Mathematics Department, Faculty of Science, Cairo University, Cairo 12613, Egypt
Automated Scheduling and Planning (ASAP) and Lab for Uncertainty in Data and Decision Making (LUCID), School of Computer Science, University of Nottingham, Nottingham NG81BB, UK
Author to whom correspondence should be addressed.
Received: 27 January 2019 / Accepted: 6 May 2019 / Published: 18 June 2019
In this paper, we propose a new approach to constructing similarity measures using the entropy measure for Interval-Valued Intuitionistic Fuzzy Sets. In addition, we provide several illustrative examples to demonstrate the practicality and effectiveness of the proposed formula. Finally, we use the new proposed similarity measure to develop a new approach for solving problems of pattern recognition and multi-criteria fuzzy decision-making.
Atanassov  introduced intuitionistic fuzzy sets (IFSs) which are characterized by both a membership function and a non-membership function. In 1989, Atanassov and Gargov  generalised the notion of intuitionistic fuzzy sets to interval-valued intuitionistic fuzzy sets (IVIFSs), in which the membership values are intervals, rather than exact numbers. IVIFSs operations, relations, and operators concerning IVIFSs were defined by Atanassov. Recently, Yager proposed the notation of a Pythagorean fuzzy set (PFS) , after which Peng developed it to a generalized form called the interval-valued Pythagorean fuzzy set (IVPFS). Han and Deng  proposed an interval-valued Pythagorean prioritized game framework in a group decision-making process.Two important topics in the theory of fuzzy sets—entropy measures and similarity measures of IFSs—have been widely proposed.
In 1965, Zadeh  introduced fuzzy entropy and defined it as the fuzziness degree of a fuzzy set. De Luca and Termini  proposed the axiomatic construction of the entropy of fuzzy sets. In 1996, the entropy measure of IFS and IVIFS was investigated by Burillo and Bustince  to measure the degree of intuitionism. A non-probabilistic entropy measure of IFS and IVIFS was proposed in terms of the ratio of intuitionistic fuzzy cardinalities by Szimidt and Kacprzyk . Hung and Yang  proposed an axiomatic definition of the entropy of IFSs and IVFSs using probability, and they proposed the axiomatic definition and its properties. In addition, they introduced two families of entropy measures and showed how their proposed measure was more reliable for measuring the degree of fuzziness. Recently, Pan and Deng  proposed a new entropy measure that can measure uncertainty of probability distribution. This measure is the generalization of the Shannon entropy.
The similarity measure of IFSs is used to measure the degree of similarity between two IFSs. In 2004, Szimidt and Kacprzyk  investigated a similarity measure between IFSs using a distance measure. They applied the new measure to assess the extent of agreement between a group of experts giving their opinions expressed by intuitionistic fuzzy preference relations . Moreover, a family of similarity measures was proposed by Szimidt and Kacprzyk , and they compared them with some existing similarity measures. Hong and Kim , Hung and Yang , and Xu  defined some other similarity measures using different distance measures for IFSs. In addition to the new measure, Hung and Yang proved some properties of the proposed measure, and compared their measure with existing measures, and the comparison showed that their proposed similarity measure is much simpler than existing measures. More recently, Garg and Kumar  discussed the weaknesses of some of the existing measures, and proposed some novel similarity measures between IFSs.
Zeng and Guo  proved that some similarity measures and entropies of IVFSs could be deduced by normalized distances of IVFSs using the axiomatic definitions. Also, they investigated some formulae to calculate the entropy and similarity measure of the IVIFSs. Zeng and Li  and Zhang et al.  showed that similarity measures and entropies of the IVFSs could be transformed by each other. Zhang and Yu et al.  put forward some entropy formulae of IFSs according to the relationship between entropies and similarity measures of IFSs. In 2011, Wei and Wang  proposed an approach to construct similarity measures using entropy measures for interval-valued intuitionistic fuzzy sets.
In this paper, we follow the procedure outlined by Wei and Wang  and propose a new similarity measure. We demonstrate the usefulness of our new measure for pattern recognition and multi-criteria decision-making through two case studies.
The structure of this paper is as follows. Section 2 reviews some concepts and definitions of IFSs and IVIFSs. Section 3 gives an axiomatic definition, and provides some proposed entropy formulas of IVIFSs. Section 4 investigates the relationship between the entropy and similarity measure of IVIFSs, and proves that similarity measures of IVIFSs can be constructed by entropy measures of IVIFSs based on the axiomatic definition. In particular, we define a new similarity measure of IVIFSs according to the entropy formula of IVIFSs defined in Section 3. This is followed by applications of the proposed similarity measure to pattern recognition and multi-criteria fuzzy decision-making in Section 5. This paper is concluded in Section 6.
Let X be a universe of discourse. An IFS in X is a triple, having the following form:
with the condition
where , denote the degrees of membership and non-membership of x in A, respectively. IFS(X) denotes the set of IFSs on X.
We call the Intuitionistic index of x in A, which denotes the hesitancy degree of x in A .
The complementary set of A is defined as
Let X be a universe of discourse, and denotes all closed subintervals of the interval . An IVIFS A in X is an object having the form:
with the condition
where , denote the degrees of membership and non-membership of x in A, let (where and , where ϵ is an infinitesimal number which is a number that is larger than each negative real number and is smaller than each positive real number), (where and , where ϵ is an infinitesimal number which is a number that is larger than each negative real number and is smaller than each positive real number). Then,
with the condition . IVIFS(X) denotes the set of IVIFSs in X .
we call the Interval-Valued Intuitionistic index of x in A, which denotes the hesitancy degree of x in A.
For two IVIFSs and ,, the following relations and operations can be defined:
if and only if , , , , for each
if and only if and
3. Entropy of IVIFSs
A real-valued function is called an entropy for IVIFSs if it satisfies the following axiomatic requirements :
if, and only if A is a fuzzy set.
if, and only if for all .
For two IVIFSs A and B on X, if when and and and for each , then .
For an IVIFS A, In [10,21,23], some examples of the entropy formula were provided and defined by, for each ,
4. New Similarity Measure between IVIFSs
In this section, we propose a new similarity measure between IVIFS by using an entropy measure of IVIFSs. Firstly, we give an extension of the axiomatic definition for similarity measures in ([16,24,25,26,27]) of IVIFSs.
A real-valued function is called a similarity measure on IVIFS(X) if it satisfies the following axiomatic requirements [28,29]:
if, and only if ;
Let be a finite universe of discourse. For , we define IVIFS by the following:
Suppose that E is an entropy measure for IVIFS. The function on IVIFS(X) defined above by is a similarity measure.
It is straightforward to prove that satisfies the four conditions – in Definition 4.
(S1) We can easily show since for all and .
(S2) With the definition of the entropy and similarity measure, we can obtain
(S3) From the definition of M(A,B), it is straightforward to show that . Thus, it is obvious that .
(S4) Since , we have , , , . Thus:
Hence we have
Similarly, we can obtain:
Besides we have ,
by the definition of . We can get:
In the same way, we can also have
Let E be the entropy measure defined by Equation (6), that is, for IVIFSs,
Then, the proposed similarity measure S for A, looks like:
where ,, and , are defined by Equations (9)–(12) □
Let be a weight vector of the elements , Then, we can define the weighted similarity measure by
where and . If .
The IFS and IVIFS theory have been applied to areas such as artificial intelligence, networking, soft decision-making, programming logic, and operational research. IVIFSs and IFSs are useful for dealing with vagueness and imprecision . Moreover, they are suitable for modeling and processing imperfect information. In this section, we present two applications.
5.1. Pattern Recognition
We apply the proposed weighted similarity measure, defined by Equation (13), to solve pattern recognition problems with IVIFS information. We adopt the same steps as in :
Suppose that there exist m patterns represented by IVIFSs
for , in the feature space , and suppose that there is a sample to be recognized, which is represented by an IVIFS
Calculate the weighted similarity degree between and B by formula in Equation (13).
Select the largest one, denoted by , from , . Then, B belongs to the pattern .
Now, we consider an example of a pattern recognition problem on the classification of building materials given in . We use the same data as in .
Assume that there are four classes of building materials , , and an unknown building material B, which are represented by the IVIFSs in the feature space with a weight vector ω:
Suppose we have the following data indicated by and for each for every variable , and for each for every variable given in Table 1 and Table 2 .
Our purpose is to distinguish which class the unknown pattern B belongs to by using the above steps and calculating the similarity degrees between each and B by (13), where we have:
We obtained the results in Table 3. Clearly, the similarity degree between and B, is the largest one. Hence, B belongs to pattern , which is the same result as in . Moreover, it is very clear that the pairs of B are closer to than the others.
5.2. Multiple-Criteria Decision-Making Problems
In this subsection, we introduce a method to solve a multi-criteria fuzzy decision making problem with weights using the proposed similarity measure [21,23].
Let be a set of options, and be a set of criteria. Assume that is the weight of the criterion , where and . The characteristics of the option in terms of criteria C are represented by the following IVIFSs:
where indicates the degree that option satisfies criterion , and indicates the degree that option does not satisfy criterion .
Using the weighted similarity measure defined by Equation (13), we introduce the following approach to solving the above multi-criteria fuzzy decision-making problem:
Find out the positive-ideal solution and negative-ideal solution :
where, for each
Calculate the similarity measure between the option and the positive-ideal solution , and the similarity measure between the option and the negative-ideal solution by using the formula in Equation (13).
Calculate the relative similarity measure of with respect to and , where
Select the largest one, denoted by , among the values , . Then, is the best choice.
().Consider a supplier selection problem for a product. Suppose that a company sets up six evaluating indices for this problem: price (), deadline (), quality (), the level of technology (), service (), and the future cooperation (), and suppose that the correspondent weight vector is . Assume that there are five suppliers . Experts evaluate these suppliers by the above indices and obtain the evaluating information in Table 4, which indicated and for each for every variable and for each for every variable :
By Step 1, we obtain the positive-ideal solution and the negative-ideal solution in Table 5.
By Step 2, we use formula in Equation (13) to calculate the similarity measure and , respectively:
Then, we obtain the relative similarity measure of with respect to and by the formula in Equation (14):
The results in Table 6 shows that , so is the proper supplier, which is the same result as in . We replicated the same procedure that was used in . We obtained the result that is the proper supplier.
Many similarity measures that were defined based on entropy measures have been applied to the problems based on intuitionistic fuzzy information, but they could not be used to deal with the problems based on interval-valued intuitionistic fuzzy information. In this paper, we introduced a new similarity measure between IVIFSs. This measure has been constructed using the entropy measure for IVIFSs. Moreover, we have verified the efficiency of the proposed similarity measure by applying it to two case studies. Two applications were introduced—one was to solve a pattern recognition problem, and the other was for a multiple-criteria fuzzy decision-making problem. A potential issue is that one measure will not be optimal for all data. Although our measure was effective in our two case studies, in future work we could explore its performance on other applications, such as multiple-attribute decision-making and edge detection, image segmentation, and fault-tree analysis. In addition, we will compare the performance of our measures and IFSs measures on IVIFS data in future.
S.S.M. conceived of the presented idea. S.S.M. developed the theory and performed the computations. A.A. and R.I.J. encouraged S.S.M. to investigate the idea and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript. S.S.M. wrote the manuscript with support from A.A. and R.I.J.; A.A. and R.I.J. helped supervise the project.
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The four classes of building materials , .
The four classes of building materials , .
The four classes of building materials , .
The four classes of building materials , .
The proposed similarity measure S results.
The proposed similarity measure S results.