Counterintuitive Test Problems for Transformed Fuzzy Number-Based Similarity Measures between Intuitionistic Fuzzy Sets

: This paper analyzes the counterintuitive behaviors of transformed fuzzy number (FN)- based similarity measures between intuitionistic fuzzy sets (IFSs). Among these transformed FN-based similarity measures, Chen and Chang’s similarity measure (2015) is a novel one. An algorithm of computing Chen and Chang’s similarity measure is proposed. We analyze the counterintuitive behaviors of Chen and Chang’s similarity measure for seven general test problems and four test problems with three inclusive IFSs. The results indicate that there are six counterintuitive test problems for Chen and Chang’s similarity measure.


Introduction
Fuzzy sets (FSs) theory, proposed by Zadeh [1], has successfully been applied in various fields.As a generalization of FSs, intuitionistic fuzzy sets (IFSs) proposed by Atanassov [2] are characterized by a membership function and a non-membership function.
In the literature, various approaches for the similarity measures between IFSs are inconsistent with our intuition [12][13][14][15][16][17][18][19][20][21].To analyze the counterintuitive behaviors of Chen and Chang's similarity measure, we present an algorithm to compute Chen and Chang's similarity measure.To illustrate the mechanics of the proposed algorithm, we present some examples.Two kinds of test examples are considered to analyze its counterintuitive behaviors.One is the six general test problems proposed by Tang and Yang [22].Furthermore, we propose a new general test problem.The other one is the four special test problems with three inclusive IFSs.
The organization of this paper is as follows.Section 2 briefly reviews the FSs and IFSs and presents the transformed FN-based similarity measures.Section 3 presents an algorithm for calculating Chen and Chang's similarity measure.We analyze the counterintuitive behaviors of Chen and Chang's similarity measure for seven general test problems in Section 4 and the four special test problems with three inclusive IFNs in Section 5. Finally, some concluding remarks and future research are presented.

IFSs and Similarity Measures
We firstly review the basic notations of FSs and IFSs.Let X = {x 1 , x 2 , . . ., x n } be a non-empty universal set of real numbers R. Definition 1.An FS A over X is defined as We use FS(X) to denote the set of all FSs over X.

Definition 2. An IFS A over X is defined as
The degree of hesitancy associated with each x i is defined as measuring the lack of information or certitude.The set of all IFSs over X is denoted by IFS(X).
We now briefly review some operations involving IFSs.
The complement of A is defined as We denote the pure intuitionistic fuzzy set by PI = (x i , 0, 0) 1 ≤ i ≤ n .
For the fuzzy set A ∈ FS(X), π A (x i ) = 0 for 1 ≤ i ≤ n, and for the pure intuitionistic fuzzy set, We now recall the definition of similarity measures between two IFSs.In the literature, the existing transformed FN-based similarity measures S(A, B) between two IFSs A(µ A (x i ), v A (x i )) and B(µ B (x i ), v B (x i )), for 1 ≤ i ≤ n, are reviewed as follows.
Zhang and Yu's similarity measure [9] is where w i is the weight of element x i , w i ∈ [0, 1], for 1 ≤ i ≤ n, n i=1 w i = 1, Chen and Chang's similarity measure [13] is where w i is the weight of element x i , Define two membership functions of the transformed right-angled triangular FNs A x i and B x i obtained from the IFS (µ A (x i ), 1 − v A (x i )) of element x i belonging to the IFS A as follows Then, the degree of similarity between A x i and B x i and the difference of the areas between A x i and B x i are respectively Among these transformed FN-based similarity measures, the most distinctive is that the form of the transformation technique is symmetric triangular FN for S ZY (A, B) and right-angled triangular FN for S CC (A, B).Chen and Chang indicated that if then Zhang and Yu's similarity measure S ZY (A, B) has the division by zero problem.Therefore, this paper focuses on Chen and Chang's similarity measure S CC (A, B).

Chen and Chang's Similarity Measure
This section will present an algorithm of computing S CC (A, B) for For simplicity, assume that n = 1, and the abbreviated notations of µ A (x 1 ), A x 1 , and µ A x 1 (z) are then denoted by µ A , A x , and µ A x (z), respectively.
Without loss of generality, assume that , we can swap A and B. Four cases are considered as follows.

Case 1: µ
From and We compute dz by decomposing it into three parts: and it follows that and We calculate dz by decomposing it into four parts, as follows: and Case 4: π A = 0 or π B = 0.
If π A = 0, then and If π B = 0, a similar argument shows that We observe that case 4 is a special one of case 1.The mechanics of computing S CC (A, B) are listed as follows (Algorithm 1). Else We illustrate some concrete examples with various A(µ A , v A ) and B(µ B , v B ).
Example 1. A(a, a) and B(a + α, a + α), a + α ≤ 0.5, a, α ≥ 0. From the definition of S CC (A, B), it follows that This result coincides with that of Algorithm 1. From ∂ ∂a S CC (A, B) ≥ 0 and ∂ ∂α S CC (A, B) ≤ 0, it implies that S CC (A, B) is an increasing function of a and a decreasing function of α for a + α ≤ 0.5.For α = 0, associated A(a, a) and B(a, a) can attain the maximum value S CC (A, B) = 1.
This result coincides with that of Algorithm 1. Comparedwith that of example 1, which is consistent with our intuition, for a + α ≤ 0.5, a, α ≥ 0. For the case of a + α = 0.5, we get S CC (A, B) = S CC (C, D) f or A(a, a), B(0.5, 0.5), C(a, 0.5) and D(0.5, a).
and .
From ∂ ∂µ S CC (A, C) ≥ 0, ∂ ∂v S CC (A, C) ≥ 0, and ∂ ∂α S CC (A, C) ≤ 0, the behaviors of S CC (A, C) are the same as those of S CC (A, B).We also have which is consistent with our intuition.
Applying an exhaustive search for α ∈ {0, 0.01, 0.02, . . ., a} and a ∈ {0.01, 0.02, . . ., 0.5}, we get S CC (A, B) − S CC (B, C) ≥ 0, which is consistent with our intuition.Additionally, S CC (A, B) is a decreasing function of a and S CC (B, C) is a decreasing function of a and α. S CC (A, B) is a decreasing function of µ.We can attain the maximum value S CC (A, B) = 0.75 for µ = 0 and minimum value S CC (A, B) = 0.25 for µ = 1, which are inconsistent with our intuition.

General Counterintuitive Test Problems
Much literature has been written on the counterintuitive examples for the similarity measures between two IFSs.Two typical counterintuitive examples are (I) S(A, B) = 1 for A B, A, B ∈ IFS(X) and (II) S(P 1 , Q) = S(P 2 , Q) for P 1 P 2 , P 1 , P 2 , Q ∈ IFS(X).Tang and Yang [22]  Test problem 3 (T3) Test problem 5 (T5) Test problem 6 (T6) For the specific S CC (A, B), we also propose the following general test problem to analyze its counterintuitive behaviors.Test problem 7 (T7) We now analyze the counterintuitive behaviors of similarity measure S CC (A, B) for seven general test problems.For test problems T1 and T2 with a < b, we apply Algorithm 1 to establish and and T1 and T2 are not counterintuitive test problems for S CC (A, B) with a < b and b 0.5.A symmetric argument shows that T1 and T2 are not counterintuitive ones for S CC (A, B) with a > b and b 0.5.Therefore, T1 and T2 are not counterintuitive test problems for S CC (A, B) with a b and b 0.5.
For test problem T3 with a < b, we get and It follows that S CC (P 1 , Q) S CC (P 2 , Q) for a < b.A similar argument shows that S CC (P 1 , Q) S CC (P 2 , Q) for a > b.Therefore, T3 is not a counterintuitive test problem for S CC (A, B) with a b.
For test problem T4, S CC (P 1 , Q) = 0.25, imply that T4 with a 1 is not a counterintuitive test problem for S CC (A, B).
For test problem T5 with a ≤ b, from and For test problem T6 with α > 0, we now use Algorithm 1 to obtain ) .
Since S CC (P 1 , Q) S CC (P 2 , Q) for α > 0, it follows that T6 is not a counterintuitive test problem for S CC (A, B).
We now apply Algorithm 1 with test problem T7 to deduce that Then, T7 is a counterintuitive test problem for S CC (A, B).Therefore, the counterintuitive test problems for Chen and Chang's similarity measure S CC (A, B) are (1)

Counterintuitive Test Problem with A ⊆ B ⊆ C
This section presents the counterintuitive test problems of Chen and Chang's similarity measure S CC (A, B) for the case that A ⊆ B ⊆ C, A, B, C ∈IFS(X).Chen, Cheng and Lan [14] proposed some counterexamples with A ⊆ B ⊆ C for S CC (A, B).More precisely, given A ⊆ B ⊆ C, we have S CC (A, C) S CC (A, B) or S CC (A, C) S CC (B, C), contradicting Definition 4. For test problems with A ⊆ B ⊆ C, this section proposes some general counterexamples satisfying S CC (A, C) S CC (A, B).A symmetric argument shows the similar results of S CC (A, C) S CC (B, C) which are omitted in this paper. From For simplicity, we assume that 1 Using Algorithm 1 yields 2 and Four cases of A(a, c + γ + δ), B(a + α, c + γ), and C(a + α + β, c) satisfying S CC (A, C) > S CC (A, B) are distinguished: (I) β = 0 and δ = α, (II) β = 0, (III) β = γ, and (IV) β > 0.
In the case of II: β = 0, we get It implies that if For the case of III: β = γ, we have It implies that if attains the maximum value For case IV, we illustrate some concrete examples with various values of (a, c).(1−a−c) 2  12 , a + c ≤ 1, a, c, β ≥ 0.
In the future, we will try to analyze the counterintuitive behaviors of similarity measures for the generalization of IFSs.In particular, the analysis can be extended to the hesitant fuzzy sets and neutrosophic sets.Thus, the counterintuitive analyses of similarity measures for the hesitant fuzzy sets and neutrosophic sets are a subject of considerable ongoing research.