**6. Applications**

In this section, the proposed similarity measure is used to solve the real life problems under the IVIFSs environment and obtained results have been compared with some existing similarity measures.

## *6.1. Pattern Recognition*

#### 6.1.1. Algorithms for Pattern Recognition

Letting *X* = {*<sup>x</sup>*1, *x*2, ... , *xn*} be a finite universe of discourse, there exists *m* patterns which are denoted by IVIFSs *Aj* = {< *x*1, [*μ*<sup>−</sup>*Aj*(*<sup>x</sup>*1), *<sup>μ</sup>*<sup>+</sup>*Aj*(*<sup>x</sup>*1)], [*ν*<sup>−</sup>*Aj*(*<sup>x</sup>*1), *<sup>ν</sup>*<sup>+</sup>*Aj*(*<sup>x</sup>*1)] >, ... , < *x*1, [*μ*<sup>−</sup>*Aj*(*xn*), *<sup>μ</sup>*<sup>+</sup>*Aj*(*xn*)], [*ν*<sup>−</sup>*Aj*(*xn*), *<sup>ν</sup>*<sup>+</sup>*Aj*(*xn*)] > |*<sup>x</sup>*1, ... , *xn* ∈ *X*} (*j* = 1, 2, ... , *m*) and there is a test sample to be classified which is denoted by an IVIFS *B* = {< *x*1, [*μ*<sup>−</sup>*B* (*<sup>x</sup>*1), *μ*+*B* (*<sup>x</sup>*1)], [*ν*<sup>−</sup>*B* (*<sup>x</sup>*1), *ν*+*B* (*<sup>x</sup>*1)] >, ... , < *x*1, [*μ*<sup>−</sup>*B* (*xn*), *μ*+*B* (*xn*)], [*ν*<sup>−</sup>*B* (*xn*), *ν*+*B* (*xn*)] > |*<sup>x</sup>*1,..., *xn* ∈ *<sup>X</sup>*}. The recognition process is as follows:

**Step 1**. Calculate the similarity measure *<sup>S</sup>*(*<sup>B</sup>*, *Aj*) between *B* and *Aj*(*j* = 1, . . . , *<sup>m</sup>*).

**Step 2**. Choose the maximum one *<sup>S</sup>*(*<sup>B</sup>*, *Aj*0 ) from *<sup>S</sup>*(*<sup>B</sup>*, *Aj*) (*j* = 1, 2, ... , *<sup>m</sup>*), i.e., *<sup>S</sup>*(*<sup>B</sup>*, *Aj*0 ) = max 1≤*j*≤*m <sup>S</sup>*(*<sup>B</sup>*, *Aj*). Then, the test sample *B* is classified the pattern *Aj*0 .

#### 6.1.2. Applications for Pattern Recognition

**Example 2.** *Assume that there are four classes of ores Ai*(*i* = 1, 2, 3, 4) *in the area developed by a coal mine company, for which the related feature information are expressed by IVIFSs, and Ai* = {< *x*1, )*μ*<sup>−</sup>*Ai*(*<sup>x</sup>*1), *<sup>μ</sup>*<sup>+</sup>*Ai*(*<sup>x</sup>*1)\* , )*ν*<sup>−</sup>*Ai*(*<sup>x</sup>*1), *<sup>ν</sup>*+*Ai*(*<sup>x</sup>*1)\* >, ... , < *x*4, )*μ*<sup>−</sup>*Ai*(*<sup>x</sup>*4), *<sup>μ</sup>*<sup>+</sup>*Ai*(*<sup>x</sup>*4)\* , )*ν*<sup>−</sup>*Ai*(*<sup>x</sup>*4), *<sup>ν</sup>*+*Ai*(*<sup>x</sup>*4)\* > |*<sup>x</sup>*1, *x*2, *x*3, *x*4 ∈ *<sup>X</sup>*}*, which are presented in Table 2. Now, there is an unknown ore B and our aim is to classify B into the four kinds of ores above.*

**Table 2.** Feature matrix of *A*1, *A*2, *A*3, *A*4 and *B*.


*Compute the similarity measures <sup>S</sup>*(*Ai*, *B*) *between B and Ai. By analyzing the computed results in Table 3, we can easily see that, if S*1 *is used for pattern recognition, we can obtain that <sup>S</sup>*1(*<sup>A</sup>*1, *B*) = *<sup>S</sup>*1(*<sup>A</sup>*2, *B*) = *<sup>S</sup>*1(*<sup>A</sup>*4, *B*) > *<sup>S</sup>*1(*<sup>A</sup>*3, *<sup>B</sup>*)*. In this way, we can not classify the sample B into a certain pattern accurately. If SW is used for pattern recognition, we can obtain that SW*(*<sup>A</sup>*2, *B*) = *SW*(*<sup>A</sup>*4, *B*) > *SW*(*<sup>A</sup>*1, *B*) = *SW*(*<sup>A</sup>*3, *<sup>B</sup>*)*. In this way, we can not make sure if the sample B belongs to one of A*2 *and A*4*. If we use SD for pattern recognition, we can get <sup>S</sup>*(*<sup>A</sup>*3, *B*) = *<sup>S</sup>*(*<sup>A</sup>*4, *B*) > *<sup>S</sup>*(*<sup>A</sup>*2, *B*) > *<sup>S</sup>*(*<sup>A</sup>*1, *<sup>B</sup>*)*. In this way, we can not classify the sample B into one of A*3 *and A*4*. If we use Sp for pattern recognition, we can get <sup>S</sup>*(*<sup>A</sup>*1, *B*) > *<sup>S</sup>*(*<sup>A</sup>*2, *B*) > *<sup>S</sup>*(*<sup>A</sup>*3, *B*) > *<sup>S</sup>*(*<sup>A</sup>*4, *<sup>B</sup>*)*. According to the principle of recognition, S*2 *and Sp can get the same recognition result, i.e., the sample B can be classified into the pattern A*3*. However, we can not distinguish which one is bigger between A*2 *and A*4 *when using S*2 *to calculate the similarity measure. Therefore, we can assign the sample B to the pattern A*3*.*

**Table 3.** Pattern recognition result under different similarity measures (counter-intuitive cases are in bold type; *p* = 1 in *S*1 and *S*2; *p* = 1, *t*1 = 2, *t*2 = 3 in *Sp*; **N.A.** means method is not applicable).


**Example 3** ([30])**.** *In this example, a pattern recognition example about classification of building materials is used to illustrate the proposed similarity measure. Suppose that there are four classes of building material, which are denoted by the IVIFSs Aj* = {< *x*1, [*μ*<sup>−</sup>*Aj*(*<sup>x</sup>*1), *<sup>μ</sup>*<sup>+</sup>*Aj*(*<sup>x</sup>*1)], [*ν*<sup>−</sup>*Aj*(*<sup>x</sup>*1), *<sup>ν</sup>*<sup>+</sup>*Aj*(*<sup>x</sup>*1)] >, ... , < *x*12, [*μ*<sup>−</sup>*Aj*(*<sup>x</sup>*12), *<sup>μ</sup>*<sup>+</sup>*Aj*(*<sup>x</sup>*12)], [*ν*<sup>−</sup>*Aj*(*<sup>x</sup>*12), *<sup>ν</sup>*<sup>+</sup>*Aj*(*<sup>x</sup>*12)] > |*<sup>x</sup>*1, ... , *x*12 ∈ *X*} (*j* = 1, ... , 4) *in the feature space X* = {*<sup>x</sup>*1, *x*2,..., *<sup>x</sup>*12}*, and there is an unknown pattern B:*

$$\begin{split} A\_{1} &= \{<\mathbf{x}\_{1}, [0.1, 0.2], [0.5, 0.6]>, <\mathbf{x}\_{2}, [0.1, 0.2], [0.7, 0.8]>, <\mathbf{x}\_{3}, [0.5, 0.6], [0.3, 0.4]>, \\ &<\mathbf{x}\_{4}, [0.8, 0.9], [0.0, 0.1]>, <\mathbf{x}\_{5}, [0.4, 0.5], [0.3, 0.4]>, <\mathbf{x}\_{6}, [0.0, 0.1], [0.8, 0.9]>, \\ &<\mathbf{x}\_{7}, [0.3, 0.4], [0.5, 0.6]>, <\mathbf{x}\_{8}, [1.0, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{9}, [0.2, 0.3], [0.6, 0.7]>, \\ &<\mathbf{x}\_{10}, [0.4, 0.5], [0.4, 0.5]>, <\mathbf{x}\_{11}, [0.7, 0.8], [0.1, 0.2]>, <\mathbf{x}\_{12}, [0.4, 0.5], [0.4, 0.5]>\}>, \end{split}$$

$$\begin{split} A\_{2} &= \{<\mathbf{x}\_{1}, [0.5, 0.6], [0.3, 0.4]>, <\mathbf{x}\_{2}, [0.6, 0.7], [0.1, 0.2]>, <\mathbf{x}\_{3}, [1.0, 1.0], [0.0, 0.0]>, \\ &<\mathbf{x}\_{4}, [0.1, 0.2], [0.6, 0.7]>, <\mathbf{x}\_{5}, [0.0, 0.1], [0.8, 0.9]>, <\mathbf{x}\_{6}, [0.7, 0.8], [0.1, 0.2]>, \\ &<\mathbf{x}\_{7}, [0.5, 0.6], [0.3, 0.4]>, <\mathbf{x}\_{8}, [0.6, 0.7], [0.2, 0.3]>, <\mathbf{x}\_{9}, [1.0, 1.0], [0.0, 0.0]>, \\ &<\mathbf{x}\_{10}, [0.1, 0.2], [0.7, 0.8]>, <\mathbf{x}\_{11}, [0.0, 0.1], [0.8, 0.9]>, <\mathbf{x}\_{12}, [0.7, 0.8], [0.1, 0.2]>\}, \end{split}$$

$$\begin{split} A\mathbf{x} &= \{<\mathbf{x}\_{1}, [0.4, 0.5], [0.3, 0.4]>, <\mathbf{x}\_{2}, [0.6, 0.7], [0.2, 0.3]>, <\mathbf{x}\_{3}, [0.9, 1.0], [0.0, 0.0]>, \\ &<\mathbf{x}\_{4}, [0.0, 0.1], [0.8, 0.9]>, <\mathbf{x}\_{5}, [0.0, 0.1], [0.8, 0.9]>, <\mathbf{x}\_{6}, [0.6, 0.7], [0.2, 0.3]>, \\ &<\mathbf{x}\_{7}, [0.1, 0.2], [0.7, 0.8]>, <\mathbf{x}\_{8}, [0.2, 0.3], [0.6, 0.7]>, <\mathbf{x}\_{9}, [0.5, 0.6], [0.2, 0.4]>, \\ &<\mathbf{x}\_{10}, [1.0, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{11}, [0.3, 0.4], [0.4, 0.5]>, <\mathbf{x}\_{12}, [0.0, 0.1], [0.8, 0.9]>>, \end{split}$$

$$\begin{split} A\_{4} &= \{<\mathbf{x}\_{1}, [1.0, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{2}, [1.0, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{3}, [0.8, 0.9], [0.0, 0.1]>, \\ &<\mathbf{x}\_{4}, [0.7, 0.8], [0.1, 0.2]>, <\mathbf{x}\_{5}, [0.0, 0.1], [0.7, 0.9]>, <\mathbf{x}\_{6}, [0.0, 0.1], [0.8, 0.9]>, \\ &<\mathbf{x}\_{7}, [0.1, 0.2], [0.7, 0.8]>, <\mathbf{x}\_{8}, [0.1, 0.2], [0.7, 0.8]>, <\mathbf{x}\_{9}, [0.4, 0.5], [0.3, 0.4]>, \\ &<\mathbf{x}\_{10}, [1.0, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{11}, [0.3, 0.4], [0.4, 0.5]>, <\mathbf{x}\_{12}, [0.0, 0.1], [0.8, 0.9]>>, \end{split}$$

$$\begin{split} B &= \{<\mathbf{x}\_{1}, [0.9, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{2}, [0.9, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{3}, [0.7, 0.8], [0.1, 0.2]>, \\ &<\mathbf{x}\_{4}, [0.6, 0.7], [0.1, 0.2]>, <\mathbf{x}\_{5}, [0.0, 0.1], [0.8, 0.9]>, <\mathbf{x}\_{6}, [0.1, 0.2], [0.7, 0.8]>, \\ &<\mathbf{x}\_{7}, [0.1, 0.2], [0.7, 0.8]>, <\mathbf{x}\_{8}, [0.1, 0.2], [0.7, 0.8]>, <\mathbf{x}\_{9}, [0.4, 0.5], [0.3, 0.4]>, \\ &<\mathbf{x}\_{10}, [1.0, 1.0], [0.0, 0.0]>, <\mathbf{x}\_{11}, [0.3, 0.4], [0.4, 0.5]>, <\mathbf{x}\_{12}, [0.0, 0.1], [0.7, 0.9]>\}. \end{split}$$

*Calculate the similarity measure <sup>S</sup>*(*Aj*, *B*) *between IVIFSs Aj* (*j* = 1, 2, 3, 4) *and B by use of Formulas (1)–(5). It is obvious that the similarity measure in the literature ([30]) is the special case of S*1 *and S*2*, and the computed result is the same as ([30]). According to Table 4 and the recognition principle, the unknown pattern can be classified properly in A*4 *by the computation of similarity measure. This conclusion coincides with that in [30].*

**Table 4.** Pattern recognition results under different similarity measures (counter-intuitive cases are in bold type; *p* = 1 in *S*1 and *S*2, *p* = 1, *t*1 = 2, *t*2 = 3 in *Sp*).


#### *6.2. Applications for Medical Diagnosis*

Researchers proposed a lot of methods from different points of view to deal with problems of medical diagnosis. Refs. [27,31–33] presented several ways to deal with the problems of medical diagnosis. In this section, the methods of pattern recognition are used for solving medical diagnosis problems, i.e., patients are unknown test samples, diseases are several patterns, and the symptom set is the set universe of discourse. Our aim is to classify patients in one of the illnesses, respectively.

**Example 4.** *Let A* = {*<sup>A</sup>*1 *(Viral fever), A*2 *(Typhoid), A*3 *(Pneumonia), A*4 *(Stomach problem)*} *be a set of diagnoses and X* = {*<sup>x</sup>*1 *(Temperature), x*2 *(Cough), x*3 *(Headache), x*4 *(Stomach pain)*} *be a set of symptoms. The disease–symptom matrix that is represented by IVIFSs is listed in Table 5.*

**Table 5.** Disease–symptom matrix.


*Suppose the patient B can be represented as:*

*B* = {< *x*1, [0.4, 0.5], [0.1, 0.2] >, < *x*2, [0.7, 0.8], [0.1, 0.2] >, < *x*3, [0.9, 0.9], [0.0, 0.1] >, < *x*4, [0.3, 0.5], [0.2, 0.4] <sup>&</sup>gt;}.

*Our aim is to classify the patient B in one of the illnesses A*1*, A*2*, A*3 *and A*4*. Then, we can have the following results in the environment of IVIFSs, which are listed in Table 6.*

**Table 6.** Computed results under different similarity measures (counter-intuitive cases are in bold type; *p* = 1 in *S*1 and *S*2; *p* = 1, *t*1 = 2, *t*2 = 3 in *Sp*).


*Considering the recognition principle of the maximum similarity degree for the IVIFSs, we can obtain the consequence that the similarity measure between A*2 *and B is the largest one. However, the similarity measures S*2 *could not distinguish which one is bigger between A*1 *and A*4*. Thus, we can classify the patient B to illness A*2 *due to the recognition principle. Therefore, we can diagnose that the patient's disease is typhoid.*
