A novel distance function of D numbers and its application in product engineering

Highlights

• A distance function between two D numbers is proposed and applied in product engineering.

• The function is effective when elements in the frame of discernment aren׳t mutually exclusive.

Abstract

The Dempster–Shafer theory is widely applied in uncertainty modelling and knowledge reasoning due to its ability of expressing uncertain information. A distance between two basic probability assignments (BPAs) presents a measure of performance for identification of algorithms based on the evidential theory of Dempster–Shafer. However, some conditions lead to limitations in practical application for the Dempster–Shafer theory, such as exclusiveness hypothesis and completeness constraint. To overcome these shortcomings, a novel theory called D numbers theory is proposed. A distance function of D numbers is proposed to measure the distance between two D numbers. The distance function of D numbers is a generalization of distance between two BPAs, which inherits the advantage of Dempster–Shafer theory and strengthens the capability of uncertainty modeling. An illustrative case about product engineering is provided to demonstrate the effectiveness of the proposed function.

Introduction

The Dempster–Shafer theory of evidence (Dempster, 1967, Shafer, 1976), also called the Dempster–Shafer theory or an evidence theory, is used to deal with uncertain information. This theory needs weaker conditions than bayesian theory of probability, so it is often regarded as an extension of the bayesian theory (Dempster, 1967). As an effective theory of evidential reasoning, the Dempster–Shafer theory has an advantage of directly expressing various uncertainties, so it has been widely used in many fields (Cuzzolin, 2008, Hu et al., 2013, Liu et al., 2013). Due to improvement of the Dempster–Shafer theory of evidence, many studies have been devoted for combination rule of evidence (Yager, 1996, Lefévre and Elouedi, 2013), confliction problem (Lefevre et al., 2002, Yang and Xu, 2013, Deng, 2015), generation of mass function (Bastian et al., 2010, Liu et al., 2013, Liu et al., 2014, Jiang et al., 2015), uncertain measure of evidence (Bachmann et al., 2010, Baker et al., 2012, Hu et al., 2013), decision making (Utkin, 2009, Deng et al., 2011, Fu and Yang, 2012, Fu and Chin, 2014) and so on (Karahan and Ozkan, 2013, Li et al., 2014, Su et al., 2015a, Su et al., 2015b).

Though the Dempster–Shafer theory has an advantage of directly expressing the “uncertainty” by assigning the probability to the subsets of the set composed of multiple objects, rather than to each of the individual objects. However, the mathematical framework of Dempster–Shafer theory is based on some strong hypotheses regarding the frame of discernment and basic probability assignment, which limit the ability of Dempster–Shafer theory to represent information in other situations. One of the hypotheses is that the elements in the frame of discernment are required to be mutually exclusive. In many situations, this hypothesis is difficult to be satisfied. For example, linguistic assessments shown as “Very Good”, “Good”, “Fair”,“Bad”, and “Very Bad”. Due to these assessments is based on human judgment, they inevitably contain intersections (Deng et al., 2011, Jiang et al., 2015). The exclusiveness between these propositions cannot be guaranteed precisely, so that the Dempster–Shafer theory is not reasonable for this situation. To overcome the existing shortcomings in the Dempster–Shafer theory, a new representation of uncertain information is proposed, which is called D numbers (Deng, 2012) and is widely used in many applications such as environmental impact assessment (Deng et al., 2014, Rikhtegar et al., 2014), failure analysis (Liu et al., 2014) and supply chain management (Deng et al., 2014).

Due to the presence of the measurement of performance for identification of algorithms based on the Dempster–Shafer theory, the concept of distance between BPAs has been proposed before (Jousselme et al., 2001, Jousselme and Maupin, 2012). In order to express the distance between two D numbers, a distance function of D numbers is proposed in this paper. The proposed distance function of D numbers is an extension for the distance function between two BPAs, which is proposed by Anne-Laure Jousselme (Jousselme et al., 2001). In the distance function of D numbers, the frame of discernment is not required to be mutually exclusive. In the situation that the discernment is mutually exclusive, the proposed distance function of D numbers is degenerated as the distance function between two BPAs.

Product engineering is a hot research field, many algorithms have been proposed to solve this problem (Palade et al., 2002, Ngai et al., 2011, Chan et al., 2011, Yuen, 2014, Li et al., 2014). In this paper, the proposed distance function between two D numbers is used in the product quality evaluation. If the distance between test data and standard data is greater than the threshold t, the product is not qualified.

The rest of this paper is organized as follows. Section 2 introduces some basic concepts about the Dempster–Shafer theory and the distance between two BPAs. In Section 3 the proposed distance function of D numbers is presented. Some examples are given to compare the differences between the distance of BPAs and the distance of the D numbers. Section 4 uses an illustrative case to show the application of the proposed function in product engineering. Conclusion is given in Section 5.