Mathematical Problems in Engineering

Volume 2018, Article ID 6025680, 10 pages

https://doi.org/10.1155/2018/6025680

## The Improved Combination Rule of D Numbers and Its Application in Radiation Source Identification

Department of Electronic and Information Engineering, Naval Aviation University, Yantai 264001, China

Correspondence should be addressed to Haiqiao Liu; moc.qq@1567217532

Received 17 April 2018; Revised 1 July 2018; Accepted 12 July 2018; Published 9 September 2018

Academic Editor: Ayed A. Salman

Copyright © 2018 Xin Guan 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.

#### Abstract

The D numbers theory is a novel theory to express uncertain information. It successfully overcomes some shortcomings of Dempster-Shafer theory, such as the conditions of exclusiveness hypothesis and completeness constraint. However, the combination rule of D numbers does not satisfy the associative property, which leads to limitations in practical application for D numbers. In this paper, the improved D numbers theory is proposed to overcome the weakness based on the analysis of D numbers’ combination rule. A new algorithm is constructed with the strict proof to simplify the combination rule. The similarities and differences among DS theory, D numbers, and the improved D numbers are introduced with the numerical analysis. An illustrative example of the radiation source identification is presented to demonstrate the effectiveness of the improved method.

#### 1. Introduction

In real applications, uncertainty reasoning is widely applied to information fusion, risk assessment, pattern recognition, artificial intelligence, decision-making, and so forth. Many methods such as fuzzy sets theory, Dempster-Shafer theory, and possibility theory, have been proposed to model uncertain information.

Dempster [1] proposed the upper and lower probabilities that are induced by a multivalued mapping in the 1960s. His student Shafer introduced the concept of belief function and published the book [2], which is the sign of the formation of evidence theory. During the application process of evidence theory, many related problems have been studied in depth, such as the acquisition of basic probability assignments [3–6], information fusion of reliable sources [7, 8], conflict representation model [9, 10], and decision rule [11, 12]. Compared with the traditional Bayesian theory, DS theory, also called an evidence theory, needs weaker conditions and handles uncertain and incomplete information effectively, so it is often regarded as an extension of Bayesian theory. However, DS theory still has some shortcomings. When evidences acquired from different sources are highly conflicting, DS combination rule is defective in resolving evidence combination problem [13]. In addition, there are some strong hypotheses on the frame of discernment. For example, the elements in the frame of discernment require mutual exclusiveness and the frame must be complete. To handle the existing shortcomings in the DS theory, a new theory is proposed by Deng [14], which is called D numbers.

The DS evidence theory combination rule cannot effectively deal with high-conflict information [9, 15]. A typical example is the zero trust paradox case proposed by Zadeh [13]; that is, 0 has one vote veto. For this problem, a large number of improved algorithms [16–21] made by relevant scholars are mainly divided into two categories: one is the improvement of combination rules and the other is the improvement of data sources [22–32]. The improvement of the combination rules is mainly divided into three categories. One is to recognize the multiplicative principle of the DS evidence theory and to study how to allocate the amount of conflict, such as Smets [25, 26], Yager [27, 28], and PCR1-6 [29–31]. The second type is the multiplicative rule of DS evidence theory, which gives additive combination rules, such as Murphy combination rules [32]. The third category is to change the recognition framework and extend to the generalized power set. On this basis, new combination rules are given, such as the DSmT combination rule [31], which extends the recognition framework to the power set framework. The D number theory is a generalization of evidence theory. It cancels the assumption that the propositions in the recognition framework are mutually independent. Related scholars have done lots of research on it. Li [33] proposed a novel distance function of D numbers. Deng [34] introduced the difference between D numbers and DS theory based on generalized evidence theory.

On the application side, Zuo [35] analyzed investment decision, Deng [36] analyzed bridge condition assessment, and Deng [37] analyzed environmental impact assessment. Meanwhile, some methods were modified or extended by D numbers and have been applied to engineering field. Fei and Bian [38, 39] analyzed human resources selection and failure mode based on D numbers and TOPSIS. Su, Deng, and Fan [40–42] proposed the dependence assessment in human reliability analysis, supplier selection, and curtain grouting efficiency assessment based on D numbers and AHP. Deng and Zhou [43, 44] proposed the D-CFPR and D-DEMATEL theory based on D numbers. Liao [45] introduced transformer condition assessment using game theory and modified evidence combination extended by D numbers. Wang [46] introduced a multicriteria decision-making method based on fuzzy entropy and evidential reasoning with linguistic D numbers.

Though D numbers theory overcomes some defects of DS theory and is widely used in different fields, the combination rule of D numbers does not satisfy the associative property. In order that multiple D numbers can be combined correctly and efficiently, a combination operation for multiple D numbers is developed in [37]. But it does not change the fact that D numbers’ combination rule does not preserve the associative property. In this paper, the improved D numbers theory is proposed based on the analysis of D numbers’ combination rule which satisfies the associative property. An illustrative example about radiation source identification is given to show the effectiveness of the proposed method.

The remainder of this paper is organized as follows. A brief introduction about the DS theory and D numbers is given in Section 2. Then, the concept of improved D numbers is depicted in Section 3. In Section 4, the similarities and differences among DS theory, D numbers, and the improved D numbers are represented. After that, an illustrative example is given to show the effectiveness of the proposed method in Section 5. Finally, conclusions are given in Section 6.

#### 2. Materials and Methods Preliminaries

##### 2.1. DS Theory

Let be a set of exhaustive and exclusive hypotheses, satisfying,The power set of is indicated by , and each element of is regarded as a proposition. It can be defined asBased on the two conceptions, mass function is defined as below.

*Definition 1. *A mass function is a mapping from to , formally defined bywhich should satisfy the condition:A mass function is also called a basic probability assignment (BPA), which measures the support degree of the proposition .

In real applications, for the same problem, there may be many different sources that acquire various evidences.

*Definition 2. *Considering two pieces of evidence from different and independent information sources, denoted by two BPAs and , the combination rule of DS theory is used to derive a new BPA from two BPAs, which is represented by , and defined as follows:withwhere is the conflict coefficient of two BPAs. Note that the combination rule of DS theory is only applicable to such two BPAs which satisfy the condition .

*Definition 2*′*.* Multiple pieces of evidence from different information sources also can be combined with the following formula. The result is a new piece of evidence, which incorporates the joint information acquired from various sources.withThe factor indicates the amount of evidential conflict. If , it shows that the sources are completely compatible. If , it shows that the sources are completely contradictory, and it is no longer possible to combine them.

##### 2.2. D Numbers

D numbers theory is more suitable to the framework and is defined as follows.

*Definition 3. *Let be a finite nonempty set, and number is a mapping , withThe structure of the expression seems to be similar to mass function. However, the elements of do not require to be mutually exclusive, and D numbers theory is suitable to incomplete information.

Let , and a special form of D numbers can be defined as follows.or simply denoted aswhere and

Like the mass function, D numbers theory also has the combination rule to combine two D numbers.

*Definition 4. *Let be two D numbers, indicated byThe combination of , indicated by , is defined bywithwhereThe combination operation does not preserve the associative property. Based on formula (15), it is clear that, for ,while, for , As a result, a combination operation for multiple D numbers should be developed. Paper [17] represents a multiple D numbers’ rule of combination.

*Definition 5. *Let be numbers, is an order variable for each D number , indicated by tuple , and then the combination operation of multiple numbers is a mapping where is of the tuple which have the lowest .

In the meanwhile, for the special D number, an aggregation operator is defined as follows.

*Definition 6. *Let = , be a D number, and the integration representation of the D number is defined as

#### 3. An Improved Combination Rule of D Numbers

As a novel theory to express uncertain information, D numbers’ combination rule does not preserve the associative property. For n D numbers, there are outcomes. Formulas (17) and (18) can be extended to multiple D numbers, and the outcome can be indicated bywhere is the parameter of the D numbers. From formula (21), it can be seen that the weight of the first combination number b is lower, which is . And the weight of the latter combination number b is higher, which is . The analysis shows that the weight is normalized, namely,The main reason for the imbalance of weight is that formula (15) involves addition and multiplication, which do not satisfy the associative property. In the meanwhile, the combination rule uses serial method to combine multiple numbers, as shown in Figure 1.