An Evidential Axiomatic Design Approach for Decision Making Using the Evaluation of Belief Structure Satisfaction to Uncertain Target Values

Axiomatic design (AD) provides a general theory for system and product development. In recent years, the principles of AD have been successfully applied to the decision‐making field, and derived a fuzzy AD approach for fuzzy decision‐making environment. In this work, the interest is paid on the theoretical developments and applications of AD in the uncertain environment expressed by Dempster–Shafer evidence theory. Based on the concept of belief structure satisfaction to uncertain target values, an evidential AD approach is proposed for decision making by combining the independence axiom and information axiom of AD with the framework of Dempster–Shafer theory. An illustrative example has demonstrated the effectiveness of the proposed approach. This work, on the one hand, has successfully generalized the principles of AD to the Dempster–Shafer uncertain environment; on the other hand, it has presented a successful application of the concept of belief structure satisfaction.     

A class of fuzzy measures generated from a Dempster–Shafer belief structure

Here the Dempster–Shafer belief structure is viewed as providing partial information about the underlying fuzzy measure associated with a uncertain variable. In this perspective there exists many possible fuzzy measures that can be associated with a Dempster–Shafer belief structure. Typically only two of these measures have been made explicit, those being the measure of belief and plausibility. Here we introduce a whole class of fuzzy measures that can be associated with a Dempster–Shafer belief structure. As an aid to choosing between these myriad of possibilities we discuss the entropy of a fuzzy measure. ©1999 John Wiley & Sons, Inc.     

Fuzzy Dempster–Shafer reasoning for rule-based classifiers

In real classification problems intrinsically vague information often coexist with conditions of “lack of specificity” originating from evidence not strong enough to induce knowledge, but only degrees of belief or credibility regarding class assignments. The problem has been addressed here by proposing a fuzzy Dempster–Shafer model (FDS) for multisource classification purposes. The salient aspect of the work is the definition of an empirical learning strategy for the automatic generation of fuzzy Dempster–Shafer classification rules from a set of exemplified training data. Dempster–Shafer measures of uncertainty are semantically related to conditions of ambiguity among the data and then automatically set during the learning process. Partial reduced beliefs in class assignments are then induced and explicitly represented when generating classification rules. The fuzzy deductive apparatus has been modified and extended to integrate the Dempster–Shafer propagation of evidence. The strategy has been applied to a standard classification problem in order to develop a sensitivity analysis in an easily controlled domain. A second experimental test has been conducted in the field of natural risk assessment, where vagueness and lack of specificity conditions are prevalent. These empirical tests show that classification benefits from the combination of the fuzzy and Dempster–Shafer models especially when conditions of lack of specifity among data are prevalent. ©1999 John Wiley & Sons, Inc.     

An intuitionistic view of the Dempster–Shafer belief structure

We discuss the Dempster–Shafer belief theory and describe its role in representing imprecise probabilistic information. In particular, we note its use of intervals for representing imprecise probabilities. We note in fuzzy set theory that there are two related approaches used for representing imprecise membership grades: interval-valued fuzzy sets and intuitionistic fuzzy sets. We indicate the first of these, interval-valued fuzzy sets, is in the same spirit as Dempster–Shafer representation, both use intervals. Using a relationship analogous to the type of relationship that exists between interval-valued fuzzy sets and intuitionistic fuzzy sets, we obtain from the interval-valued view of the Dempster–Shafer model an intuitionistic view of the Dempster–Shafer model. Central to this view is the use of an intuitionistic statement, pair of values, (Bel(A) Dis(A)), to convey information about the value of a variable lying in the set A. We suggest methods for combining intuitionistic statements and making inferences from these type propositions.     

Entailment for measure based belief structures

We discuss the Dempster–Shafer belief structure on finite universes and note its use for modeling variables that have both probabilistic uncertainty as well as imprecision. We note for these structures the probability that the variable lies in a subset cannot be precisely known but only be known to an interval value. We discuss methods for deducing this uncertainty interval. We next discuss the issue of entailment of belief structures, inferring the validity of additional belief model of a variable from an already established belief model of the variable. We next discuss a more general belief structure were the underling uncertainty rather tha0n being based on a probability distribution is based on a general measure type of uncertainty. We then extend the concept of entailment to the case where the belief structures are these more general measure based belief structures. In order to accomplish this we must extend the idea of containment from classic Dempster–Shafer belief structures to measure based belief structures.     

An interpretation of intuitionistic fuzzy sets in terms of evidence theory: Decision making aspect

This paper presents a new interpretation of intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of evidence (DST). This interpretation makes it possible to represent all mathematical operations on intuitionistic fuzzy values as the operations on belief intervals. Such approach allows us to use directly the Dempster’s rule of combination to aggregate local criteria presented by intuitionistic fuzzy values in the decision making problem. The usefulness of the developed method is illustrated with the known example of multiple criteria decision making problem. The proposed approach and a new method for interval comparison based on DST, allow us to solve multiple criteria decision making problem without intermediate defuzzification when not only criteria, but their weights are intuitionistic fuzzy values.     

时态Dempster—Shafer理论

1 引言不确定性推理是人工智能中一个重要研究方向。在不同的应用领域,基于不同的不确定性度量理论,提出了许多不确定性推理理论和方法,例如确定因子法、概率推理、模糊推理和Dempster-Sharer理论等。Dempster-Shafer理论是Shafer在Dempster提出的概率区间度量理论的基础上进一步发展的不确定性推理理论。与概率推理等相比,Demp     

Dempster–Shafer structure based fuzzy logic system for stochastic modeling

The Dempster–Shafer (D–S) theory of evidence is introduced to improve fuzzy inference under the complex stochastic environment. The Dempster–Shafer based fuzzy set (DFS) is first proposed, together with its union and intersection operations, to capture the principal stochastic uncertainties. Then, the fuzzy inference will be modified based on the extensional Dempster rule of combination. This new approach is able to capture the stochastic disturbance acting on fuzzy membership function, and provide a more effective inference under strong stochastic uncertainty. Finally, the numerical simulation and the experimental prediction of the wind speed are conducted to show the potential of the proposed method in stochastic modeling.     

On combination rule in Dempster–Shafer theory using OWA-based soft likelihood functions and its applications in environmental impact assessment

Dempster–Shafer theory (DST) was presented as an effective mathematical tool to represent uncertainty. Its significant innovation is to allow the allocation of the belief of mass to sets or intervals, and it becomes a valuable method in the field of decision making and evaluation when accurate information is not available or when knowledge is expressed subjectively by humans. A crucial research issue in DST is the combination of multi-sources of evidence. In this paper, a novel combination rule for Dempster–Shafer structures is developed based on ordered weighted average (OWA)-based soft likelihood functions proposed by Yager. First, the belief intervals, including the belief measures and plausibility measures, of all the hypotheses in the frame of discernment (FOD) are calculated. Second, the representative value of belief interval is defined based on golden rule introduced by Yager. Third, the soft likelihood value of each hypothesis is calculated based on the proposed OWA-based soft likelihood function for belief interval, which can be considered as the combined evidence. The final evaluation results can be employed for practical applications, such as decision making and evaluation. In addition, the improved evidence combination rule is presented which takes into account the weight of evidence. Several illustrative examples are conducted to manifest the use of the developed methods. Finally, an application for environmental impact assessment is given to demonstrate the usefulness of the developed combination rule in DST.     

Belief and Plausibility Functions on Intuitionistic Fuzzy Sets

Belief and plausibility functions based on Dempster–Shafer theory have been used to measure uncertainty. They are also widely studied and applied in diverse areas. Numerous studies in the literature have presented various generalizations of belief and plausibility functions to fuzzy sets. However, there are still less generalizations of belief and plausibility functions to intuitionistic fuzzy sets. Because intuitionistic fuzzy sets can present the degrees of both membership and nonmembership with a degree of hesitancy, the knowledge and semantic representation becomes more general and applicable than fuzzy sets. In this paper, we propose a generalization of belief and plausibility functions to intuitionistic fuzzy sets based on fuzzy integral. Some numerical examples show the effectiveness of the proposed generalization. Furthermore, this generalization of belief and plausibility functions to intuitionistic fuzzy sets is able to catch more information about the change of intuitionistic fuzzy focal elements.     

Decision-based fuzzy image restoration for noise reduction based on evidence theory

A novel decision-based fuzzy averaging (DFA) filter consisting of a D–S (Dempster–Shafer) noise detector and a two-pass noise filtering mechanism is presented in this paper. The proposed filter can effectively deal with impulsive noise, and a mix of Gaussian and impulsive noise. Bodies of evidence are extracted, and the basic belief assignment is developed using the simple support function, which avoids the counter-intuitive problem of Dempster’s combination rule. The combination belief value is the decision rule for the D–S noise detector. A fuzzy averaging method, where the weights are constructed using a predefined fuzzy set, is developed to achieve noise cancellation. A simple second-pass filter is employed to improve the final filtering performance. Experimental results confirm the effectiveness of the new DFA filter both in suppressing impulsive noise as well as a mix Gaussian and impulsive noise and in improving perceived image quality.     

广义证据推理融合结构

针对Dempster Shafer理论（DST）及Dezert Smarandache理论（DSmT）难以处理不确定信息的问题,定义了辨识框架中的不确定因子,提出了2种自适应通用分配法则（AUPR）.并提出了证据理论的广义融合框架,并在此基础上构建了广义证据推理机.以Pioneer 2 DXe机器人为实验平台,绘制了实验场景的信度分布图.实验结果验证了所提方法的有效性和实用性,为构建统一的信息融合框架提供了有力的依据.     

Generalization of belief and plausibility functions to fuzzy sets based on the sugeno integral

Uncertainty has been treated in science for several decades. It always exists in real systems. Probability has been traditionally used in modeling uncertainty. Belief and plausibility functions based on the Dempster–Shafer theory (DST) become another method of measuring uncertainty, as they have been widely studied and applied in diverse areas. Conversely, a fuzzy set has been successfully used as the idea of partial memberships of multiple classes for the presentation of unsharp boundaries. It is well used as the representation of human knowledge in complex systems. Nowadays, there exist several generalizations of belief and plausibility functions to fuzzy sets in the literature. In this article, we propose a new generalization of belief and plausibility functions to fuzzy sets based on the Sugeno integral. We then make comparisons of the proposed generalization with some existing methods. The results show the effectiveness of the proposed generalization, especially for being able to catch more information about the change of fuzzy focal elements. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1215–1228, 2007.     

基于扩展原理的混合型证据推理不确定决策方法

提出一种基于扩展原理的混合证据推理不确定决策模型.通过α截集将同一决策问题中各属性使用的精确数、区间数和模糊数等异构评估信度统一分解为区间结构,采用区间证据推理方法求解各隶属度下的效用区间,并按隶属度次序重组方案效用;化简模糊数质心公式,并用于模糊定量评估的信度计算和方案模糊效用的排序;最后,通过具体实例验证了所提出方法的有效性和可行性.将该方法在算例中的适用情况进行比较和分析,结果表明所提出的方法具有良好的适应性.     

Generalized belief function,plausibility function,and Dempster's combinational rule to fuzzy sets

Uncertainty always exists in nature and real systems. It is known that probability has been used traditionally in modeling uncertainty. Since a belief function was proposed as an another type of measuring uncertainty, Dempster‐Shafer theory (DST) has been widely studied and applied in diverse areas. Because of the advent of computer technology, the representation of human knowledge can be processed by a computer in complex systems. The analysis of fuzzy data becomes increasingly important. Up to date, there are several generalizations of DST to fuzzy sets proposed in the literature. In this article, we propose another generalization of belief function, plausibility function, and Dempster's combinational rule to fuzzy sets. We then make the comparisons of the proposed extension with some existing generalizations and show its effectiveness. © 2003 Wiley Periodicals, Inc.     

Evidence updating based on novel Jeffrey-like conditioning rules

Dempster–Shafer evidence theory (DST) is an important tool for uncertainty modeling and reasoning, where the uncertainty reasoning includes both the evidence combination and conditioning. New conditioning rules for evidence updating in DST are proposed in this paper. First, two new definitions, namely the weak conditional basic belief assignment (BBA) and strong conditional BBA, are proposed in the spirit of conditional probability in the probabilistic framework. Then, the corresponding Jeffrey-like conditioning rules are proposed to update evidence. The proposed methods have some desirable properties for the evidence updating. Some numerical examples are provided, where existing conditioning rules in DST are compared with newly proposed methods. Experimental results and related analyses show that the conditional BBAs and Jeffrey-like rules proposed in this paper are rational and effective.     

Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy

Multi-sensor data fusion technology plays an important role in real applications. Because of the flexibility and effectiveness in modeling and processing the uncertain information regardless of prior probabilities, Dempster–Shafer evidence theory is widely applied in a variety of fields of information fusion. However, counter-intuitive results may come out when fusing the highly conflicting evidences. In order to deal with this problem, a novel method for multi-sensor data fusion based on a new belief divergence measure of evidences and the belief entropy was proposed. First, a new Belief Jensen–Shannon divergence is devised to measure the discrepancy and conflict degree between the evidences; then, the credibility degree can be obtained to represent the reliability of the evidences. Next, considering the uncertainties of the evidences, the information volume of the evidences are measured by making use of the belief entropy to indicate the relative importance of the evidences. Afterwards, the credibility degree of each evidence is modified by taking advantage of the quantitative information volume which will be utilized to obtain an appropriate weight in terms of each evidence. Ultimately, the final weights of the evidences are applied to adjust the bodies of the evidences before using the Dempster’s combination rule. A numerical example is illustrated that the proposed method is feasible and effective in handling the conflicting evidences, where the belief value of target increases to 99.05%. Furthermore, an application in fault diagnosis is given to demonstrate the validity of the proposed method. The results show that the proposed method outperforms other related methods where the basic belief assignment (BBA) of the true target is 89.73%.     

Knowledge discovery in distributed databases using evidence theory

Distributed databases allow us to integrate data from different sources which have not previously been combined. The Dempster–Shafer theory of evidence and evidential reasoning are particularly suited to the integration of distributed databases. Evidential functions are suited to represent evidence from different sources. Evidential reasoning is carried out by the well‐known orthogonal sum. Previous work has defined linguistic summaries to discover knowledge by using fuzzy set theory and using evidence theory to define summaries. In this paper we study linguistic summaries and their applications to knowledge discovery in distributed databases. © 2000 John Wiley & Sons, Inc.     

Induced generalized intuitionistic fuzzy operators

We study the induced generalized aggregation operators under intuitionistic fuzzy environments. Choquet integral and Dempster–Shafer theory of evidence are applied to aggregate inuitionistic fuzzy information and some new types of aggregation operators are developed, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Dempster–Shafer operators. Then we investigate their various properties and some of their special cases. Additionally, we apply the developed operators to financial decision making under intuitionistic fuzzy environments. Some extensions in interval-valued intuitionistic fuzzy situations are also pointed out.     

Combining uncertainty and imprecision in models of medical diagnosis

The paper presents a unified fuzzy-probabilistic framework for modeling processes of medical diagnosis. The two basic concepts of the Dempster–Shafer theory, i.e. focal elements and a basic probability assignment, correspond to disease symptoms and the significance of an individual symptom in the diagnosis, respectively. The belief computation is related to diagnostic inference. The final conclusion of the inference is the diagnosis with the greatest belief value. Fuzzy sets are used to describe focal elements. It is shown how their membership functions and basic probability assignments are estimated on the basis of experimental data. The interpretation of focal elements as fuzzy sets along with individual consideration of evidence imprecision and uncertainty of diagnosis are the essential new aspects of the presented method. Experimental studies have demonstrated the superiority of the proposed approach over some other modeling alternatives.