1 Introduction
With the development of society, a lot of harmful substances have affected human health, which leads to a high probability of human diseases. Therefore, medical diagnosis Wang2016A is particularly important. However, there is still a serious lack of effective methods in addressing medical diagnostic problems. As a result, how to realize effectively medical diagnosis is still an open issue.
Medical diagnosis belongs to the application of computers in decisionmaking and artificial intelligence. Up to now, the study on medical diagnosis has been done by many scholars
Wang2016A ; Hajarolasvadi2016Employing ; Das2016Medical ; Juan2014Agent . In , Kathryn Z Smith2016Past examined the relationship between Opioid Use Disorder diagnosis, PTSD diagnosis with NMOU, and average monthly frequency of NMOU. Woolard et al. Woolard2016A introduced a retrospective study to show the extent of compliance with perioperative guidelines in patients. In Ahn2012A , the recent development of mobile detection instruments used for medical diagnosis was reviewed. The features of GGT in patients that improve diagnosis efficiency were tried to unravel in Wang2016The . However, these methods above do not take into account fuzzy concept and uncertainties of medicine. In fact, due to the own characteristics of medicine Ma2015Parameters ; 2015Toward ; Deng2015Newborns , more fuzzy concept and more uncertainties, some mathematics methods, which have the ability to deal with the fuzzy and uncertain information, are needed for solving medical diagnosis problems. Recently, fuzzy mathematics Zadeh1965Fuzzy and DempsterShafer (DS) evidence theory Shafer1976A ; Dempster1967Upper ; fu2015group ; Wang2016A are widely applied in medical diagnosis, since they could reasonably model uncertainty and fuzzy information and describe them. Wang et al. Wang2016A adopted fuzzy soft sets based on ambiguity measure and DempsterShafer evidence theory, and the method was applied in medical diagnosis. Recently, a theoretical model Mak2015Ahas been created to calculate the probabilities of hypothetical patients having designated diseases. And then based on the theoretical model, a fuzzy probabilistic method was presented to estimate the probability of a patient having a certain disease. In
Michalski2011Generalized , an application of GIFSS that was defined by Michalski demonstrated through a practical example of a multicriteria medical diagnosis problem. Fuzzy soft set theory was applied through wellknown Sanchez’s approach for medical diagnosis using fuzzy arithmetic operations 2013Fuzzy . In Chen2013The , an extended QUALIFLEX approach for dealing with a medical decisionmaking problem Semler2015Leveraging in the context of interval type fuzzy sets was proposed. Jose Kwon2016Medical proposed a new methodology to combine fuzzy rulebased classification systems with intervalvalued fuzzy sets Lee2012Fuzzy , which is a suitable tool to face the medical diagnosis. An approach which combines intuitionistic trapezoidal fuzzy numbers with inclusion measure for medical diagnosis was proposed by Wang Wang2015The . Yang Yang2015Dataused a linear regression model based on trapezoidal fuzzy numbers to predict which readings in the outlying data vector are suspected to be faulty for medical diagnosis. Some similarity measures
Chou2016A ; Song2015A which can applied in medical diagnosis are proposed. In Muthukumar2015A , a weighted similarity measure on intuitionistic fuzzy soft sets for medical diagnosis are presented.In fuzzy mathematics Zadeh1965Fuzzy ; Wang2016AP , fuzzy numbers Giachetti1997Analysis ; Yager1978 ; Jiang2015 , could describe human perception and subjectivity and could be able to handle uncertain or imprecise information dengadeng to some extent. But in the actual research, we found that reliability of information in decision environment such as medical diagnosis, is very important too. If only to rely on the function of the fuzzy numbers, there may exist the limitation to appropriately describing reliability of information. In order to solve this problem, Zadeh Zadeh2011A extended the concept of fuzzy numbers via introducing a new concept of Znumber. Znumber, a tuple fuzzy number, includes the restriction of the evaluation and the reliability of human judgement. Znumber, which contains two components, is an ordered pair of fuzzy numbers Zadeh2011A . The first fuzzy number is used to represent the uncertain information in evaluation, and the second fuzzy number is used to measure the reliability or confidence in truth or probability. Therefore, Znumber can describe the level of human judgment and can be more effectively applied in decisionmaking such as medical diagnosis, fault diagnosis Yang2015Datam .
However, in the procedure of applying Znumbers such as decisionmaking, we have to face an issue, that is how to address the restriction and the reliability of Znumber Zadeh2011A . Up to now, the study on Znumber has been done by some scholars. Ever since Kang et al. Kang2012A proposed an approach to convert Znumbers into fuzzy numbers, in which the second component is defuzzified to a crisp number, numerous researchers proposed some useful methods to deal with the problems in the uncertain environment by applying the approach Kang2012A . In order to address linguistic decision making problems, Kang et al. Kang2012Decision presented a MCDM method with Znumbers based on the method introduced in Kang2012A . Bakar Bakar2015Multi introduced a multilayer method to rank Znumbers, in which there are two layers, namely, Znumber conversion and fuzzy number ranking. In Mohamad2014A , Mohamad et al. proposed a decision making procedure based on Znumbers, in which Znumbers are first transformed to fuzzy numbers, and then a ranking fuzzy number method is later used to prioritize the alternatives. In all of the above described methods, a Znumber is transformed into a fuzzy number or a generalized fuzzy number via converting the second component to a crisp number. However, according to Aliev2015The , to convert the second component to a crisp number may lead to the loss of original information, and will exist an unreasonable situation, that is, two different Znumbers may be converted to the same fuzzy number. Obviously, some existing methods for addressing Znumbers still have some weaknesses, and how to deal with the relationship of restriction and the reliability of Znumber has still not been effectively processed. To address this issue, in the paper, a new ranking fuzzy numbers method is firstly introduced to process Znumbers. Then based on the new ranking method, the BPAs of Znumbers can be generated, in which the different importance of the first component and the second component of a Znumber is considered to make the results more reliable. Finally, in order to handle the problem of lack of information, Dempster’s combination rule Dempster1967Upper ; Shafer1976A , as a powerful mathematical tool for dealing with incomplete information, is applied to fuse the obtained BPAs to make the final decision. In the proposed decisionmaking method, instead of converting the second component of Znumber into a crisp number, we consider the first component and second component as two independent fuzzy numbers,which can reduce the lack of information. To get more effective and reliable diagnosis results in medical diagnosis, a new medical diagnosis method based on the proposed decisionmaking method is proposed in this paper, where Znumbers are used to represent the medical diagnostic information.
The remainder of this paper will be organized as follows. In Section 2, some definitions and concepts are introduced. In Section 3, a new ranking method for fuzzy numbers is proposed. In Section 4, a new method to determine BPA based on Znumbers is proposed. In Section 5, the proposed method is applied in medical diagnosis. In Section 6, the conclusions are made.
2 Preliminaries
In this section, some concepts used in this paper will be introduced.
2.1 Generalized trapezoidal fuzzy number Sanguansat2011 ; Chen2009
In a universe of discourse , the membership function of a fuzzy number maps each element in to a real interval . The membership function of a generalized trapezoidal fuzzy number is defined as follows:
(1) 
2.2 Znumber Zadeh2011A
A Znumber , shown as Fig. 1, contains two components. The first component is a restriction on the values which can take. The second component is a measure of reliability of the first component . According to Zadeh2011A , is an ordered pair of fuzzy numbers. Typically, and are described in a natural language Zadeh2011A . Example: (about min, verylow).
In this paper, in order to properly deal with the problems in the uncertain environment such as medical diagnosis, the first component and the second component are described in natural language applying member linguistic terms. member linguistic terms are shown in Table 1.
Linguistic Terms  the first component  the second component 

Absolutelylow  (0,0,0,0;1.0)  (0,0,0,0;1.0) 
Verylow  (0,0,0.02,0.07;1.0)  (0,0,0.02,0.07;1.0) 
Low  (0.04,0.1,0.18,0.23;1.0)  (0.04,0.1,0.18,0.23;1.0) 
Fairlylow  (0.17,0.22,0.36,0.42;1.0)  (0.17,0.22,0.36,0.42;1.0) 
Medium  (0.32,0.41,0.58,0.65;1.0)  (0.32,0.41,0.58,0.65;1.0) 
Fairlyhigh  (0.58,0.63,0.80,0.86;1.0)  (0.58,0.63,0.80,0.86;1.0) 
High  (0.72,0.78,0.92,0.97;1.0)  (0.72,0.78,0.92,0.97;1.0) 
Veryhigh  (0.93,0.98,1.0,1.0;1.0)  (0.93,0.98,1.0,1.0;1.0) 
Absolutelyhigh  (1.0,1.0,1.0,1.0;1.0)  (1.0,1.0,1.0,1.0;1.0) 
For example, an expert diagnosed a patient. The expert’s diagnoses are represented by Znumbers, shown as follows:
According to Table 1, the diagnoses above can be described as follows:
2.3 DempsterShafer evidence theory Shafer1976A ; Dempster1967Upper
Let be the frame of discernment:
The BPA of meets the following conditions:
The BPA represents the degree of evidence support for the proposition of in recognition framework. For example, represents the degree of evidence support for empty set.
The Dempster’s combination rule is defined as follows:
(2) 
where
The Dempster’s combination rule could be effectively applied in decisionmaking such as medical diagnosis Wang2016A .
2.4 Maximal entropy model O'Hagan1988
The Maximal entropy method (MEM) was presented by O’Hagan O'Hagan1988 in . The MEM can get weights of parameters by solving a maximal entropy model. In this paper, MEM is applied to obtain the weights of parameters based on the different importance of them.
3 The proposed method for ranking fuzzy numbers
In many applications of Znumbers, such as decisionmaking, how to deal with Znumbers is an important issue. In this paper, in the procedure of handling Znumbers, ranking fuzzy numbers Yager1978 ; Murakami1983 ; Chen2009 ; Bakar2014 ; jiang2016Ranking becomes an important process. In this part, a new ranking fuzzy numbers method is proposed for dealing with Znumbers.
Firstly, the three scoring factors (or three factors affecting score) of the fuzzy number, i.e., the defuzzified value, height and spread, are calculated. Then, considering the different importance of the three scoring factors, different weights are assigned to them. Finally, ranking score of fuzzy number is defined, which reflects ranking order of the fuzzy number. Let be a fuzzy number Wang2006On; Chen2012, , as shown in Fig. 2, where , , and are real values, denotes the height of the fuzzy number , and . The proposed method for ranking fuzzy numbers is now shown as follows:
 Step 1:

The defuzzified value , height and spread of fuzzy number are calculated separately, described as follows:
(4) (5) (6) where
 Step 2:

Define a vector associated with the ordered arguments.
For the three scoring factors and , the ranking order of the importance of them is . Accordingly, , are arranged in the order of their importance from large to small, namely, the vector is defined as follows:
(7)  Step 3:

Calculate the weighting vector of the three ordered elements of vector by the maximal entropy model, shown as follows:
(8) where The values of and will be respectively assigned to the three sorted elements of vector according to their importance. Based on the stated above, it can be gotten that , then that can be known in Fig. 3 with . Generally, the value of is defined as the middle value of the interval , namely, .
 Step 4:

Calculate the value of ranking score of fuzzy number , shown as follows:
(9) where is the transpose of weighting vector . The value of score reflects the value of ranking order of the fuzzy number. The greater the value of ranking score, the better the ranking order.
4 The proposed method to determine BPA
Suppose that a test number such as a decision . The first component of is , and the second component of is . and are two fuzzy numbers denoted in Fig. 2. In this section, we define the Znumber as the maximal reference number and the Znumber as the minimal reference number. Firstly, the ranking scores of two components of Znumber are calculated. Secondly, the weights of the first component and the second component are obtained by the maximal entropy model. Thirdly, the deviation degree of the test number is defined to represent the location of the test number between the maximal reference number and the minimal reference number . Then, the similarity measure between the test number and the reference number is defined to denote the confidence degree of the test number such as a decision. Finally, the BPA is generated based on the defined deviation degree of the test number from the reference number , which can be applied in decisionmaking such as diagnosis.
4.1 The steps of the proposed method
The steps of the proposed method is shown as follows:
 Step 1:

Calculate the ranking score and of fuzzy number , respectively according to Section 3.
 Step 2:

According to Section 3, calculate the ranking score and of fuzzy number , respectively and calculate the ranking score and of fuzzy number , respectively. The conclusions can be made as follows:
Obviously, when the value of ranking score is , the ranking order is the best.
 Step 3:

Calculate the weights of the first component and the second component based on the maximal entropy model, shown as follows:
(10) where . The different importance of the first component and the second component is considered in this paper. According to the definition of Znumber Zadeh2011A , the first component of Znumber is to describe the uncertainty, while the second component, as a measure of reliability of the first component, can influence but cannot decide the Znumber that can be decided by the first component. Obviously, the first component is more important than the second component . Thus the first component is assigned the larger weight.
 Step 4:

Defined the deviation degree of the test number , shown as follows:
(11) where the deviation degree of represents the location of the test number between the maximal reference number and the minimal reference number , which can be shown in Fig. 4. Obviously, the confidence degree of the maximal reference number is the largest, that is, . The larger the value of , the more far away from the reference number the test number . Namely, the confidence degree of the test number Z is lower. The location of the test number Z is shown in Fig. 5. Namely, the confidence degree of the test number is lower.
From Fig. 4, several conclusions can be obtained: firstly, represents that has been processed; secondly, the location on the circle can only be on the circle at the lower left corner, since the ranking score and ; thirdly, when the weights of the first component and the second component are not taken into consideration, the deviation degrees of two different locations on the same circle will be same, which is unreasonable; fourthly, when the different weights are assigned to and , the deviation degrees of two different locations on the same circle will be different, which is reasonable. It can be seen that the defined deviation degree is effective and reasonable.
 Step 5:

Generate the BPA of the test number . Firstly, the similarity measure between the test number such as a decision and the reference number is defined, shown as follows:
(12) Obviously, the similarity measure between the test number and the reference number denotes the confidence degree of the test number such as a decision. The larger the value of similarity measure, the higher confidence degree of the test number. On the other hand, the similarity measure between a test number and the reference number can also be regarded as the ranking score of to get the ranking order of .
Then, the BPAs can be gotten by normalizing the obtained similarity measure . Finally, in order to address the problem of lack of information, the obtained BPAs will be fused by Dempster’s combination rule to get the final decision. In the following subsection, the proposed method will be made a comparison with the existing methods for ranking Znumbers to illustrate the effectiveness and superiority of the proposed methodology.
4.2 A comparison with the existing methods for ranking Znumbers
In this part, the similarity measure defined in Step in subsection 4.1 is used as the ranking score of Znumber to rank Znumbers. In the following, we use sets of Znumbers in jiang2016Ranking to compare the ranking results obtained by the defined similarity measure and a number of existing ranking methods to show the effectiveness and superiority of the proposed method. Let all first components of Znumbers in the sets are the same, . All the of Znumbers are shown in Fig. 5 and the comparison results are shown in Table 2. According to Fig. 5 and Table 2, we can see the drawbacks of the existing ranking methods and the advantages of the proposed method:
(1) If second components of two Znumbers are and in Set of Fig. 5, the ranking result, , obtained by the defined similarity measure in this paper is reasonable and consistent with human intuition, since the truth that the ranking order of two the second components is according to section 3 and the two first components of , are the same. However, Mohamad’s method, Bakar’s method and Kang’s method can’t correctly address this situation and get an incorrect ranking order , since the fact that the different Znumbers get the same ranking order. In this case, it means that the second component doesn’t work in Mohamad’s method, Bakar’s method and Kang’s method, which is not consistent with the concept of Znumber in Zadeh2011A .
(2) If two second components of Znumbers are and in Set of Fig. 5, the proposed method can get the correctly ranking result, , since the fact that the ranking order of two the second components is according to section 3 and two the first components of , are the same. However, Mohamad’s method, Bakar’s method and Kang’s method can’t correctly address this situation and get an incorrect ranking order , since the truth that the different Znumbers get the same ranking order. In this case, it means that in Mohamad’s method, Bakar’s method and Kang’s method, the second component doesn’t work, which is not consistent with the concept of Znumber.
(3) If two second components of Znumbers are and in Set of Fig. 5, the ranking result, , obtained by the defined similarity measure in this paper is reasonable and consistent with human intuition, since the truth that the ranking order of the two second components is according to section 3 and the two first components of , are the same. However, Mohamad’s method, Bakar’s method and Kang’s method can’t correctly address this situation and get an incorrect ranking order , since the fact that the different Znumbers get the same ranking order. In this case, it means that the second component doesn’t work in Mohamad’s method, Bakar’s method and Kang’s method, which is not consistent with the concept of Znumber.
In summary, from Fig. 5 and Table 2, it is obvious that the defined similarity measure in this paper provides a reasonable ranking order and overcomes the drawbacks of the existing ranking methods. Specially, the proposed method can be applied in decisionmaking such as risk analysis and medical diagnosis, since it uses Znumber as a whole to model, and generates BPAs by taking into account the different importance of two components of Znumber, and can get reasonable ranking order. The above summary illustrates the effectiveness and superiority of the proposed methodology.
methods  Set1  Set2  Set3  

Mohamad’s method Mohamad2014A  0.0774  0.0774  0.0774  0.0774  0.0774  0.0774 
Bakar’s method Bakar2015Multi  0.0288  0.0288  0.0288  0.0288  0.0288  0.0288 
Kang’s method Kang2012Decision  0.3000  0.3000  0.3000  0.3000  0.3000  0.3000 
The proposed method  0.2610  0.2663  0.2598  0.2610  0.2278  0.2610 
4.3 An illustrative experiment of application
Manufactory  Subcomponents  Linguistic values of severity of loss 

Manufactory  Subcomponents  Linguistic values of the reliability 
In order to further illustrate the effectiveness of the method, in this part, a frequently used experiment of application in decision making Mohamad2014A ; Kahraman2008Fuzzy will be done to compared the proposed method with Mohamad et al.’ method Mohamad2014A to validate the effectiveness of the proposed method. Decision making plays an important role in our real life. Specially, decision making is the main task in the medical diagnosis. There are three manufactories , , and and each manufactory produces the components , , and , respectively. A component consists of three subcomponents, that is and , where . To assess the risk faced by each subcomponent, the evaluating items are represented by and , where . represents the severity of loss of the subcomponents. denotes the reliability of the decision maker’s opinion on each subcomponent. Therefore, the entries of decision matrix can be represented as .
The severity of the loss of the subcomponents and the reliability of the decision makers’ opinion are shown in Table 3.
Firstly, the risk assessment Dong2017Risk for manufactory is represented by Znumbers, which is shown in Table 4.
Secondly, according to Section 4.1, the BPAs of risk assessment from three subcomponents are calculated based on Eqs. (1012). The results are presented in Table 5.
0.1326  0.4336  0.3355  0.0983  
0.3413  0.2571  0.1062  0.2954  
0.1260  0.2758  0.2682  0.3300 
Finally, the obtained BPAs are fused by Dempster’s combination rule to get the final decision to address the problem of lack of information. The results denote that the manufactories has the highest risk or the highest probability of failure followed by and .
methods  

Mohamad et al.’ method Mohamad2014A  0.1049  0.2460  0.0630  0.0000 
The proposed method  0.1740  0.5103  0.2866  0.0291 
From Table (36), it can be seen that compared with Mohamad et al.’ method Mohamad2014A , the proposed method can get correct and reasonable result, that is, the ranking order of risk is , since the truth that both subcomponents and made the same assessment, that is, the manufactory has the highest risk and has the lowest risk, which illustrates the effectiveness of the proposed method. However, Mohamad et al.’ method Mohamad2014A gets the incorrect result, , that is, the manufactory has the lowest risk. On the one hand, their incorrect ranking order results from the fact that Mohamad et al.’ method does not consider the different importance of two components of Znumber, and their method converts the second component of Znumber to a crisp number, which may lead to the loss of information. On the other hand, for the three fuzzy numbers converted from the aggregated evaluation for manufactory by their method Mohamad2014A , , the reasonable ranking order should be , which is consistent with the truth that the centroid point of on the axis is larger than that of on the axis. However, Mohamad et al.’ method obtains the incorrect ranking order . In addition, Mohamad et al.’ method can not solve the situations that when Znumbers are converted to the symmetrical fuzzy numbers with axis, such as the fuzzy numbers shown in Fig. 6. About all, it is can be seen that the proposed method can overcome the weaknesses of the previous method.
5 Application of the proposed method to medical diagnosis
In this section, an application of the proposed method to medical diagnosis is illustrated. From a patient’s symptoms, he may be suffering from three diseases, namely, Commoncold, Meningitis and Measles. There are three experts , and , they respectively made three kinds of diagnoses (Commoncold, Meningitis, Measles) represented by Znumbers for the patient, which are shown in Table 7. For example, the expert diagnoses that the degree of certainty of Commoncold is Veryhigh, and the measure of reliability of his diagnosis is Veryhigh, which can be represented by . The corresponding linguistic terms are shown in Table 8.
In Table 7, denotes the degree of certainty of diagnosis made by the decisionmakers and denotes the measure of reliability of .
CommonCold  Meningitis  Measles  

expert  Znumbers represented by linguistic terms 

In order to solve the problem of the loss of information and enhance the reliability of the results, the BPAs obtained in Section 4 are fused by Dempster’s combination rule. The procedure of the proposed medical diagnosis method, as shown in Fig. 7, is detailed as follows:
 Step 1:

According to Section 3, calculate respectively the ranking score of the two components and of diagnoses shown in Table 8. Take the diagnoses of in Table 8 as an example, shown as follows:
The first component of , shown in Fig. 8. The procedure of the ranking score of is shown as follows:
namely,
In the same way, the ranking scores of all components can be calculated, shown as follows:
 Step 2:

Calculate respectively the deviation degree of diagnosis based on Eq. (11). The deviation degree of the diagnoses of can be obtained as follows:
 Step 3:

Generate BPAs of diagnoses. Firstly, the similarity measure between diagnosis and the maximal reference number is calculated based on Eq. (12). The similarity measure between the diagnosis of and the maximal reference number can be shown as follows:
This paper define that
Then, the obtained similarity measure are normalized to obtain the BPAs of the diagnoses of , shown as follows:
In the same way, the BPAs of diagnoses of and can be calculated, as shown in Table 9.
 Step 4:

Based on Eq. (2), the BPAs obtained in Step are combined by applying Dempster’s combination rule. The fusing results are shown in Table 10.
m(Commoncold) m(Meningitis) m(Measles) m(Commoncold, Meningitis, Measles) 0.6789 0.1836 0.1147 0.0238 0.4746 0.1674 0.1718 0.1862 0.1717 0.1675 0.5596 0.1012 Table 9: The BPAs of diagnoses from three experts m(Commoncold) m(Meningitis) m(Measles) m(Commoncold,Meningitis,Measles) fusing result 0.7085 0.1076 0.1814 0.0025 Table 10: The fusing results by Dempster’s rule of combination
From the linguistic values of diagnoses in Table 7, it can be known that both and tend to consider that the clinical patient is most likely to suffer from the common cold, and is less likely to suffer from measles. However, tends to consider that the clinical patient is most likely to suffer from the measles, and is less likely to suffer from the common cold, which conflicts with the diagnoses of and . In Table 9, it can be seen that the obtained BPAs are consistent with the above analysis.
The proposed method addresses the multiple sources and conflicting information by using Dempster’s combination rule, and the fusing results are shown in Table 10. From Table 10, it can be seen that the final result of diagnosis is that the patient had a common cold, which is consistent with the truth that two of three experts consider that the patient is suffer from the common cold. From the obtained result of diagnosis, it can be seen that this method can solve the issues of the uncertainty and the confliction of information and can achieve a reasonable medical diagnosis.
To illustrate the effectiveness of Znumber modeling in medical diagnosis, in the following, we will give a case that the reliability of information is not considered. Now, all second components of Znumbers are highest value , that is , and the first components are the same as Table 8.
The new diagnoses are presented in Table 11. The new BPAs of diagnoses from three experts are shown in Table 12, and the new fusing results by Dempster’s rule of combination are given in Table 13.
expert  Znumbers represented by linguistic terms 

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