , n) where
μj
+
= max{μij}, ϑj
+
= min{ϑij}, πj
+
= min{πij}.
Then, we call:
The relative spherical fuzzy ideal alternative (RSFIA).
In the following, we apply the cross- entropy to the MADM problems.
Step 1. Define the alternatives 𝐴𝑖 and 𝐴+
:
Step 2. Calculate the spherical fuzzy weighted cross- entropy between alternatives 𝐴𝑖 and 𝐴+
:
Step 3. Rank all the alternatives 𝐴𝑖 (𝑖 = 1,2, . . 𝑚) and select the best one(s) in accordance with
𝐶(𝐴𝑖,𝐴+
) (𝑖 = 1,2, … , 𝑚). The smaller the value of 𝐶(𝐴𝑖, 𝐴+
) is, the better the alternative 𝐴𝑖 is. In this case,
the alternative 𝐴𝑖 is close to the ideal alternative 𝐴+
. Through the cross-entropy 𝐶(𝐴𝑖, 𝐴+
) (𝑖 = 1,2, … , 𝑚)
between each alternative and the ideal alternative, the ranking order of all alternatives can be determined
and the best one can be easily identified as well.
A+
= (r̃1
+
, r̃2
+
, … . , r̃n
+
). (11)
A i = ((μi1, ϑi1, πi1), (μi2, ϑi2, πi2), … , (μin, ϑin, πin)), i = 1,2, … , m, (12)
A+
= ((μ1
+
, ϑ1
+
, π1
+), (μ2
+
, ϑ2
+
,π2
+), … , (μn
+
, ϑn
+
, πn
+)),
μj
+
= max{μij}, ϑj
+
= min{ϑij}, πj
+
= min{πij}, j = 1,2, … , n.

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4| Numerical Example
Thus, in this section we shall present a numerical example for potential evaluation of emerging technology
commercialization with spherical fuzzy information in order to illustrate the method proposed in this
paper. Let us suppose there is a problem to deal with the potential evaluation of emerging technology
commercialization which is classical multiple attribute decision making problems. There is a panel with
possible emerging technology enterprises 𝐴𝑖 (𝑖 = 1,2,3,4,5) to select. The experts selects six attribute to
evaluate the five possible emerging technology enterprises:
I. 𝐺1 is the technical advancement.
II. 𝐺2 is the potential market and market risk.
III. 𝐺3 is the industrialization infrastructure.
IV. 𝐺4 is the development of science and technology.
V. 𝐺5 is the financial conditions.
VI. 𝐺6 is the employment creation.
In order to avoid the influence of each other, the decision makers are required to evaluate the five possible
emerging technology enterprises 𝐴𝑖 (𝑖 = 1,2,3,4,5) under the above six attributes and the decision matrix
𝑅̃ = (𝑟̃𝑖𝑗)
5×6
is presented in Table 1, where 𝑟̃𝑖𝑗(𝑖 = 1,2,3,4,5, 𝑗 = 1,2,3,4,5,6) are in the form of SFSs. The
weight vector of 𝑥𝑖(𝑖 = 1,2, … ,6) is
ω = (0.12, 0.25, 0.09, 0.16, 0.18, 0.20)T
.
Table 1. The spherical fuzzy decision matrix.
To get the most desirable emerging technology enterprises, the following steps are involved:
Step 1. Based on the Table 1, we denote the five possible emerging technology enterprises 𝐴𝑖 (𝑖 = 1,2,3,4,5)
by:
A 1 = {(0.53,0.33,0.09), (0.89,0.08,0.03), (0.03,0.82,0.13),
C(A i,A+
) = ∑ ωj
(
μij
2
ln
μij
2
1
2
(μij
2 + μj
+2
)
+ (1 − μij
2)ln
1 − μij
2
1 −
1
2
(μij
2 + μj
+2
))
n
j=1
+
∑ ωj
(
ϑij
2
ln
ϑij
2
1
2
(ϑij
2
+ ϑj
+2
)
+ (1 − ϑij
2
)ln
1 − ϑij
2
1 −
1
2
(ϑij
2
+ ϑj
+2
))
+
n
j=1
∑ ωj
(
πij
2
ln
πij
2
1
2
(πij
2 + πj
+2
)
+ (1 − πij
2)ln
1 − πij
2
1 −
1
2
(πij
2 + πj
+2
))
n
j=1
i = 1,2, … , m.
(15)
𝐀𝟏 𝐀𝟐 𝐀𝟑 𝐀𝟒 𝐀𝟓
𝑥1 [0.53,0.33,0.09] [0.73,0.12,0.08] [0.91,0.03,0.02] [0.85,0.09,0.05] [0.90,0.05,0.02]
𝑥2 [0.89,0.08,0.03] [0.13,0.64,0.21] [0.07,0.09,0.05] [0.68,0.08,0.21] [0.74,0.16,0.10]
𝑥3 [0.03,0.82,0.13] [0.42,0.35,0.18] [0.02,0.89,0.05] [0.04,0.85,0.05] [0.05,0.87,0.10]
𝑥4 [0.33,0.12,0.15] [0.52,0.61,0.31] [0.15,0.07,0.76] [0.15,0.73,0.08] [0.16,0.71,0.05]
𝑥5 [0.73,0.15,0.08] [0.08,0.89,0.03] [0.62,0.06,0.04] [0.68,0.08,0.02] [0.13,0.75,0.09]
𝑥6 [0.17,0.53,0.13] [0.31,0.39,0.25] [0.51,0.42,0.12] [0.09,0.81,0.15] [0.91,0.03,0.05]

362
Rayappan
and
Mohana
J.
Fuzzy.
Ext.
Appl.
2(4)
(2021)
355-363
(0.33,0.12,0.15), (0.73,0.15,0.08), (0.17,0.53,0.13)};
A 2 = {(0.73,0.12,0.08), (0.13,0.64,0.21), (0.42,0.35,0.18),
(0.52,0.61,0.31), (0.08,0.89,0.03), (0.31,0.39,0.25)};
A 3 = {(0.91,0.03,0.02), (0.07,0.09,0.05), (0.02,0.89,0.05)
(0.15,0.07,0.76), (0.62,0.06,0.04), (0.51,0.42,0.12)};
A 4 = {(0.85,0.09,0.05), (0.68,0.08,0.21), (0.04,0.85,0.05),
(0.15,0.73,0.08), (0.68,0.08,0.02), (0.09,0.81,0.15)};
A 5 = {(0.90,0.05,0.02), (0.74,0.16,0.10), (0.05,0.87,0.10)
(0.16,0.71,0.05), (0.13,0.75,0.09), (0.91,0.03,0.05)}.
Step 2. Based on the Table 1 and Eqs. (12) – (14), we can get the RSFIA 𝐴+
:
A+
= {(0.91,0.03,0.02), (0.89, 0.08,0.03), (0.42, 0.35, 0.05)
(0.52, 0.07, 0.05), (0.73,0.06,0.02), (0.91, 0.03, 0.05)}.
Step 3. Calculate the cross-entropy 𝐶𝜔(𝐴𝑖,𝐴+
) between 𝐴𝑖(𝑖 = 1,2,3,4,5) and the RSFIA 𝐴+
by using
(15):
Cω(A 1, A+
) = 0.1499,
Cω(A 2, A+
) = 0.3655,
Cω(A 3, A+
) = 0.2354,
Cω(A 4, A+
) = 0.226,
Cω(A 5, A+
) = 0.1481.
Step 4. Rank the emerging technology enterprises 𝐴𝑖(𝑖 = 1,2,3,4,5) in accordance with the cross-entropy
𝐶𝜔(𝐴𝑖,𝐴+
) (𝑖 = 1,2,3,4,5):
A 5 > A 1 > A 4 > A 3 > A 2.
Thus, the most desirable emerging technology enterprise is 𝐴5.
5| Conclusion
In this paper, we investigate the multiple attribute decision making problems with spherical fuzzy
information. The advantage of spherical fuzzy set is easily reflecting the ambiguous nature of subjective
judgments because the spherical fuzzy sets are suitable for capturing imprecise, uncertain and
inconsistent information in the multiple attribute decision making analysis. Thus, the cross- entropy of
spherical fuzzy sets, called spherical fuzzy cross entropy, is proposed as an extension of the cross-
entropy of fuzzy sets. Then, a multiple attribute decision making method based on the proposed
spherical fuzzy cross entropy is established in which attribute values for alternatives are spherical fuzzy
numbers. In decision making process, we utilize the spherical fuzzy weighted cross entropy between the

363
Spherical
fuzzy
cross
entropy
for
multiple
attribute
decision
making
problems
ideal alternative and an alternative to rank the alternatives corresponding to the cross entropy values and
to select the most desirable one(s). The plan of our future work is to apply the proposed method to some
other practical decision-making problems, such as the performance evaluation, water resource schedule
and risk investment.
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