Distance and Similarity Measures for Spherical Fuzzy Sets and Their Applications in Selecting Mega Projects

A new condition on positive membership, neutral membership, and negative membership functions give us the successful extension of picture fuzzy set and Pythagorean fuzzy set and called spherical fuzzy sets (SFS). This extends the domain of positive membership, neutral membership, and negative membership functions. Keeping in mind the importance of similarity measure and application in data mining, medical diagnosis, decision making, and pattern recognition, several studies on similarity measures have been proposed in the literature. Some of those, however, cannot satisfy the axioms of similarity and provide counter-intuitive cases. In this paper, we proposed the set-theoretic similarity and distance measures. We provide some counterexamples for already proposed similarity measures in the literature and shows that how our proposed method is important and applicable to the pattern recognition problems. In the end, we provide an application of a proposed similarity measure for selecting mega projects in under developed countries.


Introduction
The membership function is used to define the fuzzy set (F S). The uncertainty model effectively by the fuzzy set theory define by Zadeh [1]. The fuzzy set theory only focuses on one aspect of information, the containment or belongingness. Attansove defines the intuitionistic fuzzy set (IF S) [2], which is the generalization of FS and model uncertainty effectively. The membership and non-membership functions are used to define IF S. Due to the consideration of non-membership function, the IF S is more effective than F S for practical applications. The membership functions Some of the proposed similarity measures for SF Ss have some problems which are pointed out in Section 4. To improve the idea of the similarity measure, we proposed the set-theoretic similarity and distance measures. The proposed similarity measure is then applied to the pattern recognition. The selection of mega projects for under developing countries is done by the proposed similarity measure.
The remaining paper is organized as follows: Introduction and preliminaries are presented in Sections 1 and 2. In Section 3, we proposed the set-theoretic similarity measures for SF Ss. In Section 4, we provide some counterexamples for already proposed similarity measures. To support the proposed similarity measure a numerical example of selecting mega projects in under developing countries is presented in Section 5. Comparison analysis and conclusion are presented in Sections 6 and 7.

Preliminaries
In this section, we provide some basic definitions of F S, IF S, P F S, and SF S. The already proposed similarity measures for SF S are discussed.
A fuzzy set is defined by Zadeh [1], which handles uncertainty based on the view of gradualness effectively. Definition 1. [1] A membership function ξÂ :Ŷ → [0, 1] defines the fuzzy setÂ over theŶ, where ξÂ(y) particularized the membership of an element y ∈Ŷ in fuzzy setÂ.
In [10], Cuong defines the P F S, which is an extension of a fuzzy set and applicable in many real-life problems. The picture fuzzy set is obtained by adding an extra membership function, namely, the degree of the neutral membership in IF S. The information regarding the situation of type: yes, abstain, no and refusal can be model by using picture fuzzy set easily. Voting can be a good example of a picture fuzzy set because it involves the situation of more answers of the type: yes, abstain, no, refusal.  1] are the degree of positive membership, neutral membership and degree of negative membership, respectively, such that 0 ≤ ξÂ(y) + ηÂ(y) + υÂ(y) ≤ 1.
In [29], Rafiq defines some similarity measures for SF Ss based on cosine and cotangent functions. Definition 4. [29] For two SF SsÂ andB inŶ, a cosine similarity measure betweenÂ andB is defined as follows: .
(2) Definition 5. [29] For two SF SsÂ andB inŶ, similarity measures using cosine function betweenÂ andB are defined as follows: where ∨ is the maximum operation. Definition 6. [29] For two SF SsÂ andB inŶ, a cotangent similarity measure betweenÂ andB are defined as follows: where ∨ is the maximum operation. Definition 7. [29] For two SF SsÂ andB inŶ, a cosine similarity measure by using degree of refusal membership betweenÂ andB are defined as follows: where ∨ is the maximum operation. Definition 8. [29] For two SF SsÂ andB inŶ, a cotangent similarity measure by using degree of refusal membership betweenÂ andB is defined as follows: where ∨ is the maximum operation.

A New Similarity Measures for SF Ss
In this section, we define new similarity and distance measures for SF Ss give their proof.

Definition 10.
A similarity measure between SF SsÂ andB is a mappingŜ : SFS × SFS → [0, 1], which satisfies the following properties: Definition 11. For two SF SsÂ andB inŶ, a new similarity measures is defined betweenÂ andB as follows: Example 1. LetŶ = {y 1 , y 2 , y 3 , y 4 , y 5 } be the universal set. We consider two SF SsÂ andB inŶ, which are given as follows: Proof. To prove S s a similarity measure, we have to verify the four conditions of Definition 10 for S s .
(S1). Since for all Therefore for all y j , By hypothesis it follows that r + s + t = 0. This implies that r = −(s + t), which is not possible.

Corollary 1. α is reflexive and symmetric.
Proof. The reflexive and symmetric part follows from Theorem 1.
The following example shows that the relation α is not transitive. Sometimes the alternatives under observations are not of equal importance, therefore, we defines weights of alternatives to signify their importance and defines weighted similarity measures between SF Ss. Definition 13. For two SF SsÂ andB inŶ, a new weighted similarity measure is defined betweenÂ andB as follows: where ω j ∈ [0, 1] are the weights of alternatives, but not all zero, Theorem 2. S s ω (Â,B) is the similarity measure between two SF SsÂ andB inŶ.
Proof. The proof is similar to the proof of Theorem 1.
On the basis of new similarity measure S s , we define distance measures for SF Ss.

Definition 14.
For two SF SsÂ andB inŶ, a new distance measures is defined betweenÂ andB as follows: Definition 15. For two SF SsÂ andB inŶ, a new weighted distance measure is defined betweenÂ andB as follows: where ω j ∈ [0, 1] are the weights of alternatives, but not all zero, Theorem 3. D s and D s ω are the distance measures between SF Ss.
Proof. The proof is similar to the proof of Theorem 1.

Applications in Pattern Recognition and Counter Examples
In this section, we provide some counter examples for already proposed similarity measures in the literature in pattern recognition. We have seen that the already proposed measures cannot classify the unknown pattern while set theoretic similarity measure classify the unknown pattern, which shows that our proposed similarity measure is applicable in pattern recognition problems.

Example 4.
In this example, we have seen that the second condition of Definition 10 (S 2 ) is not satisfied for cosine similarity measure S 1 c (Definition 4), i.e., ifÂ = {(a, a, a)/y j |y j ∈Ŷ, 1 ≤ j ≤ m} and Clearly,Â =B but S 1 c (Â,B) = 1. Hence S 1 c is not effective for these cases and not reliable to find the similarity measure between SF Ss. However, when we find the similarity measure by using S s , we get S s (Â,B) = 0.686175.

Example 5.
Let Q 1 and Q 2 be two known patterns with class labels Z 1 and Z 2 , respectively, are given. The SF Ss are used to represents the patterns inŶ = {y 1 , y 2 , y 3 } as follows: Our aim is to find out the class of unknown pattern P belongs to. However, when we use cosine similarity measure S 3 c (Definition 5), we get the same similarity measure i.e., S 3 c (P, Q 1 ) = S 3 c (P, Q 2 ) = 0.965086. Furthermore, when we use the cotangent similarity measure S 5 c (Definition 6), we get the same similarity measures i.e., S 5 c (P, Q 1 ) = S 5 c (P, Q 2 ) = 0.767857. Hence in this case we cannot decide the class of unknown pattern P by using S 3 c and S 5 c . However, when we find the similarity measure by using S s , we get S s (P, Q 1 ) = 0.555916 and S s (P, Q 2 ) = 0.575836. Since S s (P, Q 2 ) > S s (P, Q 1 ), therefore, the unknown pattern P belongs to class Z 2 . Example 6. Let Q 1 and Q 2 be two known patterns with class labels Z 1 and Z 2 , respectively, are given. The SF Ss are used to represents the patterns inŶ = {y 1 , y 2 , y 3 } as follows: Our aim is to find out the class of unknown pattern P belongs to. However, when we use cosine similarity measure S 2 c (Definition 5), we get the same similarity measure i.e., S 2 c (P, Q 1 ) = S 2 c (P, Q 2 ) = 0.888668. Furthermore, when we use the cotangent similarity measure S 4 c (Definition 6), we get the same similarity measures i.e., S 4 c (P, Q 1 ) = S 4 c (P, Q 2 ) = 0.638144. Hence in this case we cannot decide the class of unknown pattern P by using S 2 c and S 4 c . However, when we find the similarity measure by using S s , we get S s (P, Q 1 ) = 0.50385 and S s (P, Q 2 ) = 0.564666. Since S s (P, Q 2 ) > S s (P, Q 1 ), therefore, the unknown pattern P belongs to class Z 2 . Example 7. Let Q 1 and Q 2 be two known patterns with class labels Z 1 and Z 2 , respectively, are given. The SF Ss are used to represents the patterns inŶ = {y 1 , y 2 , y 3 } as follows: Our aim is to find out the class of unknown pattern P belongs to. However, when we use cosine similarity measure S 6 c (Definition 7), we get the same similarity measure i.e., S 6 c (P, Q 1 ) = S 6 c (P, Q 2 ) = 0.889689. Furthermore, when we use the cotangent similarity measure S 8 c (Definition 8), we get the same similarity measures i.e., S 8 c (P, Q 1 ) = S 8 c (P, Q 2 ) = 0.642526. Hence in this case we cannot decide the class of unknown pattern P by using S 6 c and S 8 c . However, when we find the similarity measure by using S s , we get S s (P, Q 1 ) = 0.492114 and S s (P, Q 2 ) = 0.58562. Since S s (P, Q 2 ) > S s (P, Q 1 ), therefore, the unknown pattern P belongs to class Z 2 . Example 8. Let Q 1 and Q 2 be two known patterns with class labels Z 1 and Z 2 , respectively, are given. The SF Ss are used to represents the patterns inŶ = {y 1 , y 2 , y 3 } as follows: Our aim is to find out the class of unknown pattern P belongs to. However, when we use cosine similarity measure S 7 c (Definition 7), we get the same similarity measure i.e., S 7 c (P, Q 1 ) = S 7 c (P, Q 2 ) = 0.874075. Furthermore, when we use the cotangent similarity measure S 9 c (Definition 8), we get the same similarity measures i.e., S 9 c (P, Q 1 ) = S 9 c (P, Q 2 ) = 0.597149. Hence in this case we cannot decide the class of unknown pattern P by using S 7 c and S 9 c . However, when we find the similarity measure by using S s , we get S s (P, Q 1 ) = 0.497164 and S s (P, Q 2 ) = 0.549396. Since S s (P, Q 2 ) > S s (P, Q 1 ), therefore, the unknown pattern P belongs to class Z 2 .

Selection of Mega Projects in Underdeveloping Countries
The megaprojects are characterized by vast complexity (especially in organizational terms), large investment commitment, long-lasting impact on the economy, the environment, and society. So it is important to choose the best method for the selection of mega projects for under developing countries. Because it affects the lives of millions of peoples, take much time to develop and build, involve multiple public and private stakeholders, and have a long-lasting impact on the economy, the environment, and society. As we have seen that the proposed similarity measures have counter-intuitive cases for Examples 4-8. Therefore, the selection of mega projects for under developing countries is done by the proposed similarity measure.
It is important for under developing countries to select upcoming mega projects on priority which has less effect on their economy, environment, less maintenance cost has long term benefits, fewer peoples effects from that project and generate high revenue. For example, a country has to start the mega project and they get a loan from the world bank so the country has to think before spending the money because they have to refund after some time. The government has five projects in his focus like 1 million house construction, dam construction, orange metro train, invest in industry and power sector. This set can be represented as U and the elements of U represented as E i , 1 ≤ i ≤ 5, that is U = {1 million house construction, dam construction, orange metro train, invest in industry, power sector}.
To select the project on a priority basis, there are some parameters selected by experts from different fields to check the importance of projects like long term benefits, time, impact, revenue generated, costs and short term benefits. We represents this criteria as a set W and the elements of W represented as e j , 1 ≤ j ≤ 6, that is W = {long term benefits, time, impact, revenue generated, cost, short term benefits}.
We apply proposed technique for selecting upcoming mega projects on priority basis which is the classical multi attribute decision making problem. The weight vector for each attribute e j , j ∈ {1, 2, ..., 6} isω = (0.12, 0.25, 0.09, 0.16, 0.20, 0.18) T . All the data collected in spherical fuzzy information is summarized in Table 1. In Table 1, we have seen that for each mega project E i , i ∈ {1, 2, ..., 5}, experts interpret their evaluation in the form of SFVs corresponding to each attribute (criteria).
To apply the proposed method, we calculate the ideal alternative (mega project) E + from given data as follows Then the similarity measures between each alternative and ideal alternative are calculated. Heigh values of similarity measure more closer to the ideal alternative. In this case, the ideal alternative is Then the similarity measures S s between between each alternative E i and ideal alternative E + are calculated. The details of similarity measures presented in Table 2 and the ranking of alternatives (mega projects) is given as follows: The comparison between the already proposed similarity measures and proposed similarity measure is presented in Table 2.

Comparison Analysis
A comparison between new proposed similarity measure and already proposed similarity measure for SF Ss is conducted to illustrate the superiority of the new similarity measure.
We have seen from Example 4 that the second condition of Definition 10 (S 2 ) is not satisfied for cosine similarity measure S 1 c , i.e.,Ŝ (Â,B) = 1 evenÂ =B. Furthermore, we provide a general criteria when second condition of Definition 10 (S 2 ) is not satisfied for cosine similarity measure S 1 c . In Example 5, we have seen that the S 3 c and S 5 c can not classify the unknown pattern from known pattern. In Example 6, we have seen that the S 2 c and S 4 c can not classify the unknown pattern from known pattern. In Example 7, we have seen that the S 6 c and S 8 c can not classify the unknown pattern from known pattern. In Example 8, we have seen that the S 7 c and S 9 c can not classify the unknown pattern from known pattern.
However, in all Examples 4-8, the new similarity measure S s classify the unknown pattern and hence successfully applicable to the pattern recognition problems. In Section 5, S s applied successfully to selecting the mega projects for under developing countries.
From Table 3, we have seen that for different special cases, the already proposed similarity measures are not illegible for classification of unknown pattern but S s applied successfully. For cases 1 and 2, the similarity measures S 3 c , S 5 c , S 7 c and S 9 c provide counter-intuitive cases. The similarity measures S 2 c , S 4 c , S 6 c and S 8 c provide counter-intuitive cases for 3 and 4 cases. The second axiom of similarity measure for S 1 c (Definition 4 ) is not satisfied for case 5. As we have seen in Example 4, that if we have membership, neutral and non-membership degrees for a set are equal but different from another set which has also same membership degrees, then the S 1 c has result 1. This is inconsistent with the definition of a similarity measure.

Conclusions
In this paper, we have defined new similarity measures for SF Ss called set theoretic similarity measures. We define set theoretic similarity measure, weighted set theoretic similarity measure, set theoretic distance and weighted set theoretic distance measures and provide their proofs in this paper. We discuss some special cases (Examples 4-8) where already proposed similarity measure fails to classify the unknown pattern while the proposed similarity measure successfully applied to the pattern recognition problems. Furthermore, S s applied successfully to selecting the mega projects for under developing countries.
In the future direction, we will apply the set theoretic similarity measure to data mining, medical diagnosis, decision making, complex group decision making, linguistic summarization risk analysis, pattern recognition, color image retrieval, histogram comparison and image processing.
Author Contributions: All authors contributed equally in this research paper. All authors have read and agreed to the published version of the manuscript.