Introduction

INFORMATICA

Informatica

1822-88440868-4952

0868-4952

Vilnius University

INFO1165

10.15388/Informatica.2018.160

Research Article

The Generalized Dice Similarity Measures for Picture Fuzzy Sets and Their Applications

Wei

Guiwu

weiguiwu@163.com∗

G. 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 100 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 Intelligent Systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, International Journal of Computational Intelligence Systems, Soft Computing 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.

Gao

Hui

H. Gao has an MSc degree in management sciences and engineering from school of Economics and Management at University of Electronic Science and Technology of China, China. She is an associate professor in the School of Business at Sichuan Normal University.

School of Business, Sichuan Normal University, Chengdu, 610101, China

∗Corresponding author.

2018

112018

29110712412017112017

2018

Open access article under the CC BY license.

Key words generalized Dice similarity measures Dice similarity measures picture fuzzy set asymmetric measures projection measures patterns recognition building material recognition

1 Introduction

Multiple attribute decision making is a main branch of decision theory, where (PFS) introduced by Cuong (2013) has been successfully applied in recent years. The picture fuzzy set is a generalization of an intuitionistic fuzzy set (IFS) (see Atanassov, 1986, 1989; Xu and Cai, 2008; Wei, 2011; Xu and Chen, 2008; Ye, 2011; Chen, 2016; Wei, 2015; Zhang and Xu, 2015; Li and Ren, 2015; Wei, 2009, 2010). The picture fuzzy set (Cuong, 2013) is characterized by three functions expressing the degree of membership, the degree of neutral membership and the degree of non-membership. 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 cannot be accurately expressed in the traditional FS and IFS. Recently, many researchers have applied PFSs to the decision-making problems. Various methods have been developed to solve the multiple attribute decision-making problems with pi cture fuzzy information. For example, Singh (2014) investigated the correlation coefficients for picture fuzzy set and applied the correlation coefficient to clustering analysis with picture fuzzy information. Son (2015) proposed a novel distributed picture fuzzy clustering method with picture fuzzy information. Thong and Son (2015) developed a new approach to multi-variables fuzzy forecasting by using picture fuzzy clustering and picture fuzzy rules interpolation method. 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. Wei (2017c) proposed some aggregating operators for picture fuzzy set. Wei et al. (2016a) defined the projection models for multiple attribute decision making with picture fuzzy information. Wei (2017e) developed some cosine similarity measures for picture fuzzy sets. Wei (2018) proposed some picture 2-tuple linguistic aggregation operators for multiple attribute decision making. Wei (2017b) developed some picture 2-tuple linguistic Bonferroni mean operators for multiple attribute decision making. Wu and Wei (2017) gave some picture uncertain linguistic aggregation operators for multiple attribute decision making.

The similarity measure is one of the important and useful tools for degree of similarity between objects (see Szmidt and Kacprzyk, 2000; Liu, 2005; Hung and Yang, 2007; Xu and Xia, 2010; Ye, 2011; Tian, 2013; Rajarajeswari and Uma, 2013; Szmidt, 2014; Ye, 2016b). Functions expressing the degree of similarity of items or sets are used in physical anthropology, automatic classification, ecology, psychology, citation analysis, information retrieval, patterns recognition and numerical taxonomy (Ye, 2012b). In fact, the degree of similarity or dissimilarity between the objects under study plays an important role. In vector space, especially the Jaccard, Dice, and cosine similarity measures (Jaccard, 1901; Dice, 1945a; Salton and McGill, 1987) are often used in information retrieval, citation analysis, and automatic classification. Therefore, Ye (2012b) proposed the Jaccard, Dice, and cosine similarity measures between trapezoidal intuitionistic fuzzy numbers (TIFNs) and applied them to group decision-making problems. Ye (2012a) proposed the multi-criteria decision making models by using the Dice similarity measure between expected intervals of trapezoidal fuzzy numbers. Ye (2014) developed the Dice measures for simplified neutrosophic sets. Ye (2016a) proposed the generalized Dice measures for multiple attribute decision making under simplified neutrosophic environments.

However, these similarity measures do not deal with the similarity measures for picture fuzzy information. Therefore, it is necessary to extend the Dice measure to picture fuzzy set to handle patterns recognition, citation analysis, information retrieval and multiple attribute decision making problems to satisfy the requirements of decision makers’ preference and flexible decision making. In order to do so, the main purposes of this paper are: (1) to propose two forms of the Dice measures of PFSs, (2) to present the generalized Dice measures of PFSs, and (3) to develop the generalized Dice measures-based patterns recognition methods with picture fuzzy information. In the patterns recognition process, the main advantage of the proposed methods is more general and more flexible than existing patterns recognition methods with picture fuzzy information to satisfy the practical requirements.

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 picture fuzzy sets. In Section 3, we shall propose some Dice similarity measure and some weighted Dice similarity measure between PFSs. In Section 4, the Dice similarity measures for PFSs are applied to building material recognition and minerals field recognition. Section 5 concludes the paper with some remarks.

2 Preliminaries

In the following, we introduce some basic concepts related to intuitionistic fuzzy sets and picture fuzzy sets.

Definition 1 (<italic>See</italic> Atanassov, <xref ref-type="bibr" rid="j_info1165_ref_001">1986</xref>, <xref ref-type="bibr" rid="j_info1165_ref_002">1989</xref>).

An IFS is given by (1) 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 number μ A ( x ) and ν A ( x ) represents, respectively, the membership degree and non-membership degree of the element x to the set A.

Definition 2 (<italic>See</italic> Atanassov, <xref ref-type="bibr" rid="j_info1165_ref_001">1986</xref>, <xref ref-type="bibr" rid="j_info1165_ref_002">1989</xref>).

For each IFS A in X, if (2) 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.

Although Atanassov’s intuitionistic fuzzy set theory (Atanassov, 1986, 1989) has been successfully applied in different areas, there are situations in real life which cannot be represented by Atanassov’s intuitionistic fuzzy sets. Picture fuzzy sets (Cuong, 2013) are an extension of Atanassov’s intuitionistic fuzzy sets (Atanassov, 1986, 1989). Picture fuzzy set (Cuong, 2013) 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 representing the uncertain information in the process of patterns recognition and cluster analysis. Definition 3 (<italic>See</italic> Cuong, <xref ref-type="bibr" rid="j_info1165_ref_004">2013</xref>).

A picture fuzzy set (PFS) A on the universe is X an object of the form (3) 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 ) ∈ [ 0 , 1 ] is called the “degree of neutral membership of A” and η A ( x ) ∈ [ 0 , 1 ] 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.

3 Some Dice Similarity Measures for Picture Fuzzy Sets

The Dice similarity measure cannot be computed in this undefined situation when one vector is zero, which overcomes the disadvantage of the cosine similarity measure (Dice, 1945b). Therefore, the concept of the Dice similarity measure is introduced in this section (Dice, 1945b).

Let X = ( x 1 , x 2 , … , x n ) and Y = ( y 1 , y 2 , … , y n ) be two vectors of length where all the coordinates are positive real numbers. Then the Dice similarity measure [41] is defined as follows: (4) D ( X , Y ) = 2 X · Y ‖ X ‖ 2 2 + ‖ Y ‖ 2 2 = 2 ∑ j = 1 n x j y j ∑ j = 1 n x j 2 + ∑ j = 1 n y j 2 where X · Y = ∑ j = 1 n x j y j is called the inner product of the vector X and Y and ‖ X ‖ 2 = ∑ j = 1 n x j 2 and ‖ Y ‖ 2 = ∑ j = 1 n y j 2 are the Euclidean norms of X and Y (also called the L 2 norms).

The Dice similarity measure takes value in the interval [ 0 , 1 ]. However, it is undefined if x j = y j = 0 ( j = 1 , 2 , … , n). In this case, let the Dice measure value be zero when x j = y j = 0 ( j = 1 , 2 , … , n).

3.1 Dice Similarity Measures for Picture Fuzzy Sets

Let A be an 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.

In this section, we shall propose some Dice similarity measures and some weighted Dice similarity measures between PFSs based on the concept of the Dice similarity measure (Ye, 2014, 2016a).

Let A = ( μ A ( x j ) , η A ( x j ) , ν A ( x j ) ) and B = ( μ B ( x j ) , η B ( x j ) , ν B ( x j ) ), j = 1 , 2 , … , n, be two groups of picture fuzzy numbers, a Dice similarity measure between PFSs A and B is proposed as follows: (5) D PFS 1 ( A , B ) = 1 n ∑ j = 1 n 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 ) ) ( 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 ) ) .

The Dice similarity measure between PFSs A and B also satisfies the following properties:

(1)

0 ⩽ D PFS 1 ( A , B ) ⩽ 1;

(2)

D PFS 1 ( A , B ) = D PFS 1 ( B , A );

(3)

D PFS 1 ( A , B ) = 1 if A = B i.e. μ A ( x j ) = μ B ( x j ), η A ( x j ) = η B ( x j ), ν A ( x j ) = ν B ( x j ), j = 1 , 2 , … , n.

Proof.

(1) Let us consider the jth item of the summation in Eq. (5): D PFS 1 ( A j , B j ) = 1 n ∑ j = 1 n 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 ) ) ( 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 ) ) . It is obvious that D PFS 1 ( A , B ) ⩾ 0, and ( 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 ) ) ⩾ 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 ) ) according to the inequality a 2 + b 2 ⩾ 2 a b. Thus, 0 ⩽ D PFS 1 ( A j , B j ) ⩽ 1. From Eq. (5), the summation of n terms is 0 ⩽ D PFS 1 ( A , B ) ⩽ 1.

(2) It is obvious that the proposition is true.

(3) 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, there is D PFS 1 ( A , B ) = 1 n ∑ j = 1 n 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 ) ) ( 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 ) ) = 1 n ∑ j = 1 n 2 ( μ A ( x j ) μ A ( x j ) + η A ( x j ) η A ( x j ) + ν A ( x j ) ν A ( x j ) ) ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) = 1 n ∑ j = 1 n 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) = 1 .

Therefore, we have finished the proofs. □

If we consider the weights of x j , a weighted Dice similarity measure between PFSs A and B is proposed as follows: (6) W D PFS 1 ( A , B ) = ∑ j = 1 n ω j 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 ) ) ( 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 2 , … , ω n ) T is the weight vector of x j ( j = 1 , 2 , … , n), with w j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n w j = 1.

In particular, if ω = ( 1 n , 1 n , … , 1 n ) T , then the weighted Dice similarity measure reduces to Dice similarity measure. That’s to say, if we take ω j = 1 n , j = 1 , 2 , … , n, then there is W D PFS 1 ( A , B ) = D PFS 1 ( A , B ). Obviously, the weighted Dice similarity measure of two PFSs A and B also satisfies the following properties:

(1)

0 ⩽ W D PFS 1 ( A , B ) ⩽ 1;

(2)

W D PFS 1 ( A , B ) = W D PFS 1 ( B , A );

(3)

W D PFS 1 ( A , B ) = 1 if A = B i.e. μ A ( x j ) = μ B ( x j ), η A ( x j ) = η B ( x j ), ν A ( x j ) = ν B ( x j ), j = 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 Dice similarity measure and weighted Dice similarity measure between PFSs as follows: (7) D PFS 2 ( A , B ) = 1 n ∑ j = 1 n 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 ) ) ( 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 ) ) , (8) W D PFS 2 ( A , B ) = ∑ j = 1 n ω j 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 ) ) ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) ) + ∑ j = 1 n ( 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 2 , … , 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.

3.2 Another Form of the Dice Similarity Measure for Picture Fuzzy Sets

In this section, we shall develop another form of Dice similarity measure for picture fuzzy sets, which is defined as follows.

Definition 4.

Let A = ( μ A ( x j ) , η A ( x j ) , ν A ( x j ) ) and B = ( μ B ( x j ) , η B ( x j ) , ν B ( x j ) ), j = 1 , 2 , … , n, be two groups of picture fuzzy numbers, then a Dice similarity measure between PFSs A and B is proposed as follows: (9) D PFS 3 ( A , B ) = ∑ j = 1 n 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 ) ) ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) .

The Dice similarity measure between PFSs A and B also satisfies the following properties:

(1)

0 ⩽ D PFS 3 ( A , B ) ⩽ 1;

(2)

D PFS 3 ( A , B ) = D PFS 3 ( B , A );

(3)

D PFS 3 ( A , B ) = 1 if A = B, i.e. μ A ( x j ) = μ B ( x j ), η A ( x j ) = η B ( x j ), ν A ( x j ) = ν B ( x j ), j = 1 , 2 , … , n.

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

If we consider the weights of x j , a weighted Dice similarity measure between PFSs A and B is proposed as follows: (10) W D PFS 3 ( A , B ) = 2 ∑ j = 1 n ω j 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 ) ) ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ∑ j = 1 n ω j 2 ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) , where ω = ( ω 1 , ω 2 , … , ω n ) T is the weight vector of x j ( j = 1 , 2 , … , n), with w j ∈ [ 0 , 1 ], j = 1 , 2 , … , n, ∑ j = 1 n w j = 1. In particular, ω = ( ω 1 , ω 2 , … , ω n ) T , then the weighted Dice similarity measure reduces to Dice similarity measure. That’s to say, if we take ω j = 1 n , j = 1 , 2 , … , n, then there is W D PFS 3 ( A , B ) = D PFS 3 ( A , B ).

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

(1)

0 ⩽ W D PFS 3 ( A , B ) ⩽ 1;

(2)

W D PFS 3 ( A , B ) = W D PFS 3 ( B , A );

(3)

W D PFS 3 ( A , B ) = 1 if A = B i.e. μ A ( x j ) = μ B ( x j ), η A ( x j ) = η B ( x j ), ν A ( x j ) = ν B ( x j ), j = 1 , 2 , … , n.

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 another form of Dice similarity measure and weighted Dice similarity measure between PFSs as follows: (11) D PFS 4 ( A , B ) = 2 ∑ 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 ) ) ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) ) + ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) + ρ B 2 ( x j ) ) , (12) W D PFS 4 ( A , B ) = 2 ∑ j = 1 n ω j 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 ) ) ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) ) + ∑ j = 1 n ω j 2 ( 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 2 , … , 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.

3.3 The Generalized Dice Similarity Measure for Picture Fuzzy Sets

In this section, we develop the generalized Dice similarity measure for picture fuzzy sets. As the generalization of the Dice similarity measure for picture fuzzy sets, the generalized Dice similarity measure for picture fuzzy sets are defined below.

Definition 5.

Let A = ( μ A ( x j ) , η A ( x j ) , ν A ( x j ) ) and B = ( μ B ( x j ) , η B ( x j ) , ν B ( x j ) ), j = 1 , 2 , … , n, be two groups of picture fuzzy numbers, then the generalized Dice similarity measure between PFSs A and B is defined as follows: (13) G D PFS 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 ) ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) , (14) G D PFS 2 ( A , B ) = ∑ 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 ) ) λ ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) , where λ is a positive parameter for 0 ⩽ λ ⩽ 1.

Then, the generalized Dice similarity measure includes some special cases by altering the parameter value λ.

If λ = 0.5, the two generalized Dice similarity measures (13) and (14) reduced to Dice similarity measures (9) and (10): (15) G D PFS 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 ) ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = 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 ) ) 0.5 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − 0.5 ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = 1 n ∑ j = 1 n 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 ) ) ( 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 ) ) , (16) G D PFS 2 ( A , B ) = ∑ 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 ) ) λ ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ 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 ) ) 0.5 ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − 0.5 ) ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = 2 ∑ 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 ) ) ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) .

If λ = 0 , 1, the two generalized Dice similarity measures reduced to the following asymmetric similarity measures respectively: (17) G D PFS 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 ) ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = 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 B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) for λ = 0 , (18) G D PFS 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 ) ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = 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 ) for λ = 1 , (19) G D PFS 2 ( A , B ) = ∑ 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 ) ) λ ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ 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 ) ) ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) for λ = 0 , (20) G D PFS 2 ( A , B ) = ∑ 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 ) ) λ ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ 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 ) ) ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) for λ = 1 .

From above analysis, it can be seen that the above four asymmetric similarity measures are the extension of the relative projection measure of the picture fuzzy numbers.

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. Thus, we further propose the following two weighted generalized Dice similarity measures for PFSs, respectively, as follows: (21) W D PFS 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 ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) , (22) W D PFS 2 ( A , B ) = ∑ j = 1 n ω j 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 ) ) λ ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) ( 1 − λ ) ∑ j = 1 n ω j 2 ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) , where ω = ( ω 1 , ω 2 , … , 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 w j = 1. In particular, ω = ( ω 1 , ω 2 , … , ω n ) T , then the weighted Dice similarity measure reduces to Dice similarity measure. That’s to say, if we take ω j = 1 n , j = 1 , 2 , … , n, then there is W G D PFS 1 ( A , B ) = D PFS 1 ( A , B ), W G D PFS 2 ( A , B ) = G D PFS 2 ( A , B ).

Then, the weighted generalized Dice similarity measure includes some special cases by altering the parameter value λ. If λ = 0.5, the two weighted generalized Dice similarity measures (21) and (22) reduced to weighted Dice similarity measures (6) and (10): (23) W D PFS 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 ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ 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 ) 0.5 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − 0.5 ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ j = 1 n ω j 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 ) ) ( 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 ) ) , (24) W D PFS 2 ( A , B ) = ∑ j = 1 n ω j 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 ) ) λ ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ω j 2 ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ j = 1 n ω j 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 ) ) 0.5 ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − 0.5 ) ∑ j = 1 n ω j 2 ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = 2 ∑ j = 1 n ω j 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 ) ) ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ∑ j = 1 n ω j 2 ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) .

If λ = 0 , 1, the two weighted generalized Dice similarity measures reduced to the following asymmetric weighted similarity measures respectively: (25) W G D PFS 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 ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ 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 B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) for λ = 0 , (26) W G D PFS 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 ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ 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 ) for λ = 1 , (27) W G D PFS 2 ( A , B ) = ∑ j = 1 n ω j 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 ) ) λ ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ω j 2 ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ j = 1 n ω j 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 ) ) μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) for λ = 0 , (28) W G D PFS 2 ( A , B ) = ∑ j = 1 n ω j 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 ) ) λ ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ω j 2 ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) ) = ∑ j = 1 n ω j 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 ) ) μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) for λ = 1 .

From above analysis, it can be seen that the above four asymmetric weighted similarity measures are the extension of the relative weighted projection measure of the picture fuzzy numbers. 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 generalized Dice similarity measure and weighted generalized Dice similarity measure between PFSs as follows: (29) G D PFS 3 ( 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 ) ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) + ρ B 2 ( x j ) ) , (30) G D PFS 4 ( A , B ) = ∑ 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 ) ) λ ∑ j = 1 n ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) + ρ B 2 ( x j ) ) , (31) W G D PFS 3 ( 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 ) ) λ ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) ) + ( 1 − λ ) ( μ B 2 ( x j ) + η B 2 ( x j ) + ν B 2 ( x j ) + ρ B 2 ( x j ) ) , (32) W G D PFS 4 ( A , B ) = ∑ j = 1 n ω j 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 ) ) λ ∑ j = 1 n ω j 2 ( μ A 2 ( x j ) + η A 2 ( x j ) + ν A 2 ( x j ) + ρ A 2 ( x j ) ) + ( 1 − λ ) ∑ j = 1 n ω j 2 ( 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 2 , … , 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 w j = 1, and λ is a positive parameter for 0 ⩽ λ ⩽ 1.

4 Applications

In this section, the Dice similarity measures for PFSs are applied to building material recognition (adapted from Xu and Cai, 2008). Let us consider four building materials: sealant, floor varnish, wall paint and polyvinyl chloride flooring, which are represented by the PFSs A i ( i = 1 , 2 , 3 , 4 ) in the feature space X = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 }. The weight vector of x i ( i = 1 , 2 , … , 7 ) is: w = ( 0.12 , 0.15 , 0.09 , 0.16 , 0.20 , 0.10 , 0.18 ) T .

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

Table 1

The data on building materials.

	A 1	A 2	A 3	A 4	A
x 1	(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)
x 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)
x 3	(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)
x 4	(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)
x 5	(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)
x 6	(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)
x 7	(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)

According to Eqs. (21), (22), (31), (32) and different values of the parameter λ, the weighted generalized Dice measure values between A i ( i = 1 , 2 , 3 , 4 ) can be obtained, which are shown in Tables 2, 3, 4 and 5 respectively.

Table 2

The generalized Dice similarity measures of Eq. (21) and ranking orders.

λ	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	Ranking orders
0	0.658	0.675	0.708	1.059	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.2	0.661	0.700	0.726	1.027	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.5	0.688	0.749	0.767	0.988	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.7	0.726	0.791	0.816	0.967	A 4 ≻ A 3 ≻ A 2 ≻ A 1
1.0	0.834	0.883	1.521	0.939	A 3 ≻ A 4 ≻ A 2 ≻ A 1

Table 3

The generalized Dice similarity measures of Eq. (22) and ranking orders.

λ	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	Ranking orders
0	0.604	0.713	0.754	1.055	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.2	0.633	0.743	0.794	1.027	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.5	0.683	0.792	0.860	0.988	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.7	0.722	0.829	0.911	0.964	A 4 ≻ A 3 ≻ A 2 ≻ A 1
1.0	0.788	0.891	1.001	0.930	A 3 ≻ A 4 ≻ A 2 ≻ A 1

Table 4

The generalized Dice similarity measures of Eq. (31) and ranking orders.

λ	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	Ranking orders
0	0.658	0.676	0.715	1.059	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.2	0.661	0.700	0.727	1.027	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.5	0.688	0.745	0.752	0.988	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.7	0.724	0.786	0.776	0.966	A 4 ≻ A 2 ≻ A 3 ≻ A 1
1.0	0.827	0.870	0.832	0.938	A 4 ≻ A 2 ≻ A 3 ≻ A 1

Table 5

The generalized Dice similarity measures of Eq. (32) and ranking orders.

λ	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	WGD ( A 1 , B )	Ranking orders
0	0.604	0.714	0.760	1.055	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.2	0.633	0.743	0.773	1.027	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.5	0.682	0.790	0.794	0.988	A 4 ≻ A 3 ≻ A 2 ≻ A 1
0.7	0.719	0.825	0.809	0.963	A 4 ≻ A 2 ≻ A 3 ≻ A 1
1.0	0.783	0.884	0.831	0.929	A 4 ≻ A 2 ≻ A 3 ≻ A 1

From the Tables 2, 3, 4 and 5, different ranking orders are shown by taking different values of λ and different Dice similarity measures. Then the building material A should belong to the class of building material A 3 or A 4 according to the principle of the maximum degree of Dice similarity measures between PFSs.

Furthermore, for the special cases of the four generalized Dice measures we obtain the following results:

▶ When λ = 0, the four weighted generalized Dice measures are reduced to the weighted projection measures of A i ( i = 1 , 2 , 3 , 4 ) on A. Thus, the building material A should belong to the class of building material A 4 according to the principle of the maximum degree of Dice similarity measures between PFSs.

▶ When λ = 0, the four weighted generalized Dice measures are reduced to the weighted Dice similarity measures of A i ( i = 1 , 2 , 3 , 4 ) on A. Thus, the building material A should belong to the class of building material A 4 according to the principle of the maximum degree of Dice similarity measures between PFSs.

▶ When λ = 0, the four weighted generalized Dice measures are reduced to the weighted projection measures of A i ( i = 1 , 2 , 3 , 4 ) on A. Thus, the building material A should belong to the class of building material A 3 or A 4 according to the principle of the maximum degree of Dice similarity measures between PFSs.

Therefore, according to different Dice similarity measures and different values of the parameter λ, ranking orders may be also different. Thus the proposed patterns recognition methods can be assigned some value of λ and some measure to satisfy the real requirements.

Obviously, the patterns recognition methods based on the Dice measures and the projection measures are the special cases of the proposed patterns recognition models based on generalized Dice measures. Therefore, in the patterns recognition process, the patterns recognition models developed in this paper are more general and more flexible than existing patterns recognition models under picture fuzzy environment.

5 Conclusion

The picture fuzzy set is characterized by three functions expressing the degree of membership, the degree of neutral membership and the degree of non-membership. It was proposed as a generalization of an intuitionistic fuzzy set in order to deal with indeterminate and inconsistent information. In this work, we shall present some novel Dice similarity measures of picture fuzzy sets and the generalized Dice similarity measures of picture fuzzy sets and indicate that the Dice similarity measures and asymmetric measures (projection measures) are the special cases of the generalized Dice similarity measures in some parameter values. Then, we propose the generalized Dice similarity measures-based patterns recognition models with picture fuzzy information. Then, we apply the generalized Dice similarity measures between picture fuzzy sets to building material recognition. Finally, an illustrative example is given to demonstrate the efficiency of the similarity measures for building material recognition. In the future, the application of the proposed Dice similarity measure of PFSs needs to be explored in decision making, risk analysis and many other fields under uncertain environment (see Wei et al., 2017a; Wei and Lu, 2018; Wei et al., 2017c; Zeng et al., 2016; Lu et al., 2017b; Zhang and Xu, 2014; Lu et al., 2017a; Zeng et al., 2017; Wei et al., 2017b; Hu et al., 2013; Peng and Yang, 2015; Wei, 2016a; Xu and Ma, 2016; Wei et al., 2016b; Garg, 2016; Wei, 2017e; Wei and Lu, 2017; Wei, 2017d,a; Wei and Wang, 2017).

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. 17XJA630003) and the construction plan of scientific research innovation team for colleges and universities in Sichuan Province (15TD0004).

References

Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96.

Atanassov, K. (1989). More on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 3, 37–46.

Chen, T.Y. (2016). An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis. Applied Soft Computing, 2, 390–409.

Cuong, B. (2013). Picture fuzzy sets-first results. Part 1. In: Seminar Neuro-Fuzzy Systems with Applications. Clarendon Press, Hanoi.

Dice, L.R. (1945a). Measures of the amount of ecologic association between species. Ecology, 6, 297–302.

Dice, L.R. (1945b). Measures of the amount of ecologic association between species. Ecology, 26, 297–302.

Garg, H. (2016). A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem. Journal of Intelligent and Fuzzy Systems, 1(1), 529–540.

Hu, J.H., Zhang, Y., Chen, X.H. (2013). Multi-criteria decision making method based on possibility degree of interval type-2 fuzzy number. Knowledge-Based Systems, 3, 21–29.

Hung, W.L., Yang, M.S. (2007). Similarity measures of intuitionistic fuzzy sets based on Lp metric. International Journal of Approximate Reasoning, 6, 120–136.

Jaccard, P. (1901). Distribution de la flore alpine dans le Bassin des Drouces et dans quelques regions voisines. Bulletin de la Société Sciences Nat, 7(140), 241–272.

Li, D.F., Ren, H.P. (2015). Multi-attribute decision making method considering the amount and reliability of intuitionistic fuzzy information. Journal of Intelligent and Fuzzy Systems, 8(4), 1877–1883.

Liu, H.W. (2005). New similarity measures between intuitionistic fuzzy sets and between elements. Mathematical and Computer Modelling, 2, 61–70.

Lu, M., Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2017a). Bipolar 2-tuple linguistic aggregation operators in multiple attribute decision making. Journal of Intelligent and Fuzzy Systems, 3(2), 1197–1207.

Lu, M., Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2017b). Hesitant pythagorean fuzzy hamacher aggregation operators and their application to multiple attribute decision making. Journal of Intelligent and Fuzzy Systems, 3(2), 1105–1117.

Peng, X., Yang, Y. (2015). Some results for pythagorean fuzzy sets. International Journal of Intelligent System, 30, 1133–1160.

Rajarajeswari, P., Uma, N. (2013). Intuitionistic fuzzy multi similarity measure based on cotangent function. International Journal of Engineering Research & Technology, 2(11), 1323–1329.

Salton, G., McGill, M.J. (1987). Introduction to Modern Information Retrieval. Clarendon Press, Hanoi.

Singh, P. (2014). Correlation coefficients for picture fuzzy sets. Journal of Intelligent & Fuzzy Systems, 7, 2857–2868.

Son, L. (2015). DPFCM: a novel distributed picture fuzzy clustering method on picture fuzzy sets. Expert System with Applications, 2, 51–66.

Szmidt, E. (2014). Distances and similarities in intuitionistic fuzzy sets. In: Studies in Fuzziness and Soft Computing, Vol. 307. Springer.

Szmidt, E., Kacprzyk, J. (2000). Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114, 505–518.

Thong, N.T. (2015). HIFCF: An effective hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis expert systems with applications. Expert Systems with Applications, 2(7), 3682–3701.

Thong, P.H., Son, L.H. (2015). A new approach to multi-variables fuzzy forecasting using picture fuzzy clustering and picture fuzzy rules interpolation method. In: 6th International Conference on Knowledge and Systems Engineering. Hanoi, Vietnam, pp. 679–690.

Tian, M.Y. (2013). A new fuzzy similarity based on cotangent function for medical diagnosis. Advanced Modeling and Optimization, 5(2), 151–156.

Wei, G.W. (2009). Some geometric aggregation functions and their application to dynamic multiple attribute decision making in intuitionistic fuzzy setting. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 7, 179–196.

Wei, G.W. (2010). Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Applied Soft Computing, 10, 423–431.

Wei, G.W. (2011). Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making. Expert Systems with Applications, 38(9), 11671–11677.

Wei, G.W. (2015). Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information. International Journal of Fuzzy Systems, 17(3), 484–489.

Wei, G.W. (2016a). Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making. International Journal of Machine Learning and Cybernetics, 6, 1093–1114.

Wei, G.W. (2016b). Picture fuzzy cross-entropy for multiple attribute decision making problems. Journal of Business Economics and Management, 17(4), 491–502.

Wei, G.W. (2017a). Interval-valued dual hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making. Journal of Intelligent and Fuzzy Systems, 3(3), 1881–1893.

Wei, G.W. (2017b). Picture 2-tuple linguistic Bonferroni mean operators and their application to multiple attribute decision making. International Journal of Fuzzy System, 19(4), 997–1010.

Wei, G.W. (2017c). Picture fuzzy aggregation operators and their application to multiple attribute decision making. Journal of Intelligent and Fuzzy Systems, 33(2), 713–724.

Wei, G.W. (2017d). Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making. Journal of Intelligent and Fuzzy Systems, 3(4), 2119–2132.

Wei, G.W. (2017e). Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making. Informatica, 28(3), 547–564.

Wei, G.W. (2018). Picture 2-tuple linguistic aggregation operators in multiple attribute decision making. Soft Computing, 22(3), 989–1002. 2018.

Wei, G.W., Lu, M. (2017). Dual hesitant Pythagorean fuzzy Hamacher aggregation operators in multiple attribute decision making. Archives of Control Sciences, 7(3), 365–395.

Wei, G.W., Lu, M. (2018). Pythagorean fuzzy power aggregation operators in multiple attribute decision making. International Journal of Intelligent Systems, 33(1), 169–186.

Wei, G.W., Wang, J.M. (2017). A comparative study of robust efficiency analysis and data envelopment analysis with imprecise data. Expert Systems with Applications, 1, 28–38.

Wei, G.W., Alsaadi, F.E., Hayat, T. (2016a). Projection models for multiple attribute decision making with picture fuzzy information. International Journal of Machine Learning and Cybernetics. https://doi.org/10.1007/s13042-016-0604-1.

Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2016b). Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making. Iranian Journal of Fuzzy Systems, 3(4), 1–16.

Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2017a). Hesitant bipolar fuzzy aggregation operators in multiple attribute decision making. Journal of Intelligent and Fuzzy Systems, 3(2), 1119–1128.

Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2017b). A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure. International Journal of Fuzzy Systems, 9(3), 607–614.

Wei, G.W., Lu, M., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2017c). Pythagorean 2-tuple linguistic aggregation operators in multiple attribute decision making. Journal of Intelligent and Fuzzy Systems, 33(2), 1129–1142.

Wu, S.J., Wei, G.W. (2017). Picture uncertain linguistic aggregation operators and their application to multiple attribute decision making. International Journal of Knowledge-Based and Intelligent Engineering Systems, 21(4), 243–256.

Xu, Z.S., Cai, X. (2008). Intuitionistic Fuzzy Information Aggregation: Theory and Applications. Science Press.

Xu, Z.S., Chen, J. (2008). An overview of distance and similarity measures of intuitionistic fuzzy sets. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 6(4), 529–455.

Xu, Z.S., Ma, Z.M. (2016). Symmetric pythagorean fuzzy weighted geometric averaging operators and their application in multicriteria decision-making problems. International Journal of Intelligent Systems, 31, 1198–1219.

Xu, Z.S., Xia, M.M. (2010). Some new similarity measures for intuitionistic fuzzy values and their application in group decision making. Journal of System Science and Engineering, 19, 430–452.

Ye, J. (2011). Cosine similarity measures for intuitionistic fuzzy sets and their applications. Mathematical and Computer Modelling, 53(1–2), 91–97.

Ye, J. (2012a). Multicriteria decision-making method using the Dice similarity measure between expected intervals of trapezoidal fuzzy numbers. Journal of Decision Systems, 1(4), 307–317.

Ye, J. (2012b). Multicriteria group decision-making method using vector similarity measures for trapezoidal intuitionistic fuzzy numbers. Group Decision and Negotiation, 21(4), 519–530.

Ye, J. (2014). Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making. International Journal of Fuzzy Systems, 6(2), 204–211.

Ye, J. (2016a). The generalized Dice measures for multiple attribute decision making under simplified neutrosophic environments. Journal of Intelligent and Fuzzy Systems, 1(1), 663–671.

Ye, J. (2016b). Similarity measures of intuitionistic fuzzy sets based on cosine function for the decision making of mechanical design schemes. Journal of Intelligent and Fuzzy Systems, 30(1), 151–158.

Zeng, S., Marqués, D.P., Zhu, F. (2016). A new model for interactive group decision making with intuitionistic fuzzy preference relations. Informatica, 27(4), 911–928.

Zeng, S.Z., Merigó, J.M., Marqués, D.P. (2017). Intuitionistic fuzzy induced ordered weighted averaging distance operator and its application to decision making. Journal of Intelligent and Fuzzy Systems, 2(1), 11–22.

Zhang, X.L., Xu, Z.S. (2014). Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. International Journal of Intelligent System, 9, 1061–1078.

Zhang, X.L., Xu, Z.S. (2015). Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making. Applied Soft Computing, 6, 42–56.