Advances in Fuzzy Systems
Volume 2018 (2018), Article ID 8023013, 11 pages
https://doi.org/10.1155/2018/8023013

Research Article

Pythagorean Fuzzy -Norm Information Measure for Multicriteria Decision-Making Problem

Abhishek Guleria and Rakesh Kumar Bajaj

Department of Mathematics, Jaypee University of Information Technology, Waknaghat, Solan, Pin-173 234, Himachal Pradesh, India

Correspondence should be addressed to Rakesh Kumar Bajaj; rakesh.bajaj@gmail.com

Received 19 May 2018; Accepted 13 August 2018; Published 2 September 2018

Academic Editor: Hanbo Zheng

Copyright © 2018 Abhishek Guleria and Rakesh Kumar Bajaj. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the present communication, a parametric (, )-norm information measure for the Pythagorean fuzzy set has been proposed with the proof of its validity. The monotonic behavior and maximality feature of the proposed information measure have been studied and presented. Further, an algorithm for solving the multicriteria decision-making problem with the help of the proposed information measure has been provided keeping in view of the different cases for weight criteria, when weights are unknown and other when weights are partially known. Numerical examples for each of the case have been successfully illustrated. Finally, the work has been concluded by providing the scope for future work.

1. Introduction

The concept of intuitionistic fuzzy set (IFS) (Atanassov) [1] has been widely studied and applied to deal with uncertainties and hesitancy inherent in practical circumstances. The prominent characteristic of an IFS is that it assigns a number from the unit interval to every element in the domain of discourse, a degree of membership, and a degree of nonmembership along with the degree of indeterminacy whose total sum equals unity. In literature, intuitionistic fuzzy sets comprehensively span applications in the field of decision-making problems, pattern recognition, sales analysis financial services, medical diagnosis, etc.

Pythagorean fuzzy set (PFS), proposed by Yager [2], is an efficient generalization of intuitionistic fuzzy set, characterized by a membership value and a nonmembership value satisfying the inequality that the squared sum of these values is less than or equal to 1. Yager and Abbasov [3] well stated that, in some practical multiple-criteria decision-making problems, it is viable that sum of the degree of the membership and the degree of nonmembership value of a particular alternative provided by a decision-maker may be in such a way that their sum is bigger than 1, where it would not be feasible to use intuitionistic fuzzy set. Therefore, PFS proves to be proficiently more capable of representing and handling vagueness, impreciseness, and uncertainties than IFS in various decision-making processes. It may be noted that PFS is more generalized than IFS as the span of membership degree of PFS is more than span of membership degree of IFS which enables wider applicability.

Various researchers theoretically developed the concept of Yager’s Pythagorean fuzzy sets [4] and applied it in the field of decision-making problems, medical diagnosis, and pattern recognition and in other real-world problem. In order to deal the decision-making problem with PFSs, Zhang and Xu [5] proposed a comparison method based on a score function to identify the Pythagorean fuzzy positive ideal solution (PIS) and the Pythagorean fuzzy negative ideal solution (NIS). Further they extended the TOPSIS method to compute the distances between each alternative with PIS and NIS, respectively. Peng and Yang [6] proposed some basic operations for PFSs and provided Pythagorean fuzzy aggregation operators along with their important properties. In continuation, they developed a Pythagorean superiority and inferiority ranking algorithm to solve group decision-making problems in view of uncertainty. Further, Peng et al. [7] established the relationship between the distance measure, similarity measure, entropy, and the inclusion measure and suggested the systematic transformation of information measures for PFSs. Yager [8] introduced some of the basic set operations for PFSs and established the relationship between Pythagorean membership values and complex number. In addition to this, the solutions of multicriteria decision-making with satisfactions through Pythagorean membership values have been carried out. A new method for Pythagorean fuzzy MCDM problems with the help of aggregation operators and distance measures has been developed by Zeng et al. [9]. Further, they proposed the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator and developed a hybrid TOPSIS method.

Using PFSs, Ren et al. [10] had a simulation test to study the effect of the risk attitudes of the decision makers over the solutions of decision-making problems. Zhang [11] introduced a novel closeness index-based ranking method for Pythagorean fuzzy numbers and proposed interval valued Pythagorean fuzzy set with basic operations and important properties. In addition to this, the hierarchical multicriteria decision-making problems in Pythagorean fuzzy environment have been solved by developing a closeness index-based Pythagorean fuzzy QUALIFLEX method. Liu et al. [12] developed various types of Pythagorean fuzzy aggregation operators and used them to solve decision-making problems. Zeng [13] developed a Pythagorean fuzzy multiattribute group decision-making method on the basis of a new Pythagorean fuzzy probabilistic ordered weighted averaging (OWA) operator. Though various researchers have significantly contributed in the development of the theory of PFSs as deliberated above, a seldom study on the entropy of PFSs and its applications has been found in literature. Xue et al. [14] studied the linear programming technique for multidimensional analysis of preference (LINMAP) method under Pythagorean fuzzy environment to solve multiple attribute group decision-making problem by incorporating Pythagorean fuzzy entropy along with various other applications. Vital applications of entropy and information measures based on the IFS theory have been well known in the literature. In order to deal with real-world problems more efficiently and to cater the need of the hour, generalizations of the existing approaches play an important role as they contribute more flexibility in applications; e.g., parameters may characterize various factors such as time constraint, lack of knowledge, and environmental conditions, etc.

Bajaj et al. [15] proposed a new -norm intuitionistic fuzzy entropy and a weighted -norm Intuitionistic fuzzy divergence measure with their computational applications in pattern recognition and image thresholding. Gandotra et al. [16] studied multiple-criteria decision-making problem with the help of parametric entropy under -cut and -cut based distance measures for different possible values of parameters and provided the ranking method for the available alternatives.

In this communication, we have proposed a new -norm information measure of Pythagorean fuzzy set and applied the information measure in an algorithm to solve multicriteria decision-making problem. In continuation, the implementation of the proposed algorithm by taking suitable examples has also been illustrated. The rest of this paper is organized as follows: in Section 2, we present some basic notions and preliminaries related to the proposed information measure. A new -norm information measure of Pythagorean fuzzy set has been well proposed with the proof of its validity in Section 3. Further, in Section 4, the maximality and the monotonic behavior of the proposed information measure with respect to parameters and have been studied and validated empirically. In Section 5, a new multicriteria decision-making algorithm is provided on the basis of the proposed -norm information measure of PFS in view of two cases of weights of criteria: one when weights are unknown and other when weights are partially known. In order to support and implement the proposed algorithm, an example for each case has been explicitly dealt in Section 6. The paper is finally concluded in Section 7.

2. Preliminaries

In this section, we recall and present some fundamental concepts in connection with Pythagorean fuzzy set, which are well known in literature.

Definition 1 (see [1]). An intuitionistic fuzzy set (IFS) in (universe of discourse) is given by

where and denote the degree of membership and degree of nonmembership, respectively, and for every satisfy the condition

and the degree of indeterminacy for any IFS and is given by .

Definition 2 (see [2]). A Pythagorean fuzzy set (PFS) in (universe of discourse) is given by

where and denote the degree of membership and degree of nonmembership, respectively, and for every satisfy the condition

and the degree of indeterminacy for any PFS and is given by

In case of PFS, the restriction corresponding to the degree of membership and the degree of nonmembership is

whereas the condition in case of IFS is

for . This difference in constraint conditions gives a wider coverage for information span which can be geometrically shown in Figure 1.

Figure 1: IFS versus PFS.

Some of the important binary operations on PFSs are being presented below which are available in literature.

Definition 3 (see [7]). If and are two Pythagorean fuzzy sets in , then the operations can be defined as follows:

(a)Complement: .(b)Containment: .(c)Union: .(d)Intersection: .

Definition 4 (see [10]). Let and be two PFSs, then the Euclidean distance between and is defined as follows:

Definition 5 (see [5]). Let and be two PFS, then the Hamming distance between and is defined as follows:

3. New -Norm Information Measure of PFS

Let and be the set of all probability distribution association with random variable taking finite values Joshi and Kumar [17] defined and studied a real valued function from to as -norm information measure of the distribution for and and given by

where and Or and

The most important property of this measure is that when =1 or =1, then (10) becomes the or -norm entropy studied by Boekee and Lubbe [18] and if and or and , then it gives Shannon’s [19] entropy.

Based on the axiomatic definition of entropy for intuitionistic fuzzy set, proposed by Hung and Yang (2006) [20], we analogously define a real valued function , called entropy of Pythagorean fuzzy set if and only if the following four axioms are satisfied:

(i)(PFS1) Sharpness: iff is a crisp set, i.e., ,  ; or ,  ;  .(ii)(PFS2) Maximality: is maximum iff

(iii)(PFS3) Symmetry: .(iv)(PFS4) Resolution: iff , i.e., and for or if and for

In context with Pythagorean fuzzy information, we propose the following Pythagorean fuzzy entropy analogous to measure (10):

Theorem 6. The proposed entropy measure is a valid Pythagorean fuzzy information measure.

Proof. To prove this, we shall show that it satisfies all the axioms PFS1 to PFS4.

(i) (PFS1) (Sharpness): If , then

 Since   , it is possible only in the following cases:

(a)Either , i.e., ,(b) i.e., ,(c) i.e., .

 These three cases implies that is a crisp set. Conversely, if is a crisp set then which is obvious.(ii) (PFS2)(Maximality): In Section 4, we have empirically proved that is maximum iff

 Analytically, we prove the concavity of the by calculating its hessian at the critical point, i.e., with particular values of and . The Hessian of is as :

 It may be noted that is a negative semidefinite matrix for different possible values of and which shows that it is a concave function. Hence, the concavity of the function establishes the maximality property.(iii) (PFS3)(Symmetry): It is obvious from the definition that

(iv) (PFS4)(Resolution): We have

 and

 because if and with , then ;   and which implies that the above result holds. Similarly, if and with , then also the above result holds.

Now, since is a concave function on the Pythagorean fuzzy set , therefore, if then and imply .
Therefore, by the above explained result, we conclude that satisfies condition of resolution PFS4.
Similarly, if , then and . By using the above proved result, we conclude that satisfies the condition PFS4.
Hence, satisfies all the axioms of Pythagorean fuzzy entropy and, therefore, is a valid measure of Pythagorean fuzzy information.

Theorem 7. Let and be two PFSs defined in where and such that either or . Then

Proof. Divide into two parts and such that , i.e., ,    ;  , i.e.,   . Now

which implies

Similarly,

On adding the above two terms, we get

Theorem 8. For any Pythagorean fuzzy set , we have

Proof. By definition, the proof is obvious.

4. Monotonicity of - Norm Information Measure of PFS

In this section, we carry an empirical study for investigating the maximality and monotonic nature of the proposed -norm information measure of PFS. For this, we consider the following four Pythagorean fuzzy sets , , , and over the universe of discourse :

For various values of and and using equation (12), we compute and tabulate all the values of . On the basis of the tabulated data and the plots given in Table 1 and Figure 2, it is quite clear that takes maximum value when and is a monotonically decreasing function of and .

Table 1: Values of entropy for different values of and .

SI. No.

1
0.2
254.676
218.665
184.776
194.201
54.000
46.744
39.669
41.643
20.000
17.381
14.767
15.498
16.960
14.743
12.500
13.123
16.346
14.208
12.040
12.637
16.186
14.068
11.919
12.511

2
0.3
147.744
127.021
107.410
112.867
21.030
18.328
15.608
16.369
5.134
4.526
3.855
4.044
4.068
3.590
3.017
3.171
3.864
3.408
2.854
2.996
3.811
3.360
2.811
2.951

3
0.5
107.670
92.610
78.329
82.303
12.000
10.497
8.952
9.385
2.000
1.785
1.495
1.577
1.464
1.309
1.030
1.096
1.366
1.218
0.943
0.998
1.341
1.193
0.920
0.973

4
0.8
93.073
80.064
67.717
71.154
9.340
8.180
6.972
7.311
1.264
1.134
0.906
0.970
0.877
0.790
0.534
0.584
0.809
0.722
0.468
0.503
0.792
0.701
0.451
0.483

5
1.2
86.508
74.420
62.939
66.134
8.263
7.241
6.162
6.464
1.004
0.902
0.673
0.736
0.675
0.611
0.335
0.375
0.618
0.549
0.281
0.301
0.604
0.528
0.267
0.283

6
1.7
83.052
71.448
60.420
63.489
7.727
6.773
5.753
6.038
0.884
0.796
0.552
0.614
0.584
0.529
0.236
0.261
0.532
0.467
0.192
0.192
0.519
0.444
0.179
0.176

7
2.5
80.576
69.319
58.613
61.592
7.356
6.449
5.467
5.740
0.805
0.726
0.468
0.525
0.524
0.474
0.177
0.178
0.476
0.406
0.141
0.115
0.464
0.381
0.128
0.101

8
5
78.100
67.190
56.806
59.692
6.996
6.133
5.188
5.446
0.731
0.659
0.393
0.433
0.469
0.413
0.135
0.103
0.424
0.327
0.102
0.050
0.413
0.296
0.087
0.037

9
7
77.420
66.604
56.309
59.170
6.899
6.048
5.113
5.366
0.712
0.640
0.374
0.409
0.455
0.391
0.125
0.087
0.411
0.294
0.091
0.033
0.400
0.261
0.073
0.021

10
10
76.917
66.172
55.942
58.784
6.828
5.985
5.058
5.308
0.698
0.625
0.361
0.392
0.444
0.370
0.117
0.074
0.401
0.263
0.081
0.019
0.391
0.231
0.060
0.009

11
20
76.339
65.674
55.520
58.340
6.746
5.912
4.995
5.241
0.682
0.605
0.346
0.372
0.433
0.340
0.107
0.059
0.390
0.224
0.067
0.005
0.380
0.203
0.035
0.001

12
40
76.053
65.427
55.311
58.120
6.706
5.877
4.963
5.208
0.674
0.595
0.338
0.362
0.427
0.325
0.100
0.053
0.385
0.213
0.052
0.003
0.375
0.199
0.013
0.000

13
50
75.996
65.378
55.270
58.076
6.698
5.869
4.957
5.201
0.673
0.593
0.337
0.361
0.426
0.323
0.098
0.052
0.384
0.211
0.048
0.002
0.374
0.199
0.009
0.000

14
70
75.931
65.322
55.222
58.026
6.689
5.861
4.950
5.194
0.671
0.591
0.335
0.358
0.425
0.320
0.096
0.050
0.383
0.210
0.044
0.002
0.372
0.199
0.005
0.000

15
100
75.882
65.280
55.187
57.989
6.682
5.855
4.945
5.188
0.670
0.589
0.333
0.357
0.424
0.317
0.094
0.050
0.382
0.209
0.041
0.002
0.372
0.198
0.003
0.000

16
200
75.826
65.231
55.145
57.945
6.675
5.848
4.939
5.182
0.668
0.587
0.332
0.355
0.423
0.315
0.092
0.049
0.381
0.208
0.037
0.002
0.371
0.198
0.002
0.000

17
500
75.792
65.202
55.120
57.919
6.670
5.844
4.935
5.178
0.667
0.586
0.331
0.354
0.422
0.314
0.091
0.048
0.380
0.208
0.036
0.002
0.370
0.198
0.001
0.000

Figure 2: Monotonicity of the -norm information measure.

5. MCDM Algorithm with -Norm Entropy

Suppose that there is a set of feasible alternatives, i.e., and a set of criteria The decision-making problem is to select the most suitable alternative out of these alternatives. The appraisal values of an alternative with respect to the criteria are given by , where is the degree to which the alternative satisfies criteria and is the degree to which the alternative does not satisfy attribute , satisfying ,   and with and . This problem can be modeled by representing it through the following Pythagorean fuzzy decision matrix:

Let be the weight vector of all the criteria where and is the degree of importance of the th criteria. Sometimes this criteria weight is completely unknown and sometimes it is partially known because of the lack of knowledge, time, data, and the limited expertise of the problem domain. In this section, we discuss and devise two methods to determine the weights of criteria by using the proposed entropy (12).

Case 1 (unknown weights). When the criteria weights are completely unknown, then we calculate the weights by using the proposed PFS entropy as

where , and

is the proposed Pythagorean fuzzy entropy for

Case 2 (partially known weights). In case the weights are partially known for a multiple-criteria decision-making problem, we use the minimum entropy principle (Wang and Wang [21]) to determine the weight vector of the criteria by constructing the programming model as follows.
The overall entropy of the alternative is

where ; or

Since there are fair competitive environments between each of the alternatives, the weight coefficient with respect to the same criteria should also be equal. Further, in order to get the ideal weight, we construct the following accompanying model:

;   or , subject to .

Finally, the procedure for implementing the proposed algorithm is being presented using Figure 3.

Figure 3: Flowchart of the proposed algorithm using PFS.

The steps of the proposed methodology are enumerated and detailed as follows.

Step 1. We construct the decision matrix , where the elements    are the appraisal of the alternative with respect to the criteria

Step 2. Determine the criteria weights by using (27) and (30).

Step 3. Define the most preferred solution and the least preferred solution as

where ,  ;  ; and

where ,  ;  , respectively.

Step 4. By using Definition 5, the distance measures of from and will be evaluated as follows:

and

Step 5. Determine the relative degrees of closeness as follows:

Step 6. On the basis of the relative degree of closeness obtained in Step 5, we determine the optimal ranking order of the alternatives. The alternative with the maximal degree of closeness is supposed to be the best alternative.

6. Numerical Examples

Based on two different cases considered in the proposed algorithm, we present two different examples as follows.

Example 1 (unknown weights). Suppose an automobile company produces four different cars, say, , , , and . Suppose a customer wants to buy a car based on the four given criteria, say, comfort , good mileage , safety , and interiors . Assume the appraisal values of the alternatives with respect to each criterion provided by the expert are represented by PFS as

Then the calculations for the ranking procedure are as follows:

(1)Calculate the criteria weight vector using (27):

(2)The most preferred solution and the least preferred solution are given by

 and

 respectively.(3)The distances between each of from and are given by

(4)The values of relative degree of closeness are as follows:

(5)The ranking of the alternatives as per the relative degree of closeness is and is the best available alternative. It may be noted that the above ranking is with respect to the specific values of = 3 and = 0.3. The consistency of the ranking procedure for different values of parameters and may also be observed and studied by making a simulation study over the varying values of the parameters depending on the requirement.

Example 2 (partially known weights). Suppose there are 1000 students in a college. On the basis of three selected criteria, say, (personality), (intelligence), and (communication skills), the administration wants to select a college representative. Let there be three candidates, say, , , and . The PFS decision matrix for the above problem is

Let the information about the criteria weight be partially given in the following form . The calculation for the ranking procedure for the above decision-making problem is presented as follows:

(1)Using (30), we determine the criteria weights by following linear programming model:

 By solving this linear programming problem using MATLAB software, we obtained the criteria weight vector as follows:

(2)The most preferred solution and the least preferred solution are given by and , respectively.(3)The distances between each of from and are given by

(4)The values of relative degree of closeness are

(5)The ranking of the alternatives as per the relative degree of closeness is and is the best available alternative. It may be noted that the above ranking is with respect to the specific values of = 3 and = 0.3. The consistency of the ranking procedure for different values of the parameters and may also be observed and studied by making a simulation study over the varying values of the parameters.

7. Conclusions and Future Work

We have successfully proposed a new parametric -norm information measure for Pythagorean fuzzy set along with the proof of its validity and discussed its maximality and the monotonic behavior with respect to parameters under consideration. Further, an algorithm for multicriteria decision-making problem has been well proposed and successfully implemented with the help of two different kind of numerical examples when weights are unknown and other when weights are partially known.

In the area of pattern recognition, the directed divergence measure/symmetric divergence measure explains dissimilarity between pairs of probability distribution which is generally utilized for the procedure of factual surmising. It might be noticed that these difference measures and similarity measures are dual ideas. The similarity measure might be characterized as a diminishing capacity of the difference measures, particularly when the scope of divergence measures is .

The proposed parametric -norm information measure can further be extended to the concept of the parametric directed divergence measure/symmetric divergence measure for Pythagorean fuzzy sets. Various applications including total ambiguity and information improvement measures based on the proposed divergence measures may also be discussed in detail, e.g., taking a monotonic decreasing function into account, the upper bound of the symmetric divergence measure can be calculated and the similarity measure can be subsequently defined between two PFSs. The concept of similarity based clustering method (SCM) can also be investigated and the structure of the considered data set might be examined with the assistance of the similarity measure.

Data Availability

The data for the implementation of the proposed algorithm in the numerical example are hypothetical data and have no connection with any particular agency’s data.

Disclosure

This article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.

R. R. Yager, “Pythagorean fuzzy subsets,” in Proceedings of the 9th Joint World Congress on Fuzzy Systems and NAFIPS Annual Meeting, IFSA/NAFIPS 2013, pp. 57–61, Edmonton, Canada, June 2013.

R. R. Yager and A. M. Abbasov, “Pythagorean membership grades, complex numbers, and decision making,” International Journal of Intelligent Systems, vol. 28, no. 5, pp. 436–452, 2013.

R. R. Yager, “Pythagorean membership grades in multicriteria decision making,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 4, pp. 958–965, 2014.

X. L. Zhang and Z. S. Xu, “Extension of TOPSIS to multiple criteria decision making with pythagorean fuzzy sets,” International Journal of Intelligent Systems, vol. 29, no. 12, pp. 1061–1078, 2014.

X. Peng and Y. Yang, “Some results for Pythagorean fuzzy sets,” International Journal of Intelligent Systems, vol. 30, no. 11, pp. 1133–1160, 2015.

X. Peng, H. Yuan, and Y. Yang, “Pythagorean fuzzy information measures and their applications,” International Journal of Intelligent Systems, vol. 32, no. 10, pp. 991–1029, 2017.

R. R. Yager, “Properties and applications of Pythagorean fuzzy sets,” in Imprecision and Uncertainty in Information Representation and Processing, P. Angelov and S. Sotirov, Eds., vol. 332 of Studies in Fuzziness and Soft Computing, pp. 119–136, Springer, 2016.

S. Zeng, J. Chen, and X. Li, “A hybrid method for pythagorean fuzzy multiple-criteria decision making,” International Journal of Information Technology & Decision Making, vol. 15, no. 2, pp. 403–422, 2016.

P. Ren, Z. Xu, and X. Gou, “Pythagorean fuzzy TODIM approach to multi-criteria decision making,” Applied Soft Computing, vol. 42, pp. 246–259, 2016.

X. Zhang, “Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods,” Information Sciences, vol. 330, pp. 104–124, 2016.

W.-F. Liu, J. Chang, and X. He, “Generalized Pythagorean fuzzy aggregation operators and applications in decision making,” Control and Decision, vol. 31, no. 12, pp. 2280–2286, 2016.

S. Zeng, “Pythagorean fuzzy multiattribute group decision making with probabilistic information and OWA approach,” International Journal of Intelligent Systems, vol. 32, no. 11, pp. 1136–1150, 2017.

W. Xue, Z. Xu, X. Zhang, and X. Tian, “Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making,” International Journal of Intelligent Systems, vol. 33, no. 1, pp. 93–125, 2018.

R. K. Bajaj, T. Kumar, and N. Gupta, “R-norm intuitionistic fuzzy information measures and its computational applications,” Communications in Computer and Information Science, vol. 305, pp. 372–380, 2012.

N. Gandotra, R. K. Bajaj, and J. Mathew, “On ranking in triangular intuitionistic fuzzy multi-criteria decision making under (α, β)-cut with ‘Useful’ parametric entropy,” in Proceedings of the 5th International Conference on Advances in Computing and Communications, ICACC 2015, pp. 69–74, Kochi, India, September 2015.

R. Joshi and S. Kumar, “(R, S)-norm information measure and a relation between coding and questionnaire theory,” Open Systems & Information Dynamics, vol. 23, no. 3, Article ID 1650015, 12 pages, 2016.

D. E. Boekee and J. C. van der Lubbe, “The R-norm information measure,” Information and Control, vol. 45, no. 2, pp. 136–155, 1980.

C. E. Shannon, “A mathematical theory of communication,” Bell Labs Technical Journal, vol. 27, pp. 379–423, 1948.

W. L. Hung and M. S. Yang, “Fuzzy entropy on intuitionistic fuzzy sets,” International Journal of Intelligent Systems, vol. 21, no. 4, pp. 443–451, 2006.

J. Q. Wang and P. Wang, “Intuitionistic liguistic fuzzy multi-criteria decision-making method based on intuitionistic fuzzy entropy,” Control and Decision, vol. 27, no. 11, pp. 1694–1698, 2012.