New similarity and distance measures of Pythagorean fuzzy sets and its application to selection of advertising platforms

3.3.3.1 Existing distance measures

Let α and β are two PFNs in the universe of discourse X, Zhang and Xu [34] proposed the Hamming distance measure between α and β as follows:

D 1 ( α , β ) = 1 2 ( | μ α 2 - μ β 2 | + | v α 2 - v β 2 | + | π α 2 - π β 2 | )(3.16)

Further, Ren, Xu and Gou [35] defined a new distance measure based on Euclidean distance model as follows:

D 2 ( α , β ) = 1 2 ( ( μ α 2 - μ β 2 ) 2 + ( v α 2 - v β 2 ) 2 + ( π α 2 - π β 2 ) 2 )(3.17)

Moreover, Chen [36] extends the Minkowski model to propose a more flexible distance measure as follows:

D 3 ( α , β ) = [ 1 2 ( | μ α 2 - μ β 2 | λ + | v α 2 - v β 2 | λ + | π α 2 - π β 2 | λ ) ] 1 λ(3.18) where λ is distance measure parameter, with λ ⩾ 1. When λ = 1, (3.18) reduces to (3.16). When λ = 2, (3.18) reduces to (3.17).

Considering each PFN is characterized by four parameters (μ _α , v _α , r _α , d _α ), Li and Li and Zeng [38] proposed a normalized generalized distance measure as follows:

D 4 ( α , β ) = [ 1 4 ( | μ α - μ β | λ + | v α - v β | λ + | r α - r β | λ + | d α - d β | λ ) ] 1 λ(3.19)

Furthermore, Zeng, Li and Yin [39] considering the hesitancy degree (π _α ) should be included, then they presented a revised distance measure as follows:

D 5 ( α , β ) = [ 1 5 ( | μ α - μ β | λ + | v α - v β | λ + | π α - π β | λ + | r α - r β | λ + | d α - d β | λ ) ] 1 λ(3.20)

On this basis, Chen [40] defined a new generalized distance measure as follows:

D 6 ( α , β ) = [ 1 3 ( | μ α 2 - μ β 2 | λ + | v α 2 - v β 2 | λ + | r α 2 - r β 2 | λ + | d α 2 - d β 2 | λ ) ] 1 λ(3.21)

It can be seen that existing methods only defines the distance measures of PFNs from a two-dimensional plane, which is shown in Fig. 3, and it does not take into account the angle between the corresponding vectors of the two PFNs. In the following, we establish an extended concept of the generalized distance measures based on the Spherical Coordinate System.

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Fig. 3

Area of a PFN P (μ _P , v _P , r _P , d _P ).

In the section 3.2 (New Similarity Measures), we conclude a fact that PFNs are distributed in one eighth of a unit sphere and we defined a set of vectors S. Now, for any PFN α, we make the perpendicular of the μOv plane from point α, and the foot is α′. The angle between vector O α ′ → and the positive direction of μ-axis is denoted as θ _α (Definition 2), and the angle between vector α → ∈ S and the positive direction of π-axis is denoted as φ α ( ∈ [ 0 , π 2 ] ) .

We now introduce a new way to express the PFNs by giving a pair of values d _α and l _α for any α∈ PFNs, where d _α is defined in Definition 2.2 ( d α = 1 - 2 θ α π ) and l α = 2 φ α π . Yager [10] pointed out that d _α is essentially indicating on a scale of 1 to 0 how fully the strength r _α is pointing to membership. If d _α  = 1 the direction of r _α is completely to membership while d _α  = 0 the direction of the strength is completely to nonmembership. Let us now understand the meaning of value l _α , which we called the direction of accuracy. Peng and Yang [45] proposed the accuracy function for PFNs, for any PFN α, its accuracy is defined as follows: a ( α ) = μ α 2 + v α 2

It is obvious that a ( α ) = 1 - π α 2 . On the other hand, by the geometric relation, we know that φ _α  = arccos(π _α ). Thus, in the case l _α  = 0, then φ _α  = 0 and the hesitancy degree π _α  = 1 and the accuracy degree a (α) = 0. On the contrary, in the case l _α  = 1, then φ α = π 2 and the hesitancy degree π _α  = 0 and the accuracy degree a (α) = 1. Therefore, if we take the accuracy degree of 1 as the ideal accuracy degree, then we see that l _α is essentially indicating on a scale of 1 to 0 how fully the vector α → ( = ( μ α , v α , π α ) ) is pointing to the ideal accuracy degree.

A convenient geometrical interpretation of the space of PFN α is presented in Fig. 4.

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Fig. 4

Space of a PFN P (μ _α , v _α , θ _α , φ _α ).

Based on the Spherical Coordinate System, we can obtain the spherical coordinates of PFN α as follows: { μ α = sin φ α cos θ α v α = sin φ α sin θ α π α = cos φ α(3.22)

Namely, { φ α = arccos ( π α ) θ α = arctan ( v α μ α )(3.23)

Example 3.2. Let α is a PFN, α = (0.3, 0.4), then π _α  = 0.866, according to the Equation (3.23), we obtain that φ _α  = arccos(π _α ) = arccos 0.866 = 0.5236, and θ α = arctan ( v α μ α ) = arctan ( 0.4 0.3 ) = 0.9273 , hence, d α = 1 - 2 θ α π = 0.4097 ,

l α = 2 φ α π = 0.3333 , thus, based on the new way of expressing PFNs, we have

α = (d _α , l _α ) = (0.4097, 0.3333).

Taking the influence of the five parameters (μ _α , v _α , π _α , d _α , l _α ) into account, we now presented the new distance measures of PFNs as follows.

Definition 3.4. Let α and β are two PFNs, the new Hamming distance measure between α and β is defined as follows:

D H ( α , β ) = 1 5 ( | μ α - μ β | + | v α - v β | + | π α - π β | + | d α - d β | + | l α - l β | )(3.24)

The new Euclidean distance measure between α and β is defined as follows:

D E ( α , β ) = [ 1 5 ( | μ α - μ β | 2 + | v α - v β | 2 + | π α - π β | 2 + | d α - d β | 2 + | l α - l β | 2 ) ] 1 2(3.25)

The new generalized distance measure between α and β is defined as follows:

D G ( α , β ) = [ 1 5 ( | μ α - μ β | λ + | v α - v β | λ + | π α - π β | λ + | d α - d β | λ + | l α - l β | λ ) ] 1 λ(3.26) where λ is distance measure parameter, with λ ⩾ 1. When λ = 1, (3.26) reduces to (3.24). When λ = 2, (3.26) reduces to (3.25).

Different from the existing distance measures, the new distance measures take the l _α  (l _β ) into account instead of the strength r _α  (r _β ). In fact, |r _α  - r _β | can be understood as the length expansion, while the |d _α  - d _β | and the |l _α  - l _β | can be understood as the angular variation. However, the change of angle and length is inconsistent in most cases, for example, when the value of r _α changes from 0 to 2 2 , then we know that the value of φ _α correspondingly changes from 0 to π 4 , then the value of d _α changes from 0 to 1 2 , thus, we see that the length expansion ( 2 2 ) of r _α is different from the angular variation ( 1 2 ) of d _α . The new distance measure is calculated by using the angle change uniformly (except |μ _α  - μ _β |, |v _α  - v _β |, |π _α  - π _β |), which avoiding the influence of inconsistency.

However, in addition to the above five parameters (μ _α , v _α , π _α , d _α , l _α ), the angle between the vector α → and vector β → shouldn’t be ignored. For convenience, we use the symbol ϑ (α, β) to represent this angle. According to the Equation (3.13), we know that cos ϑ ( α → , β → ) = μ α μ β + v α v β + π α π β , thus, ϑ ( α , β ) = arccos ( μ α μ β + v α v β + π α π β )(3.27)

Obviously, ϑ ( α , β ) ∈ [ 0 , π 2 ] , and according to the proof of Theorem 3.3, we see that ϑ (α, β) = 1 if and only if α = β. Now, we note ω ( α , β ) = 2 π ϑ ( α , β ) which is called the strength of ϑ (α, β). Hence, ω (α, β) ∈ [0, 1]. And in the case ω (α, β) = 0, then ϑ (α, β) = 0, thus, α = β, namely, vector α → and vector β → coincide completely. On the contrary, in the case ω (α, β) = 1, then ϑ ( α , β ) = π 2 , thus, μ _α μ _β  + v _α v _β  + π _α π _β  = 0, namely, α → β → = 0 , where symbol “·” is the inner product operation, geometrically, vector α → and vector β → are perpendicular.

Considering the angle between vector α → and vector β → , some other distance measures are given as follows:

Definition 3.5. Let α and β are two PFNs, the new Hamming distance measure between α and β is defined as follows:

D ω H ( α , β ) = 1 6 ( | μ α - μ β | + | v α - v β | + | π α - π β | + | d α - d β | + | l α - l β | + ω ( α , β ) )(3.28)

The new Euclidean distance measure between α and β is defined as follows:

D ω E ( α , β ) = [ 1 6 ( | μ α - μ β | 2 + | v α - v β | 2 + | π α - π β | 2 + | d α - d β | 2 + | l α - l β | 2 ) + ω 2 ( α , β ) ] 1 2(3.29)

The new generalizeddistance measure between α and β is defined as follows:

D ω G ( α , β ) = [ 1 6 ( | μ α - μ β | λ + | v α - v β | λ + | π α - π β | λ + | d α - d β | λ + | l α - l β | λ + ω λ ( α , β ) ) ] 1 λ(3.30)

where λ is distance measure parameter, with λ ⩾ 1. When λ = 1, (3.30) reduces to (3.28). When λ = 2, (3.30) reduces to (3.29). And ω ( α , β ) = 2 π ϑ ( α , β ) , where ϑ (α, β) is the angle between the vector α → and vector β → .

Now, we extend the new generalized distance measures to PFSs. For simplicity, we denote _μ  (x _i ) = μ _A  (x _i ) - μ _B  (x _i ) , _v  (x _i ) = v _A  (x _i ) - v _B  (x _i ) , _π  (x _i ) = π _A  (x _i ) - π _B  (x _i ) , _d  (x _i ) = d _A  (x _i ) - d _B  (x _i ) , _l  (x _i ) = l _A  (x _i ) - l _B  (x _i ), where A and B are two PFSs on X ={ x₁, x₂, …, x _n  }.

Definition 3.6. Let A and B are two PFSs on X ={ x₁, x₂, …, x _n  }. Then some new generalized distance measures between A and B are defined as follows:

D PFS ( A , B ) = [ 1 5 n ∑ i = 1 n ( | μ ( x i ) | λ + | v ( x i ) | λ + | π ( x i ) | λ + | d ( x i ) | λ + | l ( x i ) | λ ) ] 1 λ(3.31) D ω PFS ( A , B ) = [ 1 6 n ∑ i = 1 n ( | μ ( x i ) | λ + | v ( x i ) | λ + | π ( x i ) | λ + | d ( x i ) | λ + | l ( x i ) | λ + ω λ ( x i A , x i B ) ) ] 1 λ(3.32)

Where λ is distance measure parameter, with λ ⩾ 1. When λ = 1, (3.31) and (3.32) reduce to the Hamming distance measures. When λ = 2, (3.31) and (3.32) reduces to the Euclidean distance measures. And ω ( x i A , x i B ) = 2 π ϑ ( x i A , x i B ) , where ϑ ( x i A , x i B ) is the angle between the vector x i A → and vector x i B → .

Furthermore, we propose the weighted distance measures.

Definition 3.7. Let A and B are two PFSs on X ={ x₁, x₂, …, x _n  }. w = (w₁, w₂, …, w _n )  ^T be the weights corresponding to the decision criteria. Then some new generalized weighted distance measures between A and B are defined as follows:

D PFSw ( A , B ) = [ 1 5 ∑ i = 1 n w i ( | μ ( x i ) | λ + | v ( x i ) | λ + | π ( x i ) | λ + | d ( x i ) | λ + | l ( x i ) | λ ) ] 1 λ(3.33)

D ω PFSw ( A , B ) = [ 1 6 ∑ i = 1 n w i ( | μ ( x i ) | λ + | v ( x i ) | λ + | π ( x i ) | λ + | d ( x i ) | λ + | l ( x i ) | λ + ω λ ( x i A , x i B ) ) ] 1 λ(3.34)

Where λ is distance measure parameter, with λ ⩾ 1. When λ = 1, (3.33) and (3.34) reduce to the Hamming distance measures. When λ = 2, (3.33) and (3.34) reduces to the Euclidean distance measures. And ω ( x i A , x i B ) = 2 π ϑ ( x i A , x i B ) , where ϑ ( x i A , x i B ) is the angle between the vector x i A → and vector x i B → .

Remark 1. When λ→ + ∞, the new generalizeddistance measures reduce to following formulas: lim λ → + ∞ D G ( α , β ) = max { | μ α - μ β | , | v α - v β | , | π α - π β | , | d α - d β | , | l α - l β | } lim λ → + ∞ D ω G ( α , β ) = max { | μ α - μ β | , | v α - v β | , | π α - π β | , | d α - d β | , | l α - l β | , ω λ ( α , β ) } lim λ → + ∞ D PFS ( A , B ) = max i { | μ A ( x i ) - μ B ( x i ) | , | v A ( x i ) - v B ( x i ) | , | π A ( x i ) - π B ( x i ) | , | d A ( x i ) - d B ( x i ) | , | l A ( x i ) - l B ( x i ) | } lim λ → + ∞ D ω PFS ( A , B ) = max i { | μ A ( x i ) - μ B ( x i ) | , | v A ( x i ) - v B ( x i ) | , | π A ( x i ) - π B ( x i ) | , | d A ( x i ) - d B ( x i ) | , | l A ( x i ) - l B ( x i ) | , ω ( x i A , x i B ) } . lim λ → + ∞ D PFSw ( A , B ) = max i { | μ A ( x i ) - μ B ( x i ) | , | v A ( x i ) - v B ( x i ) | , | π A ( x i ) - π B ( x i ) | , | d A ( x i ) - d B ( x i ) | , | l A ( x i ) - l B ( x i ) | } lim λ → + ∞ D ω PFSw ( A , B ) = max i { | μ A ( x i ) - μ B ( x i ) | , | v A ( x i ) - v B ( x i ) | , | π A ( x i ) - π B ( x i ) | , | d A ( x i ) - d B ( x i ) | , | l A ( x i ) - l B ( x i ) | , ω ( x i A , x i B ) }

Takahashi [46] has defined the distance measures on a nonempty set.

Definition 3.8. Let X be a nonempty set. A mapping D: X × X → [0, + ∞) is called a distance measure, which satisfies three properties in the following:

D (x, y) ⩾ 0 and D (x, y) = 0 if and only if x = y. (Nonnegative)

Theorem 3.5.

The above new distance measures belong in [0, 1].

Before the proving, we first introduce a related lemma.

Lemma 1. [47] (Minkowski Inequality) Let p, x₁, x₂, …, x _n and y₁, y₂, …, y _n are real numbers, and 1⩽ p ⩽ + ∞, then ( ∑ i = 1 n | x i + y i | p ) 1 p ⩽ ( ∑ i = 1 n | x i | p ) 1 p + ( ∑ i = 1 n | y i | p ) 1 p

We only prove the distance measure D _ωPFSw  (A, B) between PFSs A and B satisfy the condition (3) in Definition 3.8.

Proof. Let A, B and C be three PFSs on X ={ x₁, x₂, …, x _n  }. w = (w₁, w₂, …, w _n )  ^T be the weights corresponding to the decision criteria. Based on the basic properties of Space Geometry, we know that ϑ ( x i A , x i B ) ⩽ ϑ ( x i A , x i C ) + ϑ ( x i B , x i C ) for any x _i  ∈ X, which is shown in Fig. 5.

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Fig. 5

ϑ ( x i A , x i B ) ⩽ ϑ ( x i A , x i C ) + ϑ ( x i B , x i C ) .

Thus, ω ( x i A , x i B ) ⩽ ω ( x i A , x i C ) + ω ( x i B , x i C ) . On the other hand, it is obvious that |μ _A  (x _i ) - μ _B    (x _i ) | = |μ _A  (x _i ) - μ _C  (x _i ) + μ _C  (x _i ) - μ _B  (x _i ) |, |v _A    (x _i ) - v _B  (x _i ) | = |v _A  (x _i ) - v _C  (x _i ) + v _C  (x _i ) - v _B    (x _i ) |, |π _A  (x _i ) - π _B  (x _i ) | = |π _A  (x _i ) - π _C  (x _i ) + π _C    (x _i ) - π _B  (x _i ) |, |d _A  (x _i ) - d _B  (x _i ) | = |d _A  (x _i ) - d _C    (x _i ) + d _C  (x _i ) - d _B  (x _i ) |, |l _A  (x _i ) - l _B  (x _i ) | = |l _A    (x _i ) - l _C  (x _i ) + l _C  (x _i ) - l _B  (x _i ) |. Hence, utilize Lemma 1, D ω PFSw ( A , B ) = [ 1 6 ∑ i = 1 n w i ( | μ A ( x i ) - μ B ( x i ) | λ + | v A ( x i ) - v B ( x i ) | λ + | π A ( x i ) - π B ( x i ) | λ | d A ( x i ) - d B ( x i ) | λ + | l A ( x i ) - l B ( x i ) | λ + ω λ ( x i A , x i B ) ) ] 1 λ = [ 1 6 ∑ i = 1 n w i ( | μ A ( x i ) - μ C ( x i ) + μ C ( x i ) - μ B ( x i ) | λ + | v A ( x i ) - v C ( x i ) + v C ( x i ) - v B ( x i ) | λ + | π A ( x i ) - π C ( x i ) + π C ( x i ) - π B ( x i ) | λ + | d A ( x i ) - d C ( x i ) + d C ( x i ) - d B ( x i ) | λ + | l A ( x i ) - l C ( x i ) + l C ( x i ) - l B ( x i ) | λ + ω λ ( x i A , x i B ) ) ] 1 λ ≤ [ 1 6 ∑ i = 1 n w i ( | μ A ( x i ) - μ C ( x i ) + μ C ( x i ) - μ B ( x i ) | λ + | v A ( x i ) - v C ( x i ) + v C ( x i ) - v B ( x i ) | λ + | π A ( x i ) - π C ( x i ) + π C ( x i ) - π B ( x i ) | λ + | d A ( x i ) - d C ( x i ) + d C ( x i ) - d B ( x i ) | λ + | l A ( x i ) - l C ( x i ) + l C ( x i ) - l B ( x i ) | λ + ( ω ( x i A , x i C ) + ω ( x i B , x i C ) ) λ ) ] 1 λ ⩽ [ 1 6 ∑ i = 1 n w i ( | μ A ( x i ) - μ C ( x i ) | λ + | v A ( x i ) - v C ( x i ) | λ + | π A ( x i ) - π C ( x i ) | λ | d A ( x i ) - d C ( x i ) | λ + | l A ( x i ) - l C ( x i ) | λ + ω λ ( x i A , x i C ) ) ] 1 λ + [ 1 6 ∑ i = 1 n w i ( | μ B ( x i ) - μ C ( x i ) | λ + | v B ( x i ) - v C ( x i ) | λ + | π B ( x i ) - π C ( x i ) | λ | d B ( x i ) - d C ( x i ) | λ + | l B ( x i ) - l C ( x i ) | λ + ω λ ( x i B , x i C ) ) ] 1 λ = D ω PFSw ( A , C ) + D ω PFSw ( B , C ) . which completes the proof. □