The Dice measure of cubic hesitant fuzzy sets and its initial evaluation method of benign prostatic hyperplasia symptoms

In order to give the initial evaluation of benign prostatic hyperplasia (BPH) symptoms regarding a patient, physician usually performs the clinical inquiry of the patient, whereas his/her responses may contain hesitant fuzzy and uncertain information. However, existing evaluation/diagnosis approaches of BPH symptoms cannot cope with the hybrid problem of both hesitant and uncertain responses of patients. Furthermore, existing evaluation approaches may lose some useful responses (e.g. hesitant fuzzy information) so as to result in the unreasonable or indeterminate evaluation/diagnosis in the evaluation process of patients. To overcome this insufficiency, this study firstly introduces the concept of a cubic hesitant fuzzy set (CHFS) based on combining uncertain/interval-valued fuzzy information with hesitant fuzzy information so as to express the hybrid fuzzy information and proposes the Dice measure between CHFSs based on the extension method of the least common multiple cardinality/number (LCMC) for the hesitant fuzzy sets in CHFS. Then, the initial evaluation approach of BPH symptoms is developed by using the Dice measure of CHFSs. Lastly, the assessment results of six BPH patients are presented as the clinical actual cases to indicate the applicability and effectiveness of the developed assessment approach in CHFS setting. The comparison with existing common evaluation methods shows that the developed evaluation/diagnosis method is superior to the existing common evaluation methods in the evaluation/diagnosis process of the clinical actual cases.

SciEntific REPORTS | (2019) 9:60 | DOI: 10.1038/s41598-018-37228-9 0.7] is given by one of three experts and the HFS {0.5, 0.6} is given by two of three experts under their uncertain and hesitant situation, and then the hybrid form of both [0.5, 0.7] (the uncertain part) and {0.5, 0.6} (the hesitant fuzzy part) cannot be expressed simultaneously by the aforementioned various fuzzy concepts. Then, Fu et al. 18 presented a cubic hesitant fuzzy set (CHFS) so as to express the hybrid fuzzy information and applied it to the evaluation/diagnosis problems of the prostate cancer in CHFS setting. However, the evaluation method in 18 can only cope with the evaluation/diagnosis problems of the prostate cancer, but cannot suit the evaluation problems of BPH patients in CHFS setting. Furthermore, existing common evaluation methods of BPH symptoms 10,19,20 can also cope with evaluation/diagnosis problems of BPH symptoms with SNSs or uncertain information, but cannot handle evaluation/diagnosis problems of BPH symptoms with both uncertain and hesitant fuzzy information in CHFS setting. As the further generalization of the cubic hesitant fuzzy evaluation/diagnosis method 18 , this study extends it to the evaluation problems of BPH patients. To do so, this paper first proposes the Dice measure of CHFSs, and then develops the initial evaluation/diagnosis of BPH symptoms based on the Dice measure of CHFSs to solve the initial evaluation/diagnosis problems of BPH symptoms in CHFS setting.
As the framework of this study, Section 2 introduces the CHFS concept based on the hybrid form of both IVFN and HFS and proposes the Dice measure between CHFSs based on the extension method of the least common multiple cardinality/number (LCMC) for the HFSs in CHFS. In Section 3, the evaluation method of BPH symptoms is developed based on the Dice measure of CHFSs under CHFS setting. Section 4 presents the evaluation of the BPH symptoms by six BPH patients as the clinical actual cases to show the effectiveness and rationality of the evaluation approach of BPH symptoms based on the Dice measure of CHFSs. Lastly, conclusions and further study are contained in Section 5.

Cubic hesitant fuzzy sets and the Dice measure of CHFSs
Cubic hesitant fuzzy sets. Regarding a hybrid form of an IVFN and a fuzzy value, Jun et al. 17 presented a (fuzzy) CS in a fixed non-empty set U by the following form: where a(u) = [a − , a + ] is an IVFN and μ(u) is a fuzzy value for u ∈ U. Then, a HFS concept 21,22 in a fixed non-empty set U is expressed as Regarding a hybrid form of both a HFS and a CS, Fu et al. 18 gave the definition of CHFS below. Definition 1. Set U as a fixed non-empty set. A CHFS R is defined as the following form 18 :    For convenient representation, Eqs (1) and (2) are also written as the following simple form: The following numerical example is given to indicate the LCMC extension method. The LCMC of both is q = 6 for t 1 = 2 and t 2 = 3 in r 1 and r 2 . By applying Eqs (1) and (2), the two CHFEs h 1 and h 2 can be extended to the following forms: The Dice measure of CHFSs. In this subsection, we propose the Dice measure of CHFSs based on the LCMC extension method for the HFSs in CHFS since the similarity measure is an important mathematical tool in pattern recognition and medical diagnosis areas.
If r 1n and r 2n are considered as the two vectors of q k + 2 dimensions, the Dice measure between R 1 and R 2 is defined as Then, the Dice measure D(R 1 , R 2 ) indicates the following proposition. Proposition 1. The Dice measure D(R 1 , R 2 ) contains the following properties: Proof: When the importance of the CHFEs r 1k and r 2k is taken into account, we set ω k for 0 ≤ ω k ≤ 1 and Thus, the weighted Dice measure D ω (R 1 , R 2 ) also has the following proposition.

Example 2.
Let us consider two CHFSs: Then, their weight vector is given as ω = (0.4, 0.6) to calculate the weighted Dice measure between R 1 and R 2 .
First, we get their LCMC q 1 = q 2 = 6 from a pair of r 11 and r 21 and a pair of r 12 and r 22 . Thus, we get the following extension forms: .  The Dice measure-based evaluation/diagnosis method of BPH symptoms Aging men commonly encounter the disease of BPH and suffer from obstructive and irritative voiding symptoms.
To assess BPH symptoms, the seven questions introduced by the American Urological Association (AUA) are considered as the AUA symptom indices 19,20 for BPH, which are scored on a scale from 0 to 5 points so as to use the evaluation/diagnosis of BPH symptoms for clinical patients. An objective documentation of BPH symptoms was offered by the international prostate symptom score (I-PSS) 19,20 , In existing common clinical evaluation/ diagnosis of BPH symptoms, the common score and evaluation method 19,20 are shown in Tables 1 and 2. Then, the total score in Table 1 was thirty-five, which can be classified into three types of evaluation grades of BPH symptoms in Table 2, along with totally scoring 0-7 as mild symptom, 8-19 as moderate symptom, and 20-35 as severe symptom for a BPH patient. However, this objective evaluation method presented in I-PSS is a traditional and non-fuzzy assessment/diagnosis, whereas the patient's response to the seven questions of BPH symptoms may imply the hybrid information of both uncertain responses and hesitant responses regarding his/her vague symptoms indicated over the past month. Obviously, CHFS is very fit for the expression of the hybrid information. Thus, the Dice measure-based evaluation/diagnosis can solve the evaluation/diagnosis problems of BPH symptoms with CHFS information. Therefore, this section proposes the Dice measure-based evaluation/diagnosis approach of BPH symptoms in CHFS setting.
Based on Table 1, this study firstly establishes the inquiry table of BPH symptoms with uncertain and hesitant arguments, as shown in Table 3. In Table 3, a collection of the seven questions is expressed by the set of attributes/ indices A = {A 1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7 } and the clinical inquiry and answer of a patient Q i (i = 1, 2, …, t) indicate the BPH symptom responses in 5 times over the past month. However, since the patient's answers may imply his/ her uncertainty and hesitancy corresponding to the seven questions, he/she can give the uncertain range and hesitant values in his/her BPH symptom responses in 5 times over the past month.
In the clinical actual application, we require that physicians ask the BPH symptoms of patients over the past month by the seven questions in Table 3 so as to obtain the uncertain and hesitant information from a BPH patient Q i .
Regarding I-PSS 19,20 , we can also sort BPH patients into the three types of symptoms: Mild symptom (R 1 ), Moderate symptom (R 2 ), and Severe symptom (R 3 ), which are constructed as a set of the three types of symptoms R = {R 1 , R 2 , R 3 }, indicating the three symptom patterns, to be used for the initial evaluation/diagnosis of BPH patients, as shown in Table 4.
In Table 4, the three symptom patterns of BPH patients regarding the seven questions can be expressed as the following CHFSs: Suppose that the clinical inquiries and answers of t BPH patients are obtained by Table 3, then we can transform both uncertain ranges and hesitant values into the form of CHFEs. For a patient Q i (i = 1, 2, …, t) corresponding to CHFE information, we can present the following evaluation approach.
The Dice measure D ω (Q i , R j ) for j = 1, 2, 3 and i = 1, 2, …, t can be calculated in order to obtain a fit evaluation/ diagnosis of a BPH patient Q i . Then, the fit evaluation R j* of the BPH patient Q i can be yielded by =

Actual evaluation cases of BPH symptoms
In this section, we consider the clinical actual cases regarding six BPH patients to show the evaluation process of BPH symptoms by the evaluation/diagnosis method of BPH symptoms based on the Dice measure of CHFSs.
First, the six BPH patients Q i (i = 1, 2, …, 6) in the clinical actual cases indicate their responses to the clinical inquiries from Table 3, which are shown in Table 5.
Based on the inquiry results in Table 5, the normalized response values are obtained corresponding to the response times (uncertain values and hesitant values) divided by 5, and then can be transformed into CHFEs, which are shown in Table 6.
From Table 7, the clinical initial evaluations of the six patients demonstrate that the patient Q 2 has mild BPH symptom, the patients Q 1 , Q 4 , and Q 5 have severe BPH symptoms, and the patient Q 3 and Q 6 have moderate BPH symptoms since the patients regarding the largest measure values indicates their fit evaluation results.
To show the effectiveness of the proposed new evaluation method for the six BPH patients, we have to compare the proposed new evaluation method with the existing common evaluation method based on I-PSS 19,20 . In this case, if we do not consider the hesitant values in Table 5 for convenient comparison with the common evaluation method, the BPH symptom response values of the six BPH patients in Table 5 are reduced to the uncertain values in Table 8. Thus, the common evaluation method in the current clinical application can be applied to the  BPH symptom evaluation problems of the six BPH patients in the clinical actual cases. Based on Tables 1 and 2, the totally scoring values regarding the six BPH patients are also shown in Table 8.
For the convenient comparison, the evaluation/diagnosis results given based on the common evaluation method 19,20 and the proposed new method are indicated in Table 9.
In Table 9, the common initial evaluation/diagnosis results of the six BPH patients contain or equal the ones of the proposed new evaluation method. However, the former cannot clearly indicate the diagnosis results of the three BPH patients Q 1 , Q 3 and Q 4 and implies the evaluation/diagnosis indeterminacy so as to difficultly evaluate/ diagnose them in this situation; while the latter can clearly indicates its evaluation results and shows its effectiveness and rationality. Therefore, the proposed new evaluation method based on the Dice measure of CHFSs can overcome the insufficiency of the existing simply scoring evaluation method with uncertain values (i.e., the common evaluation method without the hesitant information in 19,20 ).
Compared with the evaluation approach using exponential similarity measure of SNSs presented in 10 , the proposed evaluation method using the Dice measure of CHFSs contains uncertain and hesitant assessment information of patients, which the evaluation approach in 10 cannot carry out. Furthermore, the new evaluation method in this study is very fit for patients' thinking and expressing habits in their clinical inquiries and answers, which show the main advantage. In the BPH evaluation process, it is obvious that the proposed new evaluation method is more feasible and effective and superior to the existing initial evaluation methods 10,19,20 .  Table 3. BPH symptom responses in 5 times for a patient Q i over the past month.   Table 5. BPH symptom responses of the six clinical patients in 5 times over the past month.    Table 9. The evaluation/diagnosis results given based on the common evaluation method 19,20 and the proposed new method.