A New Similarity Measure of Intuitionistic Fuzzy Sets Considering Abstention Group Influence and Its Applications

3.1 Analysis on Existing Similarity Measures

Definition 3.1: (see [12]). A real-valued function S: IFS(X) × IFS(X) → [0, 1] is called a similarity measure on IFS (X), if it satisfies the following four axiomatic requirements:

1. 0 ≤ S(A, B) ≤ 1;

2. S(A, B) = 1 if A = B;

3. S(A, B) = S(B, A);

4. S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C) if A ⊆ B ⊆ C.

For two IFSs A and B, Li and Cheng [12] proposed a similarity measure between IFS A and IFS B defined as follows.

For each IFS A, letφA(xi) = μA(xi) + 1 − vA(xi)2, where x∈X = {x₁, x₂, …, x_n}. Then, the similarity measure S₁(A, B) is defined as

When p = 1, Eq. (1) is reduced to the following formula:

Example 3.1: Select A = {〈x, 0.1, 0.2〉}, B ≤ {〈x, 0.4, 0.4〉}, and C ≤ {〈x, 0.2, 0.2〉} as three IFSs. It is easy to see that IFS A is more similar to IFS C than to IFS B.

However, if Eq. (2) is used to calculate the similarity measures among three IFSs, we can find that S₂(A, B) = S₂(A, C) = 0.95, which means the similarity measure of A and B is the same as the one of A and C. Obviously, the result is counterintuitive.

For two IFSs A and B, two similarity measures between A and B proposed in Xu et al. [23, 24] are given in Eq. (3) and Eq. (4):

where α > 0.

When α → +∞ and α = 1, Eq. (3) can be reduced to the Eq. (5) and Eq. (6), respectively:

Example 3.2: Select A = {〈x, 0.3, 0.5〉}, B = {〈x, 0.4, 0.4〉} and C = {〈x, 0.4, 0.5〉} as three IFSs. It is easy to find that IFS A is more similar to IFS C than to IFS B.

However, when using the Eqs. (4), (5) and (6) to calculate the similarity measures among A, B and C, the following results can be got:

• By (4) we have S₄(A, B) = S₄(A, C) = 0.82

• By (5) we have S₅(A, B) = S₅(A, C) = 0.90

• By (6) we have S₆(A, B) = S₆(A, C) = 0.90

Again, the above results can barely be rational.

Based on entropy theory, Wei et al. [21] gave another similarity measure of IFSs defined in Eq. (7).

where μ_i = |μ_A(x_i) − μ_B(x_i)|, ν_i = |ν_A(x_i) − ν_B(x_i)|.

Example 3.3: Select A = {〈x, 0.5, 0.5〉}, B = {〈x, 0.4, 0.6〉}, and C = {〈x, 0.4, 0.4〉} as three IFSs. When Eq. (7) is used to calculate the similarity measures, we have S₇(A, B) = S₇(A, C) = 0.82. Same as Example 3.1 and Example 3.2, it is impossible to distinguish which one IFS A is more similar to.

3.2 A New Similarity Measure

The reason of the problem in example 3.3 can be explained in the view of groups voting. For an alternative A = {〈x_i, μ_A(x_i), ν_A(x_i)〉|x_i∈X}, the functions μ_A(x_i) and ν_A(x_i) denote the percentages of the supporters and the dissenters. The function π_A(x_i) = 1 − μ_A(x_i) − ν_A(x_i) denotes the percentage of the abstention group. Accordingly, the percentages of the supporters and the dissenters of alternative A are 50% and 50%, and there is no abstention group. To alternative B, the corresponding percentages are 40% and 60%, and again, no abstention group exists. To alternative C, the percentages of the supporters, the dissenters, and the abstention group are 40%, 40%, and 20%, respectively. Then, A is more similar to C than to B.

According to the above analysis, it can be seen that the abstention group influence should be considered when designing a similar measure. Here, we propose a new similarity measure definition between IFSs A and B.

Definition 3.2: Let X = {x₁, x₂, …, x_n} be a finite universe of discourse.

For each IFS A = {〈x_i, μ_A(x_i), ν_A(x_i)〉|x_i∈X}, let μ′A(xi) = μA(xi) + 1 + μA(xi) − νA(xi)2πA(xi),ν′A(xi) = νA(xi). Then, a similarity measure between A and B is defined as:

where α > 0.

In Eq. (8), 1 + μA(xi) − νA(xi)2 is the possibility that some people belonging to the abstention group tend to cast votes. First, we suppose the possibilities that people from the abstention group tend to cast votes or to be dissenters are the same, i.e. we weigh π_A(x_i) with 1/2, then, we modify the weigh by μA(xi) − νA(xi)2. In the view of group voting, if percentages of the supporters and the dissenters are 80% and 10%, respectively, then, the majority of the abstention group will tend to cast votes, which means that the more percentage the supporters hold, the larger proportion that people of the abstention group tend to cast votes.

In the following, examples 3.1−3.3 are revisited to demonstrate the validity of the proposed similarity measure.

When α = 1, Eq. (8) is simplified to the following formula:

where μ′A(xi) = μA(xi) + 1 + μA(xi) − νA(xi)2πA(xi),ν′A(xi) = νA(xi).

Example 3.1′: Select A = {〈x, 0.1, 0.2〉}, B = {〈x, 0.4, 0.4〉}, and C = {〈x, 0.2, 0.2〉} as three IFSs. Using Eq. (9) to calculate the similarity measures S(A, B) and S(A, C), we can get S¹(A, B) = 0.72, S¹(A, C) = 0.92. The result shows that IFS A is more similar to IFS C than to IFS B, which is consistent with intuition.

Example 3.2′: Select A = {〈x, 0.3, 0.5〉}, B = {〈x, 0.4, 0.4〉}, and C = {〈x, 0.4, 0.5〉} as three IFSs. Using Eq. (9) to calculate the similarity measures S(A, B) and S(A, C), we can get S¹(A, B) = 0.78, S¹(A, C) = 0.94. The result shows that IFS A is more similar to IFS C than to IFS B, which is consistent with intuition.

Example 3.3′: Select A = {〈x, 0.5, 0.5〉}, B = {〈x, 0.4, 0.6〉}, and C = {〈x, 0.4, 0.4〉} as three IFSs. Using Eq. (9) to calculate the similarity measures S(A, B) and S(A, C), we can get S¹(A, B) = 0.80, S¹(A, C) = 0.90. The result shows that IFS A is more similar to IFS C than to IFS B, and again, it is consistent with the analysis above.

4.1 Pattern Recognition

The similarity measure defined by Eq. (9) is applied to solve the pattern recognition problems with intuitionistic fuzzy information. We adopt the following steps:

Step 1: We suppose that there are m patterns, which are denoted by IFSs for a pattern recognition problem. The IFSs are defined as

Ai = {〈xj, μAi(xj), νAi(xj)〉|xj ∈ X}(i = 1, 2, …, m)

in the feature space X = {x₁, x₂, …, x_m}. Besides, assume that there is a sample to be recognized represented by an IFS

B = {〈xj, μB(xj), νB(xj)〉|xj ∈ X}

Step 2: The similarity measure S(A_i, B) is calculated by Eq. (9).

Step 3: We choose the largest value of the similarity measure between A_i and B, denoted by S(A_i0, B), from S(A_i, B)(i = 1, 2, …, m). Then, we can conclude that sample B belongs to the pattern A_i0.

Then, we consider the pattern recognition problems from Refs. [10] and [9]. As one example has 12 features, and the other two examples have only six features, we combine the three examples and set the feature number as six.

Example 4.1: Assume that there are 19 classes of building materials denoted by 19 IFSs C_i(i = 1, 2, …, 19) in the feature space X = {x₁, …, x₆} (see the following Table 1).

Besides, we also have an unknown building material which is represented by IFSs in the following.

B = {〈x1, 0.629, 0, 303〉, 〈x2, 0, 524, 0.356〉, 〈x3, 0.210, 0.689〉,〈x4, 0.218, 0.753〉, 〈x5, 0.069, 0.876〉, 〈x6, 0.658, 0.256〉}

Our purpose is to distinguish which class of building material the unknown pattern B belongs to.

According to the above steps, we calculate the similarity degree shown in Table 2. We use the similarity measure S in Eq. (9) proposed in our paper, and we also use the similarity measures S₂ (proposed by Li and Cheng [12]), S₄, S₅, S₆ (proposed by Xu et al. [23, 24]), and S₇ (proposed by Wei et al. [21]).

In Table 2, we can see that most similarity measures (S, S₂, S₆, S₇) give the right answer, i.e. the similarity degree S(C₁₉, B) between C₁₉ and B is the largest one, indicating that the unknown building material B belongs to C₁₉. However, we can also see that there are some similarity measures (S₄, S₅) that give the wrong answers, as these two similarity measures have some disadvantages, which are not consistent with the reality. In this pattern recognition application, our similarity measure performs as well as the similarity measures S₂, S₆, and S₇. However, our measure takes more factors into consideration than S₂, S₆, and S₇ do; it can be shown in examples 3.1, 3.2, and 3.3 that we can find that our measure is better than the similarity measures S₂, S₆, and S₇. Additionally, it is obvious that our similarity measure is better than the similarity measures S₄ and S₅. As our measure takes the abstention group into consideration, it is more reasonable than S₄ and S₅ in real applications.

4.2 Multi-Criteria Group Decision Making

As for a group decision-making problem, we assume that A = {a₁, a₂, …, a_m}, C = {c₁, c₂, …, c_n} is a set of alternatives and criteria, respectively. Then, we give the following steps for the decision maker to find the best alternative:

Step 1: The evaluation of the alternative a_i with respect to the criterion c_j is an intuitionistic fuzzy number represented by a_ij = {〈c_j, μ_ij(x_j), ν_ij(x_j)〉}. In this case, the alternative a_i is presented by the following IFSs:

ai = {〈cj, μij(xj), νij(xj)〉|cj ∈ C}

where 0 ≤ u ≤ 1, 0 ≤ ν ≤ 1, j = 1, 2, …, n, i = 1, 2, …, m.

Step 2: We define the ideal IFSs: for each criterion in the ideal alternative a^* as a_ij = 〈c_j, 1, 0〉 for “excellence”. Then, we can gain the similarity measure between the ideal alternative a^* and alternative a_i(i = 1, 2, …, m) for each decision maker: Sk(ai, a∗) = ∑j = 1nS(ai, a∗).

Step 3: Calculate the last value of alternatives considering each decision maker’s evaluation.

S(ai, a∗) = ∑k = 1lSk(ai, a∗)

Step 4: Determine the order of alternatives. The lager the value of S(a_i, a^*) is, the better the alternative is.

Then, in order to get a larger dataset, we combine two real examples from Refs. [27] and [6] to demonstrate the effectiveness of our measure.

Example 4.2: The information management steering committee of the Midwest American Manufacturing Corporation is comprised of the Chief Executive Officer, the Chief Information Officer, and the Chief Operating Officer and must prioritize for the development and implementation of a set of eight information technology improvement projects a_i(i = 1, 2, …, 6), which have been proposed by area managers. The information projects are quality management information, inventory control, customer order tracking, material purchasing management, fleet management and design change management. There are seven decision makers and six attributes c_i(i = 1, …, 6). The attributes are productivity, technological innovation capability, marketing capability, differentiation, management, and risk avoidance. The weight vector of the attributes is w = (0.2, 0.15, 0.25, 0.1, 0.1, 0.2). There are seven decision makers to evaluate the alternatives in terms of IFS shown in Table 3.

To calculate the similarity degree between the ideal alternative and the alternative a_i(i = 1, 2, 3, 4, 5, 6), let the ideal alternative a^* be:

a∗ = {(c1, 1, 0), (c2, 1, 0), (c3, 1, 0), (c4, 1, 0), (c5, 1, 0), (c6, 1, 0)}

Next, we calculate the value of each alternative using the similarity measure S in Eq. (9) proposed in our paper as well as the similarity measures S₂ (proposed by Li and Cheng [12]), S₄, S₅, S₆ (proposed by Xu et al. [23, 24]), and S₇ (proposed by Wei et al. [21]). The result is shown in Table 4.

In Table 4, it can be found that our proposed method gives the answer S(a₂, a^*) > S(a₆, a^*) > S(a₄, a^*) > S(a₅, a^*) > S(a₁, a^*) > S(a₃, a^*). The answer is most similar with the result obtained in Ref. [27]. In Ref. [27], the largest similarity is S(a₂, a^*), and the next is S(a₆, a^*). But there are also some differences with the results, which are calculated by our method. Because the data are combined, the examples are from Refs. [27] and [6]. So it has some difference. Our method gives the right answer that the value of S(a₂, a^*) between a₂ and a^* is the largest. We can also find that the similarity measures (S₄, S₅, S₇) can also get the largest similarity degree S(a₂, a^*). But they get a different order for the six projects. Our measure considers more factors than S₄, S₅, S₇, and our measure is better than the similarity measures S₄, S₅, S₇. Moreover, we can clearly see that our similarity measure performs better than the similarity measures S₂ and S₆. Because the abstention group influence is considered, our measure is more suitable to process the imperfect information in the reality.

4.3 Medical Diagnosis

The application of the IFSs theory in medical diagnosis has been found in Refs. [16, 19, 21]. In the following example, we will show how to solve a medical diagnosis problem with intuitionistic fuzzy information using our similarity measure in the form of Eq. (9).

Example 4.3: We consider the same data as in [16, 19]. Assume that there are a set of diagnoses D, a set of symptoms S, and a set of patients P, where

D = {Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}S = {Temperature, Headache, Stomach pain, Cough, Chest pain}, P = {Al, Bob, Joe, Ted}

Table 5 presents the characteristic symptoms of each considered diagnose, and Table 6 gives each patient’s symptoms. Each cell of the tables is presented by a pair of numbers named the membership μ and the non-membership ν, respectively. The aim is to find a proper diagnosis for each patient p_i, i = 1, 2, 3, 4.

To derive a proper diagnosis for each patient p_i, i = 1, 2, 3, 4 by utilizing the proposed similarity measure, i.e. Eq. (9), we first compute the similarity degree S(p_i, d_k) between symptoms of each patient p_i, i = 1, 2, 3, 4 and the set of symptoms for each diagnosis d_k∈D, k = 1, 2, 3, 4, 5 shown in Table 5. From Eq. (9), we have

where

Computing μ′(x_i) and ν′(x_i) by Eq. (11) and then replacing μ(x_i) and ν(x_i) in Tables 5 and 6 with μ′(x_i) and ν′(x_i), respectively, we get Tables 7 and 8. Now each cell of the tables is presented by a pair of numbers named the membership μ′ and non-membership ν′, respectively. Using Eq. (10), we can compute the similarity degree S(p_i, d_k) between symptoms of each patient p_i, i = 1, 2, 3, 4 and the set of symptoms for each diagnosis d_k∈D, k = 1, 2, 3, 4, 5. For example, we can get S(p₁, d₁) by Eq. (10):

S(p1, d1) = 1 − 15[|0.885 − 0.82| + |0.1 − 0.0| + |0.675 − 0.38| + |0.1 − 0.5| + |0.2 − 0.14| + |0.8 − 0.7| + |0.675 − 0.565| + |0.1 − 0.3| + |0.175 − 0.14| + |0.6 − 0.7|] = 0.707

Finally, we can get all S(p_i, d_k) shown in Table 9 that presents the diagnosis results for the considered patients.

To get the proper diagnosis d_k for the patient p_i, we should choose the biggest numerical value from the set S(p_i, d_k), (k = 1, 2, 3, 4, 5) included in Table 9.

In Table 9, it can be seen that Al suffers from Viral fever, Bob from Stomach problem, Joe from Typhoid, and Ted from Viral fever. These results are consistent with the ones obtained by Vlachos and Sergiadis [19]. Compared with the results given by Ref. [16], the diagnoses for Bob, Joe, and Ted are the same, while the diagnosis for Al is different.