Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis

A new divergence measure for belief functions in D–S evidence theory for multisensor data fusion

Dempster–Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster–Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster–Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster’s combination rule is exploited. In contrast to the classical Dempster’s combination rule, the condition in terms of the conflict coefficient between the evidences is released in the generalized Dempster’s combination rule. Hence, it is more general and applicable than the classical Dempster’s combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences is less than 1. In a word, this generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Thanks to this advantage, an algorithm for decision-making is devised based on the generalized Dempster–Shafer evidence theory. Finally, an application in a medical diagnosis illustrates the efficiency and practicability of the proposed algorithm.

Generalization of Dempster–Shafer theory: A complex mass function

Pythagorean fuzzy set, initially extended by Yager from intuitionistic fuzzy set, is capable of modeling information with more uncertainties in the process of multi-criteria decision making (MCDM), thus can be used on wider range of conditions. The fuzzy decision analysis of this paper is mainly built upon two expressions in Pythagorean fuzzy environment, named Pythagorean fuzzy number (PFN) and interval-valued Pythagorean fuzzy number (IVPFN), respectively. We initiate a novel axiomatic definition of Pythagorean fuzzy distance measurement, including PFNs and IVPFNs. After that, corresponding theorems are put forward and then proved. Based on the defined distance measurements, the closeness indexes are developed for both expressions, inspired by the idea of technique for order preference by similarity to ideal solution (TOPSIS) approach. After these basic definitions have been established, the hierarchical decision approach is presented to handle MCDM problems under Pythagorean fuzzy environment. To address hierarchical decision issues, the closeness index-based score function is defined to calculate the score of each permutation for the optimal alternative. To determine criterion weights, a new method based on the proposed similarity measure and aggregation operator of PFNs and IVPFNs is presented according to Pythagorean fuzzy information from decision matrix, rather than being provided in advance by decision makers, which can effectively reduce human subjectivity. An experimental case is then conducted to demonstrate the applicability and flexibility of the proposed decision approach. Finally, extension forms of Pythagorean fuzzy decision approach for heterogeneous information are briefly introduced to show its potentials on further applications in other processing fields with information uncertainties.

Multi-criteria decision making in Pythagorean fuzzy environment

Dempster–Shafer evidence (DSE) theory, which allows combining pieces of evidence from different data sources to derive a degree of belief function that is a type of fuzzy measure, is a general framework for reasoning with uncertainty. In this framework, how to optimally manage the conflicts of multiple pieces of evidence in DSE remains an open issue to support decision making. The existing conflict measurement approaches can achieve acceptable outcomes but do not fully consider the optimization at the decision-making level using the novel measurement of conflicts. In this article, we propose a novel evidential correlation coefficient (ECC) for belief functions by measuring the conflict between two pieces of evidence in decision making. Then, we investigate the properties of our proposed evidential correlation and conflict coefficients, which are all proven to satisfy the desirable properties for conflict measurement, including nonnegativity, symmetry, boundedness, extreme consistency, and insensitivity to refinement. We also present several examples and comparisons to demonstrate the superiority of our proposed ECC method. Finally, we apply the proposed ECC in a decision-making application of motor rotor fault diagnosis, which verifies the practicability and effectiveness of our proposed novel measurement.

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A Novel Conflict Measurement in Decision-Making and Its Application in Fault Diagnosis

Environmental assessment and decision making is complex leading to uncertainty due to multiple criteria involved with uncertain information. Uncertainty is an unavoidable and inevitable element of any environmental evaluation process. The published literatures rarely include the studies on uncertain data with variable fuzzy reliabilities. This research has proposed an environmental evaluation framework based on Dempster–Shafer theory and Z-numbers. Of which a new notion of the utility of fuzzy number is proposed to generate the basic probability assignment of Z-numbers. The framework can effectively aggregate uncertain data with different fuzzy reliabilities to obtain a comprehensive evaluation measure. The proposed model has been applied to two case studies to illustrate the proposed framework and show its effectiveness in environmental evaluations. Results show that the proposed framework can improve the previous methods with comparability considering the reliability of information using Z-numbers. The proposed method is more flexible comparing with previous work.

Environmental assessment under uncertainty using Dempster–Shafer theory and Z-numbers

Using Z-number to measure the reliability of new information fusion method and its application in pattern recognition

Dempster-Shafer evidence theory has been widely applied to solving data fusion problems. However, it is still an open issue about how to combine the evidences effectively when the high conflict evidences are collected. Many scholars have made improvements to solve this problem, but there are new problems such as violation of the theoretical attributes of D-S combination rules and limitations of application scope of improvement methods. Considering these shortcomings, a new evidence synthesis formula based on correlation coefficient of belief functions is proposed in this paper. Our contribution is that the proposed formula can solve the highly conflict issues mentioned above effectively. Moreover, the various types of evidences collected can be well combined. One of the advantages of the proposed model is that conflict coefficient ${{\mathrm{K}}_{\mathrm{r}}}$ is the coefficient of the fusion formula which represents the degree of conflict about evidences. So the fusion process is more flexible and useful. Several examples and comparative experimental simulation are used to illustrate the effectiveness of the proposed methodology.

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A New Synthesis Combination Rule Based on Evidential Correlation Coefficient

Multi-attribute decision-making (MADM) is an important part of modern decision-making science. Fuzzy Analytic Hierarchy Process (Fuzzy AHP) is a popular model to deal with the issue of MADM for its flexible and effective advantages. However, The traditional Fuzzy AHP with some limitations does not consider the preference (attitude) of decision makers (DMs). In addition, some ideas of combining Ordered Weighted Average (OWA) and Fuzzy AHP don’t investigated the MADM well. Some programs are only applicable to a few examples, and more general cases do not result in effective decision making. Considering these shortcomings, an OWA-Fuzzy AHP decision model using OWA weights and Fuzzy AHP is proposed in this paper. Our contribution is that the proposed method can handle situations where the degree of fuzzy synthesis is not intersected. Moreover, the loss of information can be reduced in the process of applying the proposed method, so that the decision result is more reasonable than the previous methods. Several examples and comparative experimental simulation are given to illustrate the effectiveness and superiority of the proposed model.

An improved OWA-Fuzzy AHP decision model for multi-attribute decision making problem

Likelihood function has significant advantages in the fields of statistical inference. Based on this theory, Yager proposed a soft likelihood function to make it more widely used. However, Yager’s method can only deal with probabilities expressed by crisp values, and has strict restrictions on the form of data. Due to human subjectivity and lack of effective information, it is inevitable that data uncertainty will be involved. In order to deal with the uncertain data more flexibly and intuitively and solve the complex problems faced in real-world applications, a generalized soft likelihood function in combining multi-source belief distribution functions is proposed in this paper. Different from other existing methods, this paper uses a distribution function to represent uncertain information, which can retain more original information and improve the credibility of the results. The expectation and variance are used to rank the obtained evidences, and the evidence that contributes more to the results is ranked higher. Finally, the reliable likelihood results are obtained. The proposed method extends the method of Yager and can work well in more uncertain environment. Several numerical examples and comparative experimental simulation are used to illustrate the efficiency of the proposed soft likelihood function.

Pengdan Zhang

Papers

Environmental assessment under uncertainty using Dempster–Shafer theory and Z-numbers

Using Z-number to measure the reliability of new information fusion method and its application in pattern recognition

A New Synthesis Combination Rule Based on Evidential Correlation Coefficient

An improved OWA-Fuzzy AHP decision model for multi-attribute decision making problem

A generalized soft likelihood function in combining multi-source belief distribution functions