Abstract
Xu (2017) published a paper in Journal of Intelligent and Fuzzy Systems in which he constructs a new distance measure that not only satisfies the axiom of intuitionistic fuzzy sets, but also fulfills the axiom for traditional distance. However, several questionable results arise in Xu (2017). Thus, the purpose of this paper is threefold. First, his proof is improved. Second, his criticism for two previously published distance measures are amended. Third, it is shown that in his numerical examples, there are several poorly-founded discussions. The refinement will help readers understand Xu (2017) and then apply his new distance measure to pattern recognition problems and medical diagnosis problems.
Introduction
The selection of a distance measure (or a similarity measure) to solve decision making problems under imperfect data and information has been a hot research topic since Zadeh [1] introduced fuzzy sets and Atanassov [2] developed intuitionistic fuzzy sets (IFSs). Xu [3] points out several existing distance measures that do not satisfy axioms for distance measures with IFSs proposed by Li and Cheng [4] and then constructs a new distance measure to fulfill not only axioms for IFSs but also satisfy axioms for traditional distance. Xu [3] converts an IFS expressed as (μ, v, π = 1 - μ - v) to a vector in
However, there are several questionable findings in Xu [3] that may hinder ordinary readers to absorb his distance measures. We will first point out those unreasonable results and then provide our improvement. Xu [3] already provides an excellent literature review for IFSs and interested readers are invited refer to Xu [3]. In this paper, concentrates on Xu [3].
Brief review of Xu [3]
Xu [3] considers an intuitionistic fuzzy set (IFS) A ={ 〈 x i , μ A (x i ) , v A (x i ) 〉 |x i ∈ X } where X is a universe of discourse, with X ={ x1, x2, . . . , x n }; μ A and v A are the membership and non-membership functions, 0 ≤ μ A (x i ) ≤ 1, 0 ≤ v A (x i ) ≤ 1, and μ A (x i ) + v A (x i ) ≤ 1; the hesitancy π A (x i ) = 1 - μ A (x i ) - v A (x i ).
Xu [3] uniformly distributes hesitancy to membership and non-membership to convert 〈μ
A
(x) , v
A
(x)〉 of an IFS to a vector in
Xu [3] claims that his new distance satisfies definitions of (a) the traditional definition for a distance, and (b) distance restriction for IFS proposed by Wang and Xin [5].
First recall the traditional definition for a distance. (Reproduction of Definition 2 of Xu [3])
A metric distance D in a nonempty set X is a real function D : X × X → [0, ∞), which satisfies the following conditions, ∀x, y, z ∈ X: (MD1) D (x, y) = 0 if and only if x = y
(MD2) D (x, y) = D (y, x); (MD3) D (x, y) + D (y, z) ≥ D (x, z).
Next, recall the definition for distances between IFSs proposed by Wang and Xin [5]. (Reproduction of Definition 4 of Xu [3])
Wang and Xin [5]. Let D be a mapping: IFSs (X) × IFSs (X) → [0, 1]. For A, B, C ∈ IFSs (X), D (A, B) is a distance measure between IFSs A and B, if D satisfies the following properties: 0 ≤ D (A, B) ≤ 1 ; D (A, B) = 0 if and only if A = B; D (A, B) = D (B, A); If A ⊆ B ⊆ C, then D (A, C) ≥ D (A, B), D (A, C) ≥ D (B, C).
Next, a proof of Xu [3] is founded to be questionable. To prove Property (MD3) for the traditional distance, Xu [3] claimed that
It is agreeable that
From the definition proposed by Xu [3], we know that
In Section 3, a proof to verify that (a) the distance measure proposed by Xu [3], and (b) extended distance measure, both satisfy Property (MD3) will be provided.
In Xu [3], several numerical examples are provided for us. In Section 5.1 of Xu [3], he examines a comparative analysis, cited from Papakostas et al. [6] and Li et al. [7], to point out some unreasonable derived by other distance measures. In Section 5.2 of Xu [3], he deals with pattern recognition problems. For Example 2, cited from Li and Cheng [4], Hatzimichailidis et al. [8] and Mitchell [9], Xu [3] used four different distance measures:
In the following we refer to some material from Xu [3] for our later discussion in Section 5. In Section 5.1, Xu [3] mentions that “The usage of artificial IFSs can highlight the deficiencies of the measurement methods, although they do not have any practical meaning. The example is adapted from Papakostas et al. [6] and Li et al. [7],” and, “In Table 2, the counter-intuitive cases have been highlighted by bold characters, which are used to describe the distance measures that are contrary to what seems correct from the optical analysis of the IFSs A
i
and B
i
(i = 1,2, ... ,6) being compared.” Xu [3] mentions that “ D (A3, B3) = 0 in
It is cited from Xu [3], “There are special notes amongst the measures
We must point out for those 4 measures,
The main purpose of this paper is threefold. First, the distance proposed by Xu [3] to an abstract setting is extended such that Xu [3] becomes a special case of the proposed distance. We will verify that the extended distance satisfies two definitions: (a) Traditional distance, and (b) Distance for IFS proposed by Wang and Xin [5]. By the way, the questionable results of Xu [3] for the proof with respect to traditional distance is improved. Second, Xu [3] mentions that distance measures proposed by Szmidt and Kacprzyk [12] and Yang and Chiclana [13] did not satisfy axioms for IFSs. It will be proved that Szmidt and Kacprzyk [12] and Yang and Chiclana [13] both satisfied Property (DP4) proposed by Wang and Xin [5]. Third, Table 2 of Xu [3] will be examined to point out several questionable results and then provide our comments.
The distance measure of Xu [3] is generalized from a uniformed partition to an arbitrary rate to distribute hesitancy to membership and non- membership to convert 〈μ
A
(x) , v
A
(x)〉 of an IFS to a vector in
For two IFSs, A ={ 〈 x
i
, μ
A
(x
i
) , v
A
(x
i
) 〉 |x
i
∈ X } and A ={ 〈 x
i
, μ
A
(x
i
) , v
A
(x
i
) 〉 |x
i
∈ X }, a new distance measure, denoted as
We begin to verify that the extended distance measure satisfying the Property (DP4) proposed by [5], since proofs for other Properties (DP1), (DP2) and (DP3) are straightforward.
(Proof) The definition of A ⊆ B ⊆ C is assumed as μ A (x i ) ≤ μ B (x i ) ≤ μ C (x i ) and v A (x i ) ≥ v B (x i ) ≥ v C (x i ), for x i ∈ X, with i = 1, 2, . . . , n.
Owing to we will prove that for i = 1, 2, . . . , n,
Given an i belonging to {1, 2, . . . , n}, and then we derive that
From μ
A
(x
i
) ≤ μ
B
(x
i
) ≤ μ
C
(x
i
), we know that
From v
A
(x
i
) ≥ v
B
(x
i
) ≥ v
C
(x
i
), we know that
We find that
From μ
C
(x
i
) - μ
A
(x
i
) ≥ μ
B
(x
i
) - μ
A
(x
i
) ≥0 and v
A
(x
i
) - v
C
(x
i
) v
A
(x
i
) - v
B
(x
i
) ≥0, we find that
By the same argument, we evaluate that
From μ
C
(x
i
) - μ
A
(x
i
) ≥ μ
B
(x
i
) - μ
A
(x
i
) ≥0 and v
A
(x
i
) - v
C
(x
i
) ≥ v
A
(x
i
) - v
B
(x
i
) ≥0, we obtain that
Now we compare Δ
AB
(i) and Δ
AC
(i). Recalling Equation (13) and (14), and inequalities of Equation (16), (18), (21) and (24), we prove that the assertion of Equation (12) is valid such that we verify
From the above discussions of Theorem 1, we prove that our extended distance measure, under the condition, 0 ≤ w ≤ 1, satisfies Property (DP4). Thus, the original distance measure proposed by Xu [3] becomes a special case with w = 1/ - 2.
We begin to provide a simple proof to show our extended distance measure satisfying Property (MD3). We can treat (μ, v, μ + (1 - w) π, v + wπ) as a point in
The p-norm for two vectors in
Hence, we can treat
From the well-known Minkowski’s inequality, “the triangle inequality” holds for p-norm to imply that
We can derive that
We summarize our results in the next theorem.
The distance measure of Xu [3] is a special case with w = 0.5 for our extended distance measure, such that the distance measure of Xu [3] also fulfills Property (DM3).
Next, we consider our second purpose to show that the assertions of Xu [3] to criticize Szmidt and Kacprzyk [12], and Yang and Chiclana [13] are invalid.
For later discussion, we must point out several questionable results in Xu [3]. At Page 1566, Line 7, left column, Xu [3] mentions that
We derive that
Our derivation is supported by the equality of Equation (32).
Xu [3] discusses the distance proposed by Szmidt and Kacprzyk [12] as
If A ⊆ B ⊆ C, Xu [3] mentions that “then μ
A
≤ μ
B
≤ μ
C
and v
A
≥ v
B
≤ v
C
, we therefore have
(Proof) Under the restrictions of μ A ≤ μ B ≤ μ C and v A ≥ v B ≥ v C , we will verify that the inequality of Equation (36) is valid.
Since
By the same argument, we know that
We recall that A ⊆ B ⊆ C to imply
Based on inequalities of Equation (39) and (40), we derive that
Following the same way, we can show that
Based on our Theorem 3, the assertion of Xu [3] that “If A ⊆ B ⊆ C, then the relationship between
In the following, we begin to discuss the criticism of Xu [3] related to Yang and Chiclana [13]. At page 1567 of Xu [3], right column, Lines 5–7, he mentions that “the expressions (14)–(16) may not satisfy the Definition 4.”
We recall expressions (14)–(16) of Xu [3] that are related to Yang and Chiclana [13]. We recall their first distance,
Properties of (DP1-DP3) are easily to prove, so we concentrate our discussion on Property (DP4).
(Proof) If A ⊆ B ⊆ C, we will verify that
We will show that for i = 1, 2, . . . , n,
Hence, we will further simplify our proof to show that
We recall that in our Theorem 3, we derive that
By the same argument, we find that
Let’s recall Equation (39) and (40) to imply that the inequality of Equation (45) is verified and then we finish the proof of
The other distances of Yang and Chiclana [13] can also be verified to satisfy the Property (DP4). Therefore, Xu’s concern [3] concerning the distances of Yang and Chiclana [13] is unwarranted.
For our third purpose, we begin to discuss numerical examples in Xu [3].
Based on the assertion of Xu [3] cited in Section 2, to imply that for D (A3, B3) and D (A4, B4), the measure, denoted as D, did not make a distinction, then Xu [3] will claimed that D has problems.
We recall the reason for Xu [3] to abandon
For the other 21 distances, with respect to two pairs: (i) A5 and B5, and (ii) A6 and B6, all imply that
Xu [3] claims that for p = 0.5 and p = 1, two of his proposed distances, then
Consequently, for the rest of his paper, Xu [3] would discuss 1< p < ∞ and then he only considers
By the same argument, we consider another pair: (i) A1 and B1, and (ii) A2 and B2, all the other 21 distances proposed by other researchers derived that
Table 1 (Table 2 of Xu [3]) reveals that
By the same argument, Xu [3] gives up
We begin to point out the bold faced items of D (A
i
, B
i
) = D (A
j
, B
j
), for i ≠ j, and i, j∈ { 1, 2, . . . , 6 }. D (A
i
, B
i
) = 0, for i∈ { 1, 2, . . . , 6 }. The sign of D (A2k+1, B2k+1) - D (A2k+2, B2k+2) is contrary to the findings of other measures.
From Table 1 (that is Table 2 of Xu [3]), we find that Rule 1 and Rule 2 do not apply to the above mentioned six measures with fold face. For Rule 3, for total 25 measures, eleven of them have D (A3, B3) > D (A4, B4), and three of them have D (A3, B3) < D (A4, B4), and then the remaining eleven distances have ties as D (A3, B3) = D (A4, B4). Hence, by Rule 3, the majority distance claims that D (A3, B3) ≥ D (A4, B4).
Our above discussion can explain why
Furthermore, if our above discussion is acceptable, and then
We recall that in Sections 5.2 and 5.3 of Xu [3], he applies four distance measures:
(Proof) We derive that
For Tables 4, 6, 8 and 11, Xu [3] computes
We provide an example with a sample A ={ 〈 0, 1 〉 } that μ
A
= 0, v
A
= 1 with π
A
= 0 such that μ
A
+ (π
A
/ - 2) = 0 and v
A
+ (π
A
/ - 2) = 1, and two patterns: B ={ 〈 0.6, 0.35 〉 } and C ={ 〈 0, 0 〉 }. We imply that μ
B
= 0.6, v
B
= 0.35 with π
B
= 0.05 such that μ
B
+ (π
B
/ - 2) = 0.625 and v
B
+ (π
B
/ - 2) = 0.375, and μ
C
= 0, v
C
= 0 with π
C
= 1 to derive such that μ
C
+ (π
C
/ - 2) = 0.5 and v
V
+ (π
C
/ - 2) = 0.5. Hence,
Based on our counter example, we point out that providing
Deciding how to select the value of w for our new distance measure is an important issue. On Page 3061 of Hung et al. [14], they discussed the selection for the ration, w, to distribute hesitancy (abstention) to membership and non-membership functions, “When λ = 0, it shows that the decision-maker is the most pessimistic, because it can’t obtain anything from the part of abstention, when λ = 0.5, it shows that the decision-maker is fair and can obtain a half of abstention and λ = 1, it shows that the decision-maker is in the most optimistic situation, and can get completely support from abstention.”
In Xu [3], he selects w = 0.5 to indicate the decision-maker is fair to membership and non-membership functions. We recall Example 2 of Xu [3] for a pattern recognition problem, with one sample, S and three patterns, P1 and P3 that were discussed in [4], [8] and [9]. We cite Table 3 of Xu [3] for their IFS data.
Comparison between w = 0.5 and w = 1 for Example 2 of Xu [3]
Comparison between w = 0.5 and w = 1 for Example 2 of Xu [3]
From an optimistic point of view, we select w = 1 to compute the distance among the sample, S and three patterns, P1, P2 and P3 to list our result in Table 3. We also cite the findings of Xu [3], with w = 0.5 and p = 2. After we change the value of w, from w = 0.5, to w = 1, we still obtain that the sample S belongs to pattern P3. However, we find that
We also compute the relative ratio between (a) the best (smallest) distance, and (b) the second best distance, for w = 0.5 and w = 1, to imply that
Next, we recall Example 3 of Xu [3] for a pattern recognition problem, with one sample, S and three patterns, P1, P2 and P3 that discussed in [8], [10] and [11]. We cite Table 5 of Xu [3] for their IFS data.
Comparison between w = 0.5 and w = 0 for Example 3 of Xu [3]
From a pessimistic point of view, we select w = 0 to compute the distance among the sample, S and three patterns, P1, P2 and P3 to list our result in the Table 5. We also cite the findings of Xu [3], with w = 0.5 and p = 2.
After we change the value of wfrom w = 0.5, to w = 0, we still obtain that the sample S belongs to pattern P2. However, we find that
We also compute the relative ratio between (a) the best (smallest) distance, and (b) the second best distance, for w = 0.5 and w = 0, to imply that
From the above discussion, we demonstrate that sometimes selecting w ≠ 0.5 can help decision makers to solve the pattern recognition problems. Hence, for the future research, how to decide the optimal value of w to help decision makers will be an interesting research topic.
Last, but not least, we will provide a reasonable explanation for why Xu [3] and our generalization can pass the Property (DP4) of the axiom for a distance measure. By our generalization, we extend distance of Xu [3] from (μ, v, μ + (π/2) , v + (π/2)) to (μ, v, μ + (1 - w) π, v + wπ) such that distance of Xu [3] becomes a special case with w = 1/2 for our extension.
We rewrite (μ, v, μ + (1 - w) π, v + wπ) as
We recall that when A ⊆ B ⊆ C that means for the membership μ A ≤ μ B ≤ μ C and for the non-membership v A ≥ v B ≥ v C . We observe that the order relation among A, B and C with respect to μ and v, then the ordering is opposite. Hence, the ordering relation among A, B and C is the same for (i) μ and 1 - v, and (ii) 1-μ and v. Consequently, the proof for Property (DP4) can be verified in our Theorem 1.
Xu [3] converts an IFS into a vector in
Footnotes
Acknowledgments
This research is partially supported by Ministry of Science and Technology, R.O.C. with Grant no. MOST 107-2410-H-156-010.
