Extension and revisions for Xu’s proposed distance measure

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.

Keywords

Distance measure similarity measure intuitionistic fuzzy sets

1 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 $R^{4}$ as (μ, v, μ + (π/2) , v + (π/2)). He adopts p-norm to construct his normalized distance measure. The new distance measure in Xu [3] provides an important method applicable by future practitioners to solve operational problems with respect to decision making, image segmentation, pattern recognition, edge detection, and fault-tree analysis.

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].

2 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 $R^{4}$ as $(μ_{A} (x), v_{A} (x), μ_{A} (x) + \frac{π_{A} (x)}{2}, v_{A} (x) + \frac{π_{A} (x)}{2})$ (1) and then refers to the p-norm in $R^{4}$ to construct new distance measures as follows $\begin{matrix} D_{IFS s}^{p} (A, B) = \\ {\frac{1}{4} \sum_{i = 1}^{n} [\begin{matrix} {| Δ_{μ} (i) |}^{p} + {| Δ_{v} (i) |}^{p} \\ + {| Δ_{π μ} (i) |}^{p} + {| Δ_{π v} (i) |}^{p} \end{matrix}]}^{\frac{1}{p}} \end{matrix}$ (2) and $\begin{matrix} D_{IFS s}^{np} (A, B) = \\ {\frac{1}{4 n} \sum_{i = 1}^{n} [\begin{matrix} {| Δ_{μ} (i) |}^{p} + {| Δ_{v} (i) |}^{p} \\ + {| Δ_{π μ} (i) |}^{p} + {| Δ_{π v} (i) |}^{p} \end{matrix}]}^{\frac{1}{p}}, \end{matrix}$ (3) where $\begin{matrix} Δ_{μ} (i) = μ_{A} (x_{i}) - μ_{B} (x_{i}), Δ_{v} (i) = v_{A} (x_{i}) - v_{B} (x_{i}), \\ Δ_{π μ} (i) = {Assign}_{A}^{π μ} (x_{i}) - {Assign}_{B}^{π μ} (x_{i}), \\ Δ_{π v} (i) = {Assign}_{A}^{π v} (x_{i}) - {Assign}_{B}^{π v} (x_{i}) \end{matrix}$ ${Assign}_{A}^{π μ} (x_{i}) = μ_{A} (x_{i}) + \frac{1}{2} π_{A} (x_{i}),$ ${Assign}_{B}^{π μ} (x_{i}) = μ_{B} (x_{i}) + \frac{1}{2} π_{B} (x_{i}),$ ${Assign}_{A}^{π v} (x_{i}) = v_{A} (x_{i}) + \frac{1}{2} π_{A} (x_{i}),$ ${Assign}_{B}^{π v} (x_{i}) = v_{B} (x_{i}) + \frac{1}{2} π_{B} (x_{i}) .$

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 ${(Δ_{μ}^{AC})}^{2} = {(Δ_{μ}^{AB} + Δ_{μ}^{BC})}^{2} \leq {(Δ_{μ}^{AB})}^{2} + {(Δ_{μ}^{BC})}^{2},$ (4) ${(Δ_{v}^{AC})}^{2} = {(Δ_{v}^{AB} + Δ_{v}^{BC})}^{2} \leq {(Δ_{v}^{AB})}^{2} + {(Δ_{v}^{BC})}^{2},$ (5) ${(Δ_{π μ}^{AC})}^{2} = {(Δ_{π μ}^{AB} + Δ_{π μ}^{BC})}^{2} \leq {(Δ_{π μ}^{AB})}^{2} + {(Δ_{π μ}^{BC})}^{2},$ (6) ${(Δ_{π v}^{AC})}^{2} = {(Δ_{π v}^{AB} + Δ_{π v}^{BC})}^{2} \leq {(Δ_{π v}^{AB})}^{2} + {(Δ_{π v}^{BC})}^{2} .$ (7)

It is agreeable that ${(Δ_{μ}^{AC})}^{2} = {(Δ_{μ}^{AB} + Δ_{μ}^{BC})}^{2}$ , owing to $Δ_{μ}^{AC} = Δ_{μ}^{AB} + Δ_{μ}^{BC}$ . However, his assertion of ${(Δ_{μ}^{AB} + Δ_{μ}^{BC})}^{2} \leq {(Δ_{μ}^{AB})}^{2} + {(Δ_{μ}^{BC})}^{2}$ is questionable. It can be derived that ${(Δ_{μ}^{AB} + Δ_{μ}^{BC})}^{2} \leq {(Δ_{μ}^{AB})}^{2} + {(Δ_{μ}^{BC})}^{2}$ if and only if $\begin{matrix} 2 Δ_{μ}^{AB} Δ_{μ}^{BC} + {(Δ_{μ}^{AB})}^{2} + {(Δ_{μ}^{BC})}^{2} \leq \\ {(Δ_{μ}^{AB})}^{2} + {(Δ_{μ}^{BC})}^{2}, \end{matrix}$ (8) that is equivalent to $Δ_{μ}^{AB} Δ_{μ}^{BC} \leq 0 .$ (9)

From the definition proposed by Xu [3], we know that $Δ_{μ}^{AB} = μ_{A} - μ_{B}$ and $Δ_{μ}^{BC} = μ_{B} - μ_{C}$ . For three intuitionistic fuzzy sets, A, B, C ∈ IFSs (X), we cannot ensure that μ_A - μ_B and μ_B - μ_C have two different signs. Hence, the proof provided by Xu [3] to prove his new distance satisfying Property (MD3) contains questionable results.

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: $D_{I F S s}^{p} |_{p = 2}, D_{I F S s}^{n p} |_{p = 2}, D_{I F S s}^{p} |_{p = 3} and D_{I F S s}^{n p} |_{p = 3}$ to locate the closest pattern for the given sample. By the same argument, for Example 3, cited from Hung and Yang [10, 11] and Hatzimichailidis et al. [8], and for Example 4, cited from Wang and Xin [5] and Hatzimichailidis et al. [8], Xu [3] also applied four different distance measures: $D_{I F S s}^{p} |_{p = 2}, D_{I F S s}^{n p} |_{p = 2}, D_{I F S s}^{p} |_{p = 3} and D_{I F S s}^{n p} |_{p = 3}$ to find the closest pattern for the given sample. In Section 5.3 of Xu [3], he considered medical diagnosis problems by four different distance measures: $D_{I F S s}^{p} |_{p = 2}, D_{I F S s}^{n p} |_{p = 2}, D_{I F S s}^{p} |_{p = 3} and D_{I F S s}^{n p} |_{p = 3}$ for four patients – Al, Bob, Joe and Ted – to search for possible diagnoses. For the sake of completeness, Table 2 of Xu [3] is cited in the Table 1.

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 $D_{R}^{imp}$ , which is inconsistent with the Property (DP2) of Definition 2, since A₃ ≠ B₃”.

It is cited from Xu [3], “There are special notes amongst the measures $D_{H}^{Ata}$ , $D_{E}^{Ata}$ , $D_{nE}^{Liu}$ (p = 2), $D_{H}^{G - Hu - C}$ , $D_{E}^{G - H - C}$ , D^Wa-Xi, $D_{1}^{Wa - Xi}$ , $D_{2}^{Wa - Xi}$ and $D_{H}^{Sz - Ka}$ , $D_{E}^{Sz - Ka}$ , $D_{H}^{Ya - Ch}$ , $D_{E}^{Ya - Ch}$ , $D_{H}^{Lu - Xi}$ . The distance between A₂ and B₂ amongst these measures is equal to 0.1, but it is clear that the distances between A₃ and B₃, A₄ and B₄, A₅ and B₅, and A₆ and B₆, may result in contradictory results from Table 2. The reason behind this different resides again in the implementation or not of the hesitation margins in the distance function.”

We must point out for those 4 measures, $D_{IFSs}^{0.5}$ , $D_{IFSs}^{1}$ , $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ proposed by Xu [3], all derived that D (A₂, B₂) = 0.1000 to reveal his argument about 0.1 is questionable. Moreover, Xu [3] mentions, “the distance between A₃ and B₃ is equal to the distance between A₄ and B₄ in measures $D_{H}^{Sz - Ka}$ , $D_{H}^{Ya - Ch}$ , $D_{E}^{Ya - Ch}$ and $D_{H}^{Lu - Xi}$ . This is obviously contradictory against the intuition, and hence is incorrect. However, the new distances $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ avoid the occurrence of these contradictions.”

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.

Table 1

The comparison of distances between IFSs A_i and B_i (i = 1, 2, . . . , 6) (Reproduction of Table 2 of Xu [3])

	A₁ & B₁	A₂ & B₂	A₃ & B₃	A₄ & B₄	A₅ & B₅	A₆ & B₆
$D_{H}^{Ata}$	0.10	0.10	0.50	0.50	0.10	0.05
$D_{E}^{Ata}$	0.10	0.10	0.71	0.50	0.10	0.07
$D_{H}^{Sz - Ka}$	0.20	0.10	1.00	1.00	0.20	0.10
$D_{E}^{Sz - Ka}$	0.17	0.10	1.00	0.87	0.17	0.10
$D_{nE}^{Liu} (p = 2)$	0.17	0.10	1.00	0.87	0.17	0.10
$D_{H}^{G - H - C}$	0.10	0.10	1.00	0.50	0.10	0.10
$D_{E}^{G - H - C}$	0.10	0.10	1.00	0.50	0.10	0.10
$D_{H}^{Ya - Ch}$	0.20	0.10	1.00	1.00	0.20	0.10
$D_{E}^{Ya - Ch}$	0.20	0.10	1.00	1.00	0.20	0.10
$D_{H}^{Lu - Xi}$	0.20	0.10	1.00	1.00	0.20	0.10
D ^Wa-Xi	0.10	0.10	0.75	0.50	0.10	0.075
$D_{1}^{Wa - Xi}$	0.10	0.10	0.50	0.50	0.10	0.05
$D_{2}^{Wa - Xi}$	0.10	0.10	0.50	0.50	0.10	0.05
$D_{R}^{imp}$	0.06	0.06	0.00	0.50	0.06	0.01
$D_{G}^{imp}$	0.00	0.00	0.00	0.00	0.00	0.00
$D_{L}^{imp}$	0.00	0.00	0.00	0.00	0.00	0.00
$D_{K D}^{imp}$	0.20	0.20	0.00	1.00	0.20	0.10
$D_{M}^{imp}$	0.20	0.20	1.00	1.00	0.20	0.10
$D_{L a}^{imp}$	0.14	0.14	1.00	0.50	0.14	0.09
$D_{T}^{imp}$	0.00	0.00	0.00	0.00	0.00	0.00
$D_{D}^{imp}$	0.20	0.20	0.00	1.00	0.20	0.10
$D_{IFSs}^{0.5}$	0.0250	0.1000	0.3643	0.1250	0.0250	0.0364
$D_{IFSs}^{1}$	0.0500	0.1000	0.5000	0.2500	0.0500	0.0500
$D_{IFSs}^{2}$	0.0707	0.1000	0.6124	0.3536	0.0707	0.0612
$D_{IFSs}^{3}$	0.0794	0.1000	0.6786	0.3969	0.0794	0.0679

Table 2

3 Classes/3 attributes problem, Patterns and Test sample (Reproduction of Table 3 of of Xu [3])

Patterns &Test sample	Attributes 〈x_i, μ (x_i) , v (x_i)〉 (i = 1, 2, 3)
	x ₁	x ₂	x ₃
P ₁	〈x₁, 1.0, 0.0〉	〈x₂, 0.8, 0.0〉	〈x₃, 0.7, 0.1〉
P ₂	〈x₁, 0.8, 0.1〉	〈x₂, 1.0, 0.0〉	〈x₃, 0.9, 0.0〉
P ₃	〈x₁, 0.6, 0.2〉	〈x₂, 0.8, 0.0〉	〈x₃, 1.0, 0.0〉
S	〈x₁, 0.5, 0.3〉	〈x₂, 0.6, 0.2〉	〈x₃, 0.8, 0.1〉

3 Generalization of distance measure

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 $R^{4}$ as $\begin{matrix} (μ_{A} (x), v_{A} (x), μ_{A} (x) + (1 - w) π_{A} (x), v_{A} (x) + \\ w π_{A} (x)) \end{matrix}$ (10)

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 $D_{IFSs}^{w}$ , is defined. $D_{IFS s}^{w} (A, B) = {(\frac{1}{4 n} \sum_{i = 1}^{n} Δ^{AB} (i))}^{1 / 2}$ (11) where $\begin{matrix} Δ^{AB} (i) & = & {| Δ_{μ}^{AB} (i) |}^{2} + {| Δ_{v}^{AB} (i) |}^{2} + {| Δ_{π μ}^{AB} (i) |}^{2} \\ + {| Δ_{π v}^{AB} (i) |}^{2} \\ Δ_{μ}^{AB} (i) & = & μ_{A} (x_{i}) - μ_{B} (x_{i}), \\ Δ_{v}^{AB} (i) & = & v_{A} (x_{i}) - v_{B} (x_{i}), \\ Δ_{π μ}^{AB} (i) & = & (μ_{A} (x_{i}) + (1 - w) π_{A} (x_{i})) \\ - & (μ_{B} (x_{i}) + (1 - w) π_{B} (x_{i})), \\ Δ_{π v}^{AB} (i) & = & (v_{A} (x_{i}) + (1 - w) π_{A} (x_{i})) - (v_{B} (x_{i}) \\ + & (1 - w) π_{B} (x_{i})) . \end{matrix}$

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.

Theorem 1. When three IFSs, A ⊆ B ⊆ C, then $D_{IFS s}^{w} (A, B) \leq D_{IFS s}^{w} (A, C)$ and $D_{IFS s}^{w} (B, C) \leq D_{IFS s}^{w} (A, C)$ .

(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, $Δ^{AB} (i) \leq Δ^{AC} (i) .$ (12)

Given an i belonging to {1, 2, . . . , n}, and then we derive that $\begin{matrix} Δ^{AB} (i) = {(Δ_{μ}^{AB} (i))}^{2} + {(Δ_{v}^{AB} (i))}^{2} \\ + {(Δ_{π μ}^{AB} (i))}^{2} + {(Δ_{π v}^{AB} (i))}^{2} \end{matrix}$ (13) and $\begin{matrix} Δ^{AC} (i) = {(Δ_{μ}^{AC} (i))}^{2} + {(Δ_{v}^{AC} (i))}^{2} \\ + {(Δ_{π μ}^{AC} (i))}^{2} + {(Δ_{π v}^{AC} (i))}^{2} . \end{matrix}$ (14)

From μ_A (x_i) ≤ μ_B (x_i) ≤ μ_C (x_i), we know that $\begin{matrix} Δ_{μ}^{AC} (i) = μ_{A} (x_{i}) - μ_{C} (x_{i}) \\ \leq Δ_{μ}^{AB} (i) = μ_{A} (x_{i}) - μ_{B} (x_{i}) \leq 0 \end{matrix}$ (15) to imply that ${(Δ_{μ}^{AC} (i))}^{2} \geq {(Δ_{μ}^{AB} (i))}^{2} .$ (16)

From v_A (x_i) ≥ v_B (x_i) ≥ v_C (x_i), we know that $\begin{matrix} Δ_{v}^{AC} (i) = v_{A} (x_{i}) - v_{C} (x_{i}) \geq \\ Δ_{v}^{AB} (i) = v_{A} (x_{i}) - v_{B} (x_{i}) \geq 0, \end{matrix}$ (17) to imply that ${(Δ_{v}^{AC} (i))}^{2} \geq {(Δ_{v}^{AB} (i))}^{2} .$ (18)

We find that $\begin{matrix} Δ_{π μ}^{AB} (i) = [μ_{A} (x_{i}) + (1 - w) (1 - μ_{A} (x_{i}) - v_{A} (x_{i}))] \\ - [μ_{B} (x_{i}) + (1 - w) (1 - μ_{B} (x_{i}) - v_{B} (x_{i}))] \leq 0 \\ = (- 1) [\begin{matrix} w (μ_{B} (x_{i}) - μ_{A} (x_{i})) \\ + (1 - w) (v_{A} (x_{i}) - v_{B} (x_{i})) \end{matrix}] \end{matrix}$ (19) and $\begin{matrix} Δ_{π μ}^{AC} (i) = [μ_{A} (x_{i}) + (1 - w) (1 - μ_{A} (x_{i}) - v_{A} (x_{i}))] \\ - [μ_{C} (x_{i}) + (1 - w) (1 - μ_{C} (x_{i}) - v_{C} (x_{i}))] \\ = [μ_{A} (x_{i}) + (1 - w) (1 - μ_{A} (x_{i}) - v_{A} (x_{i}))] \\ - [μ_{C} (x_{i}) + (1 - w) (1 - μ_{C} (x_{i}) - v_{C} (x_{i}))] \\ = (- 1) [\begin{matrix} w (μ_{C} (x_{i}) - μ_{A} (x_{i})) \\ + (1 - w) (v_{A} (x_{i}) - v_{C} (x_{i})) \\ \leq 0 \end{matrix}], \end{matrix}$ (20)

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 $Δ_{π μ}^{AC} (i) \leq Δ_{π μ}^{AB} (i) \leq 0$ such that ${(Δ_{π μ}^{AC} (i))}^{2} \geq {(Δ_{π μ}^{AB} (i))}^{2} .$ (21)

By the same argument, we evaluate that $\begin{matrix} Δ_{π v}^{AB} (i) = (μ_{A} (x_{i}) + w (1 - μ_{A} (x_{i}) - v_{A} (x_{i}))) \\ - (μ_{B} (x_{i}) + w (1 - μ_{B} (x_{i}) - v_{B} (x_{i}))) \\ = (- 1) [\begin{matrix} (1 - w) (μ_{B} (x_{i}) - μ_{A} (x_{i})) \\ + w (v_{A} (x_{i}) - v_{B} (x_{i})) \end{matrix}], \end{matrix}$ (22) and $\begin{matrix} Δ_{π v}^{AC} (i) = (μ_{A} (x_{i}) + w (1 - μ_{A} (x_{i}) - v_{A} (x_{i}))) \\ - (μ_{C} (x_{i}) + w (1 - μ_{C} (x_{i}) - v_{C} (x_{i}))) \\ = (- 1) [\begin{matrix} (1 - w) (μ_{C} (x_{i}) - μ_{A} (x_{i})) \\ + w (v_{A} (x_{i}) - v_{C} (x_{i})) \end{matrix}] . \end{matrix}$ (23)

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 $Δ_{π v}^{AC} (i) \leq Δ_{π v}^{AB} (i) \leq 0$ such that we derive ${(Δ_{π v}^{AC} (i))}^{2} \geq {(Δ_{π v}^{AB} (i))}^{2} .$ (24)

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 $D_{IFS s}^{w} (A, B) \leq D_{IFS s}^{w} (A, C)$ . By the same argument, we also can prove $D_{IFS s}^{w} (B, C) \leq D_{IFS s}^{w} (A, C)$ .

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 $R^{4}$ and then we consider $((\begin{matrix} μ (x), v (x), w μ (x) + (1 - w) (1 - v (x)), \\ w (1 - μ (x)) + (1 - w) v (x) \end{matrix}) : x \in X) .$ (25) with X ={ x₁, x₂, . . . , x_n }, as a vector in $R^{4 n}$ . We convert A ∈ IFSs (X), with X ={ x₁, x₂, . . . , x_n } to a vector in $R^{4 n}$ . We clearly denote our transformation as $R^{4 n} (A)$ for the IFS, A.

The p-norm for two vectors in $R^{4 n}$ as R = (r₁, r₂, . . . , r_4n) and S = (s₁, s₂, . . . , s_4n) is defined as $d_{p} (R, S) = {(\sum_{j = 1}^{4 n} {| r_{j} - s_{j} |}^{p})}^{\frac{1}{p}} .$ (26)

Hence, we can treat $4^{\frac{1}{p}} D_{IFSs}^{p} (A, B) = d_{p} (R^{4 n} (A), R^{4 n} (B)) .$ (27)

From the well-known Minkowski’s inequality, “the triangle inequality” holds for p-norm to imply that $\begin{matrix} d_{p} (R^{4 n} (A), R^{4 n} (C)) \leq \\ d_{p} (R^{4 n} (A), R^{4 n} (B)) + d_{p} (R^{4 n} (B), R^{4 n} (C)) . \end{matrix}$ (28)

We can derive that $\begin{matrix} 4^{\frac{1}{p}} D_{IFSs}^{p} (A, C) \leq \\ 4^{\frac{1}{p}} D_{IFSs}^{p} (A, B) + 4^{\frac{1}{p}} D_{IFSs}^{p} (B, C) \end{matrix}$ (29) to obtain the wanted Property (DM3) as $D_{IFSs}^{p} (A, C) \leq D_{IFSs}^{p} (A, B) + D_{IFSs}^{p} (B, C) .$ (30)

We summarize our results in the next theorem.

Theorem 2. Our extended distance measure satisfies Property (DM3).

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).

4 Our discussion with respect to Szmidt and Kacprzyk [12], and Yang and Chiclana [13]

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 $“ | Δ_{π} (i) | = | Δ_{μ} (i) - Δ_{v} (i) | \leq | Δ_{μ} (i) | + | Δ_{v} (i) |$ (31) and ${(Δ_{π} (i))}^{2} = {(Δ_{μ} (i))}^{2} + {(Δ_{v} (i))}^{2} + 2 Δ_{μ} (i) \cdot Δ_{v} (i)^{″}$ (32) as Equation (7) of Xu [3]. However, we evaluate that $\begin{matrix} Δ_{π} = π_{A} - π_{B} = 1 - μ_{A} - v_{A} - (1 - μ_{B} - v_{B}) \\ = - (μ_{A} - μ_{B}) - (v_{A} - v_{B}) \\ = - Δ_{μ} - Δ_{v} . \end{matrix}$ (33)

We derive that $| Δ_{π} (i) | = | Δ_{μ} (i) + Δ_{v} (i) | \leq | Δ_{μ} (i) | + | Δ_{v} (i) | .$ (34)

Our derivation is supported by the equality of Equation (32).

Xu [3] discusses the distance proposed by Szmidt and Kacprzyk [12] as $D_{H}^{Sz - Ka} (A, B) = \frac{1}{2} \sum_{i = 1}^{n} (| Δ_{μ} (i) | + | Δ_{v} (i) | + | Δ_{π} (i) |),$ (35) where Δ_μ (i) = μ_A (x_i) - μ_B (x_i) , Δ_v (i) = v_A (x_i) - v_B (x_i) , and Δ_π (i) = π_A (x_i) - π_B (x_i).

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 $| Δ_{μ}^{AB} | \leq | Δ_{μ}^{AC} |$ , $| Δ_{v}^{AB} | \leq | Δ_{v}^{AC} |$ However, it is difficult to judge $| Δ_{π}^{AB} | \leq | Δ_{π}^{AC} |$ or $| Δ_{π}^{AB} | \geq | Δ_{π}^{AC} |$ ... ”. First, we revise the typo of Xu [3] in the above citation, to improve the expression from v_A ≥ v_B ≤ v_C to v_A ≥ v_B ≥ v_C. Next, we must point out that whether or not $| Δ_{π}^{AB} | \leq | Δ_{π}^{AC} |$ or $| Δ_{π}^{AB} | \geq | Δ_{π}^{AC} |$ is not the key issue. The key issue is whether or not $| Δ_{μ}^{AB} | + | Δ_{v}^{AB} | + | Δ_{π}^{AB} | \leq | Δ_{μ}^{AC} | + | Δ_{v}^{AC} | + | Δ_{π}^{AC} |,$ (36) that is, to verify the inequality of Equation (36), we do not need the following three inequalities: $| Δ_{μ}^{AB} | \leq | Δ_{μ}^{AC} |$ , $| Δ_{v}^{AB} | \leq | Δ_{v}^{AC} |$ , and $| Δ_{π}^{AB} | \leq | Δ_{π}^{AC} |$ all are valid.

Theorem 3. $D_{H}^{Sz - Ka}$ satisfies (DP4).

(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 $Δ_{μ}^{AB} = μ_{A} - μ_{B} \leq 0$ , we imply that $| Δ_{μ}^{AB} | = μ_{B} - μ_{A}$ . Owing to $Δ_{v}^{AB} = v_{A} - v_{B} \geq 0$ , we derive that $| Δ_{v}^{AB} | = v_{A} - v_{B}$ . From $Δ_{π}^{AB} = μ_{B} + v_{B} - μ_{A} - v_{A} = (μ_{B} - μ_{A}) - (v_{A} - v_{B})$ , then we obtain that $\begin{matrix} | Δ_{μ}^{AB} | + | Δ_{v}^{AB} | + | Δ_{π}^{AB} | \\ = (μ_{B} - μ_{A}) + (v_{A} - v_{B}) + | (μ_{B} - μ_{A}) - (v_{A} - v_{B}) | \\ = 2 max {(μ_{B} - μ_{A}), (v_{A} - v_{B})} . \end{matrix}$ (37)

By the same argument, we know that $Δ_{μ}^{AC} = μ_{A} - μ_{C} \leq 0$ , to imply $| Δ_{μ}^{AC} | = μ_{C} - μ_{A}$ , and because of $Δ_{v}^{AC} = v_{A} - v_{C} \geq 0$ to derive $| Δ_{v}^{AC} | = v_{A} - v_{C}$ . We compute that $Δ_{π}^{AC} = μ_{C} + v_{C} - μ_{A} - v_{A} = (μ_{C} - μ_{A}) - (v_{A} - v_{C})$ , to obtain $\begin{matrix} | Δ_{μ}^{AC} | + | Δ_{v}^{AC} | + | Δ_{π}^{AC} | \\ = (μ_{C} - μ_{A}) + (v_{A} - v_{C}) + | (μ_{C} - μ_{A}) - (v_{A} - v_{C}) | \\ = 2 max {(μ_{C} - μ_{A}), (v_{A} - v_{C})} . \end{matrix}$ (38)

We recall that A ⊆ B ⊆ C to imply $μ_{C} - μ_{A} \geq μ_{B} - μ_{A} \geq 0,$ (39) and $v_{A} - v_{C} \geq v_{A} - v_{B} \geq 0 .$ (40)

Based on inequalities of Equation (39) and (40), we derive that $\begin{matrix} max {(μ_{C} - μ_{A}), (v_{A} - v_{C})} \geq \\ max {(μ_{B} - μ_{A}), (v_{A} - v_{B})}, \end{matrix}$ (41) such that the inequality of Equation (36) is verified.

Following the same way, we can show that $D_{H}^{Sz - Ka} (B, C) \leq D_{H}^{Sz - Ka} (A, C)$ . Hence, we prove that the distance proposed by Szmidt and Kacprzyk [12] satisfying (DP4).

Based on our Theorem 3, the assertion of Xu [3] that “If A ⊆ B ⊆ C, then the relationship between $| Δ_{π}^{AB} |$ and $| Δ_{π}^{AC} |$ cannot be strictly determined. Thus, it is inappropriate to take Δ_π (i) into the distances between IFSs directly. This is mainly because π_A (x_i) and π_B (x_i) have no fixed relationships if A ⊆ B or B ⊆ A.” is misleading. We agree on those concerns of Xu [3] but his concerns are not related to proving that the distance proposed by Szmidt and Kacprzyk [12] satisfies property (DP4).

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, $D_{H}^{Ya - Ch} (A, B) = \sum_{i = 1}^{n} max {| Δ_{μ} (i) |, | Δ_{v} (i) |, | Δ_{π} (i) |} .$ (42)

Properties of (DP1-DP3) are easily to prove, so we concentrate our discussion on Property (DP4).

Theorem 4. $D_{H}^{Ya - Ch}$ satisfies property (DP4).

(Proof) If A ⊆ B ⊆ C, we will verify that $D_{H}^{Ya - Ch} (A, B)$ $\leq D_{H}^{Ya - Ch} (A, C)$ that is $\begin{matrix} \sum_{i = 1}^{n} max {| Δ_{μ}^{AB} (i) |, | Δ_{v}^{AB} (i) |, | Δ_{π}^{AB} (i) |} \\ \leq \sum_{i = 1}^{n} max {| Δ_{μ}^{AC} (i) |, | Δ_{v}^{AC} (i) |, | Δ_{π}^{AC} (i) |} . \end{matrix}$ (43)

We will show that for i = 1, 2, . . . , n, $\begin{matrix} max {| Δ_{μ}^{AB} (i) |, | Δ_{v}^{AB} (i) |, | Δ_{π}^{AB} (i) |} \\ \leq max {| Δ_{μ}^{AC} (i) |, | Δ_{v}^{AC} (i) |, | Δ_{π}^{AC} (i) |} . \end{matrix}$ (44)

Hence, we will further simplify our proof to show that $\begin{matrix} max {| Δ_{μ}^{AB} |, | Δ_{v}^{AB} |, | Δ_{π}^{AB} |} \\ \leq max {| Δ_{μ}^{AC} |, | Δ_{v}^{AC} |, | Δ_{π}^{AC} |}, \end{matrix}$ (45) that is independent of the i.

We recall that in our Theorem 3, we derive that $\begin{matrix} max {| Δ_{μ}^{AB} |, | Δ_{v}^{AB} |, | Δ_{π}^{AB} |} \\ = max {μ_{B} - μ_{A}, v_{A} - v_{B}, | (μ_{B} - μ_{A}) - (v_{A} - v_{B}) |} \\ = max {μ_{B} - μ_{A}, v_{A} - v_{B}} . \end{matrix}$ (46)

By the same argument, we find that $\begin{matrix} max {| Δ_{μ}^{AC} |, | Δ_{v}^{AC} |, | Δ_{π}^{AC} |} \\ = max {μ_{C} - μ_{A}, v_{A} - v_{C}} . \end{matrix}$ (47)

Let’s recall Equation (39) and (40) to imply that the inequality of Equation (45) is verified and then we finish the proof of $D_{H}^{Ya - Ch} (A, B) \leq D_{H}^{Ya - Ch} (A, C)$ . Similarly, we can prove that $D_{H}^{Ya - Ch} (B, C)$ ≤ $D_{H}^{Ya - Ch} (A, C)$ .

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.

5 Our improvement for numerical examples of Xu [3]

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 $D_{IFSs}^{0.5}$ and $D_{IFSs}^{1}$ as follows. There are 25 distance measures in Table 1 (Table 2 of Xu [3]), where four of them are developed by Xu [3].

For the other 21 distances, with respect to two pairs: (i) A₅ and B₅, and (ii) A₆ and B₆, all imply that $D (A_{5}, B_{5}) \geq D (A_{6}, B_{6})$ (48)

Xu [3] claims that for p = 0.5 and p = 1, two of his proposed distances, then $D_{IFSs}^{0.5} (A_{5}, B_{5}) < D_{IFSs}^{0.5} (A_{6}, B_{6}),$ (49) and $D_{IFSs}^{1} (A_{5}, B_{5}) < D_{IFSs}^{1} (A_{6}, B_{6}) .$ (50)

Consequently, for the rest of his paper, Xu [3] would discuss 1< p < ∞ and then he only considers $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ .

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 $D (A_{1}, B_{1}) \geq D (A_{2}, B_{2}) .$ (51)

Table 1 (Table 2 of Xu [3]) reveals that $D_{IFSs}^{2} (A_{1}, B_{1}) = 0.0707 < D_{IFSs}^{2} (A_{2}, B_{2}) = 0.1 .$ (52) and $D_{IFSs}^{3} (A_{1}, B_{1}) = 0.0794 < D_{IFSs}^{3} (A_{2}, B_{2}) = 0.1 .$ (53)

By the same argument, Xu [3] gives up $D_{IFSs}^{0.5}$ and $D_{IFSs}^{1}$ , then Xu [3] should abandon his $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ , too. Therefore, his argument to discard $D_{IFSs}^{0.5}$ and $D_{IFSs}^{1}$ is questionable.

We begin to point out the bold faced items of $D_{E}^{Sz - Ka}$ (A₃, B₃), $D_{nE}^{Liu}$ (p = 2) (A₃, B₃), $D_{H}^{G - H - C}$ (A₃, B₃), $D_{E}^{G - H - C}$ (A₃, B₃), and $D_{La}^{imp}$ (A₃, B₃) are difficult to accept. As there are three rules proposed by Xu [3], then the distance, say D, will be marked by bold face.

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 (A_2k+1, B_2k+1) - D (A_2k+2, B_2k+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 $D_{D}^{imp} (A_{4}, B_{4}) = 1.00$ is marked by bold face. However, we point that $D_{E}^{Sz - Ka} (A_{3}, B_{3})$ , $D_{nE}^{Liu} (p = 2) (A_{3}, B_{3})$ , $D_{H}^{G - H - C} (A_{3}, B_{3})$ , $D_{E}^{G - H - C} (A_{3}, B_{3})$ , and $D_{La}^{imp} (A_{3}, B_{3})$ were marked by bold face and need further explanations.

Furthermore, if our above discussion is acceptable, and then $D_{R}^{imp} (A_{4}, B_{4}) = 0.50$ and $D_{K D}^{imp} (A_{4}, B_{4}) = 1.00$ should be marked by bold face. Hence, we point out that the argument proposed by Xu [3] for his Table 2 contains questionable results.

We recall that in Sections 5.2 and 5.3 of Xu [3], he applies four distance measures: $D_{I F S s}^{p} |_{p = 2}, D_{I F S s}^{p} |_{p = 3}, D_{I F S s}^{n p} |_{p = 2} and D_{I F S s}^{n p} |_{p = 3}$ . We will show that after Xu [3] computs $D_{I F S s}^{p} |_{p = 2} and D_{I F S s}^{p} |_{p = 3}$ , then his computation of $D_{I F S s}^{n p} |_{p = 2} and D_{I F S s}^{n p} |_{p = 3}$ becomes unnecessary.

Theorem 5. $D_{IFSs}^{p} (P, S_{1}) \geq D_{IFSs}^{p} (P, S_{1})$ if and only if $D_{IFSs}^{np} (P, S_{1}) \geq D_{IFSs}^{np} (P, S_{1})$ .

(Proof) We derive that $\begin{matrix} D_{IFS s}^{np} (A, B) = \\ {\frac{1}{4} \sum_{i = 1}^{n} [\begin{matrix} {| Δ_{μ} (i) |}^{p} + {| Δ_{v} (i) |}^{p} \\ + {| Δ_{π μ} (i) |}^{p} + {| Δ_{π v} (i) |}^{p} \end{matrix}]}^{\frac{1}{p}} \end{matrix}$ (54) and $\begin{matrix} D_{IFS s}^{np} (A, B) = \\ {\frac{1}{4 n} \sum_{i = 1}^{n} [\begin{matrix} {| Δ_{μ} (i) |}^{p} + {| Δ_{v} (i) |}^{p} + \\ {| Δ_{π μ} (i) |}^{p} + {| Δ_{π v} (i) |}^{p} \end{matrix}]}^{\frac{1}{p}} \end{matrix}$ (55) to imply that $D_{IFS s}^{np} (A, B) = \frac{1}{n^{1 / p}} D_{IFS s}^{p} (A, B)$ such that $D_{IFS s}^{p} (A, B)$ and $D_{IFS s}^{np} (A, B)$ will imply the same order during comparison. Hence, after Xu [3] finishs his evaluations for $D_{IFS s}^{p} (A, B)$ , then it is unnecessary to compute $D_{IFS s}^{np} (A, B)$ .

For Tables 4, 6, 8 and 11, Xu [3] computes $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ . The results derived by $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ , implied the same findings in Xu [3]. However, in the following, we will construct a pattern recognition problem such that $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ will derive different findings.

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, $D_{IFSs}^{2} (A, B) = 0.6252$ and $D_{IFSs}^{2} (A, C) = 0.6124$ to imply that sample A should be assigned to pattern C by the distance $D_{IFSs}^{2}$ . On the other hand, $D_{IFSs}^{3} (A, B) = 0.6255$ and $D_{IFSs}^{3} (A, C) = 0.6786$ to imply that sample A should be assigned to pattern B by the distance $D_{IFSs}^{3}$ .

Based on our counter example, we point out that providing $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ for the same application problem means that, sometimes, different results will be proposed by $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ . Hence, we suggest using only one of them: $D_{IFSs}^{2}$ or $D_{IFSs}^{3}$ , not only simplify the computation process but also avoid possible different results proposed by $D_{IFSs}^{2}$ and $D_{IFSs}^{3}$ .

6 Directions for future researchers

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.

Table 3
Comparison between w = 0.5 and w = 1 for Example 2 of Xu [3]

P = 2 $D_{IFSs}^{n p} (S, P_{1})$ $D_{IFSs}^{n p} (S, P_{2})$ $D_{IFSs}^{n p} (S, P_{3})$

w = 0.5 0.2638 0.2372 0.1568

w = 1 0.2929 0.2693 0.1658

P = 2	$D_{IFSs}^{n p} (S, P_{1})$	$D_{IFSs}^{n p} (S, P_{2})$	$D_{IFSs}^{n p} (S, P_{3})$
w = 0.5	0.2638	0.2372	0.1568
w = 1	0.2929	0.2693	0.1658

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 $D_{IFSs}^{n p} {(S, P_{2})}_{w = 0.5} - D_{IFSs}^{n p} {(S, P_{3})}_{w = 0.5} = 0.0804$ (56) and $D_{IFSs}^{n p} {(S, P_{2})}_{w = 1} - D_{IFSs}^{n p} {(S, P_{3})}_{w = 1} = 0.1035$ (57) to reveal that taking w = 1, the distance between (a) the best (smallest) distance, and (b) the second best distance is enlarged such that decision makers are more easier to decide that the sample, S, belonged to the pattern P₃.

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 $\begin{matrix} \frac{D_{IFSs}^{n p} {(S, P_{2})}_{w = 0.5}}{D_{IFSs}^{n p} {(S, P_{3})}_{w = 0.5}} = 1.513 \\ < \frac{D_{IFSs}^{n p} {(S, P_{2})}_{w = 1}}{D_{IFSs}^{n p} {(S, P_{3})}_{w = 1}} = 1.624, \end{matrix}$ (58) to indicate that selecting w = 1 will help decision makers to decide the minimum distance among ${D_{IFSs}^{n p}$ (S, P₁) _w=1, $D_{IFSs}^{n p}$ (S, P₂) _w=1, $D_{IFSs}^{n p}$ (S, P₃) _w=1}.

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.

Table 4

3 Classes/3 attributes problem, Patterns and Test sample (Reproduction of Table 5 of Xu [3])

Patterns & Test sample	Attributes 〈x_i, μ (x_i) , v (x_i)〉(i = 1, 2, 3)
	x ₁	x ₂	x ₃
P ₁	〈x₁, 0.1, 0.1〉	〈x₂, 0.5, 0.1〉	〈x₃, 0.1, 0.9〉
P ₂	〈x₁, 0.5, 0.5〉	〈x₂, 0.7, 0.3〉	〈x₃, 0.0, 0.8〉
P ₃	〈x₁, 0.7, 0.2〉	〈x₂, 0.1, 0.8〉	〈x₃, 0.4, 0.4〉
S	〈x₁, 0.4, 0.4〉	〈x₂, 0.6, 0.2〉	〈x₃, 0.0, 0.8〉

Table 5

Comparison between w = 0.5 and w = 0 for Example 3 of Xu [3]

P = 2	$D_{IFSs}^{n p} (S, P_{1})$	$D_{IFSs}^{n p} (S, P_{2})$	$D_{IFSs}^{n p} (S, P_{3})$
w = 0.5	0.1354	0.0577	0.4193
w = 1	0.1915	0.0816	0.4262

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 $D_{IFSs}^{n p} {(S, P_{1})}_{w = 0.5} - D_{IFSs}^{n p} {(S, P_{2})}_{w = 0.5} = 0.0777,$ (59) and $D_{IFSs}^{n p} {(S, P_{1})}_{w = 1} - D_{IFSs}^{n p} {(S, P_{2})}_{w = 1} = 0.1099,$ (60) to reveal that taking w = 0, the distance between (a) the best (smallest) distance, and (b) the second best distance becomes enlarged and then decision makers can assign the sample, S, to the pattern P₃ more easily.

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 $\begin{matrix} \frac{D_{IFSs}^{n p} {(S, P_{1})}_{w = 0.5}}{D_{IFSs}^{n p} {(S, P_{2})}_{w = 0.5}} = 2.345 \\ < \frac{D_{IFSs}^{n p} {(S, P_{1})}_{w = 0}}{D_{IFSs}^{n p} {(S, P_{2})}_{w = 0}} = 2.346, \end{matrix}$ (61) to show that selecting w = 0 not only enlarged the distance between a) the best (smallest) distance, and (b) the second best distance, but also increased the ratio between a) the best (smallest) distance, and (b) the second best distance.

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 $(μ, v, w μ + (1 - w) (1 - v), w (1 - μ) + (1 - w) v) .$ (62)

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.

7 Conclusion

Xu [3] converts an IFS into a vector in $R^{4}$ , and then shows his new measure satisfies assumptions of (a) Traditional distance, and (b) IFS distance. This study extends his findings to a more general setting and then revises his proof for the traditional distance. Moreover, his challenges for Szmidt and Kacprzyk [12] and Yang and Chiclana [13] are improved. At last, the questionable results in his numerical examples are pointed out and revised. The findings provide a patch work to help readers realize the importance of Xu’s works.

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.

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Extension and revisions for Xu’s proposed distance measure

Abstract

Keywords

1 Introduction

2 Brief review of Xu [3]

Table 3 Comparison between w = 0.5 and w = 1 for Example 2 of Xu [3] P = 2 D IFSs n p ( S , P 1 ) D IFSs n p ( S , P 2 ) D IFSs n p ( S , P 3 ) w = 0.5 0.2638 0.2372 0.1568 w = 1 0.2929 0.2693 0.1658

Footnotes

Acknowledgments

References

Table 3
Comparison between w = 0.5 and w = 1 for Example 2 of Xu [3]

P = 2 $D_{IFSs}^{n p} (S, P_{1})$ $D_{IFSs}^{n p} (S, P_{2})$ $D_{IFSs}^{n p} (S, P_{3})$

w = 0.5 0.2638 0.2372 0.1568

w = 1 0.2929 0.2693 0.1658