# Simplified Neutrosophic Exponential Similarity Measures for the Initial Evaluation/Diagnosis of Benign Prostatic Hyperplasia Symptoms

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−}0, 1

^{+}[. Therefore, Smarandache [6] and Wang et al. [7,8] proposed the concepts of a single-valued neutrosophic set (SVNS) and an interval neutrosophic set (INS), where the truth, indeterminacy, and falsity functions are constrained in the real standard interval [0, 1] as the subclasses of the neutrosophic set. Further, Ye [9] introduced the concept of a simplified neutrosophic set (SNS), which is a subclass of the neutrosophic set including the concepts of SVNS and INS. SNSs are very suitable for handling medical diagnosis problems, since a symptom may imply a lot of incomplete, uncertain, and inconsistent information for a disease, which characterizes a relation between symptoms and a disease. Recently, SNSs have been applied to medical diagnosis problems. Ye [10] presented the improved cosine similarity measures between SNSs for medical diagnosis. As a generalization of SVNS, Ye et al. [11,12] introduced a single-valued neutrosophic multiset and the Dice similarity measure and distance-based similarity measures of single-valued neutrosophic multisets, and then applied them to medical diagnosis. Broumi and Deli [13] presented a correlation measure of neutrosophic refined sets (neutrosophic multisets) and their application in medical diagnosis. Broumi and Smarandache [14] introduced an extended Hausdorff distance and its similarity measure of refined neutrosophic sets (neutrosophic multisets) and applied the similarity measure to medical diagnosis. Furthermore, Ye and Fu [15] put forward a single-valued neutrosophic similarity measure based on tangent function and the tangent similarity measure-based multi-period medical diagnosis method (a dynamic medical diagnosis method).

## 2. Basic Concepts of SNSs

**Definition**

**1.**

_{N}(x), an indeterminacy-membership function I

_{N}(x), and a falsity-membership function F

_{N}(x). Then, the SNS N can be expressed as $N=\left\{\langle x,{T}_{N}(x),{I}_{N}(x),{F}_{N}(x)\rangle |x\in X\right\}$, where T

_{N}(x), I

_{N}(x), and F

_{N}(x) are singleton subintervals/subsets in the real standard [0, 1], such that T

_{N}(x): X → [0, 1], I

_{N}(x): X → [0, 1], and F

_{N}(x): X → [0, 1].

_{N}(x), I

_{N}(x), F

_{N}(x) ∈ [0, 1], and 0 ≤ T

_{N}(x) + I

_{N}(x) + F

_{N}(x) ≤ 3, and INS for T

_{N}(x), I

_{N}(x), F

_{N}(x) ⊆ [0, 1,] and 0 ≤ sup T

_{N}(x) + sup I

_{N}(x) + sup F

_{N}(x) ≤ 3 for each point x in X.

_{M}(x), I

_{M}(x), F

_{M}(x) ∈ [0, 1], 0 ≤ T

_{M}(x) + I

_{M}(x) + F

_{M}(x) ≤ 3, T

_{N}(x), I

_{N}(x), F

_{N}(x) ∈ [0, 1], and 0 ≤ T

_{N}(x) + I

_{N}(x) + F

_{N}(x) ≤ 3 for each point x in X, then M and N are reduced to two SVNSs. Thus, the inclusion, equation, and complement for SNSs M and N are defined, respectively, as follows [9]:

- (1)
- N ⊆ M if and only if T
_{N}(x) ≤ T_{M}(x), I_{N}(x) ≥ I_{M}(x), F_{N}(x) ≥ F_{M}(x) for any x in X; - (2)
- N = M if and only if N ⊆ M and M ⊆ N;
- (3)
- ${M}^{c}=\left\{\langle x,{F}_{M}(x),1-{I}_{M}(x),{T}_{M}(x)\rangle |x\in X\right\}$ and ${N}^{c}=\left\{\langle x,{F}_{N}(x),1-{I}_{N}(x),{T}_{N}(x)\rangle |x\in X\right\}$.

_{M}(x), I

_{M}(x), F

_{M}(x) ⊆ [0, 1], 0 ≤ sup T

_{M}(x) + sup I

_{M}(x) + sup F

_{M}(x) ≤ 3, T

_{N}(x), I

_{N}(x), F

_{N}(x) ⊆ [0, 1], and 0 ≤ sup T

_{N}(x) + sup I

_{N}(x) + sup F

_{N}(x) ≤ 3 for each point x in X, then M and N are reduced to two INSs. Thus, the inclusion, equation, and complement for SNSs N and M are defined, respectively, as follows [9]:

- (1)
- N ⊆ M if and only if inf T
_{N}(x) ≤ inf T_{M}(x), inf I_{N}(x) ≥ inf I_{M}(x), inf F_{N}(x) ≥ inf F_{M}(x), sup T_{N}(x) ≤ sup T_{M}(x), sup I_{N}(x) ≥ sup I_{M}(x), sup F_{N}(x) ≥ sup F_{M}(x) for any x in X; - (2)
- N = M if and only if N ⊆ M and M ⊆ N;
- (3)
- ${M}^{c}=\left\{\langle x,[\mathrm{inf}{F}_{M}(x),\mathrm{sup}{F}_{M}(x)],[1-\mathrm{sup}{I}_{M}(x),1-\mathrm{inf}{I}_{M}(x)],[\mathrm{inf}{T}_{M}(x),\mathrm{sup}{T}_{M}(x)]\rangle |x\in X\right\}$ and ${N}^{c}=\left\{\langle x,[\mathrm{inf}{F}_{N}(x),\mathrm{sup}{F}_{N}(x)],[1-\mathrm{sup}{I}_{N}(x),1-\mathrm{inf}{I}_{N}(x)],[\mathrm{inf}{T}_{N}(x),\mathrm{sup}{T}_{N}(x)]\rangle |x\in X\right\}$.

_{M}(x), I

_{M}(x), F

_{M}(x) in M and T

_{N}(x), I

_{N}(x), F

_{N}(x) in N are equal, the INSs M and N are reduced to the SVNSs M and N. Therefore, SVNSs are the special cases of INSs, and also SVNSs and INSs are also the special cases of SNSs.

## 3. ESMs of SNSs

**Definition**

**2.**

_{j}, T

_{M}(x

_{j}), I

_{M}(x

_{j}), F

_{M}(x

_{j})〉| x

_{j}∈ X} and N = {〈x

_{j}, T

_{N}(x

_{j}), I

_{N}(x

_{j}), F

_{N}(x

_{j})〉| x

_{j}∈ X} be any two SVNSs in X = {x

_{1}, x

_{2}, …, x

_{n}}. Thus, we can define an ESM between N and M as follows:

**Proposition**

**1.**

_{1}, x

_{2}, …, x

_{n}}, the ESM E

_{1}(M, N) should satisfy the following properties (1)–(4):

- (1)
- 0 ≤ E
_{1}(M, N) ≤ 1; - (2)
- E
_{1}(M, N) = 1 if and only if M = N, i.e., T_{M}(x_{j}) = T_{N}(x_{j}), I_{M}(x_{j}) = I_{N}(x_{j}), and F_{M}(x_{j}) = F_{N}(x_{j}) for x_{j}∈ X (j = 1, 2, …, n); - (3)
- E
_{1}(M, N) = E_{1}(N, M); - (4)
- If P is an SVNS in X and M ⊆ N ⊆ P, then E
_{1}(M, P) ≤ E_{1}(M, N) and E_{1}(M, P) ≤ E_{1}(N, P).

**Proof.**

_{M}(x

_{j}) I

_{M}(x

_{j}) F

_{M}(x

_{j}) ∈ [0, 1] and T

_{N}(x

_{j}) I

_{N}(x

_{j}) F

_{N}(x

_{j}) ∈ [0, 1] in the two SVNSs M and N, the distance $\left(\left|{T}_{M}({x}_{j})-{T}_{N}({x}_{j})\right|+\left|{I}_{M}({x}_{j})-{I}_{N}({x}_{j})\right|+\left|{F}_{M}({x}_{j})-{F}_{N}({x}_{j})\right|\right)/3$ lies between 0 and 1. By applying Equation (1), ESM also lies between 0 and 1. Hence, there is 0 ≤ E

_{1}(M, N) ≤ 1.

_{M}(x

_{j}) = T

_{N}(x

_{j}), I

_{M}(x

_{j}) = I

_{N}(x

_{j}), F

_{M}(x

_{j}) = F

_{N}(x

_{j}) for x

_{j}∈ X and j = 1, 2, …, n. Hence, there are $\left|{T}_{M}^{}({x}_{j})-{T}_{N}^{}({x}_{j})\right|=0$, $\left|{I}_{M}^{}({x}_{j})-{I}_{N}^{}({x}_{j})\right|=0$, and $\left|{F}_{M}^{}({x}_{j})-{F}_{N}^{}({x}_{j})\right|=0$. Thus, we can obtain the following result:

_{1}(M, N) = 1, we have the following equation:

_{M}(x

_{j}) = T

_{N}(x

_{j}), I

_{M}(x

_{j}) = I

_{N}(x

_{j}), and F

_{M}(x

_{j}) = F

_{N}(x

_{j}) for x

_{j}∈ X and j = 1, 2, …, n. Hence M = N.

_{M}(x

_{j}) ≤ T

_{N}(x

_{j}) ≤ T

_{P}(x

_{j}), I

_{M}(x

_{j}) ≥ I

_{N}(x

_{j}) ≥ I

_{P}(x

_{j}), F

_{M}(x

_{j}) ≥ F

_{N}(x

_{j}) ≥ F

_{P}(x

_{j}) for x

_{j}∈ X and j = 1, 2, …, n. Then, we have

_{1}(M, P) ≤ E

_{1}(M, N) and E

_{1}(M, P) ≤ E

_{1}(N, P) since the exponential function $\mathrm{exp}\left(-\frac{1}{3}\left(\left|{T}_{M}^{}({x}_{j})-{T}_{N}^{}({x}_{j})\right|+\left|{I}_{M}^{}({x}_{j})-{I}_{N}^{}({x}_{j})\right|+\left|{F}_{M}^{}({x}_{j})-{F}_{N}^{}({x}_{j})\right|\right)\right)$ is a decreasing function.

_{j}for x

_{j}∈ X into account and assumes that the weight of an element x

_{j}is w

_{j}(j = 1, 2, …, n) with w

_{j}∈ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$. Hence, we can introduce the following weighted ESM between M and N:

_{1}(M, N) should satisfy the properties (1)–(4) in Proposition 1. Especially when w

_{j}= 1/n for j = 1, 2, …, n, Equation (2) reduces to Equation (1).

_{j}, T

_{M}(x

_{j}), I

_{M}(x

_{j}), F

_{M}(x

_{j})〉| x

_{j}∈ X} and N = {〈x

_{j}, T

_{N}(x

_{j}), I

_{N}(x

_{j}), F

_{N}(x

_{j})〉| x

_{j}∈ X} be any two INSs in X = {x

_{1}, x

_{2}, …, x

_{n}}, where T

_{M}(x

_{j}) = [inf T

_{M}(x

_{j}), sup T

_{M}(x

_{j})] ⊆ [0, 1], I

_{M}(x

_{j}) = [inf I

_{M}(x

_{j}), sup I

_{M}(x

_{j})] ⊆ [0, 1], and F

_{M}(x

_{j}) = [inf F

_{M}(x

_{j}), sup F

_{M}(x

_{j})] ⊆ [0, 1] in M for any x

_{j}∈ X are denoted by ${T}_{M}({x}_{i})=[{T}_{M}^{L}({x}_{i}),{T}_{M}^{U}({x}_{i})]$, ${I}_{M}({x}_{i})=[{I}_{M}^{L}({x}_{i}),{I}_{M}^{U}({x}_{i})]$, and ${F}_{M}({x}_{i})=[{F}_{M}^{L}({x}_{i}),{F}_{M}^{U}({x}_{i})]$, respectively, and T

_{N}(x

_{j}) = [inf T

_{N}(x

_{j}), sup T

_{N}(x

_{j})] ⊆ [0, 1], I

_{N}(x

_{j}) = [inf I

_{N}(x

_{j}), sup I

_{N}(x

_{j})] ⊆ [0, 1], and F

_{N}(x

_{j}) = [inf F

_{N}(x

_{j}), sup F

_{N}(x

_{j})] ⊆ [0, 1] in N for any x

_{j}∈ X are denoted by ${T}_{N}({x}_{i})=[{T}_{N}^{L}({x}_{i}),{T}_{N}^{U}({x}_{i})]$, ${I}_{N}({x}_{i})=[{I}_{N}^{L}({x}_{i}),{I}_{N}^{U}({x}_{i})]$, and ${F}_{N}({x}_{i})=[{F}_{N}^{L}({x}_{i}),{F}_{N}^{U}({x}_{i})]$, respectively, for convenience. Then, based on the extension of the above similarity measures Equations (1) and (2), we can introduce the following two ESMs between M and N:

_{j}is the weight of an element x

_{j}(j = 1, 2, …, n) with w

_{j}∈ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$.

_{M}(x

_{j}), I

_{M}(x

_{j}), F

_{M}(x

_{j}) in M and T

_{N}(x

_{j}), I

_{N}(x

_{j}), F

_{N}(x

_{j}) in N are equal. Therefore, the above ESMs of INSs also satisfy properties (1)–(4) in Proposition 1. The proof is similar to that of Proposition 1, and thus it is not repeated here.

## 4. Initial Evaluation/Diagnosis Method of BPH Using the ESMs

_{1}(Over the past month, how often have you had a sensation of not emptying your bladder completely after you finished urinating?), Q

_{2}(Over the past month, how often have you had to urinate again less than two hours after you finished urinating?), Q

_{3}(Over the past month, how often have you found you stopped and started again several times when you urinated?), Q

_{4}(Over the past month, how often have you found it difficult to postpone urination?), Q

_{5}(Over the past month, how often have you had a weak urinary stream?), Q

_{6}(Over the past month, how often have you had to push or strain to begin urination?), Q

_{7}(Over the past month, how many times did you most typically get up to urinate from the time you went to bed at night until the time you got up in the morning?)} for a physician to survey the patients’ BPH symptoms. The clinical survey of the number of BPH symptoms in the 5 times for a patient P

_{k}(k = 1, 2, …, t) can be constructed by Table 1, where T, I, and F denote truth, indeterminacy, and falsity, respectively.

_{1}(Normal symptom), S

_{2}(Mild symptom), S

_{3}(Moderate symptom), S

_{4}(Severe symptom)} as the symptom knowledge for the initial evaluation of BPH patients, as shown in Table 2.

_{1}= {〈Q

_{1}, 0, 0, 1〉, 〈Q

_{2}, 0, 0, 1〉, 〈Q

_{3}, 0, 0, 1〉, 〈Q

_{4}, 0, 0, 1〉, 〈Q

_{5}, 0, 0, 1〉, 〈Q

_{6}, 0, 0, 1〉, 〈Q

_{7}, 0, 0, 1〉},

_{2}= {〈Q

_{1}, 0, 0.2, 0.8〉, 〈Q

_{2}, 0, 0.2, 0.8〉, 〈Q

_{3}, 0, 0.2, 0.8〉, 〈Q

_{4}, 0, 0.2, 0.8〉, 〈Q

_{5}, 0, 0.2, 0.8〉, 〈Q

_{6}, 0, 0.2, 0.8〉, 〈Q

_{7}, 0, 0.2, 0.8〉},

_{3}= {〈Q

_{1}, 0.2, 0.4, 0.4〉, 〈Q

_{2}, 0.2, 0.4, 0.4〉, 〈Q

_{3}, 0.2, 0.4, 0.4〉, 〈Q

_{4}, 0.2, 0.4, 0.4〉, 〈Q

_{5}, 0.2, 0.4, 0.4〉, 〈Q

_{6}, 0.2, 0.4, 0.4〉, 〈Q

_{7}, 0.2, 0.4, 0.4〉}.

_{4}= {〈Q

_{1}, 0.6, 0.4, 0〉, 〈Q

_{2}, 0.6, 0.4, 0〉, 〈Q

_{3}, 0.6, 0.4, 0〉, 〈Q

_{4}, 0.6, 0.4, 0〉, 〈Q

_{5}, 0.6, 0.4, 0〉, 〈Q

_{6}, 0.6, 0.4, 0〉, 〈Q

_{7}, 0.6, 0.4, 0〉}.

_{k}(k = 1, 2, …, t) with SNS information, we can give the following evaluation/diagnosis method.

_{k}with BPH symptoms, we can calculate the similarity measure W

_{q}(P

_{k}, S

_{i}) for q = 1 or 2, i = 1, 2, 3, 4 and k = 1, 2, …, t. The proper BPH symptom evaluation S

_{i*}for patient P

_{k}is derived by ${i}^{*}=\mathrm{arg}\underset{1\le i\le 4}{\mathrm{max}}\{{W}_{q}({P}_{k},{S}_{i})\}$.

#### 4.1. Initial Evaluation of the BPH Symptoms Under a Single-Valued Neutrosophic Environment

**Example**

**1.**

_{k}(k = 1, 2, 3) with respect to all the questions can be represented by the following SVNS information:

_{1}= {〈Q

_{1}, 0.4, 0.2, 0.4〉, 〈Q

_{2}, 0.4, 0.4, 0.2〉, 〈Q

_{3}, 0.4, 0.2, 0.4〉, 〈Q

_{4}, 0.4, 0.2, 0.4〉, 〈Q

_{5}, 0.6, 0.4, 0.0〉, 〈Q

_{6}, 0.4, 0.0, 0.6〉, 〈Q

_{7}, 0.6, 0.0, 0.4〉},

_{2}= {〈Q

_{1}, 0.2, 0.2, 0.6〉, 〈Q

_{2}, 0.4, 0.2, 0.4〉, 〈Q

_{3}, 0.2, 0.0, 0.8〉, 〈Q

_{4}, 0.4, 0.2, 0.4〉, 〈Q

_{5}, 0.2, 0.4, 0.4〉, 〈Q

_{6}, 0.4, 0.0, 0.6〉, 〈Q

_{7}, 0.2, 0.2, 0.6〉},

_{3}= {〈Q

_{1}, 0.6, 0.0, 0.4〉, 〈Q

_{2}, 0.6, 0.2, 0.2〉, 〈Q

_{3}, 0.6, 0.2, 0.2〉, 〈Q

_{4}, 0.8, 0.2, 0.0〉, 〈Q

_{5}, 0.6, 0.2, 0.2〉, 〈Q

_{6}, 0.8, 0.2, 0.0〉, 〈Q

_{7}, 0.4, 0.4, 0.2〉}.

_{j}is w

_{j}= 1/7 for j = 1, 2, …, 7. By applying Equation (2), we can obtain the results of the similarity measure between the patient P

_{k}(k = 1, 2, 3) and the considered symptom S

_{i}(i = 1, 2, 3, 4), as shown in Table 4.

_{1}and P

_{2}have moderate symptoms, and Patient P

_{3}has severe symptoms.

#### 4.2. Initial Evaluation of the BPH Symptoms Under an Interval Neutrosophic Environment

**Example**

**2.**

_{k}(k = 1, 2, 3) with respect to all the questions can be represented by the following INS information:

_{1}= {〈Q

_{1}, [0.4, 0.6], [0, 0.2], [0.2, 0.4]〉, 〈Q

_{2}, [0.2, 0.4], [0.4, 0.6], [0, 0.2]〉, 〈Q

_{3}, [0.4, 0.6], [0.2, 0.4], [0.2, 0.4]〉, 〈Q

_{4}, [0.4, 0.6], [0, 0.2], [0.2, 0.4]〉, 〈Q

_{5}, [0.6, 0.8], [0.2, 0.4], [0, 0]〉, 〈Q

_{6}, [0.4, 0.6], [0, 0.2], [0.4, 0.6]〉, 〈Q

_{7}, [0.4, 0.6], [0, 0.2], [0.2, 0.4]〉},

_{2}= {〈Q

_{1}, [0.2, 0.4], [0, 0.2], [0.4, 0.6]〉, 〈Q

_{2}, [0.4, 0.4], [0, 0.2], [0.2, 0.4]〉, 〈Q

_{3}, [0.2, 0.4], [0, 0.2], [0.6, 0.8]〉, 〈Q

_{4}, [0.4, 0.6], [0.2, 0.4], [0, 0.4]〉, 〈Q

_{5}, [0.2, 0.4], [0.4, 0.6], [0, 0.2]〉, 〈Q

_{6}, [0.4, 0.6], [0, 0.2], [0.4, 0.6]〉, 〈Q

_{7}, [0.2, 0.4], [0.2, 0.4], [0.2, 0.4]〉},

_{3}= {〈Q

_{1}, [0.6, 0.8], [0, 0], [0.2, 0.4]〉, 〈Q

_{2}, [0.6, 0.8], [0.2, 0.2], [0, 0.2]〉, 〈Q

_{3}, [0.6, 0.8], [0.2, 0.4], [0, 0]〉, 〈Q

_{4}, [0.6, 0.8], [0.2, 0.4], [0, 0]〉, 〈Q

_{5}, [0.6, 0.8], [0.2, 0.4], [0, 0.2]〉, 〈Q

_{6}, [0.6, 0.8,] [0.2, 0.4], [0, 0]〉, 〈Q

_{7}, [0.4, 0.6], [0.4, 0.6], [0, 0.2]〉}.

_{j}is w

_{j}= 1/7 for j = 1, 2, …, 7. By applying Equation (4), we can obtain the results of the similarity measure between Patient P

_{k}(k = 1, 2, 3) and the considered symptom S

_{i}(i = 1, 2, 3, 4), as shown in Table 6.

_{1}and P

_{3}have severe symptoms, and Patient P

_{2}has moderate symptoms.

#### 4.3. Comparison and Analysis

_{k}(k = 1, 2, 3) with respect to all the questions of Examples 1 and 2, which are shown in Table 7 and Table 8, respectively. According to the common evaluation method of I-PSS [1,2], where one time means one score in I-PSS [1,2], we can give the clinical initial evaluation results of three patients (P

_{1}, P

_{2}, P

_{3}), which are also shown in Table 7 and Table 8, respectively.

_{1}is difficult to determine the moderate and/or severe symptoms in the common evaluation/diagnosis method based on I-PSS [1,2]. However, P

_{1}has severe symptoms based on the comprehensive evaluation/diagnosis results in this study. Therefore, the diagnosis results of the two examples demonstrate the effectiveness of the proposed diagnosis method under simplified neutrosophic environments.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Table 1.**The number of benign prostatic hyperplasia (BPH) symptoms in the 5 times for a patient P

_{k}. T: true; I: indeterminate; F: false.

Question | T | I | F |
---|---|---|---|

Q_{1}: Over the past month, how often have you had a sensation of not emptying your bladder completely after you finished urinating? | |||

Q_{2}: Over the past month, how often have you had to urinating again less than two hours after you finished urinating? | |||

Q_{3}: Over the past month, how often have you found you stopped and started again several times when you urinated? | |||

Q_{4}: Over the past month, how often have you found it difficult to postpone urination? | |||

Q_{5}: Over the past month, how often have you had a week urinary stream? | |||

Q_{6}: Over the past month, how often have you had to push or strain to begin urination? | |||

Q_{7}: Over the past month, how many times did you most typically get up to urinate from the time you went to bed at night until the time you got up in the morning? |

S_{i} (Symptom Type) | Q_{1} | Q_{2} | Q_{3} | Q_{4} | Q_{5} | Q_{6} | Q_{7} |
---|---|---|---|---|---|---|---|

S_{1} (Normal symptom) | <0, 0, 1> | <0, 0, 1> | <0, 0, 1> | <0, 0, 1> | <0, 0, 1> | <0, 0, 1> | <0, 0, 1> |

S_{2} (Mild symptom) | <0, 0.2, 0.8> | <0, 0.2, 0.8> | <0, 0.2, 0.8> | <0, 0.2, 0.8> | <0, 0.2, 0.8> | <0, 0.2, 0.8> | <0, 0.2, 0.8> |

S_{3} (Moderate symptom) | <0.2, 0.4, 0.4> | <0.2, 0.4, 0.4> | <0.2, 0.4, 0.4> | <0.2, 0.4, 0.4> | <0.2, 0.4, 0.4> | <0.2, 0.4, 0.4> | <0.2, 0.4, 0.4> |

S_{4} (Severe symptom) | <0.6, 0.4, 0> | <0.6, 0.4, 0> | <0.6, 0.4, 0> | <0.6, 0.4, 0> | <0.6, 0.4, 0> | <0.6, 0.4, 0> | <0.6, 0.4, 0> |

Question | P_{1} | P_{2} | P_{3} | ||||||
---|---|---|---|---|---|---|---|---|---|

T | I | F | T | I | F | T | I | F | |

Q_{1} | 2/5 | 1/5 | 2/5 | 1/5 | 1/5 | 3/5 | 3/5 | 0/5 | 2/5 |

Q_{2} | 2/5 | 2/5 | 1/5 | 2/5 | 1/5 | 2/5 | 3/5 | 1/5 | 1/5 |

Q_{3} | 2/5 | 1/5 | 2/5 | 1/5 | 0/5 | 4/5 | 3/5 | 1/5 | 1/5 |

Q_{4} | 2/5 | 1/5 | 2/5 | 2/5 | 1/5 | 2/5 | 4/5 | 1/5 | 0/5 |

Q_{5} | 3/5 | 2/5 | 0/5 | 1/5 | 2/5 | 2/5 | 3/5 | 1/5 | 1/5 |

Q_{6} | 2/5 | 0/5 | 3/5 | 2/5 | 0/5 | 3/5 | 4/5 | 1/5 | 0/5 |

Q_{7} | 3/5 | 0/5 | 2/5 | 1/5 | 1/5 | 3/5 | 2/5 | 2/5 | 1/5 |

**Table 4.**Similarity measure values between P

_{k}and S

_{i}with single-valued neutrosophic sets (SVNSs).

S_{1} | S_{2} | S_{3} | S_{4} | |
---|---|---|---|---|

W_{1}(P_{1}, S_{i}) | 0.4457 | 0.5460 | 0.7285 | 0.6857 |

W_{1}(P_{2}, S_{i}) | 0.5896 | 0.7038 | 0.7814 | 0.5244 |

W_{1}(P_{3}, S_{i}) | 0.3319 | 0.4406 | 0.6112 | 0.7778 |

Question | P_{1} | P_{2} | P_{3} | ||||||
---|---|---|---|---|---|---|---|---|---|

T | I | F | T | I | F | T | I | F | |

Q_{1} | [2/5, 3/5] | [0/5, 1/5] | [1/5, 2/5] | [1/5, 2/5] | [0/5, 1/5] | [2/5, 3/5] | [3/5, 4/5] | [0/5, 0/5] | [1/5, 2/5] |

Q_{2} | [2/5, 3/5] | [2/5, 3/5] | [0/5, 1/5] | [2/5, 2/5] | [0/5, 1/5] | [1/5, 2/5] | [3/5, 4/5] | [1/5, 1/5] | [0/5, 1/5] |

Q_{3} | [2/5, 3/5] | [1/5, 2/5] | [1/5, 2/5] | [1/5, 2/5] | [0/5, 1/5] | [3/5, 4/5] | [3/5, 4/5] | [1/5, 2/5] | [0/5, 0/5] |

Q_{4} | [2/5, 3/5] | [0/5, 1/5] | [1/5, 2/5] | [2/5, 3/5] | [1/5, 2/5] | [0/5, 2/5] | [3/5, 4/5] | [1/5, 2/5] | [0/5, 0/5] |

Q_{5} | [3/5, 4/5] | [1/5, 2/5] | [0/5, 0/5] | [1/5, 2/5] | [2/5, 3/5] | [0/5, 1/5] | [3/5, 4/5] | [1/5, 2/5] | [0/5, 1/5] |

Q_{6} | [2/5, 3/5] | [0/5, 1/5] | [2/5, 3/5] | [2/5, 3/5] | [0/5, 1/5] | [2/5, 3/5] | [3/5, 4/5] | [1/5, 2/5] | [0/5, 0/5] |

Q_{7} | [2/5, 3/5] | [0/5, 1/5] | [1/5, 2/5] | [1/5, 2/5] | [1/5, 2/5] | [1/5, 2/5] | [2/5, 3/5] | [2/5, 3/5] | [0/5, 1/5] |

**Table 6.**Similarity measure values of between P

_{k}and S

_{i}with interval neutrosophic sets (INSs).

S_{1} | S_{2} | S_{3} | S_{4} | |
---|---|---|---|---|

W_{2}(P_{1}, S_{i}) | 0.3956 | 0.4910 | 0.6784 | 0.7164 |

W_{2}(P_{2}, S_{i}) | 0.4799 | 0.5831 | 0.7331 | 0.6297 |

W_{2}(P_{3}, S_{i}) | 0.2718 | 0.3734 | 0.5741 | 0.8322 |

**Table 7.**The number of BPH symptoms (single values of T) in the 5 times for three patients, where one time means one score in the international prostate symptom score (I-PSS).

Question | P_{1} | P_{2} | P_{3} | ||||||
---|---|---|---|---|---|---|---|---|---|

T (time) | I | F | T (time) | I | F | T (time) | I | F | |

Q_{1} | 2 | / | / | 1 | / | / | 3 | / | / |

Q_{2} | 2 | / | / | 2 | / | / | 3 | / | / |

Q_{3} | 2 | / | / | 1 | / | / | 3 | / | / |

Q_{4} | 2 | / | / | 2 | / | / | 4 | / | / |

Q_{5} | 3 | / | / | 1 | / | / | 3 | / | / |

Q_{6} | 2 | / | / | 2 | / | / | 4 | / | / |

Q_{7} | 3 | / | / | 1 | / | / | 2 | / | / |

Total score | 16 | 10 | 22 | ||||||

BPH symptom | Moderate | Moderate | Severe |

Question | P_{1} | P_{2} | P_{3} | ||||||
---|---|---|---|---|---|---|---|---|---|

T (time) | I | F | T (time) | I | F | T (time) | I | F | |

Q_{1} | [2, 3] | / | / | [1,2] | / | / | [3, 4] | / | / |

Q_{2} | [2, 3] | / | / | [2, 2] | / | / | [3, 4] | / | / |

Q_{3} | [2, 3] | / | / | [1,2] | / | / | [3, 4] | / | / |

Q_{4} | [2, 3] | / | / | [2, 3] | / | / | [3, 4] | / | / |

Q_{5} | [3, 4] | / | / | [1,2] | / | / | [3, 4] | / | / |

Q_{6} | [2, 3] | / | / | [2, 3] | / | / | [3, 4] | / | / |

Q_{7} | [2, 3] | / | / | [1,2] | / | / | [2, 3] | / | / |

Total score | [15, 22] | [10, 16] | [20, 27] | ||||||

BPH symptom | Moderate and/or severe | Moderate | Severe |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fu, J.; Ye, J.
Simplified Neutrosophic Exponential Similarity Measures for the Initial Evaluation/Diagnosis of Benign Prostatic Hyperplasia Symptoms. *Symmetry* **2017**, *9*, 154.
https://doi.org/10.3390/sym9080154

**AMA Style**

Fu J, Ye J.
Simplified Neutrosophic Exponential Similarity Measures for the Initial Evaluation/Diagnosis of Benign Prostatic Hyperplasia Symptoms. *Symmetry*. 2017; 9(8):154.
https://doi.org/10.3390/sym9080154

**Chicago/Turabian Style**

Fu, Jing, and Jun Ye.
2017. "Simplified Neutrosophic Exponential Similarity Measures for the Initial Evaluation/Diagnosis of Benign Prostatic Hyperplasia Symptoms" *Symmetry* 9, no. 8: 154.
https://doi.org/10.3390/sym9080154