Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3596517, 10 pages

http://dx.doi.org/10.1155/2016/3596517

## A Modified Combination Rule for Numbers Theory

^{1}School of Science, Hubei University for Nationalities, Enshi, Hubei 445000, China^{2}School of Engineering, Vanderbilt University, Nashville, TN 37235, USA

Received 30 April 2016; Accepted 3 October 2016

Academic Editor: Peide Liu

Copyright © 2016 Ningkui Wang 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

numbers theory is an appropriate method to deal with the information of uncertainty and incompleteness when making a reasonable decision. Previous numbers theory provides a rule to combine multiple numbers. However, the commutative law is not satisfied in the rule of combining multiple numbers. In this paper, a modified method for multiple numbers combination is proposed. The proposed method defines a new function for multiple numbers combination which is mainly determined by the original value of numbers. Then the proposed combination rule is applied to environmental impact assessment (EIA); our results show that the proposed method is efficient for multiple numbers combination and it is useful when dealing with uncertainty and incompleteness.

#### 1. Introduction

In the real world, it is difficult but necessary to make a comprehensive assessment to make a reasonable decision because much uncertainty and incompleteness are often involved in the assessments [1–4]. Several methods, such as probability theory [5, 6], fuzzy theory [7–13], rough set [14–17], uncertainty theory [18–20], and Dempster-Shafer theory of evidence (DST) [21–23], are widely used to deal with these problems. These methods have widely been used in kinds of fields, like supplier selection [24, 25], risk assessments [26–29], and so on [30, 31].

The DST needs weaker conditions than the Bayesian theory of probability, it is often regarded as an extension of the Bayesian theory [32, 33]. For the frame of discernment, which consists of mutually exclusive and collective elements, the basic probability assignment (BPA) can distribute confident degree to the power set of the frame of discernment. Furthermore, an overall assessment can be obtained by combining pairs of BPAs in the DST. Therefore, the DST has been widely applied to multiple criteria decision-making [34–44]. However, some strong hypotheses obviously exist in the DST because of the definitions of the frame of discernment and the BPA. Firstly, the elements in the frame of discernment require being mutually exclusive, but it is hard to be satisfied in the real life especially in linguistic assessments, such as the evaluation on the subjects; “good” and “very good” are two common linguistic evaluations, but they are not completely mutually exclusive so that the DST is unable to handle them. At the same time, the sum of all the BPAs must be equal to 1. However, lacking of some professional knowledge and inadequacy judgements may lead to incompleteness everywhere in the real word. These shortcomings have limited its usage in some fields [45, 46].

Regarded as the generalization of DST, numbers theory is proposed by Deng [47, 48]. It removes these hypotheses reasonably; the elements in the framework of numbers theory do not need to be mutually exclusive and incomplete assessments can also exist in numbers theory. Because numbers theory has the ability to deal with uncertainty and incompleteness, it has been used in EIA [48], failure mode and effects analysis [49], supplier selection [50], and curtain grouting efficiency assessment [51]. Nevertheless, associative property is not satisfied in the previous numbers’ combination rule. In [48], Deng et al. do some work for multiple numbers combination in special circumstances. However, the associative property is not addressed in a general condition. In this paper, a modified method for multiple numbers combination is proposed.

The remainder of this paper is organized as follows. In Section 2, preliminaries about DST and numbers theory are described in detail. The problem of the previous numbers combination rule and the proposed method is shown in Section 3. An illustrative numerical example is presented in Section 4. Conclusions are given in Section 5.

#### 2. Problem Statement and Preliminaries

##### 2.1. Dempster-Shafer Theory

DST is proposed by Dempster and Shafer; some basic concepts are introduced as follows [21, 22].

*Definition 1. *Establish that is a set of mutually exclusive and collectively exhaustive elements which can be represented as follows:The power set of is denoted as ; any element belongs to the power set is said to be a proposition. For a frame of discernment , a mass function is a mapping, which is denoted as follows:in which the following conditions are satisfied:where is an empty set and is a subset of ; the function represents how strongly the evidence supports .

*Definition 2 (Dempster’s rule of combination). *Given two BPAs and , Dempster’s rule of combination donated as is defined as follows:withwhere , , and are the elements of and is a normalization constant which means the conflict coefficient of two BPAs.

Note that Dempster’s rule of combination is feasible only when because means that the two BPAs are one hundred percent conflicted. Associative property is well satisfied in Dempster’s rule of combination.

##### 2.2. Numbers Theory

There are some strong hypotheses in DST which have limited its wide usage in some fields especially in linguistic assessments. numbers theory is proposed in [47, 48] and it has overcome these hypotheses. The details about numbers theory are introduced as follows.

*Definition 3. *Let be a finite nonempty set; numbers is a mapping:where the following conditions are satisfied:where is an empty set and is a subset of . The elements in the set of numbers do not require mutual exclusiveness and the sum of the assessments can be less than 1 in numbers theory.

Suppose that five linguistic assessments “extremely poor (EP),” “poor (P),” “average (A),” “good (G),” and “very good (VG)” are used for the evaluation of a car. The framework of DST must be mutually exclusive and numbers theory providing the framework with nonexclusive hypotheses is more tallying with the actual situation. The differences of their framework of DST and numbers are shown in Figure 1 [48]. In (7), numbers theory is acceptable for incomplete information since which is more close to the real situation.