INFORMATICA Informatica 1822-88440868-49520868-4952 Vilnius University INFO1150 10.15388/Informatica.2017.144 Research Article Some Cosine Similarity Measures for Picture Fuzzy Sets and Their Applications to Strategic Decision Making WeiGuiwuweiguiwu@163.com

G.W. Wei has an MSc and a PhD degree in applied mathematics from SouthWest Petroleum University, Business Administration from school of Economics and Management at SouthWest Jiaotong University, China, respectively. From May 2010 to April 2012, he was a postdoctoral researcher with the School of Economics and Management, Tsinghua University, Beijing, China. He is a professor in the School of Business at Sichuan Normal University. He has published more than 90 papers in journals, books and conference proceedings including journals such as Omega, Decision Support Systems, Expert Systems with Applications, Applied Soft Computing, Knowledge and Information Systems, Computers & Industrial Engineering, Knowledge-Based Systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, International Journal of Computational Intelligence Systems and Information: An International Interdisciplinary Journal. He has published 1 book. He has participated in several scientific committees and serves as a reviewer in a wide range of journals including Computers & Industrial Engineering, International Journal of Information Technology and Decision Making, Knowledge-Based Systems, Information Sciences, International Journal of Computational Intelligence Systems and European Journal of Operational Research. He is currently interested in aggregation operators, decision making and computing with words.

 School of Business, Sichuan Normal University, Chengdu, 610101, China 201711201728354756410201632017 © 2017 Vilnius University2017 Open access article under the CC BY license.

In this paper, we presented another form of eight similarity measures between PFSs based on the cosine function between PFSs by considering the degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership in PFSs. Then, we applied these weighted cosine function similarity measures between PFSs to strategic decision making. Finally, an illustrative example for selecting the optimal production strategy is given to demonstrate the efficiency of the similarity measures for strategic decision making problem.

 strategic decision making; picture fuzzy set cosine function cosine similarity measure optimal production strategy Introduction

The similarity measures are important and useful tools for determining the degree of similarity between two objects. Measures of similarity between fuzzy sets have gained attention from researchers for their wide applications in various fields, such as pattern recognition, machine learning, decision making and image processing, many measures of similarity between fuzzy sets have been proposed and researched in recent years (see, Bustince et al., 2006, 2007, 2008; Lee et al., 2009). Fuzzy set theory, introduced by Zadeh (1965), has been widely used to model uncertainty present in real-world applications. Atanassov (1986) extended fuzzy sets to Atanassov’s intuitionistic fuzzy sets (IFSs), many different similarity measures between IFSs have been investigated in Li et al. (2007). Li and Cheng (2002) proposed a suitable similarity measure between IFSs and applied it to pattern recognition problems. Liang and Shi (2003) defined some similarity measures to differentiate different IFSs and discussed the relationships between them. Furthermore, Mitchell (2003) modified Li and Cheng’s measures. Based on the extension of the Hamming distance on fuzzy sets, Szmidt and Kacprzyk (2000) developed a similarity measure between IFSs based on the Hamming distance. Hung and Yang (2004) calculated the distance between IFSs based on the Hausdorff distance and generated some similarity measures between IFSs. Liu (2005) developed some new similarity measures between IFSs and between elements. Hung and Yang (2007) proposed a similarity measure between IFSs based on the Lp metric. Xu and Xia (2010) defined the geometric distance and similarity measures of IFSs for group decision making problems. Ye (2011) proposed the cosine similarity measure between IFSs. Hung (2012) developed the likelihood-based measurement of IFSs for the medical diagnosis and bacteria classification problems. Shi and Ye (2013) further improved the cosine similarity measure of IFSs. Tian (2013) proposed the cotangent similarity measure between IFSs for medical diagnosis. Rajarajeswari and Uma (2013) further introduced the cotangent similarity measure which considers membership, nonmembership and hesitation degrees in IFSs. Furthermore, Szmidt (2014) discussed distances between IFSs and introduced a family of similarity measures which considered the membership, nonmembership and hesitation degrees described in IFSs. Ye (2016) proposed two new cosine similarity measures and weighted cosine similarity measures based on cosine function and the information carried by the membership degrees, nonmembership degree and hesitancy degree in intuitionistic fuzzy sets (IFSs). Son and Phong (2016) gave the intuitionistic vector similarity measures for medical diagnosis.

Recently, Cuong (2014) proposed picture fuzzy set (PFS) and investigated some basic operations and properties of PFS. The picture fuzzy set is characterized by three functions expressing the degree of membership, the degree of neutral membership and the degree of nonmembership. The only constraint is that the sum of the three degrees must not exceed 1. Basically, PFS based models can be applied to situations requiring human opinions involving more answers of types: yes, abstain, no, refusal, which can’t be accurately expressed in the traditional FS and IFS. Until now, some progress has been made in the research of the PFS theory. Singh (2014) investigated the correlation coefficients for picture fuzzy set and applied the correlation coefficient to clustering analysis with picture fuzzy information. Son etc. introduced several novel fuzzy clustering algorithms on the basis of picture fuzzy sets and applications to time series forecasting and weather forecasting (see, Son, 2015; Thong and Son, 2015). Thong (2015) developed a novel hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis and application to health care support systems. Wei (2016b) proposed picture fuzzy cross-entropy model for multiple attribute decision making problems.

Although Atanassov’s intuitionistic fuzzy set theory and similarity measures have been successfully applied in different areas (see Tang et al., 2017; Wei, 2015, 2008, 2009, 2010a, 2010b, 2011a; Wei et al., 2011, 2013a, 2013b; Wei and Zhao, 2012b; Zhao and Wei, 2013; Zhao et al., 2014), but there are situations in real life which can’t be represented by Atanassov’s intuitionistic fuzzy sets. Voting can be a good example of such situation as the human voters may be divided into four groups of those who: vote for, abstain, refuse to vote. Basically, picture fuzzy sets (see Cuong, 2014) based models may be adequate in situations when we face human opinions involving more answers of the type: yes, abstain, no, refusal. Therefore, in order to deal with these types of situations, in this paper we introduce the concept of similarity measures for picture fuzzy sets based on the cosine functions, which is a new extension of the similarity measure of IFSs based on the cosine functions. In order to do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to intuitionistic fuzzy set and some similarity measure between IFSs and picture fuzzy sets. In Section 3, we shall propose some similarity measure and some weighted similarity measure between PFSs based on the concept of the cosine function. In Section 4, the similarity measures for PFSs are applied to strategic decision making problem for selecting the optimal production strategy. Section 5 concludes the paper with some remarks.

 Preliminaries

In the following, we introduce some basic concepts related to intuitionistic fuzzy sets and some similarity measure between IFSs.

(<italic>See</italic> Atanassov, <xref ref-type="bibr" rid="j_info1150_ref_001">1986</xref><italic>,</italic> <xref ref-type="bibr" rid="j_info1150_ref_002">1989</xref>).

An IFS is given by A = { ⟨ x , μ A ( x ) , ν A ( x ) ⟩ | x ∈ X } , where μ A : X → [ 0 , 1 ] and ν A : X → [ 0 , 1 ], where, 0 ⩽ μ A ( x ) + ν A ( x ) ⩽ 1, ∀ x ∈ X. The numbers μ A ( x ) and ν A ( x ) represent, respectively, the membership degree and non- membership degree of the element x to the set A.

 (<italic>See</italic> Atanassov, <xref ref-type="bibr" rid="j_info1150_ref_002">1989</xref>).

For each IFS A in X, if π A ( x ) = 1 − μ A ( x ) − ν A ( x ) , ∀ x ∈ X . Then π A ( x ) is called the degree of indeterminacy of x to A.

Suppose that there are two IFSs: A = { ⟨ x j , μ A ( x j ) , ν A ( x j ) ⟩ | x j ∈ X } and B = { ⟨ x j , μ B ( x j ) , ν B ( x j ) ⟩ | x j ∈ X } in the universe of discourse X = { x 1 , x 2 , … , x n }.

Ye (2011) proposed the cosine similarity measure between IFSs and as following: IFC 1 ( A , B ) = 1 n ∑ j = 1 n μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) μ A 2 ( x j ) + ν A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) . Shi and Ye (2013) further presented the cosine similarity measure by considering membership degree, nonmembership degree and hesitancy degree in IFSs as the vector space of the three terms: IFC 2 ( A , B ) = 1 n ∑ j = 1 n μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) + π A ( x j ) π B ( x j ) μ A 2 ( x j ) + ν A 2 ( x j ) + π A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) + π B 2 ( x j ) . Based on cosine function, Ye (2016) proposed two cosine similarity measures between IFSs A and B. IFCS 1 ( A , B ) = 1 n ∑ j = 1 n cos { π 2 [ | μ A ( x j ) − μ B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | π A ( x j ) − π B ( x j ) | ] } , IFCS 1 ( A , B ) = 1 n ∑ j = 1 n cos { π 4 [ | μ A ( x j ) − μ B ( x j ) | + | ν A ( x j ) − ν B ( x j ) | + | π A ( x j ) − π B ( x j ) | ] } .

On the other hand, Tian (2013) proposed a cotangent similarity measure between IFSs and as following: IFCT 1 ( A , B ) = 1 n ∑ j = 1 n cot [ π 4 + π 4 ( | μ A ( x j ) − μ B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ) ] , where the symbol “∨” is the maximum operation. When the three terms like membership degree, nonmembership degree and hesitancy degree are considered in IFSs, Rajarajeswari and Uma (2013) defined the cotangent similarity measure of IFSs: IFCT 2 ( A , B ) = 1 n ∑ j = 1 n cot [ π 4 + π 4 ( | μ A ( x j ) − μ B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | π A ( x j ) − π B ( x j ) | ) ] .

In the following, we introduced the weighted cosine and cotangent similarity measures between IFSs and, respectively (see Ye, 2011; Shi and Ye, 2013; Rajarajeswari and Uma, 2013; Ye, 2016): IFC 1 ( A , B ) = ∑ j = 1 n ω j μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) μ A 2 ( x j ) + ν A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) , IFC 2 ( A , B ) = ∑ j = 1 n ω j μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) + π A ( x j ) π B ( x j ) μ A 2 ( x j ) + ν A 2 ( x j ) + π A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) + π B 2 ( x j ) , WIFCS 1 ( A , B ) = ∑ j = 1 n ω j cos { π 2 [ | μ A ( x j ) − μ B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | π A ( x j ) − π B ( x j ) | ] } , WIFCS 2 ( A , B ) = ∑ j = 1 n ω j cos { π 4 [ | μ A ( x j ) − μ B ( x j ) | + | ν A ( x j ) − ν B ( x j ) | + | π A ( x j ) − π B ( x j ) | ] } , WIFCT 1 ( A , B ) = ∑ j = 1 n ω j cot [ π 4 + π 4 ( | μ A ( x j ) − μ B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ) ] , WIFCT 2 ( A , B ) = ∑ j = 1 n ω j cot [ π 4 + π 4 ( | μ A ( x j ) − μ B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | π A ( x j ) − π B ( x j ) | ) ] , where ω j ( j = 1 , 2 , … , n ) is the weight of an element x j , ω j ∈ [ 0 , 1 ] and ∑ j = 1 n = 1 and the symbol “∨” is the maximum operation.

 Some Similarity Measure Based on Cosine Function for Picture Fuzzy Sets

Although Atanassov’s intuitionistic fuzzy set theory (see Atanassov, 1986, 1989) has been successfully applied in different areas, there are situations in real life which can’t be represented by Atanassov’s intuitionistic fuzzy sets. Picture fuzzy sets are extension of Atanassov’s intuitionistic fuzzy sets. Picture fuzzy set (see Cuong, 2014) based models may be adequate in situations when we face human opinions involving more answers of types: yes, abstain, no, refusal. It can be considered as a powerful tool to represent the uncertain information in the process of patterns recognition and cluster analysis. (<italic>See</italic> Cuong, <xref ref-type="bibr" rid="j_info1150_ref_008">2014</xref>).

A picture fuzzy set (PFS) A on the universes an object of the form A = { ⟨ x , μ A ( x ) , η A ( x ) , ν A ( x ) ⟩ | x ∈ X } , where μ A ( x ) ∈ [ 0 , 1 ] is called the “degree of positive membership of A”, η A ( x ) is called the “degree of neutral membership of A” and μ A ( x ) is called the “degree of negative membership of A”, and μ A ( x ) , η A ( x ) , ν A ( x ) satisfy the following condition: 0 ⩽ μ A ( x ) + η A ( x ) + ν A ( x ) ⩽ 1 , ∀ x ∈ X . Then for x ∈ X, ρ A ( x ) = 1 − ( μ A ( x ) + η A ( x ) + ν A ( x ) ) could be called the degree of refusal membership of x in A.

Cosine Similarity Measure for Picture Fuzzy Sets

Let A be a PFS in an universe of discourse X = { x }, the PFS is characterized by the degree of positive membership μ A ( x ), the degree of neutral membership η A ( x ) and the degree of negative membership ν A ( x ) which can be considered as a vector representation with the three elements. Therefore, a cosine similarity measure and a weighted cosine similarity measure for PFSs are proposed in an analogous manner to the cosine similarity measure based on Bhattacharya’s distance (see Salton and Mcgill, 1983; Bhattacharya, 1946) and cosine similarity measure for intuitionistic fuzzy set (see Ye, 2011).

Suppose that there are two PFSs: A = { ⟨ x j , μ A ( x j ) , η A ( x j ) , ν A ( x j ) ⟩ | x j ∈ X } and B = { ⟨ x j , μ B ( x j ) , η B ( x j ) , ν B ( x j ) ⟩ | x j ∈ X } in the universe of discourse X = { x 1 , x 2 , … , x n }.

A cosine similarity measure between PIFSs and is proposed as follows: PFC 1 ( A , B ) = 1 n ∑ j = 1 n μ A ( x j ) μ B ( x j ) + η A ( x j ) η B ( x j ) + ν A ( x j ) ν B ( x j ) μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) . If we take n = 1, then the cosine similarity measure between PFSs A and B becomes the correlation coefficient between PFSs A and B, i.e. C PFS ( A , B ) = K PFS ( A , B ). Therefore, the cosine similarity measure between PFSs A and B also satisfies the following properties:

0 ⩽ PFC 1 ( A , B ) ⩽ 1;

PFC 1 ( A , B ) = PFC 1 ( B , A );

PFC 1 ( A , B ) = 1, if A = B, i = 1 , 2 , … , n.

It is obvious that the proposition is true according to the cosine value.

It is obvious that the proposition is true.

When A = B, there are μ A ( x j ) = μ B ( x j ), η A ( x j ) = η B ( x j ) and ν A ( x j ) = ν B ( x j ) for j = 1 , 2 , … , n. So C PFS 1 ( A , B ) = 1. Therefore, we have finished the proofs.

  □

In the following, we shall investigate the distance measure of the angle as d ( A , B ) = arccos ( C PFS 1 ( A , B ) ). It satisfies the following properties:

d ( A , B ) ⩾ 0, if 0 ⩽ C PFS ( A , B ) ⩽ 1;

d ( A , B ) = arccos ( 1 ) = 0, if C PFS ( A , A ) = 1;

d ( A , B ) = d ( B , A ), if C PFS ( A , B ) = C PFS ( B , A );

d ( A , C ) ⩽ d ( A , B ) + d ( B , C ), if A ⊆ B ⊆ C for any PFS C.

Obviously, d ( A , B ) satisfies the properties (1)–(3). In the following, d ( A , B ) will be proved to satisfy the property (4).

For any C = { ⟨ x j , μ C ( x j ) , η C ( x j ) , ν C ( x j ) ⟩ | x j ∈ X }, A ⊆ B ⊆ C, Since Eq. (16) is the sum of terms, let us investigate the distance measures of the angle between the vectors: d j ( A ( x j ) , B ( x j ) ) = arccos ( PFC i 1 ( A ( x i ) , B ( x i ) ) ) , d j ( B ( x j ) , C ( x j ) ) = arccos ( PFC i 1 ( B ( x i ) , C ( x i ) ) ) , d j ( A ( x j ) , C ( x j ) ) = arccos ( PFC i 1 ( A ( x i ) , C ( x i ) ) ) , for j = 1 , 2 , … , n, where PFC j 1 ( A , B ) = μ A ( x j ) μ B ( x j ) + η A ( x j ) η B ( x j ) + ν A ( x j ) ν B ( x j ) μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) , PFC j 1 ( B , C ) = μ B ( x j ) μ C ( x j ) + η B ( x j ) η C ( x j ) + ν B ( x j ) ν C ( x j ) μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) μ C 2 ( x j ) + η C 2 ( x j ) + ν C 2 ( x j ) , PFC j 1 ( A , C ) = μ A ( x j ) μ C ( x j ) + η A ( x j ) η C ( x j ) + ν A ( x j ) ν C ( x j ) μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) μ C 2 ( x j ) + η C 2 ( x j ) + ν C 2 ( x j ) .

For three vectors A ( x j ) = ⟨ μ A ( x j ) , η A ( x j ) , ν A ( x j ) ⟩ , B ( x j ) = ⟨ μ B ( x j ) , η B ( x j ) , ν B ( x j ) ⟩ , C ( x j ) = ⟨ μ C ( x j ) , η C ( x j ) , ν C ( x j ) ⟩ in one plane, if A ( x j ) ⊆ B ( x j ) ⊆ C ( x j ), j = 1 , 2 , … , n. Then, it is obvious that d j ( A ( x j ) , C ( x j ) ) ⩽ d j ( A ( x j ) , B ( x j ) ) + d j ( B ( x j ) , C ( x j ) ) , according to the triangle inequality. Combining the inequality with Eq. (16), we can obtain d ( A , C ) ⩽ d ( A , B ) + d ( B , C ) . Thus d ( A , B ) satisfies the property (4). So we finished the proof.  □

If we consider the weights of x j , a weighted cosine similarity measure between PFSs A and B is proposed as follows: WPFC 1 ( A , B ) = ∑ j = 1 n ω j μ A ( x j ) μ B ( x j ) + η A ( x j ) η B ( x j ) + ν A ( x j ) ν B ( x j ) μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) , where ω = ( ω 1 , ω 1 , … , ω n ) T is the weight vector of x j ( j = 1 , 2 , … , n ), with ω j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n ω j = 1. In particular, if ω = ( 1 / n , 1 / n , … , 1 / n ) T , then the weighted cosine similarity measure reduces to cosine similarity measure. That’s to say, if we take ω i = 1 / n, i = 1 , 2 , … , n, then there is WPFC 1 ( A , B ) = PFC 1 ( A , B ).

Obviously, the weighted cosine similarity measure of two PFSs A and B also satisfies the following properties:

0 ⩽ WPFC 1 ( A , B ) ⩽ 1;

WPFC 1 ( A , B ) = WPFC 1 ( B , A );

WPFC 1 ( A , B ) = 1, if A = B, i = 1 , 2 , … , n.

Similar to the previous proof method, we can prove the above three properties.

When the four terms like degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership are considered in PFSs, we further propose the cosine similarity measure and weighted cosine similarity measure between PFSs as follows: PFC 2 ( A , B ) = 1 n ∑ j = 1 n μ A ( x j ) μ B ( x j ) + η A ( x j ) η B ( x j ) + ν A ( x j ) ν B ( x j ) + ρ A ( x j ) ρ B ( x j ) μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) + ρ B 2 ( x j ) , WPFC 2 ( A , B ) = ∑ j = 1 n ω j μ A ( x j ) μ B ( x j ) + η A ( x j ) η B ( x j ) + ν A ( x j ) ν B ( x j ) + ρ A ( x j ) ρ B ( x j ) μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) + ρ B 2 ( x j ) , where ω = ( ω 1 , ω 1 , … , ω n ) T is the weight vector of x i ( i = 1 , 2 , … , n ), with ω j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n ω j = 1.

 Similarity Measures of Picture Fuzzy Sets Based on Cosine Function

Based on the cosine function, in this section, we shall propose two cosine similarity measures between PFSs and analyse their properties.

Let A = { ⟨ x j , ( μ A ( x j ) , η A ( x j ) , ν A ( x j ) ) ⟩ | x j ∈ X } and B = { ⟨ x j , ( μ B ( x j ) , η B ( x j ) , ν B ( x j ) ) ⟩ | x j ∈ X } be any two PFSs in X = { x 1 , x 2 , … , x n }. Then, we shall define four cosine similarity measures between PFSs, respectively, as follows: PFCS 1 ( A , B ) = 1 n ∑ j = 1 n cos { π 2 [ | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ] } , PFCS 2 ( A , B ) = 1 n ∑ j = 1 n cos { π 4 [ | μ A ( x j ) − μ B ( x j ) | + | η A ( x j ) − η B ( x j ) | + | ν A ( x j ) − ν B ( x j ) | ] } , where the symbol “∨” is the maximum operation.

When the four terms like degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership are considered in PFSs, we further propose two cosine similarity measures between PFSs as follows: PFCS 3 ( A , B ) = 1 n ∑ j = 1 n cos { π 2 | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | ρ A ( x j ) − ρ B ( x j ) | } , PFCS 4 ( A , B ) = 1 n ∑ j = 1 n cos { π 4 | μ A ( x j ) − μ B ( x j ) | + | η A ( x j ) − η B ( x j ) | + | ν A ( x j ) − ν B ( x j ) | + | ρ A ( x j ) − ρ B ( x j ) | } .

For two PFSs A and B in X = { x 1 , x 2 , … , x n }, the cosine similarity measures PFCS k ( A , B ) , k = 1 , 2 , 3 , 4 , should satisfy the following properties (1)–(4):

0 ⩽ PFCS k ( A , B ) ⩽ 1;

PFCS k ( A , B ) = 1 if and only if A = B;

PFCS k ( A , B ) = PFCS k ( B , A );

If C is a PFS in X and A ⊆ B ⊆ C, then PFCS k ( A , C ) ⩽ PFCS k ( A , B ) and PFCS k ( A , C ) ⩽ PFCS k ( B , C ) .

(1) Since the value of the cosine function is within [ 0 , 1 ], the similarity measure based on the cosine function is also within [ 0 , 1 ]. Thus, there is 0 ⩽ PFCS k ( A , B ) ⩽ 1.

(2) For two PFSs A and B in X = { x 1 , x 2 , … , x n }, if A = B, then μ A ( x j ) = μ B ( x j ), η A ( x j ) = η B ( x j ), ν A ( x j ) = ν B ( x j ), ρ A ( x j ) = ρ B ( x j ) for j = 1 , 2 , … , n. Thus, | μ A ( x j ) − μ B ( x j ) | = 0, | η A ( x j ) − η B ( x j ) | = 0, | ν A ( x j ) − ν B ( x j ) | = 0, | ρ A ( x j ) − ρ B ( x j ) | = 0. So, PFCS k ( A , B ) = 1, k = 1 , 2 , 3 , 4.

If PFCS k ( A , B ) = 1, k = 1 , 2 , 3 , 4, this implies | μ A ( x j ) − μ B ( x j ) | = 0 , | η A ( x j ) − η B ( x j ) | = 0 , | ν A ( x j ) − ν B ( x j ) | = 0 , | ρ A ( x j ) − ρ B ( x j ) | = 0 , for j = 1 , 2 , 3 , 4. Since cos ( 0 ) = 1. Then, there are μ A ( x j ) = μ B ( x j ) , η A ( x j ) = η B ( x j ) , ν A ( x j ) = ν B ( x j ) , ρ A ( x j ) = ρ B ( x j ) , for j = 1 , 2 , 3 , 4. Hence A = B.

(3) Proof is straightforward.

(4) If A ⊆ B ⊆ C, then there are μ A ( x j ) ⩽ μ B ( x j ) ⩽ μ C ( x j ) , η A ( x j ) ⩽ η B ( x j ) ⩽ η C ( x j ) , ν A ( x j ) ⩾ ν B ( x j ) ⩾ ν C ( x j ) , for j = 1 , 2 , … , n. Then, we have | μ A ( x j ) − μ B ( x j ) | ⩽ | μ A ( x j ) − μ C ( x j ) | , | μ B ( x j ) − μ C ( x j ) | ⩽ | μ A ( x j ) − μ C ( x j ) | , | η A ( x j ) − η B ( x j ) | ⩽ | η A ( x j ) − η C ( x j ) | , | η B ( x j ) − η C ( x j ) | ⩽ | η A ( x j ) − η C ( x j ) | , | ν A ( x j ) − ν B ( x j ) | ⩽ | ν A ( x j ) − ν C ( x j ) | , | ν B ( x j ) − ν B ( x j ) | ⩽ | ν A ( x j ) − ν C ( x j ) | , | ρ A ( x j ) − ρ B ( x j ) | ⩽ | ρ A ( x j ) − ρ C ( x j ) | , | ρ B ( x j ) − ρ C ( x j ) | ⩽ | ρ A ( x j ) − ρ C ( x j ) | . Hence, PFCS k ( A , C ) ⩽ PFCS k ( A , B ) and PFCS k ( A , C ) ⩽ PFCS k ( B , C ) for k = 1 , 2 , 3 , 4 as the cosine function is a decreasing function with the interval [ 0 , π / 2 ]. Thus, the proofs of these properties are completed.  □

In many situations, the weight of the elements x j ∈ X should be taken into account. For example, in multiple attribute decision making, the considered attributes usually have different importance, and thus need to be assigned different weights. As a result, four weighted cosine similarity measure between PFSs A and B is proposed as follows: WPFCS 1 ( A , B ) = ∑ j = 1 n ω j cos { π 2 [ | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ] } , WPFCS 2 ( A , B ) = ∑ j = 1 n ω j cos { π 4 [ | μ A ( x j ) − μ B ( x j ) | + | η A ( x j ) − η B ( x j ) | + | ν A ( x j ) − ν B ( x j ) | ] } , WPFCS 3 ( A , B ) = ∑ j = 1 n ω j cos { π 2 | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | ρ A ( x j ) − ρ B ( x j ) | } , WPFCS 4 ( A , B ) = ∑ j = 1 n ω j cos { π 4 | μ A ( x j ) − μ B ( x j ) | + | η A ( x j ) − η B ( x j ) | + | ν A ( x j ) − ν B ( x j ) | + | ρ A ( x j ) − ρ B ( x j ) | } , where ω = ( ω 1 , ω 1 , … , ω n ) T is the weight vector of x i , i = 1 , 2 , … , n, with ω j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n ω j = 1 and the symbol “∨” is the maximum operation. In particular, if ω = ( 1 / n , 1 / n , … , 1 / n ) T , then the weighted cosine similarity measure reduces to cosine similarity measure. That’s to say, if we take ω j = 1 / n, j = 1 , 2 , … , n, then there is WPFCS k ( A , B ) = PFCS k ( B , A ), k = 1 , 2 , 3 , 4. Obviously, the weighted cosine similarity measures also satisfy the axiomatic requirements of similarity measures in Proposition 2.

For two PFSs A and B in X = { x 1 , x 2 , … , x n }, the weighted cosine similarity measures WPFCS k ( A , B ), k = 1 , 2 , 3 , 4, satisfy the following properties (1)–(4):

0 ⩽ WPFCS k ( A , B ) ⩽ 1;

WPFCS k ( A , B ) = 1 if and only if A = B;

WPFCS k ( A , B ) = WPFCS k ( B , A );

If C is a PFS in X and A ⊆ B ⊆ C, then WPFCS k ( A , C ) ⩽ WPFCS k ( A , B ) and WPFCS k ( A , C ) ⩽ WPFCS k ( B , C ) .

By using similar proof in Proposition 1, we can give the proofs of these properties (1)–(4).

 Similarity Measures of Picture Fuzzy Sets Based on Cotangent Function

In this section, we shall propose a cotangent similarity measures between PFSs as follows: PFCT 1 ( A , B ) = 1 n ∑ j = 1 n cot [ π 4 + π 4 ( | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ) ] , where the symbol “∨” is the maximum operation. When the four terms like degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership are considered in PFSs, we further propose a cotangent similarity measures between PFSs as follows: PFCT 2 ( A , B ) = 1 n ∑ j = 1 n cot [ π 4 + π 4 | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | ρ A ( x j ) − ρ B ( x j ) | ] . In many situations, the weight of the elements x i ∈ X should be taken into account. For example, in multiple attribute decision making, the considered attributes usually have different importance, and thus need to be assigned different weights. As a result, four weighted cotangent similarity measure between PFSs A and B is proposed as follows: WPFCT 1 ( A , B ) = ∑ j = 1 n ω j cot [ π 4 + π 4 ( | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ) ] , WPFCT 2 ( A , B ) = ∑ j = 1 n ω j cot [ π 4 + π 4 | μ A ( x j ) − μ B ( x j ) | ∨ | η A ( x j ) − η B ( x j ) | ∨ | ν A ( x j ) − ν B ( x j ) | ∨ | ρ A ( x j ) − ρ B ( x j ) | ] , where ω = ( ω 1 , ω 1 , … , ω n ) T is the weight vector of x i , i = 1 , 2 , … , n ,with ω j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n ω j = 1 and the symbol “∨” is the maximum operation. In particular, if ω = ( 1 / n , 1 / n , … , 1 / n ) T , then the weighted cotangent similarity measure reduces to cotangent similarity measure.

 Numerical Example

In this section, the cosine similarity measures for PFSs are applied to strategic decision making problems (adapted from Wei and Merigó, 2012). In the following, we shall analyse a strategic decision-making problem about the selection of the optimal production strategy. Assume a company wants to create a new product and they are analysing the optimal target in order to obtain the highest benefits. After analysing the market they consider four possible strategies to follow: ①A 1 : create a new product oriented to the rich customers; ②A 2 : create a new product oriented to the mid-level and low-level customers; ③A 3 : create a new product adapted to all the customers; ④A 4 : do not create any product. After careful review of the information, the decision makers have summarized the information of the strategies in six general characteristics: ①S 1 : benefits in the short term; ②S 2 : benefits in the mid term; ③S 3 : benefits in the long term; ④S 4 : risk of the production strategy; ⑤S 5 : potential market and market risk; ⑥S 6 : industrialization infrastructure, human resources and financial conditions. The decision makers are required to evaluate the four possible production strategies A i ( i = 1 , 2 , 3 , 4 ) under six general characteristics and the decision information is represented by PFSs and is presented in Table 1. Each of which is featured by the content of six characteristics in the feature space S = { S 1 , S 2 , S 3 , S 4 , S 5 , S 6 }. The weight vector of S i ( i = 1 , 2 , … , 6 ) is: ω = ( 0.12 , 0.25 , 0.09 , 0.16 , 0.20 , 0.18 ) T .

Now, we consider another kind of unknown production strategy A, with data as listed in Table 1. Based on the weight vector and the data in Table 1, we can use the above similarity measures to identify to which type the unknown production strategy A should belong.

The data on production strategies.

A 1 A 2 A 3 A 4 A
S 1 (0.53,0.33,0.09) (1.00,0.00,0.00) (0.91,0.03,0.02) (0.85,0.09,0.05) (0.90,0.05,0.02)
S 2 (0.89,0.08,0.03) (0.13,0.64,0.21) (0.07,0.09,0.05) (0.74,0.16,0.10) (0.68,0.08,0.21)
S 3 (0.42,0.35,0.18) (0.03,0.82,0.13) (0.04,0.85,0.10) (0.02,0.89,0.05) (0.05,0.87,0.06)
S 4 (0.08,0.89,0.02) (0.73,0.15,0.08) (0.68,0.26,0.06) (0.08,0.84,0.06) (0.13,0.75,0.09)
S 5 (0.33,0.51,0.12) (0.52,0.31,0.16) (0.15,0.76,0.07) (0.16,0.71,0.05) (0.15,0.73,0.08)
S 6 (0.17,0.53,0.13) (0.51,0.24,0.21) (0.31,0.39,0.25) (1.00,0.00,0.00) (0.91,0.03,0.05)

The similarity measures between A i ( i = 1 , 2 , 3 , 4 ) and A.

Similarity measures ( A 1 , A ) ( A 2 , A ) ( A 3 , A ) ( A 4 , A )
WPFC 1 ( A i , A ) 0.813 0.656 0.787 0.994
WPFC 2 ( A i , A ) 0.810 0.656 0.638 0.993
WPFCS 1 ( A i , A ) 0.813 0.765 0.762 0.992
WPFCS 2 ( A i , A ) 0.840 0.765 0.831 0.991
WPFCS 3 ( A i , A ) 0.813 0.765 0.709 0.992
WPFCS 4 ( A i , A ) 0.813 0.757 0.707 0.989
WPFCT 2 ( A i , A ) 0.486 0.442 0.469 0.666
WPFCT 2 ( A i , A ) 0.486 0.442 0.440 0.665

From the above numerical results in Table 2, we know that the degree of similarity between A 4 and A is the largest one as derived by eight similarity measures. That is, all the eight similarity measures assign the unknown production strategy A to the class of production strategy A 4 according to the principle of the maximum degree of similarity between PFSs. Yet, there exist two slightly different ranking results: for the similarity measures WPFC 1 ( A i , A ), WPFCS 2 ( A i , A ) and WPFCT 1 ( A i , A ), i = 1 , 2 , 3 , 4, all these three similarity measures derive the same ranking of the production strategies, in which the degree of similarity between A 1 and A ranks the second, the degree of similarity between A 3 and A ranks the third, the degree of similarity between A 2 and A is the smallest one. While for the other five similarity measures, the degree of similarity between A 1 and A ranks the second, the degree of similarity between A 2 and A ranks the third, the degree of similarity between A 3 and A is the smallest one.

 Conclusion

In this paper, we presented another form of eight similarity measures between PFSs based on the cosine function between PFSs by considering the degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership in PFSs. Then, we applied these weighted cosine function similarity measures between PFSs to strategic decision making problem. Finally, an illustrative example for selection of the optimal production strategy is given to demonstrate the efficiency of the similarity measures for strategic decision making problem. In the future, the application of the proposed cosine similarity measures of PFSs needs to be explored in complex group decision making, risk analysis and many other fields under uncertain environments, such as dual hesitant fuzzy linguistic sets, dual hesitant fuzzy uncertain linguistic sets, interval-valued dual hesitant fuzzy linguistic sets, and so on (see Wei et al., 2016a; Lu and Wei, 2016; Wei et al., 2016b; Zhou et al., 2013; Lin et al., 2014; Wei and Zhao, 2012a; Wei, 2016a, 2012, 2011c, 2011b; Park et al., 2009; Ye, 2010; Wu and Chiclana, 2014; Chen, 2014; Liu et al., 2015; Wei et al., 2013c; Meng et al., 2016; Wei et al., 2017; Wei and Wang, 2017; Lu et al., 2017a, 2017b; Wei, 2017a, 2017b).

 Acknowledgements

The work was supported by the National Natural Science Foundation of China under Grant Nos. 61174149 and 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (No. 15YJCZH138) and the construction plan of scientific research innovation team for colleges and universities in Sichuan Province (15TD0004).

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