Multisensor Data Fusion in Testability Evaluation of Equipment

Di, Peng; Wang, Xuan; Chen, Tong; Hu, Bin

doi:https://doi.org/10.1155/2020/7821070

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Fusion with Information from Multiple Sources in Real World Applications

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 7821070 | https://doi.org/10.1155/2020/7821070

Multisensor Data Fusion in Testability Evaluation of Equipment

Peng Di,¹Xuan Wang,¹Tong Chen,¹and Bin Hu¹

Academic Editor: Filippo Cacace

Received17 Aug 2020

Revised17 Sept 2020

Accepted17 Nov 2020

Published01 Dec 2020

Abstract

The multisensor data fusion method has been extensively utilized in many practical applications involving testability evaluation. Due to the flexibility and effectiveness of Dempster–Shafer evidence theory in modeling and processing uncertain information, this theory has been widely used in various fields of multisensor data fusion method. However, it may lead to wrong results when fusing conflicting multisensor data. In order to deal with this problem, a testability evaluation method of equipment based on multisensor data fusion method is proposed. First, a novel multisensor data fusion method, based on the improvement of Dempster–Shafer evidence theory via the Lance distance and the belief entropy, is proposed. Next, based on the analysis of testability multisensor data, such as testability virtual test data, testability test data of replaceable unit, and testability growth test data, the corresponding prior distribution conversion schemes of testability multisensor data are formulated according to their different characteristics. Finally, the testability evaluation method of equipment based on the multisensor data fusion method is proposed. The result of experiment illustrated that the proposed method is feasible and effective in handling the conflicting evidence; besides, the accuracy of fusion of the proposed method is higher and the result of evaluation is more reliable than other testability evaluation methods, which shows that the basic probability assignment of the true target is 94.71%.

1. Introduction

Testability evaluation, as an important part of testability design of equipment, is often used to test and evaluate whether the equipment meets the development requirements [1]. Through the evaluation of testability level, the defects of testability design can be found, which is related to the progress of finalization process of the equipment design. Since the 21st century, testability, as one of the general quality characteristics of equipment, has been paid more and more attention by equipment manufacturers and users. With the development of equipment testability engineering, testability evalution method has become one of the hot topics in testability field.

At present, many scholars have studied the testability evaluation method of equipment. Lin et al. [2, 3], Yang et al. [4], Liu et al. [5], and Zhou and Liu [6] proposed a method based on correlation and multisignal model for testability analysis and research by using the method of model prediction. However, in the actual testability evaluation work, it is difficult to carry out a comprehensive testability test, so the accuracy of the evaluation results obtained by model prediction method is low when the test data is small.

To solve this problem, Li et al. [7–9], Zhou et al. [10, 11], and Chang et al. [12, 13] proposed a testability evaluation method based on Bayes theory. Although it can effectively use prior information and expand the available data volume of testability assessment, it uses less information sources, and it is difficult to conduct testability assessment comprehensively and accurately.

In order to make full use of testability prior information, Liang and Zhang [14] proposed a testability evaluation method in the development stage based on information fusion. The classical Dempster–Shafer evidence theory was used to fuse multisource prior information, and the testability level was evaluated by analyzing the fusion results. Deng et al. [15] used evidence discount combination method to evaluate the current testability level of equipment by modifying the weight of prior information in Dempster–Shafer evidence fusion. The above two methods make full use of testable data from different sources, but considering that prior information such as expert experience information has certain subjectivity and uncertainty, it is easy to cause conflicts in the process of data fusion. Therefore, how to solve the conflict fusion problem of testability data becomes the focus of this paper.

To summarize all the different reviewed methods, testability evaluation method based on mathematical model prediction can effectively describe the relationship between test and fault of equipment, but the simplification of logical relationship between them will reduce the accuracy of prediction. Testability evaluation method based on Bayes theory can effectively use prior information and expand the available data of testability assessment, but it uses fewer information sources, so it is difficult to evaluate comprehensively and accurately. Although the testability evaluation method based on Dempster–Shafer evidence theory can make full use of testable information, increase the sources of evaluation data, and improve the confidence of testability assessment to a certain extent, due to the uncertainty and subjectivity of testability information and the conflict between data, it may have adverse effects on the final evaluation results. Therefore, this paper intends to solve the conflict problem in the process of testable information fusion and carry out the next research.

The technology of multisensor data fusion has been extensively utilized in many practical applications, such as the risk analysis [16, 17], decision-making [18, 19], fault diagnosis [20–23], wireless sensor networks [24–28], health prognosis [29], image processing [30], target tracking [31, 32], surveillance [33, 34], and so forth [35–38]. Multisensor data fusion can automatically analyze and synthesize the information and data from multiple sensors or multiple sources under certain criteria, so as to complete the information processing process for the required decision-making and estimation. In addition, the technology of data fusion based on the Dempster–Shafer evidence theory and maximum-entropy methods has been applied in presence of uncertainty [39–41]. Hence, the technology of multisensor data fusion is widely applied in many fields of real applications [42, 43]. Multisensor data fusion methods are based on many mathematical approaches: the rough sets theory [44, 45], fuzzy sets theory [46–53], Dempster–Shafer evidence theory [54–58], Z numbers [59, 60], D numbers theory [61–65], evidential reasoning [66–68], and so on [69–72].

Dempster–Shafer evidence theory is a powerful tool to represent and deal with uncertain information. It has been widely used in practical problems related to uncertainty modeling and reasoning, such as information fusion [49, 73–75], fault diagnosis [22, 76–81], decision-making [65, 82–85], risk assessment [86–90], multicriteria decision-making [91, 92], and pattern recognition [95–96]. However, when Dempster–Shafer evidence theory is used for fusion, due to the uncertainty of evidence, it is difficult to avoid the problem of evidence conflict.

In this paper, multisensor data fusion method is based on the improved Dempster–Shafer evidence theory which is first proposed by measuring the Lance distance between the bodies of the evidence and the belief entropy of the evidence. Based on that, the testability evaluation method of equipment is proposed by the multisensor data fusion method. The proposed method considers both of the reliability degree among the multisensor data and the uncertainty measure of the multisensor data on the weight, so that it can measure the degree of conflict of the multisensor data and fuse the conflicting multisensor data effectively. Consequently, the proposed method consists of the following procedures. Firstly, an improvement of Dempster–Shafer evidence theory is proposed, which is improved by the Lance distance function and the belief theory function, and the results of simulation illustrate that the fusion method has priority in the fusion of conflicting evidence. After that, according to the characteristics of testability multisensor data, the corresponding conversion schemes are proposed, which can be used directly in the framework of Dempster–Shafer evidence theory. Finally, the process steps of the testability evaluation of equipment based on the multisensor data fusion method are proposed. According to the analysis of the experimental results, it is found that the proposed method can effectively reduce the adverse effects of conflicting multisensor data fusion compared with other testability evaluation methods; in addition, the fusion accuracy of the proposed method is higher and the evaluation result is more reliable.

The rest of this paper is organized as follows. Section 2 briefly introduces the preliminaries of this paper. In Section 3, the multisensor data fusion method is proposed, which is based on the improvement of the Dempster–Shafer evidence theory. The focus of Section 4 is to show the conversion schemes of testability multisensor data. In Section 5, the testability evaluation model of equipment based on the multisensor data fusion method is proposed. Section 6 concludes this paper.

The contributions of this research are summarized as follows:(1)Aiming at different types of testability data, the corresponding conversion scheme of prior distribution is proposed(2)Aiming at the conflict phenomenon in the process of evidence fusion, the multisensor data fusion method is proposed, which is based on the Dempster–Shafer evidence theory and is improved by introducing Lance distance and belief entropy(3)This paper proposes a new multisensor data fusion method in testability evaluation of equipment; it can effectively reduce the conflict in the process of evidence fusion and obviously improve the reliability and accuracy of testability evaluation results

2. Preliminaries

2.1. Dempster–Shafer Evidence Theory

Evidence theory was first put forward by Professor A. P. Dempster in 1967 and then perfected by G. Shafer and became a complete and independent data system, so it is also called Dempster–Shafer evidence theory. With the continuous improvement of P. Smets and T. Denoeux, evidence theory has developed into the theory of reliability function, which can model the uncertainty of data and deal with the uncertainty and imprecision of data well. Therefore, it is widely used to solve the problem of information fusion.

In order to explain the basic principles of evidence theory in detail, this section will introduce the four core parts of evidence theory in detail: the frame of discernment, mass function, belief function and plausibility function, and Dempster’s rule of combination.

Definition 1. (the frame of discernment).
Let be a set of mutually exclusive and collectively exhaustive events, indicted byThe set is defined as a frame of discernment. A power set is the set of all possible subsets of .where is an empty set.
If , is called a proposition or hypothesis.

Definition 2. (mass function).
Suppose that is a frame of discernment; denotes any subset in the frame of discernment , . Mass function is a mapping from to , defined asIn Dempster–Shafer evidence theory, a mass function can be also called a basic probability assignment (BPA). If , is a focal element, and the set of all focal elements is called the core of evidence.

Definition 3. (belief function and plausibility function).
Let be a hypothesis in the frame of discernment . Belief function is defined asPlausibility function is defined aswhere .
and , respectively, represent the lowest and highest trust degree of proposition , corresponding to the lower bound and upper bound of probability. , the D-value of reflects the uncertainty of proposition .

Definition 4. (Dempster’s rule of combination).
Assume that the system’s frame is and two independent mass functions and are in the frame of discernment . Dempster’s rule of combination, represented in the form , is defined as follows:withwhere and is the conflict coefficient, which is used to measure the degree of conflict between BPAs and . The greater the conflict between two bodies of evidence is, the greater the value of conflict coefficient will be. Note that Dempster’s rule of combination is feasible with the condition that .
Dempster’s rule of combination can be regarded as the operation of orthogonal sum of two bodies of evidence, which meanwhile satisfies the law of exchange and combination, and can be extended to the combination of bodies of evidence.

2.2. Lance Distance

With the increasing application of Dempster–Shafer evidence theory, the research based on evidence distance is favored by scholars at home and abroad. The distance measurement between the bodies of evidence indicates the mutual support between them. Therefore, there are many methods to measure the distance between the bodies of evidence, including Minkowski distance and Mahalanobis distance. However, Minkowski distance function requires the same attribute dimension, which will lose the consistency of evidence, while Mahalanobis distance function requires matrix covariance calculation, which is not suitable for large-scale data processing. Therefore, this paper introduces a more ideal distance measurement as the Lance distance function.

Lance distance is a common method used to determine the distance between samples in cluster analysis, which was first proposed by Lance and Williams. Lance distance is a dimensionless quantity, which overcomes the shortcoming that Minkowski distance is related to the dimension of each index, and it is not sensitive to large singular values, which makes it particularly suitable for height offset and data. The main concepts are introduced as blow.

Definition 5. (Lance distance function).
Assuming that the system’s frame is and there are pieces of evidence in system, the mass functions of evidence are , where . Suppose that there are two pieces of evidence and the Lance distance between the pieces of evidence iswhere and .
The Lance distance function is very sensitive to small changes close to and . It is generally considered to be a generalization of binary data dissimilarity measure [97]. According to (8), is a dimensionless attribute and is not sensitive to large singular values. Therefore, overcomes the defects of Minkowski distance function and Mahalanobis distance function and is more suitable for evidence revision.

2.3. Belief Entropy

Information entropy is defined as the amount of information needed to describe the state of random variables. Intuitively, the wider the distribution range of random variables, the more the information needed to describe them, because the values of random variables need to be specified in a larger range. Information entropy can also be used as the discrete degree of variables or as a measure of uncertainty. In Dempster–Shafer evidence theory, Shannon proposed Shannon entropy to measure the degree of uncertainty of evidence on the basis of information entropy, while Deng Yong’s belief entropy is a general improvement of Shannon entropy [98]. Therefore, this paper introduces the belief entropy to measure the uncertainty of evidence. The basic concepts are introduced as follows.

Definition 6. (belief entropy function).
Assume that the system’s frame is , is a hypothesis of the BPA , and is the cardinality of set . Belief entropy of set is defined as follows:When the belief value is only assigned to a single element, that is, , the belief entropy degenerates to Shannon entropy [99], i.e.,It indicates that the greater the belief entropy is, the more information the evidence contains, and the greater the uncertainty of evidence is. When the evidence has large belief entropy, it is supposed to have more support from other evidence, which indicates that this evidence will play an important role in the final evidence combination.

3. The Improvement of Dempster–Shafer Evidence Theory

To improve the accuracy, validity, and reliability of multisensor data fusion, the proposed method adopts the Lance distance function and the belief entropy to improve the Dempster–Shafer evidence theory. In addition, the algorithm of the proposed method is presented in this section.

3.1. Process Steps

Step 1. Measure the reliability degree of the evidence. According to (8), the Lance distance between the bodies of evidence and can be expressed in the form of Lance distance matrix , which is defined as follows: The reliability degree of the body of evidence is defined as follows: where satisfies Step 2. Measure the uncertainty degree of the bodies of evidence. In order to avoid entrusting zero weight to the evidence in some cases, this paper determines the uncertainty degree of the body of evidence by calculating the exponential form of belief entropy as follows: The uncertainty of evidence, which is denoted as , is normalized as follows: Step 3. Generate and fuse the final weight of evidence based on the reliability degree and uncertainty degree. Based on the reliability degree , the uncertainty degree of the evidence , denoted as , will be adjusted: The weight that is considered as the final weight in terms of each evidence is normalized: Step 4. Determine the revised evidence. On the basis of the weighting factor of mass function of the evidence , mass function of the revised evidence can be obtained as follows: Step 5. Evidence fusion. According to Dempster’s rule of combination, the mass function is synthesized times ( is the number of pieces of evidence); follows Dempster’s rule of combination. The result of evidence fusion is as follows:

3.2. Algorithm

When there are pieces of evidence collected, the mass functions of evidence are . After collecting pieces of evidence, a fusion result is expected to be generated for the later research. The pseudocode for the proposed method of evidence fusion is presented in Algorithm 1.

	Input: A set of multiple pieces of evidence ;
	Output: Fusion result ;
(1)	/∗ Step 1 ∗/
(2)	for; do
(3)	for; do
(4)	Calculate the Lance distance between evidence using equation (8);
(5)	end
(6)	end
(7)	Construct the Lance distance matrix using equation (11);
(8)	/∗ Step 2 ∗/
(9)	for; do
(10)	Obtain using equation (13);
(11)	end
(12)	/∗ Step 3 ∗/
(13)	for; do
(14)	Calculate the reliability degree using equation (12);
(15)	end
(16)	/∗ Step 4 ∗/
(17)	for; do
(18)	Calculate the exponential form of belief entropy using equation (14);
(19)	end
(20)	/∗ Step 5 ∗/
(21)	for; do
(22)	Normalise the modified uncertainty degree using equation (15);
(23)	end
(24)	/∗ Step 6 ∗/
(25)	for; do
(26)	Obtain the weight of reliability and uncertainty degree using equation (16);
(27)	end
(28)	/∗ Step 7 ∗/
(29)	for; do
(30)	Normalise the final weighted of evidence using equation (17);
(31)	end
(32)	/∗ Step 8 ∗/
(33)	Obtain mass function of the revised evidence using equation (18);
(34)	/∗ Step 9 ∗/
(35)	for; do
(36)	Calculate the final fusion result using equation (19);
(37)	end

3.3. Simulation and Result

In order to verify the efficiency and effectiveness of the improved evidence fusion method based on Lance distance and belief entropy proposed in this paper, the complete example of [100] is used for experiment, the analysis and comparison with Dempster’s combination method and other combination methods are carried out, and the final fusion results are analyzed in detail under the conditions of consistent evidence and conflicting evidence.

3.3.1. Evidence Fusion with Consistent Evidence

Assume that the frame of evidence system is , the evidence set is , and the mass function set of evidence is . Suppose that is the true target; the mass function of consistent evidence is shown in Table 1.

It is evident in Table 1 that all evidence is relatively consistent in identifying target . Thus, the target should be given the biggest support by the effective fusion method.

Then, the fusion results of different methods are obtained in Table 2.

According the fusion results of different methods, BPAs of different methods with consistent evidence are shown in Figure 1.

(a)

(b)

According to Table 2 and Figure 1, we can get the following analyses about the existing modified methods:(1)In the fusion results of Dempster’s combination method, although the BPA for target is the biggest, the BPAs for targets and are 0, which do not match the original evidence source.(2)In the fusion results of Deng et al.’s method, the BPA for target is 0.2760, which means that the probability assigned to unknown items is too high in evidence fusion, leading to low accuracy of fusion.

These four methods all get the reasonable fusion results and recognize the true target precisely, but the fusion accuracy of the proposed method is higher and more reliable.

3.3.2. Evidence Fusion with Conflicting Evidence

In order to verify the effectiveness and reliability of the proposed method, conflict evidence is modified and generated based on the original evidence source in Section 3.3.1. The mass function of conflicting evidence is shown in Table 3.

Analysis of the data in Table 3 shows that the bodies of evidence support the target , while the body of evidence supports the target . Obviously, evidence is the conflicting evidence. After that, the fusion results of the different methods are exhibited in Table 4.

According the fusion results of different methods, BPAs of different methods with conflicting evidence are shown in Figure 2.

(a)

(b)

(c)

(d)

According to Table 4 and Figure 2, we can get the following analyses about the existing modified methods:(1)The fusion result of Dempster’s combination method completely believes that is the true target, which is contrary to intuition judgement. Obviously, Dempster’s combination method cannot effectively handle the fusion of conflicting evidence.(2)In the fusion result of Deng et al.’s combination method, the BPA for target is 0.3863, which illustrates that the uncertainty in the fusion result is still high. Thus, Deng et al.’s combination method may lead to the error in the fusion of conflicting evidence.(3)Compared with Ye et al.’s combination method, the BPA for target of proposed method is higher. Thus, the fusion result of the proposed method gets higher accuracy.

Compared to other methods, the proposed method has a better convergence effect that can deal with conflicting situations more effectively. Thus, it has priority in the fusion of conflicting evidence.

The fusion times of all methods are the same, all of which are . In order to obtain the value of mass function for fusion, the proposed method in this paper calculates Lance distance and belief entropy, and it has more steps than other methods, but this work is to measure the conflict degree of evidence, so the complexity of calculation is inevitable and acceptable.

4. Conversion Schemes of Testability Multisensor Data

There are many kinds of testability data types in equipment design, production, and use, such as testability virtual test data, testability test data of replaceable unit, testability growth test data, and other multisensor data. In order to ensure that the testability multisensor data can be used in the framework of Dempster–Shafer evidence theory, the various testability data is converted into a unified prior distribution form, that is, to solve the prior distribution parameters , .

4.1. Testability Virtual Test Data

With the development of testable modeling and simulation technology, the application of virtual test technology has been paid attention to. Some achievements have been made in the construction of testable virtual prototype and the implementation of virtual test. In the process of testable virtual test, considering that there are different types and levels of units in the system, the difficulty of modeling is different, so there are corresponding testable virtual tests.

In the current research, the data type of testable virtual test is unit level data, and the data type is success or failure type. According to the characteristics of this kind of data, this paper introduces the information entropy theory to solve the conversion problem from unit level data to system level data.

Suppose a virtual test system is composed of M independent units. According to the theory of information entropy, each unit corresponds to a source. Suppose the test data expression form of the unit is , where is the total number of tests of the unit, and is the number of test failures in the unit. Assuming that the probability of success in each test in the unit is , the probability of failure is , and the average amount of information provided by the unit in a test is

Then the amount of information provided by the unit is , and for a system composed of units, the total amount of information provided by it is

Assuming that is the number of unit tests after equivalent reduction, is the number of failures, is the probability of the unit succeeding in the test, and is the probability of failure, then the equivalent amount of information of the unit in tests is

According to the principle of information equivalence, the total information provided by each unit test data before equivalent conversion is equal to the total information provided by system data after equivalent conversion, i.e., . For the success probability before and after the data conversion, take the corresponding maximum likelihood estimation value; that is, for each unit, take , and for the system, take ; then, we can get the formula of conversion from each unit level to the system level:

Since the coupling relationship between the units will affect the system level indicators, in order to simplify the calculation process, this factor is not considered in this paper.

4.2. Testability Test Data of Replaceable Unit

The system level testability test of equipment needs the cooperation and integration of each replaceable unit, sharing the model and parameters of testability design, and considering the interaction of environment, which is often difficult to achieve. Therefore, in the design and development stage, there is a lack of corresponding test means to evaluate the FDR/FIR level of the whole equipment, and the testability test information of the whole equipment is less. However, the testability data of replaceable unit which as a common replaceable unit in equipment (SRU, LRU, etc.), has less difficult to carry out testability tests and more data can be collected. A certain method can be used to convert the testability test data of replaceable unit into the testability data of the system to obtain the system testability prior value.

In engineering practice, testability test with replaceable unit as test object obeys binomial distribution, and beta distribution can be used as its prior distribution. If there are replaceable units in a system, if the testability index of the replaceable unit is , then its prior distribution can be expressed as follows:

We use a small amount of testability test data collected from the replaceable unit, where is the number of fault samples injected by the replaceable unit and is the corresponding failure times of fault detection/isolation. By substituting the data into Bayes formula, the posterior distribution of can be obtained. Then, the posterior expectation and variance of are

By introducing the failure rate of the replaceable unit, the testability index value of the unit level can be converted into the system level. Let denote the failure rate of the replaceable unit; then, the testability index of the system level is

In engineering practice, testability index is affected by structure model, reliability, and maintainability of equipment. For example, in reliability, it is necessary to choose different methods to calculate the value according to the model relationship of series parallel connection and k/N (G). In this paper, we do not consider the influence of the coupling relationship of each unit on the testability index of the system level, but we use the weighted method based on the unit failure rate to get the testability index of the system level by weighting the testability index of the unit level.

The expectation and variance of testability index at system level are obtained as follows:t

According to (25) and (27), the testability prior distribution parameters , of replaceable units before data fusion can be solved.

4.3. Testability Growth Test Data

Testability growth test is to inject fault into equipment through testability design to make it run under specified environmental stress, observe and count the performance of fault detection/isolation in the test system, find out the cause of failure detection/isolation failure or failure of testability index to achieve the goal, and then improve the testability design and verify the improvement measures of a type of special test. Testability growth test mainly includes three processes: identifying design defects, detecting feedback problems, and improving design, which can be summarized as the process of “Test-Analysis-Improvement-Test.”

The Gompertz model is often used to describe the growth process. It can not only assess the reliability of products, but also be used to solve test growth curves. Its mathematical model is [102]where , , , and indicates that the equipment is in the testability growth test in the development stage.

By logarithmic transformation of (28), the following results can be obtained:

Suppose that a total of testability growth tests are carried out for a certain equipment, where , is a positive integer. Since the testability growth test data is a success or failure type, the testability growth test data can be recorded as , where , and the estimated test index points of the testability growth test are

Substituting the test data of times into (29), we can get the following results:

From (31), the Gompertz parameter values , , can be obtained, and then, substituting into the Gompertz model, the Gompertz formula for testability growth test can be obtained as

Through the testability growth test data, the testability index points of the final testability growth test can be estimated as follows:

The prior distribution of the testability growth test data can be expressed by , and the entropy function of the prior distribution determined by the testability growth test is known to be

Therefore, the optimal solution of the prior distribution parameters , determined by the testability growth test can be transformed into the following programming problems:

5. The Testability Evaluation Method of Equipment

In this paper, a multisensor data fusion method in testability evaluation of equipment is proposed. The multisensor data fusion method is based on the Dempster–Shafer evidence theory which is improved by introducing Lance distance and belief entropy, and the testability evaluation model of equipment based on multisensor data fusion is established, which can effectively enhance the reliability and accuracy of testability evaluation results.

5.1. Process Steps

In order to make effective use of the collected testability data of equipment from sensor and formulate a reasonable testability evaluation scheme, this paper proposes a concrete conversion scheme of testability data and uses an improvement of Dempster–Shafer evidence theory based on Lance distance and belief entropy to conduct multisensor data fusion, so as to further carry out testability evaluation of equipment. The flowchart of the proposed method is shown in Figure 3.

Step 1. Collect testability data of equipment from sensor. Step 2. The conversion of testability multisensor data. According to conversion schemes, the prior distributions of the three testability multisensor data are shown in Table 5. Step 3. Construct mass functions of prior distributions. The testability evaluation method proposed in this paper is based on Dempster–Shafer evidence theory. Therefore, the identification framework of testability evaluation system can be regarded as three parts: testability virtual test data, testability test data of replaceable unit, and testability growth test data. Each part can be divided into three focal elements. Assuming that the target value of testability index given by the manufacturer is and the minimum acceptable value specified by the user is , then when testability index , it is the first focal element, defined as ; when , it is the second focal element, defined as ; when , it is the third focal element, defined as . As for the testability virtual test data, can be regarded as evidence on the identification framework , and the mass function of the evidence on the identification framework is established: In the same way, the mass functions of testability test data of replaceable unit and testability growth test data are as follows: The mass functions of testability prior distributions can be obtained by (36)∼(38), as shown in Table 6. Step 4. Evidence fusion. According to Algorithm 1, the final result of evidence fusion is . Step 5. Testability evaluation. According to the mass function of the final results of evidence fusion, combined with the testability index values agreed on by the manufacturer and the user, the testability evaluation results are analyzed.

5.2. Experiment Analysis

Taking the fault detection rate (FDR) of a certain type of steam turbine generator set as the research object, the analysis of testability evaluation is carried out. After a joint agreement between the contractor and the user, the target value of FDR is and the minimum acceptable value is .

The specific steps of testability evaluation are as follows: Step 1. Collect three types of testability data from sensor at different stages of equipment. Step 2. The conversion of testability multisensor data.(1)Testability Virtual Test Data. Through the establishment of the testability virtual prototype model of the generating set, the testability virtual test is carried out, and the five groups of binary data are collected, in which , as shown in Table 7. According to (23), the prior distribution of testability virtual test data can be calculated, .(2)Testability Test Data of Replaceable Unit. The testability test is carried out on six replaceable units of this type of steam turbine generator set, and the test data are shown in Table 8. Substituting the posterior data of each replaceable unit and its fault rate into (26) and (27), the prior distribution of testability test data of replaceable unit can be calculated, .(3)Testability Growth Test Data. By collecting the data of three stages in the test growth test, that is, , the prior distribution of testability growth test data can be calculated, . In summary, the prior distributions of the testability multisensor data are shown in Table 9. Step 3. Mass function of testability prior information as shown in Table 10. Step 4. Evidence fusion. According to the Algorithm 1, the final results of evidence fusion are , , and . Step 5. Testability evaluation.

The final fusion results are compared with the testability evaluation method based on the traditional Bayes theory [103] and the testability evaluation method based on the Dempster–Shafer evidence theory [50]; the results are shown in Table 11.

It can be seen from Table 11 that when using the testability evaluation method based on the traditional Bayes theory, the probability of corresponding value of FDR of a certain type of steam turbine generator unit reaching the target value is , which is 1.35%; the probability of corresponding value meeting the lowest acceptable value is , which is 34.63%; and the probability of corresponding value that does not meet the minimum acceptable value is , which is 65.37%. Similarly, when using the testability evaluation method based on the Dempster–Shafer evidence theory, the probability of FDR reaching the target value is 0, the probability of meeting the minimum acceptable value is 12.98%, and the probability of not meeting the minimum acceptable value is 87.02%. When using the proposed testability evaluation method, the probability of reaching the target value is 4.13%, the probability of meeting the minimum acceptable value is 5.29%, and the probability of not meeting the minimum acceptable value is 94.71%.

In conclusion, compared with the other testability evaluation methods, the proposed method can effectively reduce the adverse impact of conflicting evidence on the fusion results and final decision-making in terms of processing the conflicting data. Moreover, the fusion accuracy of this method is higher and the evaluation results are more reliable. On the other hand, the user can reject the equipment with higher confidence.

6. Conclusions

In this paper, a testability evaluation method of equipment based on improved Dempster–Shafer evidence theory is proposed. Firstly, by collecting testability virtual test data, replaceable unit test data, and testability growth test data existing in the process of equipment design, production, and use, the corresponding prior distribution reduction scheme is proposed. Then, through considering the reliability and uncertainty of various types of data, an improved Dempster–Shafer evidence method is proposed. Finally, the testability evaluation of equipment is carried out on this basis. Compared with the traditional testability evaluation method, the proposed method can effectively solve the problem of conflict data fusion in the evaluation process, significantly improve the reliability of testability evaluation results, and provide a new technical way for testability evaluation of equipment.

The research results of this paper provide effective theoretical and technical support for scientific and reasonable testability assessment and have important theoretical and engineering application value. This paper studies the evaluation of fault detection rate and fault isolation rate in testability index of traditional testability design. The verification and evaluation of testability design oriented to prognostic and health management have not been carried out, and further research is needed in the future.

Data Availability

All data and models generated or used during the study can be obtained from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

X. W. contributed most of the work in this paper. P. D. contributed to manuscript revising. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This research was funded by Navy Equipment Technology Foundation Funding Project, under grant no. 191HJ35003.

References

J. Y. Shi, Testability Design Analysis and Verification, National Defense Industry Press, Beijing, China, 2011.
Z. W. Lin, Z. He, and S. Y. Yang, “Multi-signal model based method for radar testability analysis,” Systems Engineering and Electronics, vol. 31, pp. 2781–2784, 2009.
View at: Google Scholar
Z. W. Lin, Z. He, and S. F. Liu, “Analysis and evaluation of system testability based on multi-signal model,” Computer Measurement & Control, vol. 14, pp. 222–226, 2006.
View at: Google Scholar
Z. Y. Yang, A. Q. Xu, and S. C. Niu, “Model and analysis of system testability based on multi-signal model,” Journal of Engineering Design, vol. 14, pp. 364–369, 2007.
View at: Google Scholar
G. Liu, F. Li, and B. Hu, “Test and modeling of vessel equipment based on relevant model,” Journal of Naval Unversity of Engineering, vol. 24, pp. 47–52, 2012.
View at: Publisher Site | Google Scholar
P. Zhou and D. F. Liu, “Research on testability of marine diesel engine based on multi-signal model,” Journal of Test and Measurement Technology, vol. 25, pp. 95–99, 2011.
View at: Google Scholar
T. M. Li, Research on Optimization Design and Integrated Evaluation of Testability Verification Test for Equipments, National University of Defense Technology, Changsha, China, 2010.
T. Li, C. H. Hu, and X. Zhou, “Research on testability integrated evaluation model based on Bayes inference theory of dynamic population and analysis of the robustness,” Journal of Mechanical Engineering, vol. 48, no. 6, pp. 180–187, 2012.
View at: Publisher Site | Google Scholar
T. M. Li, J. Qiu, and G. J. Liu, “Research on testability field statistic verification based on Bayes inference theory of dynamic population,” Acta Aeronautica et Astronautica Sinica, vol. 31, pp. 335–341, 2010.
View at: Google Scholar
H. Zhou, X. P. Zhao, and W. J. Wu, “Study on the method of testability assessment of some launcher based on Bayes theory,” Journal of Projects Rockets Missiles and Guidance, vol. 30, pp. 253–256, 2010.
View at: Google Scholar
H. Zhou, X. P. Zhao, and W. J. Wu, “Bayes estimation on fault detection rate of certain type missile launcher,” Ordnance Industry Automation, vol. 28, pp. 16-17, 2009.
View at: Google Scholar
C. H. Chang, P. J. Cao, and J. P. Yang, “Testability evaluation of complex equipment based on test data in development stages,” China Mechanical Engineering, vol. 23, pp. 1577–1581, 2011.
View at: Google Scholar
C. H. Chang, J. P. Yang, and L. Hu, “Study on testability evaluation method for complex equipment based on Bayes theory,” Fire Control & Command Control, vol. 37, pp. 173–176, 2012.
View at: Google Scholar
Q. D. Liang and L. Zhang, “Testability evaluation of equipment based on information fusion,” Fire Control & Command Control, vol. 43, pp. 177–180, 2018.
View at: Google Scholar
L. Deng, A. Xu, and X. Liang, “Testability evaluation method based on multi-source information weighted fusion in development phase,” Computer Measurement & Control, vol. 22, pp. 2508–2511, 2014.
View at: Google Scholar
F. Vandecasteele, B. Merci, and S. Verstockt, “Reasoning on multi-sensor geographic smoke spread data for fire development and risk analysis,” Fire Safety Journal, vol. 86, pp. 65–74, 2016.
View at: Publisher Site | Google Scholar
L. Zhang, X. Wu, H. Zhu, and S. M. AbouRizk, “Perceiving safety risk of buildings adjacent to tunneling excavation: an information fusion approach,” Automation in Construction, vol. 73, pp. 88–101, 2017.
View at: Publisher Site | Google Scholar
L. Dymova and P. Sevastjanov, “An interpretation of intuitionistic fuzzy sets in terms of evidence theory: decision making aspect,” Knowledge-Based Systems, vol. 23, no. 8, pp. 772–782, 2010.
View at: Publisher Site | Google Scholar
J. Kang, Y. B. Gu, and Y. B. Li, “Multi-sensor information fusion algorithm based on DS evidence theory,” Journal of Chinese Inertial Technology, vol. 20, no. 6, pp. 670–673, 2012.
View at: Google Scholar
J. Hang, M. Cheng, and J. Zhang, “Fault diagnosis of wind turbine based on multi-sensors information fusion technology,” IET Renewable Power Generation, vol. 8, no. 3, pp. 289–298, 2014.
View at: Publisher Site | Google Scholar
W. Jiang, C. Xie, M. Zhuang, Y. Shou, and Y. Tang, “Sensor data fusion with Z-numbers and its application in fault diagnosis,” Sensors, vol. 16, no. 9, p. 1509, 2016.
View at: Publisher Site | Google Scholar
G. Cheng, X.-h. Chen, X.-l. Shan, H.-g. Liu, and C.-f. Zhou, “A new method of gear fault diagnosis in strong noise based on multi-sensor information fusion,” Journal of Vibration and Control, vol. 22, no. 6, pp. 1504–1515, 2016.
View at: Publisher Site | Google Scholar
H. Geng, Y. Liang, F. Yang, L. Xu, and Q. Pan, “Model-reduced fault detection for multi-rate sensor fusion with unknown inputs,” Information Fusion, vol. 33, pp. 1–14, 2017.
View at: Publisher Site | Google Scholar
Z.-J. Zhang, C.-F. Lai, and H.-C. Chao, “A green data transmission mechanism for wireless multimedia sensor networks using information fusion,” IEEE Wireless Communications, vol. 21, no. 4, pp. 14–19, 2014.
View at: Publisher Site | Google Scholar
W. Zhang and Z. Zhang, “Belief function based decision fusion for decentralized target classification in wireless sensor networks,” Sensors, vol. 15, no. 8, pp. 20524–20540, 2015.
View at: Publisher Site | Google Scholar
Z. Zhang, W. Zhang, H.-C. Chao, and C.-F. Lai, “Toward belief function-based cooperative sensing for interference resistant industrial wireless sensor networks,” IEEE Transactions on Industrial Informatics, vol. 12, no. 6, pp. 2115–2126, 2016.
View at: Publisher Site | Google Scholar
Z. Zhang, Z. Hao, S. Zeadally, J. Zhang, B. Han, and H.-C. Chao, “Multiple attributes decision fusion for wireless sensor networks based on intuitionistic fuzzy set,” IEEE Access, vol. 5, pp. 12798–12809, 2017.
View at: Publisher Site | Google Scholar
Z. Zhang, T. Liu, and W. Zhang, “Novel paradigm for constructing masses in dempster-shafer evidence theory for wireless sensor network’s multisource data fusion,” Sensors, vol. 14, no. 4, pp. 7049–7065, 2014.
View at: Publisher Site | Google Scholar
M. Dong and D. He, “Hidden semi-markov model-based methodology for multi-sensor equipment health diagnosis and prognosis,” European Journal of Operational Research, vol. 178, no. 3, pp. 858–878, 2007.
View at: Publisher Site | Google Scholar
F. Yang and H. Wei, “Fusion of infrared polarization and intensity images using support value transform and fuzzy combination rules,” Infrared Physics & Technology, vol. 60, pp. 235–243, 2013.
View at: Publisher Site | Google Scholar
X. Shen and P. K. Varshney, “Sensor selection based on generalized information gain for target tracking in large sensor networks,” IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 363–375, 2014.
View at: Publisher Site | Google Scholar
G. Dong and G. Kuang, “Target recognition via information aggregation through dempster–shafer’s evidence theory,” IEEE Geoscience and Remote Sensing Letters, vol. 12, no. 6, pp. 1247–1251, 2015.
View at: Publisher Site | Google Scholar
R. Niu and P. K. Varshney, “Performance analysis of distributed detection in a random sensor field,” IEEE Transactions on Signal Processing, vol. 56, no. 1, pp. 339–349, 2007.
View at: Publisher Site | Google Scholar
D. Ciuonzo, P. Rossi, and P. Willett, “Generalized rao test for decentralized detection of an uncooperative target,” IEEE Signal Processing Letters, vol. 24, no. 5, pp. 678–682, 2017.
View at: Publisher Site | Google Scholar
D. Peralta, I. Triguero, S. García, Y. Saeys, J. M. Benitez, and F. Herrera, “Distributed incremental fingerprint identification with reduced database penetration rate using a hierarchical classification based on feature fusion and selection,” Knowledge-Based Systems, vol. 126, pp. 91–103, 2017.
View at: Publisher Site | Google Scholar
D. Peralta, I. Triguero, S. García, F. Herrera, and J. M. Benitez, “DPD-DFF: a dual phase distributed scheme with double fingerprint fusion for fast and accurate identification in large databases,” Information Fusion, vol. 32, pp. 40–51, 2016.
View at: Publisher Site | Google Scholar
W. Li, J. Bao, X. Fu, G. Fortino, and S. Galzarano, “Human postures recognition based on DEMPSTER-SHAFER evidence theory and multi-sensor data fusion,” in Proceedings of the IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing, pp. 912–917, Ottawa, Canada, May 2012.
View at: Google Scholar
G. Fortino, S. Galzarano, R. Gravina, and W. Li, “A framework for collaborative computing and multi-sensor data fusion in body sensor networks,” Information Fusion, vol. 22, pp. 50–70, 2015.
View at: Publisher Site | Google Scholar
P. Francesco and C. Domenico, “Objective priors from maximum entropy in data classification,” Information Fusion, vol. 14, no. 2, pp. 186–198, 2013.
View at: Publisher Site | Google Scholar
B. Ristic and P. Smets, “Target classification approach based on the belief function theory,” IEEE Transactions on Aerospace and Electronic Systems, vol. 41, no. 2, pp. 574–583, 2005.
View at: Publisher Site | Google Scholar
P. Francesco and C. Domenico, Proceedings of the 20th Italian Workshop on Neural Networks, WIRN, Frontiers in Artificial Intelligence and Applications, 2011.
R. Gravina, P. Alinia, H. Ghasemzadeh, and G. Fortino, “Multi-sensor fusion in body sensor networks: state-of-the-art and research challenges,” Information Fusion, vol. 35, pp. 68–80, 2017.
View at: Publisher Site | Google Scholar
Y. Duan, X. Fu, W. Li, Y. Zhang, and G. Fortino, “Evolution of scale-free wireless sensor networks with feature of small-world networks,” Complexity, vol. 2017, Article ID 2516742, 15 pages, 2017.
View at: Publisher Site | Google Scholar
B. Walczak and D. L. Massart, “Rough sets theory,” Chemometrics and Intelligent Laboratory Systems, vol. 47, no. 1, pp. 1–16, 1999.
View at: Publisher Site | Google Scholar
L. Shen, F. E. H. Tay, L. Qu, and Y. Shen, “Fault diagnosis using rough sets theory,” Computers in Industry, vol. 43, no. 1, pp. 61–72, 2000.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
L. Fei, H. Wang, L. Chen, and Y. Deng, “A new vector valued similarity measure for intuitionistic fuzzy sets based on OWA operators,” Iranian Journal of Fuzzy Systems, vol. 15, pp. 31–49, 2017.
View at: Google Scholar
F. Sabahi and M.-R. Akbarzadeh-T, “A qualified description of extended fuzzy logic,” Information Sciences, vol. 244, pp. 60–74, 2013.
View at: Publisher Site | Google Scholar
H.-C. Liu, L. Liu, and Q.-L. Lin, “Fuzzy failure mode and effects analysis using fuzzy evidential reasoning and belief rule-based methodology,” IEEE Transactions on Reliability, vol. 62, no. 1, pp. 23–36, 2013.
View at: Publisher Site | Google Scholar
R. Zhang, B. Ashuri, and Y. Deng, “A novel method for forecasting time series based on fuzzy logic and visibility graph,” Advances in Data Analysis and Classification, vol. 11, pp. 759–783, 2018.
View at: Google Scholar
F. Sabahi and M.-R. Akbarzadeh-T, “Introducing validity in fuzzy probability for judicial decision-making,” International Journal of Approximate Reasoning, vol. 55, no. 6, pp. 1383–1403, 2014.
View at: Publisher Site | Google Scholar
W. Jiang and B. Wei, “Intuitionistic fuzzy evidential power aggregation operator and its application in multiple criteria decision-making,” International Journal of Systems Science, vol. 49, no. 3, pp. 582–594, 2018.
View at: Publisher Site | Google Scholar
A. Mardani, A. Jusoh, and E. K. Zavadskas, “Fuzzy multiple criteria decision-making techniques and applications—two decades review from 1994 to 2014,” Expert Systems with Applications, vol. 42, no. 8, pp. 4126–4148, 2015.
View at: Publisher Site | Google Scholar
A. P. Dempster, “Upper and lower probabilities induced by a multivalued mapping,” The Annals of Mathematical Statistics, vol. 38, no. 2, pp. 325–339, 1967.
View at: Publisher Site | Google Scholar
G. Shafer, “A mathematical theory of evidence,” Technometrics, vol. 20, p. 106, 1978.
View at: Google Scholar
X. Deng, D. Han, J. Dezert, Y. Deng, and Y. Shyr, “Evidence combination from an evolutionary game theory perspective,” IEEE Transactions on Cybernetics, vol. 46, no. 9, pp. 2070–2082, 2016.
View at: Publisher Site | Google Scholar
W. Jiang, Y. Chang, and S. Wang, “A method to identify the incomplete framework of discernment in evidence theory,” Mathematical Problems in Engineering, vol. 2017, Article ID 7635972, 15 pages, 2017.
View at: Publisher Site | Google Scholar
W. Jiang and W. Hu, “An improved soft likelihood function for Dempster-Shafer belief structures,” International Journal of Intelligent Systems, vol. 33, no. 6, pp. 1264–1282, 2018.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “A note on Z-numbers,” Information Sciences, vol. 181, no. 14, pp. 2923–2932, 2011.
View at: Publisher Site | Google Scholar
B. Kang, G. Chhipi-Shrestha, Y. Deng, K. Hewage, and R. Sadiq, “Stable strategies analysis based on the utility of Z-number in the evolutionary games,” Applied Mathematics and Computation, vol. 324, pp. 202–217, 2018.
View at: Publisher Site | Google Scholar
X. Deng, Y. Hu, Y. Deng, and S. Mahadevan, “Environmental impact assessment based on D numbers,” Expert Systems with Applications, vol. 41, no. 2, pp. 635–643, 2014.
View at: Publisher Site | Google Scholar
T. Bian, H. Zheng, L. Yin, and Y. Deng, “Failure mode and effects analysis based on D numbers and TOPSIS,” Quality and Reliability Engineering International, vol. 34, no. 4, pp. 501–515, 2018.
View at: Publisher Site | Google Scholar
F. Xiao, “An intelligent complex event processing with D numbers under fuzzy environment,” Mathematical Problems in Engineering, vol. 2016, Article ID 3713518, 10 pages, 2016.
View at: Publisher Site | Google Scholar
X. Zhou, X. Deng, Y. Deng, and S. Mahadevan, “Dependence assessment in human reliability analysis based on D numbers and AHP,” Nuclear Engineering and Design, vol. 313, pp. 243–252, 2017.
View at: Publisher Site | Google Scholar
X. Deng and Y. D.-A. H. P. Deng, “Method with different credibility of information,” Soft Computing, 2018.
View at: Publisher Site | Google Scholar
J.-B. Yang and D.-L. Xu, “Evidential reasoning rule for evidence combination,” Artificial Intelligence, vol. 205, pp. 1–29, 2013.
View at: Publisher Site | Google Scholar
C. Fu and D.-L. Xu, “Determining attribute weights to improve solution reliability and its application to selecting leading industries,” Annals of Operations Research, vol. 245, no. 1-2, pp. 401–426, 2014.
View at: Google Scholar
C. Fu, J.-B. Yang, and S.-L. Yang, “A group evidential reasoning approach based on expert reliability,” European Journal of Operational Research, vol. 246, no. 3, pp. 886–893, 2015.
View at: Publisher Site | Google Scholar
S. Xu, W. Jiang, X. Deng, and Y. Shou, “A modified physarum-inspired model for the user equilibrium traffic assignment problem,” Applied Mathematical Modelling, vol. 55, pp. 340–353, 2018.
View at: Publisher Site | Google Scholar
Q. Zhang, M. Li, and Y. Deng, “Measure the structure similarity of nodes in complex networks based on relative entropy,” Physica A: Statistical Mechanics and Its Applications, vol. 491, pp. 749–763, 2018.
View at: Publisher Site | Google Scholar
X. Zhou, Y. Hu, Y. Deng, F. T. S. Chan, and A. Ishizaka, “A DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP,” Annals of Operations Research, vol. 271, no. 2, pp. 1045–1066, 2018.
View at: Publisher Site | Google Scholar
C. Fu, D.-L. Xu, and S.-L. Yang, “Distributed preference relations for multiple attribute decision analysis,” Journal of the Operational Research Society, vol. 67, no. 3, pp. 457–473, 2016.
View at: Publisher Site | Google Scholar
G. Lin, J. Liang, and Y. Qian, “An information fusion approach by combining multigranulation rough sets and evidence theory,” Information Sciences, vol. 314, pp. 184–199, 2015.
View at: Publisher Site | Google Scholar
F. Xiao, “An improved method for combining conflicting evidences based on the similarity measure and belief function entropy,” International Journal of Fuzzy Systems, vol. 20, no. 4, pp. 1256–1266, 2018.
View at: Publisher Site | Google Scholar
H. Zheng and Y. Deng, “Evaluation method based on fuzzy relations between Dempster-Shafer belief structure,” International Journal of Intelligent Systems, vol. 33, pp. 1343–1363, 2017.
View at: Google Scholar
F. Xiao, “A novel evidence theory and fuzzy preference approach-based multi-sensor data fusion technique for fault diagnosis,” Sensors, vol. 17, no. 11, p. 2504, 2017.
View at: Publisher Site | Google Scholar
H. Luo, S.-l. Yang, X.-j. Hu, and X.-x. Hu, “Agent oriented intelligent fault diagnosis system using evidence theory,” Expert Systems with Applications, vol. 39, no. 3, pp. 2524–2531, 2012.
View at: Publisher Site | Google Scholar
Y. Du, X. Lu, X. Su, Y. Hu, and Y. Deng, “New failure mode and effects analysis: an evidential downscaling method,” Quality and Reliability Engineering International, vol. 32, no. 2, pp. 737–746, 2016.
View at: Publisher Site | Google Scholar
W. Jiang, C. Xie, M. Zhuang, and Y. Tang, “Failure mode and effects analysis based on a novel fuzzy evidential method,” Applied Soft Computing, vol. 57, pp. 672–683, 2017.
View at: Publisher Site | Google Scholar
K. Yuan, F. Xiao, L. Fei, B. Kang, and Y. Deng, “Modeling sensor reliability in fault diagnosis based on evidence theory,” Sensors, vol. 16, no. 1, p. 113, 2016.
View at: Publisher Site | Google Scholar
J. An, M. Hu, L. Fu, and J. Zhan, “A novel fuzzy approach for combining uncertain conflict evidences in the Dempster-Shafer theory,” IEEE Access, vol. 7, pp. 7481–7501, 2019.
View at: Publisher Site | Google Scholar
R. R. Yager and N. Alajlan, “Dempster-Shafer belief structures for decision making under uncertainty,” Knowledge-Based Systems, vol. 80, pp. 58–66, 2015.
View at: Publisher Site | Google Scholar
F. Chao and S. L. Yang, “Conjunctive combination of belief functions from dependent sources using positive and negative weight functions,” Expert Systems With Applications, vol. 14, pp. 1964–1972, 2014.
View at: Google Scholar
C. Fu and D.-L. Xu, “Determining attribute weights to improve solution reliability and its application to selecting leading industries,” Annals of Operations Research, vol. 245, no. 1-2, pp. 401–426, 2016.
View at: Publisher Site | Google Scholar
X. Deng and W. Jiang, “An evidential axiomatic design approach for decision making using the evaluation of belief structure satisfaction to uncertain target values,” International Journal of Intelligent Systems, vol. 33, no. 1, pp. 15–32, 2017.
View at: Publisher Site | Google Scholar
G. Kabir, S. Tesfamariam, A. Francisque, and R. Sadiq, “Evaluating risk of water mains failure using a Bayesian belief network model,” European Journal of Operational Research, vol. 240, no. 1, pp. 220–234, 2015.
View at: Publisher Site | Google Scholar
P. Dutta, “Uncertainty modeling in risk assessment based on Dempster-Shafer theory of evidence with generalized fuzzy focal elements,” Fuzzy Information and Engineering, vol. 7, no. 1, pp. 15–30, 2015.
View at: Publisher Site | Google Scholar
X. Deng and W. Jiang, “Fuzzy risk evaluation in failure mode and effects analysis using a D numbers based multi-sensor information fusion method,” Sensors, vol. 17, no. 9, p. 2086, 2017.
View at: Publisher Site | Google Scholar
X. Zheng and Y. Deng, “Dependence assessment in human reliability analysis based on evidence credibility decay model and Iowa operator,” Annals of Nuclear Energy, vol. 112, pp. 673–684, 2017.
View at: Publisher Site | Google Scholar
L. Zhang, L. Ding, X. Wu, and M. J. Skibniewski, “An improved Dempster-Shafer approach to construction safety risk perception,” Knowledge-Based Systems, vol. 132, pp. 30–46, 2017.
View at: Publisher Site | Google Scholar
L. Fei, J. Xia, Y. Feng, and L. Liu, “An ELECTRE-based multiple criteria decision making method for supplier selection using Dempster-Shafer theory,” IEEE Access, vol. 7, pp. 84701–84716, 2019.
View at: Publisher Site | Google Scholar
L. Fei, Y. Deng, and Y. Hu, “DS-VIKOR: a new multi-criteria decision-making method for supplier selection,” International Journal of Fuzzy Systems, vol. 21, no. 1, pp. 157–175, 2019.
View at: Publisher Site | Google Scholar
Z.-g. Liu, Q. Pan, J. Dezert, and A. Martin, “Adaptive imputation of missing values for incomplete pattern classification,” Pattern Recognition, vol. 52, pp. 85–95, 2016.
View at: Publisher Site | Google Scholar
G. Dong and G. Kuang, “Target recognition via information aggregation through Dempster-Shafer’s evidence theory,” IEEE Geoscience and Remote Sensing Letters, vol. 12, pp. 1247–1251, 2015.
View at: Google Scholar
T. Denoeux, “A k-nearest neighbor classification rule based on Dempster-Shafer theory,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 25, no. 5, pp. 804–813, 1995.
View at: Publisher Site | Google Scholar
J. Ma, W. Liu, P. Miller, and H. Zhou, “An evidential fusion approach for gender profiling,” Information Sciences, vol. 333, pp. 10–20, 2016.
View at: Publisher Site | Google Scholar
B. S. Everitt, S. Landau, M. Leese, and D. Stahl, Cluster Analysis, John Wiley & Sons, London, UK, 5th edition, 2011.
Y. Deng, “Deng entropy: a generalized Shannon entropy to measure uncertainty,” Manuscript, vol. 91, pp. 549–553, 2015.
View at: Google Scholar
F. Xiao, “Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy,” Information Fusion, vol. 46, pp. 23–32, 2019.
View at: Publisher Site | Google Scholar
Y. Lin, C. Wang, C. Ma, Z. Dou, and X. Ma, “A new combination method for multisensor conflict information,” The Journal of Supercomputing, vol. 72, no. 7, pp. 2874–2890, 2016.
View at: Publisher Site | Google Scholar
F. Ye, J. Chen, and Y. B. Li, “Improvement of DS evidence theory for multi-sensor conflicting information,” Symmetry, vol. 9, pp. 1–15, 2017.
View at: Google Scholar
Q. Liu, J. Feng, and J. L. Zhou, “The reliability growth analysis for liquid rocket engine according to the Gompertz model,” Journal of Aerospace Power, vol. 19, pp. 419–423, 2004.
View at: Google Scholar
H. Liu and B. Guo, “Fusion of information of multiple sources in bayesian assessment for reliability of success or failure product,” Journal of Projectiles, Rockets, Missiles and Guidance, vol. 27, pp. 38–341, 2007.
View at: Google Scholar

Copyright

Copyright © 2020 Peng Di et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

441

Downloads

638

Citations