1. Introduction
Benign prostatic hyperplasia (BPH) is a common medical problem encountered in aging men, and leads to obstructive and irritative voiding symptoms. The American Urological Association (AUA) uses seven questions as the AUA symptom indices [
1,
2] for BPH scored on a scale from 0 to 5 points. The international prostate symptom score (I-PSS) [
1,
2] offers an objective documentation of symptoms: a total score of 0–7 is mildly symptomatic, 8–19 moderately symptomatic, and 20–35 severely symptomatic. However, the objective evaluation is a non-fuzzy evaluation method (a common evaluation method) in I-PSS.
The initial evaluation of the BPH symptoms is obtained by means of clinical survey for a patient to select further examinations (e.g., creatinine, intravenous urography, urethrogram, urodynamics, urethrocystoscopy, etc.) and suitable treatment alternatives (e.g., watchful waiting, medical, surgical, or minimally invasive surgical treatments, etc.). Then, the choice of treatment is reached in a shared decision-making process between the physician and the patient. When the physician carries out the clinical survey of a patient to reach the initial evaluation of the BPH symptoms, the patient gives the responses to the seven questions which may contain a “grey zone” of the uncertainty for the patient about the BPH symptoms. Thus, the clinical data of the BPH symptoms obtained by the physician are incomplete, uncertain, or contradictory. In this case, fuzzy expression is a suitable tool. Zadeh [
3] first introduced the membership/truth degree in 1965 and defined a fuzzy set. Based on a generalization of the fuzzy set, Atanassov [
4] introduced the nonmembership/falsity degree in 1986 and defined an intuitionistic fuzzy set (IFS) as a generalization of the fuzzy set. Then, Atanassov and Gargov [
5] introduced an interval-valued IFS as a generalization of IFS. Further, Smarandache [
6] introduced the degree of indeterminacy/neutrality as an independent component in 1995 and defined a neutrosophic set as the generalization of IFS and the interval-valued IFS. He coined the terms “neutrosophy” and “neutrosophic”. From a philosophical point of view, the neutrosophic set can represent uncertain, imprecise, incomplete, and inconsistent information. Its advantage is that the neutrosophic set can express indeterminate and inconsistent information, but IFS and the interval-valued IFS cannot. From a science and engineering point of view, the neutrosophic set will be difficult to apply in real science and engineering fields [
7,
8] because the truth, indeterminacy, and falsity functions in a neutrosophic set belong to the non-standard interval ]
−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).
However, it is difficult to adapt the above-mentioned diagnosis methods to deal with the evaluation problems of BPH with simplified neutrosophic information. Generally, the existing initial evaluation method of BPH is commonly based on I-PSS [
1,
2] and uses the objective evaluation/diagnosis method with crisp values without considering fuzzy information. Hence, this common evaluation/diagnosis method may lose a lot of incomplete, uncertain, and inconsistent information in the clinical survey and initial evaluation process of the BPH symptoms for a patient, resulting in unreasonable evaluation and diagnosis distortion of the BPH symptoms. To overcome this drawback, this paper aims to propose new exponential similarity measures (ESMs) between SNSs, including single-valued neutrosophic ESMs and interval neutrosophic ESMs, and their initial evaluation/diagnosis method of the BPH symptoms with simplified neutrosophic information.
The rest of the article is structured as follows. In
Section 2, we briefly introduce some basic concepts of SNSs.
Section 3 proposes ESMs between SNSs based on exponential function, including single-valued neutrosophic ESMs and interval neutrosophic ESMs, and investigates their properties. In
Section 4, the initial evaluation/diagnosis methods of the BPH symptoms are presented based on the ESMs under a simplified neutrosophic environment, and then two examples of the evaluation of BPH symptoms are given to show the effectiveness and rationality of the proposed evaluation method. Conclusions and further research are given in
Section 5.
3. ESMs of SNSs
Based on an exponential function, this section proposes ESMs between SNSs, including single-valued neutrosophic ESMs and interval neutrosophic ESMs, and investigates their properties.
Definition 2. Let M = {〈xj, TM(xj), IM(xj), FM(xj)〉| xj ∈ X} and N = {〈xj, TN(xj), IN(xj), FN(xj)〉| xj ∈ X} be any two SVNSs in X = {x1, x2, …, xn}. Thus, we can define an ESM between N and M as follows: Obviously, ESM has the following proposition.
Proposition 1. For two SVNSs M and N in X = {x1, x2, …, xn}, the ESM E1(M, N) should satisfy the following properties (1)–(4):
- (1)
0 ≤ E1(M, N) ≤ 1;
- (2)
E1(M, N) = 1 if and only if M = N, i.e., TM(xj) = TN(xj), IM(xj) = IN(xj), and FM(xj) = FN(xj) for xj ∈ X (j = 1, 2, …, n);
- (3)
E1(M, N) = E1(N, M);
- (4)
If P is an SVNS in X and M ⊆ N ⊆ P, then E1(M, P) ≤ E1(M, N) and E1(M, P) ≤ E1(N, P).
Proof. (1) Since there are TM(xj) IM(xj) FM(xj) ∈ [0, 1] and TN(xj) IN(xj) FN(xj) ∈ [0, 1] in the two SVNSs M and N, the distance lies between 0 and 1. By applying Equation (1), ESM also lies between 0 and 1. Hence, there is 0 ≤ E1(M, N) ≤ 1.
(2) For the two SVNSs
M and
N, if
M =
N, this implies
TM(
xj) =
TN(
xj),
IM(
xj) =
IN(
xj),
FM(
xj) =
FN(
xj) for
xj ∈
X and
j = 1, 2, …,
n. Hence, there are
,
, and
. Thus, we can obtain the following result:
If
E1(
M,
N) = 1, we have the following equation:
Then, there exists the following result:
This implies , and then there are , , and . Thus, these equalities indicate that TM(xj) = TN(xj), IM(xj) = IN(xj), and FM(xj) = FN(xj) for xj ∈ X and j = 1, 2, …, n. Hence M = N.
(3) Proof is straightforward.
(4) If
M ⊆
N ⊆
P, then this implies
TM(
xj) ≤
TN(
xj) ≤
TP(
xj),
IM(
xj) ≥
IN(
xj) ≥
IP(
xj),
FM(
xj) ≥
FN(
xj) ≥
FP(
xj) for
xj ∈
X and
j = 1, 2, …,
n. Then, we have
Hence, E1(M, P) ≤ E1(M, N) and E1(M, P) ≤ E1(N, P) since the exponential function is a decreasing function.
Therefore, the proofs of these properties are completed. ☐
Generally, one takes the weight of each element
xj for
xj ∈
X into account and assumes that the weight of an element
xj is
wj (
j = 1, 2, …,
n) with
wj ∈ [0, 1] and
. Hence, we can introduce the following weighted ESM between
M and
N:
Clearly, the ESM W1(M, N) should satisfy the properties (1)–(4) in Proposition 1. Especially when wj = 1/n for j = 1, 2, …, n, Equation (2) reduces to Equation (1).
Similarly, we can extend the ESMs of SVNSs to propose ESMs between INSs.
Let
M = {〈
xj,
TM(
xj),
IM(
xj),
FM(
xj)〉|
xj ∈
X} and
N = {〈
xj,
TN(
xj),
IN(
xj),
FN(
xj)〉|
xj ∈
X} be any two INSs in
X = {
x1,
x2, …,
xn}, where
TM(
xj) = [inf
TM(
xj), sup
TM(
xj)] ⊆ [0, 1],
IM(
xj) = [inf
IM(
xj), sup
IM(
xj)] ⊆ [0, 1], and
FM(
xj) = [inf
FM(
xj), sup
FM(
xj)] ⊆ [0, 1] in
M for any
xj ∈
X are denoted by
,
, and
, respectively, and
TN(
xj) = [inf
TN(
xj), sup
TN(
xj)] ⊆ [0, 1],
IN(
xj) = [inf
IN(
xj), sup
IN(
xj)] ⊆ [0, 1], and
FN(
xj) = [inf
FN(
xj), sup
FN(
xj)] ⊆ [0, 1] in
N for any
xj ∈
X are denoted by
,
, and
, 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:
where
wj is the weight of an element
xj (
j = 1, 2, …,
n) with
wj ∈ [0, 1] and
.
Obviously, Equations (1) and (2) are the special cases of Equations (3) and (4) when the upper and lower ends of the interval numbers TM(xj), IM(xj), FM(xj) in M and TN(xj), IN(xj), FN(xj) 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
According to the seven questions in the AUA symptom indexes [
1,
2] for BPH, we can consider a set of the seven questions
Q = {
Q1 (Over the past month, how often have you had a sensation of not emptying your bladder completely after you finished urinating?),
Q2 (Over the past month, how often have you had to urinate again less than two hours after you finished urinating?),
Q3 (Over the past month, how often have you found you stopped and started again several times when you urinated?),
Q4 (Over the past month, how often have you found it difficult to postpone urination?),
Q5 (Over the past month, how often have you had a weak urinary stream?),
Q6 (Over the past month, how often have you had to push or strain to begin urination?),
Q7 (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
Pk (
k = 1, 2, …,
t) can be constructed by
Table 1, where
T,
I, and
F denote truth, indeterminacy, and falsity, respectively.
Based on I-PSS [
1,
2], BPH can be divided into the four types of symptoms, which are represented by a set of the four types of symptoms
S = {
S1 (Normal symptom),
S2 (Mild symptom),
S3 (Moderate symptom),
S4 (Severe symptom)} as the symptom knowledge for the initial evaluation of BPH patients, as shown in
Table 2.
From
Table 2, the BPH symptom types of patients with respect to all the questions can be represented by the following SNS information:
S1 = {〈Q1, 0, 0, 1〉, 〈Q2, 0, 0, 1〉, 〈Q3, 0, 0, 1〉, 〈Q4, 0, 0, 1〉, 〈Q5, 0, 0, 1〉, 〈Q6, 0, 0, 1〉, 〈Q7, 0, 0, 1〉},
S2 = {〈Q1, 0, 0.2, 0.8〉, 〈Q2, 0, 0.2, 0.8〉, 〈Q3, 0, 0.2, 0.8〉, 〈Q4, 0, 0.2, 0.8〉, 〈Q5, 0, 0.2, 0.8〉, 〈Q6, 0, 0.2, 0.8〉, 〈Q7, 0, 0.2, 0.8〉},
S3 = {〈Q1, 0.2, 0.4, 0.4〉, 〈Q2, 0.2, 0.4, 0.4〉, 〈Q3, 0.2, 0.4, 0.4〉, 〈Q4, 0.2, 0.4, 0.4〉, 〈Q5, 0.2, 0.4, 0.4〉, 〈Q6, 0.2, 0.4, 0.4〉, 〈Q7, 0.2, 0.4, 0.4〉}.
S4 = {〈Q1, 0.6, 0.4, 0〉, 〈Q2, 0.6, 0.4, 0〉, 〈Q3, 0.6, 0.4, 0〉, 〈Q4, 0.6, 0.4, 0〉, 〈Q5, 0.6, 0.4, 0〉, 〈Q6, 0.6, 0.4, 0〉, 〈Q7, 0.6, 0.4, 0〉}.
Assume that we give the clinical survey for
t BPH patients by using
Table 1 to obtain the
t patients’ responses of the BPH symptom which are represented by the form of truth, indeterminacy, and falsity values. For a patient
Pk (
k = 1, 2, …,
t) with SNS information, we can give the following evaluation/diagnosis method.
To give a proper evaluation/diagnosis for a patient Pk with BPH symptoms, we can calculate the similarity measure Wq(Pk, Si) for q = 1 or 2, i = 1, 2, 3, 4 and k = 1, 2, …, t. The proper BPH symptom evaluation Si* for patient Pk is derived by .
To illustrate the evaluation/diagnosis process of the BPH symptoms, we provide two evaluation/diagnosis examples of the BPH symptoms to demonstrate the applications and effectiveness of the proposed evaluation/diagnosis method under simplified neutrosophic (single-valued neutrosophic and interval neutrosophic) environments.
4.1. Initial Evaluation of the BPH Symptoms Under a Single-Valued Neutrosophic Environment
In some cases, we can obtain that data collected from the clinical survey of patients are single values rather than interval values for T, I, and F. In this case, ESM of SVNSs is a better tool to give a proper initial evaluation of a patient’s BPH symptoms.
Example 1. Assume that we give a clinical survey for three BPH patients by using Table 1, and then we can obtain the three patients’ responses of the BPH symptoms which are represented by the form of truth, indeterminacy, and falsity values, as shown in Table 3.
From
Table 3, the BPH symptom responses of the patient
Pk (
k = 1, 2, 3) with respect to all the questions can be represented by the following SVNS information:
P1 = {〈Q1, 0.4, 0.2, 0.4〉, 〈Q2, 0.4, 0.4, 0.2〉, 〈Q3, 0.4, 0.2, 0.4〉, 〈Q4, 0.4, 0.2, 0.4〉, 〈Q5, 0.6, 0.4, 0.0〉, 〈Q6, 0.4, 0.0, 0.6〉, 〈Q7, 0.6, 0.0, 0.4〉},
P2 = {〈Q1, 0.2, 0.2, 0.6〉, 〈Q2, 0.4, 0.2, 0.4〉, 〈Q3, 0.2, 0.0, 0.8〉, 〈Q4, 0.4, 0.2, 0.4〉, 〈Q5, 0.2, 0.4, 0.4〉, 〈Q6, 0.4, 0.0, 0.6〉, 〈Q7, 0.2, 0.2, 0.6〉},
P3 = {〈Q1, 0.6, 0.0, 0.4〉, 〈Q2, 0.6, 0.2, 0.2〉, 〈Q3, 0.6, 0.2, 0.2〉, 〈Q4, 0.8, 0.2, 0.0〉, 〈Q5, 0.6, 0.2, 0.2〉, 〈Q6, 0.8, 0.2, 0.0〉, 〈Q7, 0.4, 0.4, 0.2〉}.
Assume that the weight of each element
Qj is
wj = 1/7 for
j = 1, 2, …, 7. By applying Equation (2), we can obtain the results of the similarity measure between the patient
Pk (
k = 1, 2, 3) and the considered symptom
Si (
i = 1, 2, 3, 4), as shown in
Table 4.
In
Table 4, the largest similarity measure indicates the proper evaluation/diagnosis. Therefore, in initial clinical evaluations for the three patients, Patients
P1 and
P2 have moderate symptoms, and Patient
P3 has severe symptoms.
4.2. Initial Evaluation of the BPH Symptoms Under an Interval Neutrosophic Environment
In some cases, we can obtain that data collected from the clinical survey of patients are interval values rather than single values for T, I, and F, since patients easily express real situations by using the interval values. In this case, ESM of INSs is a better tool to give a proper initial evaluation of the BPH symptoms.
Example 2. Assume that we give the clinical survey for three BPH patients by using Table 1, and then we can obtain the three patients’ responses of the BPH symptoms, which are represented by the interval values of T, I, and F, as shown in Table 5.
From
Table 5, the BPH symptom responses of patient
Pk (
k = 1, 2, 3) with respect to all the questions can be represented by the following INS information:
P1 = {〈Q1, [0.4, 0.6], [0, 0.2], [0.2, 0.4]〉, 〈Q2, [0.2, 0.4], [0.4, 0.6], [0, 0.2]〉, 〈Q3, [0.4, 0.6], [0.2, 0.4], [0.2, 0.4]〉, 〈Q4, [0.4, 0.6], [0, 0.2], [0.2, 0.4]〉, 〈Q5, [0.6, 0.8], [0.2, 0.4], [0, 0]〉, 〈Q6, [0.4, 0.6], [0, 0.2], [0.4, 0.6]〉, 〈Q7, [0.4, 0.6], [0, 0.2], [0.2, 0.4]〉},
P2 = {〈Q1, [0.2, 0.4], [0, 0.2], [0.4, 0.6]〉, 〈Q2, [0.4, 0.4], [0, 0.2], [0.2, 0.4]〉, 〈Q3, [0.2, 0.4], [0, 0.2], [0.6, 0.8]〉, 〈Q4, [0.4, 0.6], [0.2, 0.4], [0, 0.4]〉, 〈Q5, [0.2, 0.4], [0.4, 0.6], [0, 0.2]〉, 〈Q6, [0.4, 0.6], [0, 0.2], [0.4, 0.6]〉, 〈Q7, [0.2, 0.4], [0.2, 0.4], [0.2, 0.4]〉},
P3 = {〈Q1, [0.6, 0.8], [0, 0], [0.2, 0.4]〉, 〈Q2, [0.6, 0.8], [0.2, 0.2], [0, 0.2]〉, 〈Q3, [0.6, 0.8], [0.2, 0.4], [0, 0]〉, 〈Q4, [0.6, 0.8], [0.2, 0.4], [0, 0]〉, 〈Q5, [0.6, 0.8], [0.2, 0.4], [0, 0.2]〉, 〈Q6, [0.6, 0.8,] [0.2, 0.4], [0, 0]〉, 〈Q7, [0.4, 0.6], [0.4, 0.6], [0, 0.2]〉}.
Assume that the weight of each element
Qj is
wj = 1/7 for
j = 1, 2, …, 7. By applying Equation (4), we can obtain the results of the similarity measure between Patient
Pk (
k = 1, 2, 3) and the considered symptom
Si (
i = 1, 2, 3, 4), as shown in
Table 6.
In
Table 6, the largest similarity measure indicates the proper evaluation/diagnosis. Therefore, in initial clinical evaluations for the three patients, Patients
P1 and
P3 have severe symptoms, and Patient
P2 has moderate symptoms.
4.3. Comparison and Analysis
For convenient comparison with the common evaluation method [
1,
2], based on
Table 3 and
Table 5 we only give the BPH symptom responses with the values of
T (without the values of
I and
F in I-PSS) of the patient
Pk (
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 (
P1,
P2,
P3), which are also shown in
Table 7 and
Table 8, respectively.
In
Table 7, all the initial evaluation/diagnosis results of Example 1 are the same as the ones of the new evaluation method proposed in this paper. Then, all the initial evaluation/diagnosis results of Example 2 in
Table 8 are almost the same as the ones of the new evaluation method, but the evaluation/diagnosis of
P1 is difficult to determine the moderate and/or severe symptoms in the common evaluation/diagnosis method based on I-PSS [
1,
2]. However,
P1 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.
Compared with the existing initial evaluation method based on I-PSS [
1,
2], the proposed evaluation method demonstrates their effectiveness and rationality because the developed initial evaluation method with simplified neutrosophic information contains much more evaluation information (truth, indeterminacy, and falsity information) than the existing initial evaluation method based on I-PSS (crisp values) without indeterminacy and falsity information [
1,
2]. Obviously, the common initial evaluation method based on I-PSS may lose much useful information (indeterminacy and falsity information), resulting in the unreasonable/difficult evaluation and diagnosis distortion for some patients. Therefore, the developed diagnosis method is more suitable and more practical in the initial evaluation of BPH symptoms, is superior to the existing initial evaluation method [
1,
2], and also shows an advantage in terms of the effectiveness and rationality of the proposed diagnosis method.