An evidential evaluation of nuclear safeguards

Abstract

Nuclear safeguards evaluation is a complicated issue with many missing values and uncertainties. By invoking Dempster–Shafer theory of evidence, the missing values are assigned to a subset of a set of multiple objects, at the same time, by combining different evaluation values, and the effect of uncertainty will be decreased. In this way, both the missing values and uncertainties are considered in the final evaluations. This method has been used in considering the International Atomic Energy Agency experts’ evaluation for nuclear safeguards. The result shows that ( $s_{2}$ , 0.1897) is the biggest belief degree.

Keywords

Nuclear safeguards evaluation Dempster–Shafer Theory uncertainty modeling belief function

Introduction

Nowadays, nuclear system safety has attracted much attention from all over the world. Many researches have been done on nuclear power plant,^1–8 nuclear wastes, and especially nuclear weapon. Since improper use of nuclear energy may cause mass destruction, it is necessary to evaluate nuclear safeguards and limit the development of nuclear detonations. Nuclear safeguards evaluations^9–14 are invoked to test and verify that states meet with the international treaty, which put forward by International Atomic Energy Agency (IAEA) and make sure that those nuclear materials will not be used to manufacture nuclear weapons. In order to make the evaluation of indicators and then make the final assessment of the States’ declarations to the institution concerning any activity which are related to nuclear power, the IAEA experts collect relevant information from some main sources: information provided by the State itself, Internet and newspapers, and non-safeguards IAEA databases.¹⁵

However, there still exists much uncertainty and complexity on the information they collected, and the number of indicators is huge. As a result, it is difficult for IAEA experts to conduct a comprehensive assessment of all indicators. In addition, some indicators in the assessment framework are missing values;¹⁶ it may cause some incomplete evaluation of the security. At the same time, when we want to take those missing values into consideration, uncertainties usually come as which part these missing values should belong to. Therefore, many researchers have focused on these problems, and some methods have been put forward to handle them, such as deletion and imputation.^17–19 At the same time, Liu et al.¹⁵ put forward a framework to deal with the information by introducing a newly belief rule. Kabak and Ruan^20–22 presented a method for estimating the cumulative belief in nuclear safeguards. Rodríguez et al.^16,23 proposed to manage missing values with an imputation process by means of a collaborative filtering model^24,25 based on the k-nearest neighbors (K-nn) scheme. Based on the hierarchical analysis and the IAEA model, Pei et al.²⁶ treat and handle nuclear safeguards evaluations by using a 2-tuple fuzzy linguistic aggregation model²⁷ and weighted ordered weighted averaging (WOWA) operator.²⁸

In the present work, a new method for evaluating nuclear safeguards is proposed. The method is based on Dempster–Shafer theory of evidence, which has been widely invoked to handle uncertain information, such as fault diagnosis,^29–32 human reliability analysis,³³ pattern recognition,^34–37 uncertainty modeling,^38–40 evidential reasoning,^41,42 decision making,⁴³ and environmental assessment.^44,45 In this article, first, according to the belief degrees of linguistic evaluation values given by IAEA experts and weights of IAEA experts about indicators, we obtain the basic probability assignment (BPA) of each indicator. Then, the BPA of each indicator is combined to get a general evaluation result. At the same time, some adjustments will be made on our results so that the output data can be used for other analysis. Finally, by taking the strengths of indicators into consideration, we aggregate the evaluation results of indicators and compare them with some results from other works.

Preliminaries

The IAEA physical model

In order to have an accurate evaluation of nuclear energy, the IAEA proposes the physical model to consider lots of indicators in hierarchical and multilayer structure, which take all those linguistic evaluation values into account.

The values of this model are shown in Table 1, with totally 914 indicators in it. Different levels are clear and directly address this assessment issue. At the same time, in practice, the strength of each indicator is different. Let us take a strong indicator as an example. When an indicator is connected to nuclear process or nuclear activities, it can be viewed as a strong indicator. Conversely, an indicator is weak if it is for many other reasons or for many other activities. In the process of combining indicators with different strengths during the evaluation, Liu et al.¹⁵ put forward a rule to regulate the difference with the ratio of strong indicators:medium indicators:weak indicators being 9:3:1, which means the importance of strong indicators is nine times the importance of weak indicators.

Table 1.

Stratification of the multilayer evaluation.²⁶

Level 0	Level 1	Level 2	Level 3
Overall (state)	Mining/milling	U from ores	$I_{1} - I_{83}$
		U from seawater
		U from monazite
		Th from monazite
		Th from U ore
	Conversion	To UF6	$I_{84} - I_{199}$
		To UF4
		To UCl4
	Enrichment	Gas centrifuge	$I_{200} - I_{415}$
		Gaseous diffusion
		Aerodynamic
		Molecular laser
		EMLIS
		Electromagnetic
		Chemical exchange
		Ion exchange atomic
		Vapor laser
		Plasma separation
	Fuel fabrication	Umet, UO2, MOX	$I_{496} - I_{583}$
	Reactors	GCR, AGR, HTGR	$I_{594} - I_{790}$
		LWR, PWR, BWR
		FBR
	Reprocessing		$I_{841} - I_{914}$

The evaluation of nuclear safeguards is shown in Table 2. As shown in the table

{e_{j} | j = 1, 2, 3, \dots, n}

(1)

represents all the IAEA experts

υ_{i j} (i = 1, 2, 3, \dots, 914)

(2)

which is the vector of evaluation value for each indicator I_i given by IAEA expert $e_{j}$ . $ω_{i}$ is the strength of indicator $I_{i}$ , and $ω_{i} \in ω_{strong} (= 9), ω_{medium} (= 3), ω_{weak} (= 1)$ .

Table 2.

Nuclear safeguards evaluation by IAEA experts.

	$e_{1}$	$e_{2}$	$e_{3}$	…	$e_{n}$
$I_{1} (ω_{1})$	$(υ_{11}, θ_{11})$	$(υ_{12}, θ_{12})$	$(υ_{13}, θ_{13})$	…	$(υ_{1 n}, θ_{1 n})$
$I_{2} (ω_{2})$	$(υ_{21}, θ_{21})$	$(υ_{22}, θ_{22})$	$(υ_{23}, θ_{23})$	…	$(υ_{2 n}, θ_{2 n})$
$I_{3} (ω_{3})$	$(υ_{31}, θ_{31})$	$(υ_{32}, θ_{32})$	$(υ_{33}, θ_{33})$	…	$(υ_{3 n}, θ_{3 n})$
⋮	⋮	⋮	⋮	…	⋮
$I_{914} (ω_{914})$	$(υ_{9141}, θ_{9141})$	$(υ_{9142}, θ_{9142})$	$(υ_{9143}, θ_{9143})$	…	$(υ_{914 n}, θ_{914 n})$

IAEA: International Atomic Energy Agency.

The weight of each IAEA expert $e_{j}$ about indicator $I_{i}$ is represented by $θ_{ij}$ . Based on the assumption, the evaluation for the indicator $I_{i}$ by an expert $e_{j}$ will be simplified as

\begin{matrix} (υ_{ij}, θ_{ij}) = \\ {({(β_{ij}^{k}, s_{k}) | k = 0, 1, \dots, 6}, θ_{ij}) | i = 1, \dots, 914, j = 1, \dots, n} \end{matrix}

(3)

({((β_{ij}^{0}, β_{ij}^{1}, \dots, β_{ij}^{6}), θ_{ij}) | i = 1, \dots, 914, j = 1, \dots, n})

(4)

Here, $β_{ij}^{k}$ is a value which is used to represent the belief degree associated to the linguistic term $s_{k}$ of indicator $I_{i}$ by IAEA expert $e_{j}$ , and $s_{0}$ , $s_{1}$ , $s_{2}$ , $s_{3}$ , $s_{4}$ , $s_{5}$ , and $s_{6}$ represent different levels of belief degree which are equal to none, very low, low, medium, high, very high, and perfect, respectively, and they are the initial values of the evaluation.

Here, we limit $β_{ij}^{0} + β_{ij}^{1} + β_{ij}^{2} + β_{ij}^{3} + β_{ij}^{4} + β_{ij}^{5} + β_{ij}^{6} \leq 1$ , it means that the information is incomplete sometimes. When it comes to the uncertainty, Dempster–Shafer theory of evidence can be used in the evaluation process.

Dempster–Shafer theory of evidence

Uncertainty information has always been a hot topic, and how to deal with uncertainty is still an open issue.⁴⁶ Many mathematical methods and models have been invoked, such as fuzzy sets,⁴⁷ belief structure,^48–50D numbers,^51–55Z numbers,^56,57 and so on. Dempster–Shafer theory of evidence,^58,59 as one of the widely used methods, is often used to solve uncertainty information problems. Compared with other tools, Dempster–Shafer theory can express the missing values and uncertainty by assigning a probability directly to a subset of a set of multiple objects. At the same time, the total belief of the objects will be measured separately. Due to these advantages, it has been widely used in many areas, such as information fusion,^60,61 game theory,⁶² decision making,^54,63,64 and risk and reliability analysis.^65,66

To complete the description, we introduce some basic concepts in the following parts.

Let $Ω$ be a set of mutually exclusive and collectively detailed events, indicated by

Ω = {E_{1}, E_{2}, \dots, E_{i}, \dots, E_{N}}

(5)

where $Ω$ is called a frame of discernment, and $N$ is the number of events.

The power set of $Ω$ is indicated by $2^{Ω}$ , namely

\begin{matrix} 2^{Ω} = & {\emptyset, {E_{1}}, \dots, {E_{N}}, {E_{1}, E_{2}}, \dots, \\ {E_{1}, E_{2}, \dots, E_{i}}, \dots, Ω} \end{matrix}

(6)

The elements of $2^{Ω}$ or subset of $Ω$ are called propositions.

When it comes to a frame of discernment $Ω = {E_{1}, E_{2}, \dots, E_{N}}$ , a mass function is defined as a mapping $m$ from $2^{Ω}$ to $[0, 1]$ , formally defined by

m : 2^{Ω} \to [0, 1]

(7)

which satisfies the following condition

m (\emptyset) = 0 and \sum_{A \in 2^{Ω}} m (A) = 1

(8)

In the theory of Dempster–Shafer, a mass function is usually defined as a BPA of the frame of discernment $Ω$ . The assigned probability $m (A)$ is the belief exactly assigned to $A$ and can be used to represent the strength of the evidence which supports $A$ . If $m (A) > 0$ , $A$ is called a focal element, and the union of all focal elements is called the core of the mass function, which is the most important concept in evidence theory and many operations about this function, for example, divergence measure^67–69 and negation.^70,71

Taking two pieces of evidence into consideration, they can be indicated by two BPAs, $m_{1}$ and $m_{2}$ , with Dempster’s rule of combination used to combine them.

With two belief structures $m_{1}$ and $m_{2}$ , the Dempster’s rule of combination, denoted by $m = m_{1} \oplus m_{2}$ , is defined as follows

m (A) = {\begin{matrix} \frac{1}{1 - K} \sum_{B \cap C = A} m_{1} (B) m_{2} (C), & A \neq \emptyset \\ 0, & A = \emptyset \end{matrix}

(9)

with

K = \sum_{B \cap C = \emptyset} m_{1} (B) m_{2} (C)

(10)

where $K$ is a normalization constant called the collision coefficient of two BPAs.

Dempster’s combination rule can be seen as one of the most important and widely used parts of Dempster–Shafer theory, and it also satisfies the exchange and association properties. Therefore, if there exist multiple belief structures, the calculation of their combination results can be performed in pairs or in any order. When we talk about the Dempster–Shafer theory, how to manage the conflict evidence is of great importance.^72–76 In addition, for the purpose of facilitating decision making, many methods have been developed to do the transformation or directly convert belief structures into probability distributions. For example, the pianistic transformation⁷⁷ and plausibility transformation⁷⁸ are two well-known methods.

Method

A new method is proposed in this section to evaluate the nuclear safeguards, which is based on Dempster–Shafer theory of evidence, and the details about this method will also be provided. Also, the belief degree^79,80 to aggregate the values of Table 2 is proposed.

Step 1: Notice that $υ = (β_{ij}^{0}, β_{ij}^{1}, β_{ij}^{2}, β_{ij}^{3}, β_{ij}^{4}, β_{ij}^{5}, β_{ij}^{6})$ and limit $β_{ij}^{0} + β_{ij}^{1} + β_{ij}^{2} + β_{ij}^{3} + β_{ij}^{4} + β_{ij}^{5} + β_{ij}^{6} \leq 1$ , when $β_{ij}^{0} + β_{ij}^{1} + β_{ij}^{2} + β_{ij}^{3} + β_{ij}^{4} + β_{ij}^{5} + β_{ij}^{6} < 1$ , it means there exist some missing values and these missing values are hard to assign, which may bring some uncertainties.

For each evaluation given by IAEA expert, the corresponding weight will be multiplied. $α_{ij}$ is defined as the normalized weighting factor that each expert has compared with all the experts. Then, $ξ_{ij} = α_{ij} \times β_{ij}$ .

Then a new evaluation is constructed as follows

υ^{*} = (ξ_{ij}^{0}, ξ_{ij}^{1}, ξ_{ij}^{2}, ξ_{ij}^{3}, ξ_{ij}^{4}, ξ_{ij}^{5}, ξ_{ij}^{6})

(11)

Step 2: BPAs are constructed by assigning the missing values and uncertainty to the subsets of the set composed of multiple objects. In our case, the set is constructed as ( $s_{0}$ , $s_{1}$ , $s_{2}$ , $s_{3}$ , $s_{4}$ , $s_{5}$ , $s_{6}$ ), which is reasonable considering the existence of uncertainty. Here, it can be easily seen that $\sum_{n = 0}^{6} ξ_{ij}^{n} < 1$ . Then, the results of modified BPAs are shown as follows

(ξ_{ij}^{0}, ξ_{ij}^{1}, ξ_{ij}^{2}, ξ_{ij}^{3}, ξ_{ij}^{4}, ξ_{ij}^{5}, ξ_{ij}^{6}, 1 - S) \cdot S = \sum_{n = 0}^{6} ξ_{ij}^{n}

(12)

Here, 1 – S represents evaluation value of the missing information and the uncertainty.

Step 3: If there are n IAEA experts making their evaluations on each indicator, we would have n BPAs on each indicator.

\begin{matrix} m {(υ)}_{I_{i}}^{1 *} = (ξ_{i 1}^{0}, ξ_{i 1}^{1}, ξ_{i 1}^{2}, ξ_{i 1}^{3}, ξ_{i 1}^{4}, ξ_{i 1}^{5}, ξ_{i 1}^{6}, 1 - S) \\ m {(υ)}_{I_{i}}^{2 *} = (ξ_{i 2}^{0}, ξ_{i 2}^{1}, ξ_{i 2}^{2}, ξ_{i 2}^{3}, ξ_{i 2}^{4}, ξ_{i 2}^{5}, ξ_{i 2}^{6}, 1 - S) \\ m {(υ)}_{I_{i}}^{3 *} = (ξ_{i 3}^{0}, ξ_{i 3}^{1}, ξ_{i 3}^{2}, ξ_{i 3}^{3}, ξ_{i 3}^{4}, ξ_{i 3}^{5}, ξ_{i 3}^{6}, 1 - S) \\ \dots \\ m {(υ)}_{I_{i}}^{n *} = (ξ_{in}^{0}, ξ_{in}^{1}, ξ_{in}^{2}, ξ_{in}^{3}, ξ_{in}^{4}, ξ_{in}^{5}, ξ_{in}^{6}, 1 - S) \end{matrix}

(13)

Using the combination rule of Dempster–Shafer theory of evidence to aggregate the evaluation experts made on each indicator, composite evaluation on each indicator can be obtained

m {(υ)}_{I_{i}} = m {(υ)}_{I_{i}}^{1 *} \oplus m {(υ)}_{I_{i}}^{2 *} \oplus m {(υ)}_{I_{i}}^{3 *} \oplus \dots \oplus m {(υ)}_{I_{i}}^{n *}

(14)

Step 4: Based on the strength of an indicator, the indicators are divided into three classes—the strong indicators, the medium indicators, and the weak indicators. They should be considered independently to get the corresponding evaluation results. Then, we combine them, respectively, under the rule of Dempster–Shafer theory of evidence and the results are noted as

\begin{matrix} M_{s} = (β_{s}^{0}, β_{s}^{1}, β_{s}^{2}, β_{s}^{3}, β_{s}^{4}, β_{s}^{5}, β_{s}^{6}, 1 - S^{s}) \\ M_{m} = (β_{m}^{0}, β_{m}^{1}, β_{m}^{2}, β_{m}^{3}, β_{m}^{4}, β_{m}^{5}, β_{m}^{6}, 1 - S^{m}) \\ M_{w} = (β_{w}^{0}, β_{w}^{1}, β_{w}^{2}, β_{w}^{3}, β_{w}^{4}, β_{w}^{5}, β_{w}^{6}, 1 - S^{w}) \end{matrix}

(15)

where $S^{s} = \sum_{n = 0}^{6} β_{s}^{n}$ , $S^{m} = \sum_{n = 0}^{6} β_{m}^{n}$ , and $S^{w} = \sum_{n = 0}^{6} β_{w}^{n}$ .

Step 5: Following step 4, the combination results of strong indicators, medium indicators, and weak indicators will be calculated, and the final results can be achieved on the basis of the three individual classes. We continue to combine them by using the combination rule of Dempster–Shafer theory of evidence whose results needed to be aggregated. Because of the different weights of three kinds of indicators, which is $ω_{strong} (= 9), ω_{medium} (= 3), ω_{weak} (= 1)$ , we multiply $M_{s}$ , $M_{m}$ , and $M_{w}$ with the normalized corresponding weights. Hence, the modified $M_{s}^{*}$ , $M_{m}^{*}$ , and $M_{w}^{*}$ are obtained.

Finally, $M_{s}^{*}$ , $M_{m}^{*}$ , and $M_{w}^{*}$ are combined to get the final results. In this method, although there may exist uncertainty in the evaluations, some adjustments are made so that the combination rule of Dempster–Shafer theory of evidence can be used to combine the evaluations. Even when there exist missing values in the evaluation, by assigning the missing values to a new element, the missing values will be taken into consideration and the uncertainties can be effectively reduced.

Application

In this section, we illustrate an example to demonstrate how our method works and show the advantages of it by comparing with other works. The example is based on the gaseous diffusion enrichment process, which consists of 22 indicators, shown in Table 3.²¹ It is a general assumption that each IAEA expert has the same weight for any indicator, which means in this example, for any $i$ , $θ_{i 1} = 3$ , $θ_{i 2} = 5$ , $θ_{i 3} = 4$ , and $θ_{i 4} = 2$ .

Table 3.

Specific indicators of gaseous diffusion enrichment.

Symbol	Name	Strength
$I_{1}$	Compressor for pure $UF 6$	Strong
$I_{2}$	Gaseous diffusion barrier	Strong
$I_{3}$	Heat exchanger for cooling pure $U F 6$	Strong
$I_{4}$	Diffuser housing/vessel	Medium
$I_{5}$	Gas blower for $UF 6$	Medium
$I_{6}$	Rotary shaft seal	Medium
$I_{7}$	Special control value (large aperture)	Medium
$I_{8}$	Special shut-off value (large aperture)	Medium
$I_{9}$	Chlorine trifluoride	Medium
$I_{10}$	Nickel powder, high purity	Medium
$I_{11}$	Gasket, large	Weak
$I_{12}$	Feed system/product and tails withdrawal	Weak
$I_{13}$	Expansion bellows	Weak
$I_{14}$	Header piping system	Weak
$I_{15}$	Vacuum system and pump	Weak
$I_{16}$	Aluminum oxide powder	Weak
$I_{17}$	Nickel powder	Weak
$I_{18}$	Polytetrafluoroethylene (Teflon)	Weak
$I_{19}$	Large electrical switching yard	Weak
$I_{20}$	Large heat increase in air or water	Weak
$I_{21}$	Larger specific power consumption	Weak
$I_{22}$	Larger cooling requirements (towers)	Weak

Values of the evaluation for the 22 indicators are shown in Table 4.²¹

Table 4.

Evaluations of 22 indicators by four IAEA experts.

$e_{1} (3)$	$e_{2} (5)$	$e_{3} (4)$	$e_{4} (2)$
$I_{1} (0, 0, 0, 0.7, 0.3, 0, 0)$	$(0, 0, 1, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0.3, 0.2, 0.5)$
$I_{2} (0, 0.8, 0.2, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0, 1, 0)$	$(0, 0, 0, 0.1, 0.6, 0.2, 0)$	$(0, 0, 0, 0, 0, 0, 1)$
$I_{3} (0, 0, 0.6, 0.4, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 0)$	$(0, 0, 0, 0.2, 0.5, 0.3, 0)$	$(0, 0, 0.4, 0.6, 0, 0, 0)$
$I_{4} (0, 0, 0, 0.7, 0.3, 0, 0)$	$(\frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7})$	$(0, 0, 0, 0, 0, 0.9, 0.1)$	$(0, 0, 0, 0, 0.8, 0.2, 0)$
$I_{5} (0, 0, 0, 0.7, 0.3, 0, 0)$	$(0, 0, 1, 0, 0, 0, 0)$	$(\frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7})$	$(0, 0, 0.1, 0, 0, 0.1, 0.9)$
$I_{6} (0, 0, 0, 0.1, 0.9, 0, 0)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0, 0, 0.2, 0.6, 0.2)$	$(0, 0, 0.1, 0.8, 0.1, 0, 0)$
$I_{7} (0, 0, 0.6, 0.4, 0, 0, 0)$	$(0, 0, 1, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0.2, 0.7, 0)$	$(0, 0, 0, 0, 0, 0.9, 0.1)$
$I_{8} (0, 0, 0, 0, 0, 0.3, 0.7)$	$(0, 0, 0, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0.4, 0.6, 0)$	$(0, 0, 0, 0, 0.3, 0.4, 0.1)$
$I_{9} (0, 0, 0, 0, 0, 0, 0)$	$(0, 0, 1, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0.3, 0.6, 0.1)$	$(0, 0, 0, 0, 0.7, 0.3, 0)$
$I_{10} (0, 0.2, 0.8, 0, 0, 0, 0)$	$(0, 0, 1, 0, 0, 0, 0)$	$(0, 0, 0.2, 0.5, 0.3, 0, 0)$	$(0, 0, 0, 0, 0.7, 0.3, 0)$
$I_{11} (0, 0, 0.8, 0.2, 0, 0, 0)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0.9, 0.1)$	$(0, 0, 0.2, 0.6, 0.2, 0, 0)$
$I_{12} (0.1, 0.9, 0, 0, 0, 0, 0)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0.8, 0.1, 0, 0, 0)$	$(0, 0, 0, 0.1, 0.9, 0, 0)$
$I_{13} (0, 0, 0, 0, 0, 0, 1)$	$(0, 0, 0, 0, 0, 0, 1)$	$(0, 0, 0, 0, 0, 0.4, 0.6)$	$(0, 0, 0, 0, 0.1, 0.7, 0.1)$
$I_{14} (0, 0, 0, 0, 0, 0.1, 0.9)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 0.9)$	$(0, 0, 0, 0.3, 0.5, 0.2, 0)$
$I_{15} (0, 0, 0, 0.7, 0.3, 0, 0)$	$(0, 0, 1, 0, 0, 0, 0)$	$(0, 0.9, 0.1, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 0)$
$I_{16} (0, 0.2, 0.8, 0, 0, 0, 0)$	$(\frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7})$	$(0, 0.4, 0.5, 0, 0, 0, 0)$	$(0, 0, 0, 0.9, 0.1, 0, 0)$
$I_{17} (0, 0, 0, 0.1, 0.9, 0, 0)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 1)$	$(0, 0, 0, 0, 0.9, 0.1, 0)$
$I_{18} (0, 0, 0, 0, 0, 0, 0)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0.2, 0.6, 0.2, 0, 0)$	$(0, 0, 0.8, 0.2, 0, 0, 0)$
$I_{19} (0, 0, 0.3, 0.7, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 1)$	$(0, 0, 0, 0, 0, 0.8, 0.1)$	$(0, 0, 0, 0, 0.1, 0.8, 0)$
$I_{20} (0, 0, 0, 0, 0, 0.3, 0.7)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0.6, 0.4, 0)$
$I_{21} (0, 0, 0, 0.4, 0.6, 0, 0)$	$(0, 0, 0, 1, 0, 0, 0)$	$(0, 0, 0, 0, 0.1, 0.8, 0)$	$(0, 0, 0, 0, 0, 0.4, 0.6)$
$I_{22} (0, 0, 0, 1, 0, 0, 0)$	$(0, 1, 0, 0, 0, 0, 0)$	$(0, 0, 0, 0, 0, 0, 0)$	$(0.1, 0.8, 0.1, 0, 0, 0, 0)$

IAEA: International Atomic Energy Agency.

For $I_{2}$ , it can be obtained that

\begin{matrix} m {(υ)}_{I_{2}}^{1} = (0.0, 0.8, 0.2, 0.0, 0.0, 0.0, 0.0, 0.0) \\ m {(υ)}_{I_{2}}^{2} = (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0) \\ m {(υ)}_{I_{2}}^{3} = (0.0, 0.0, 0.0, 0.1, 0.6, 0.2, 0.0, 0.1) \\ m {(υ)}_{I_{2}}^{4} = (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0) \end{matrix}

Calculate the normalized corresponding weights as mentioned in step 1, which is $α = (\frac{3}{14}, \frac{5}{14}, \frac{4}{14}, \frac{2}{14})$ . Then modified BPA of indicator $I_{2}$ can be obtained as follows

\begin{matrix} m {(υ)}_{I_{2}}^{1 *} = (0, \frac{6}{35}, \frac{3}{70}, 0, 0, 0, 0, \frac{11}{14}) \\ m {(υ)}_{I_{2}}^{2 *} = (0, 0, 0, 0, 0, \frac{5}{14}, 0, \frac{9}{14}) \\ m {(υ)}_{I_{2}}^{3 *} = (0, 0, 0, \frac{1}{35}, \frac{6}{35}, \frac{2}{35}, 0, \frac{26}{35}) \\ m {(υ)}_{I_{2}}^{4 *} = (0, 0, 0, 0, 0, 0, \frac{1}{7}, \frac{6}{7}) \end{matrix}

Therefore, the belief degree of indicator $I_{2}$ can be calculated as

\begin{matrix} m {(υ)}_{I_{2}} & = m {(υ)}_{I_{2}}^{1 *} \oplus_{m} {(υ)}_{I_{2}}^{2 *} \oplus_{m} {(υ)}_{I_{2}}^{3 *} \oplus_{m} {(υ)}_{I_{4}}^{n *} \\ = (0, 0.0915, 0.0229, 0.0161, 0.0968, 0.2833, 0.0699, 0.4195) \end{matrix}

Then, the evaluation of all indicators and the results are shown in Table 5.

Table 5.

Evaluation results of 22 indicators.

Indicators	Evaluations
$I_{1}$ (strong)	(0.0000, 0.0000, 0.2779, 0.0955, 0.0680, 0.0167, 0.0417, 0.5002)
$I_{2}$ (strong)	(0.0000, 0.0915, 0.0229, 0.0161, 0.0968, 0.2833, 0.0699, 0.4204)
$I_{3}$ (strong)	(0.0000, 0.0000, 0.1284, 0.1690, 0.1064, 0.0639, 0.0000, 0.5323)
$I_{4}$ (medium)	(0.0319, 0.0319, 0.0319, 0.1148, 0.1301, 0.2079, 0.0493, 0.4022)
$I_{5}$ (medium)	(0.0233, 0.0233, 0.2620, 0.1056, 0.0587, 0.0305, 0.0880, 0.4086)
$I_{6}$ (medium)	(0.0000, 0.0000, 0.0066, 0.3225, 0.1461, 0.0955, 0.0319, 0.3975)
$I_{7}$ (medium)	(0.0000, 0.0000, 0.3325, 0.0447, 0.0316, 0.1192, 0.0616, 0.4104)
$I_{8}$ (medium)	(0.0000, 0.0000, 0.0000, 0.0000, 0.1165, 0.2309, 0.1133, 0.5393)
$I_{9}$ (medium)	(0.0000, 0.0000, 0.2586, 0.0000, 0.1167, 0.1406, 0.0187, 0.4654)
$I_{10}$ (medium)	(0.0000, 0.0209, 0.4023, 0.0769, 0.0964, 0.0192, 0.0000, 0.3843)
$I_{11}$ (weak)	(0.0000, 0.0000, 0.1037, 0.3223, 0.0134, 0.1442, 0.0160, 0.4004)
$I_{12}$ (weak)	(0.0115, 0.1035, 0.1297, 0.2707, 0.0632, 0.0000, 0.0000, 0.4214)
$I_{13}$ (weak)	(0.0000, 0.0000, 0.0000, 0.0000, 0.0058, 0.1045, 0.5331, 0.3566)
$I_{14}$ (weak)	(0.0000, 0.0000, 0.0000, 0.2580, 0.0339, 0.0251, 0.2757, 0.4073)
$I_{15}$ (weak)	(0.0000, 0.1600, 0.2745, 0.0848, 0.0364, 0.0000, 0.0000, 0.4443)
$I_{16}$ (weak)	(0.0323, 0.1274, 0.2310, 0.0981, 0.0396, 0.0323, 0.0323, 0.4070)
$I_{17}$ (weak)	(0.0000, 0.0000, 0.0000, 0.2444, 0.1768, 0.0068, 0.1635, 0.4086)
$I_{18}$ (weak)	(0.0000, 0.0000, 0.0975, 0.4323, 0.0348, 0.0000, 0.0000, 0.4354)
$I_{19}$ (weak)	(0.0000, 0.0000, 0.0344, 0.0801, 0.0069, 0.2010, 0.2582, 0.4195)
$I_{20}$ (weak)	(0.0000, 0.0000, 0.0000, 0.2778, 0.0500, 0.0770, 0.0954, 0.4998)
$I_{21}$ (weak)	(0.0000, 0.0000, 0.0000, 0.2987, 0.0857, 0.1626, 0.0412, 0.4118)
$I_{22}$ (weak)	(0.0081, 0.3688, 0.0081, 0.1318, 0.0000, 0.0000, 0.0000, 0.4833)

The indicators are of the same values, and there are 3 strong indicators, 7 medium indicators, and 12 weak indicators. Hence, the aggregations of three kinds of indicators will be computed, respectively, according to the combination rule of Dempster–Shafer theory of evidence, noted as $M_{s}$ , $M_{m}$ , and $M_{w}$ , shown in Table 6.

Table 6.

Evaluations of three kinds of indicators.

Indicators	Evaluations
$M_{s}$	(0.0000, 0.0458, 0.2179, 0.1323, 0.1424, 0.1972, 0.0554, 0.2104)
$M_{m}$	(0.0064, 0.0092, 0.4256, 0.1316, 0.1470, 0.1858, 0.0514, 0.0453)
$M_{w}$	(0.0010, 0.0236, 0.0373, 0.7394, 0.0181, 0.0313, 0.1041, 0.0081)

Finally, we multiply $M_{s}$ , $M_{m}$ , and $M_{w}$ with the normalized corresponding weights, which is $ω = (\frac{9}{13}, \frac{3}{13}, \frac{1}{13})$ , and then we get the modified $M_{s}^{*}$ , $M_{m}^{*}$ , and $M_{w}^{*}$ , which is

\begin{matrix} M_{s}^{*} = (0.0000, 0.0317, 0.1509, 0.0916, 0.0986, 0.1365, 0.0384, 0.4523) \\ M_{m}^{*} = (0.0015, 0.0021, 0.0982, 0.0304, 0.0339, 0.0429, 0.0119, 0.7791) \\ M_{w}^{*} = (0.0000, 0.0018, 0.0029, 0.0569, 0.0014, 0.0024, 0.0080, 0.9266) \end{matrix}

Combining $M_{s}^{*}$ , $M_{m}^{*}$ , and $M_{w}^{*}$ $(M_{f} = M_{s}^{*} \oplus M_{m}^{*} \oplus M_{w}^{*})$ , the result is

\begin{matrix} M_{f} = (7.2096 e - 04, 0.0281, 0.1897, 0.1222, 0.1022, 0.1412, 0.0416, 0.3745) \end{matrix}

(16)

Therefore, 0.1897, which is ( $s_{2}$ , 0.1897), is the biggest belief degree of the final evaluation and here 0.3745 is the value of uncertainty. To compare with other methods, we list the results of their methods, and they are shown in Table 7.²¹

Table 7.

Comparison of the result of several existing methods.

Names	Liu’s results	Kabak’s results	Pei’s results	Our results
$S_{s}$	( $s_{4}$ , 0.42)	( $s_{4}$ , –0.3)	( $s_{5}$ , –0.45)	( $s_{1}$ , 0.2179)
$S_{m}$	( $s_{3}$ , 0.27)	( $s_{4}$ , 0.15)	( $s_{4}$ , –0.22)	( $s_{2}$ , 0.4256)
$S_{w}$	( $s_{3}$ , 0.33)	( $s_{2}$ , 0.34)	( $s_{4}$ , –0.35)	( $s_{3}$ , 0.7394)
$S_{f}$	( $s_{4}$ , 0.07)	( $s_{4}$ , –0.03)	( $s_{4}$ , 0.3)	( $s_{2}$ , 0.1897)

In Liu et al.’s¹⁵ method, although it takes into account the strengths of indicators, it does not consider the degree of belief in the value of linguistic evaluation, the weight of IAEA experts, and the “missing value in nuclear safeguards evaluation.” In Kabak and Ruan’s²¹ method, despite the degree of belief that the indicator k at $s_{i}$ level is a weighted sum of expert beliefs, the linguistic evaluations are not taken into consideration in the aggregation. Pei et al.²⁶ truly take the missing information and conflict information into consideration, but the uncertainty is still a problem that needed to be solved. In a word, Liu’s method, Kabak’s method, and Pei’s method all have some weaknesses.

Based on the advantages of Dempster–Shafer theory of evidence, our method can handle uncertainty information and missing values more efficiently by assigning the probability to the subsets of the set composed of multiple objects. At the same time, our approach rationally takes into account the information on the indicators and the weighting factors. In the final evaluation part, our result can not only provide the value of $s_{i}$ but also give the value of uncertainty. Thus, our method has a better performance in managing missing values and uncertainties in nuclear safeguards assessments.

Conclusion

Decreasing the effect of uncertainty and missing values is critical for nuclear evaluation. Because nuclear safeguards information may be incomplete and there are various uncertainties in nuclear safeguards assessment, this article deals with these uncertainties and missing values by assigning probabilities to a subset of a set of multiple objects and in this way the uncertainty and missing information will both be taken into the evaluation process. In the proposed method, Dempster–Shafer evidence theory is invoked to express uncertainty and missing values. At the same time, the weighting factors of the indicators and experts who took part in the evaluation process have been made full use of. As a result, our method can handle the missing values and the uncertainty better than other methods, and the limitations of it are much less than other methods. The results prove that it is of great importance to consider uncertainty and missing information.

Footnotes

Acknowledgements

The authors greatly appreciate the reviewers’ suggestions and the editor’s encouragement.

Handling Editor: Zehong Cao

Data availability

The data used to support the findings of this study are included within the article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work is partially supported by the National Natural Science Foundation of China (Grant Nos 61573290 and 61973332).

ORCID iD

Yong Deng

References

Baraldi

Compare

Zio

. Uncertainty treatment in expert information systems for maintenance policy assessment. Appl Soft Comput 2014; 22(22): 297C310.

Zhou

Sun

. Probabilistic safety assessment model in consideration of human factors based on object-oriented Bayesian networks. Nucl Power Eng 2007; 28(3): 107–112.

Baraldi

Compare

Zio

. Maintenance policy performance assessment in presence of imprecision based on Dempster–Shafer theory of evidence. Inform Sciences 2013; 245(24): 112–131.

Liu

Xia

Xie

. Research of nuclear power plants diagnosis method based on data fusion. In: International conference on nuclear engineering, Brussels, 12–16 July 2009. New York: ASME.

Wang

Maio

Zio

. Component- and system-level degradation modeling of digital instrumentation and control systems based on a multi-state physics modeling approach. Ann Nucl Energy 2016; 95: 135–147.

Maio

Colli

Zio

, et al. A multi-state physics modeling approach for the reliability assessment of nuclear power plants piping systems. Ann Nucl Energy 2015; 80: 151–165.

Chaudri

Tian

, et al. Optimization study for thermal efficiency of supercritical water reactor nuclear power plant. Ann Nucl Energy 2014; 63(1): 541–547.

Baraldi

Zio

. A Comparison between probabilistic and Dempster-Shafer theory approaches to model uncertainty analysis in the performance assessment of radioactive waste repositories. Hoboken, NJ: Wiley, 2010.

Liang

Cheng

, et al. Risk-informed analysis of the large break loss of coolant accident and PCT margin evaluation with the RISMC methodology. Nucl Eng Des 2016; 308: 214–221.

10.

Catton

Duffey

Shaw

, et al. Quantifying reactor safety margins part 6: a physically based method of estimating PWR large break loss of coolant accident pct. Nucl Eng Des 1990; 119(1): 109–117.

11.

Croft

Favalli

. Incorporating delayed neutrons into the point-model equations routinely used for neutron coincidence counting in nuclear safeguards. Ann Nucl Energy 2017; 99: 36–39.

12.

Yoo

Lee

Ham

, et al. Methodology for analyzing risk at nuclear facilities. Ann Nucl Energy 2015; 81: 213–218.

13.

Zakariya

Kahn

MTE

. Safety, security and safeguard. Ann Nucl Energy 2015; 75: 292–302.

14.

DLD

Dieulle

Vasseur

, et al. An alternative comprehensive framework using belief functions for parameter and model uncertainty analysis in nuclear probabilistic risk assessment applications. Proc IMechE, Part O: J Risk & Reliability 2013; 227(5): 471–490.

15.

Liu

Ruan

Carchon

. Synthesis and evaluation analysis of the indicator information in nuclear safeguards applications by computing with words. Int J Appl Math Comp 2002; 12(3): 449–449.

16.

Rodríguez

Martinez

Ruan

, et al. Using collaborative filtering for dealing with missing values in nuclear safeguards evaluation. Int J Uncertain Fuzz 2010; 18(04): 431–449.

17.

Jiang

Gruenwald

. Estimating missing data in data streams. In: International conference on database systems for advanced applications, Bangkok, Thailand, 9–12 April 2007, pp.981–987. Berlin: Springer.

18.

Othman

Yahia

. Yet another approach for completing missing values. In: Yahia

Nguifo

Belohlavek

(eds) Concept lattices and their applications. Berlin: Springer, 2008, pp.155–169.

19.

Pawlak

. Kernel classification rules from missing data. IEEE T Inform Theory 1993; 39(3): 979–988.

20.

Kabak

Ruan

. Analyzing environmental samples in nuclear safeguards evaluation by a cumulative belief degrees approach. In: Ruan

, et al. (eds) 9th international FLINS conference on computational intelligence: foundations and applications, vol. 4, Chengdu, China, 2–4 August 2010, pp.797–805. Singapore: World Scientific.

21.

Kabak

Ruan

. A cumulative belief degree-based approach for missing values in nuclear safeguards evaluation. IEEE T Knowl Data En 2011; 23(10): 1441–1454.

22.

Kabak

Ruan

. Dealing with missing values in nuclear safeguards evaluation. In: Proceedings of the 4th international ISKE conference intelligent decision making systems, Hasselt, 27–28 November 2009, pp.145–152. Singapore: World Scientific.

23.

Rodríguez

Ruan

Liu

, et al. Imputing missing values in nuclear safeguards evaluation by a 2-tuple computational model. In: International conference on artificial intelligence and soft computing, Zakopane, 13–17 June 2010. Berlin: Springer, pp.202–209.

24.

Castellano

Martínez

. A web-decision support system based on collaborative filtering for academic orientation. Case study of the Spanish secondary school. J Univers Comput Sci 2009; 15(14): 2786–2807.

25.

Herlocker

Konstan

Terveen

, et al. Evaluating collaborative filtering recommender systems. ACM T Inform Syst 2004; 22(1): 5–53.

26.

Pei

Ruan

Liu

, et al. A linguistic aggregation operator with three kinds of weights for nuclear safeguards evaluation. Knowl-Based Syst 2012; 28: 19–26.

27.

Herrera

Martnez

. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE T Fuzzy Syst 2000; 8(6): 746–752.

28.

Torra

. The weighted OWA operator. Int J Intell Syst 1997; 12(2): 153–166.

29.

Zhang

Deng

. Weighted belief function of sensor data fusion in engine fault diagnosis. Soft Comput. Epub ahead of print 16 may 2019. DOI: 10.1007/s00500-019-04063-7.

30.

Gong

Qian

, et al. Research on fault diagnosis methods for the reactor coolant system of nuclear power plant based on D-S evidence theory. Ann Nucl Energy 2018: 395–399.

31.

Seiti

Hafezalkotob

Najafi

, et al. A risk-based fuzzy evidential framework for FMEA analysis under uncertainty: an interval-valued DS approach. J Intell Fuzzy Syst 2018; 35: 1419–1430.

32.

Song

Jiang

. Engine fault diagnosis based on sensor data fusion using evidence theory. Adv Mech Eng 2016; 8(10): 1–9.

33.

Mahadevan

, et al. Dependence assessment in human reliability analysis using evidence theory and AHP. Risk Anal 2015; 35(7): 1296–1316.

34.

Liu

Pan

Dezert

, et al. Combination of classifiers with optimal weight based on evidential reasoning. IEEE T Fuzzy Syst 2017; PP(99): 1–15.

35.

Zhang

Liu

Zhang

, et al. Collaborative fusion for distributed target classification using evidence theory in IOT environment. IEEE Access 2018; 6: 62314–62323.

36.

Sun

Deng

. A new method to determine generalized basic probability assignment in the open world. IEEE Access 2019; 7(1): 52827–52835.

37.

Liu

Pan

Dezert

, et al. Classifier fusion with contextual reliability evaluation. IEEE T Cybernetics 2017; PP(99): 1–14.

38.

Cui

Liu

Zhang

, et al. An improved Deng entropy and its application in pattern recognition. IEEE Access 2019; 7: 18284–18292.

39.

Abelln

. Analyzing properties of Deng entropy in the theory of evidence. Chaos Soliton Fract 2017; 95: 195–199.

40.

Yang

Han

. A new distance-based total uncertainty measure in the theory of belief functions. Knowl-Based Syst 2016; 94: 114–123.

41.

Jaunzemis

Holzinger

Chan

, et al. Evidence gathering for hypothesis resolution using judicial evidential reasoning. Inform Fusion 2019; 49: 26–45.

42.

Vandoni

Aldea

Le Hégarat-Mascle

. Evidential query-by-committee active learning for pedestrian detection in high-density crowds. Int J Approx Reason 2019; 104: 166–184.

43.

Seiti

Hafezalkotob

. Developing the R-TOPSIS methodology for risk-based preventive maintenance planning: a case study in rolling mill company. Comput Ind Eng 2019; 128: 622–636.

44.

Kang

Zhang

Gao

, et al. Environmental assessment under uncertainty using Dempster–Shafer theory and Z-numbers. J Amb Intel Hum Comp. Epub ahead of print 6 February 2019. DOI: 10.1007/s12652-019-01228-y.

45.

MacDermott

Shi

Kifayat

. Collaborative intrusion detection in a federated cloud environment using the Dempster–Shafer theory of evidence. In: European conference on information warfare and security (ECCWS), vol. 1, University of Hertfordshire, Hatfield, 2–3 July 2015, pp.195–203.

46.

Yager

. On using the Shapley value to approximate the Choquet integral in cases of uncertain arguments. IEEE T Fuzzy Syst 2018; 26(3): 1303–1310.

47.

Xiao

. A hybrid fuzzy soft sets decision making method in medical diagnosis. IEEE Access 2018; 6: 25300–25312.

48.

Yager

. Fuzzy rule bases with generalized belief structure inputs. Eng Appl Artif Intel 2018; 72: 93–98.

49.

Yager

Alajlan

. Maxitive belief structures and imprecise possibility distributions. IEEE T Fuzzy Syst 2017; 25(4): 768–774.

50.

Han

Deng

. A novel matrix game with payoffs of Maxitive Belief Structure. Int J Intell Syst 2019; 34(4): 690–706.

51.

Xiao

. A novel multi-criteria decision making method for assessing health-care waste treatment technologies based on D numbers. Eng Appl Artif Intel 2018; 71(2018): 216–225.

52.

Deng

. An evaluation for sustainable mobility extended by D numbers. Technol Econ Dev Eco 2019; 25(5): 802–819.

53.

Xiao

. A multiple criteria decision-making method based on D numbers and belief entropy. Int J Fuzzy Syst 2019; 21: 1144–1153.

54.

Zhao

Deng

. Performer selection in human reliability analysis: D numbers approach. Int J Comput Commun 2019; 14(3): 437–452.

55.

Liu

Deng

. Risk evaluation in failure mode and effects analysis based on D numbers theory. Int J Comput Commun 2019; 14(5): 672–691.

56.

Yager

. On z-valuations using Zadeh’s Z-numbers. Int J Intell Syst 2012; 27(3): 259–278.

57.

Kang

Deng

Hewage

, et al. A method of measuring uncertainty for Z-number. IEEE T Fuzzy Syst 2019; 27(4): 731–738.

58.

Dempster

. Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 1967; 38(2): 325–339.

59.

Shafer

. A mathematical theory of evidence. Princeton, NJ: Princeton University Press, 1976.

60.

Deng

. Dependent evidence combination based on decision-making trial and evaluation laboratory method. Int J Intell Syst 2019; 34(7): 1555–1571.

61.

Shi

, et al. Research on the fusion of dependent evidence based on mutual information. IEEE Access 2018; 6: 71839–71845.

62.

Deng

Jiang

Wang

. Zero-sum polymatrix games with link uncertainty: a Dempster–Shafer theory solution. Appl Math Comput 2019; 340: 101–112.

63.

Jiang

. An evidential dynamical model to predict the interference effect of categorization on decision making. Knowl-Based Syst 2018; 150: 139–149.

64.

Deng

. Evidential decision tree based on belief entropy. Entropy 2019; 21(9): 897.

65.

Chen

. A novel evidential FMEA method by integrating fuzzy belief structure and grey relational projection method. Eng Appl Artif Intel 2019; 77: 136–147.

66.

Cao

Deng

. A new geometric mean FMEA method based on information quality. IEEE Access 2019; 7(1): 95547–95554.

67.

Fei

Deng

. A new divergence measure for basic probability assignment and its applications in extremely uncertain environments. Int J Intell Syst 2019; 34(4): 584–600.

68.

Xiao

. Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Inform Fusion 2019; 46(2019): 23–32.

69.

Song

Deng

. A new method to measure the divergence in evidential sensor data fusion. Int J Distrib Sens N 2019; 15: 1–8.

70.

Gao

Deng

. The negation of basic probability assignment. IEEE Access 2019; 7(1): 107006–107014.

71.

Luo

Deng

. A matrix method of basic belief assignment’s negation in Dempster–Shafer theory. IEEE T Fuzzy Syst. Epub ahead of print 22 July 2019. DOI: 10.1109/TFUZZ.2019.2930027.

72.

Murphy

. Combining belief functions when evidence conflicts. Decis Support Syst 2000; 29(1): 1–9.

73.

Sun

Deng

. A new method to identify incomplete frame of discernment in evidence theory. IEEE Access 2019; 7(1): 15547–15555.

74.

Wang

Qiao

Zhang

, et al. A new conflict management method in Dempster–Shafer theory. Int J Distrib Sens N 2017; 13(3). 1550147717696506

75.

Dong

Zhang

, et al. Combination of evidential sensor reports with distance function and belief entropy in fault diagnosis. Int J Comput Commun 2019; 14(3): 329–343.

76.

Wang

Zhang

Deng

. Base belief function: an efficient method of conflict management. J Amb Intel Hum Comp 2019; 10(9): 3427–3437.

77.

Smets

Kennes

. The transferable belief model. Artif Intell 1994; 66(2): 191–234.

78.

Cobb

Shenoy

. On the plausibility transformation method for translating belief function models to probability models. Int J Approx Reason 2006; 41(3): 314–330.

79.

Yang

. On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty. New York: IEEE Press, 2002.

80.

Yang

. Nonlinear information aggregation via evidential reasoning in multiattribute decision analysis under uncertainty. IEEE T Syst Man Cy A 2002; 32(3): 376–393.