The law of total probability plays an essential role in Bayesian reasoning, which has been used in many fields. However, some experiments show the law of total probability can be violated. In recent years, researchers have tried to explain this paradox with the interference effect in quantum theory, and they think the main reason for interference effects is the uncertain information in the decision-making process. Therefore, how to effectively model and process the uncertain information in the decision-making process is very important to understand and predict the interference effects. Zadeh proposed the fuzzy set by considering the fuzziness of information. Later, Atanassov proposed the intuitionistic fuzzy sets (IFS). IFS better describes the fuzzy information from the view of membership, nonmembership than fuzzy sets, which can also more flexibly simulate human decision making. Hence, the article proposed the fuzzy Markov decision-making model (FDM) under the framework of IFS to explain and predict the interference effects of decision-making process. In FDM, intuitionistic fuzzy number can be generated by using the negation operation of probability. In addition, the transition matrix can be obtained by using the Kolmogorov equation, which can consider the evolution time in the decision-making process. The transition matrix establishes the relationship between different stages to get the fuzzy numbers of final states. Finally, the article used the Dempster–Shafer evidence theory to transform fuzzy number into the probability. In summary, the proposed FDM can provide a novel idea to explore and explain the interference effects in the decision-making process, which is helpful to promote the development of artificial intelligence.

Dempster–Shafer (D-S) evidence theory is an effective methodology to handle unknown and imprecise information because it can assign probability into the power set. However, the process of obtaining information is a complex task, which can consider the rational, conscious, objective evaluation of utility with behavioral effects. Besides, in most cases, information can be obtained from different angles at the same time. The quantum model of mass function (QM) uses amplitude and phase angle to easily express those properties of information that can extend D-S evidence theory to the unit circle in a complex plane. Moreover, everything in nature will have its opposite, which is a kind of universality. The Bayes theorem is essentially the process of negation. However, in most cases, decisions can be made by only fully considering the known information without considering the other side of the information. Hence, considering the negation of information is a question to be investigated deeply, which can analyze information from the other point. This article proposes negation of QM by using the subtraction of vectors in the unit circle, which can degenerate into negation proposed by Yager in standard probability theory and negation proposed by Yinet al.in D-S evidence theory. Negation can provide us more information to consider the problem from both positive and negative aspects. In this article, negation can be understood information, which does not belong to event$A$, that is to say, negation can be regarded as nonmembership by using the fuzzy terms. Based on the above discussion, this article proposes the quantum pythagorean fuzzy evidence theory (QPFET), which is the novel work to consider QPFET from the point of negation. Besides, there are some numerical examples to explain the proposed method. In order to explore the applications of QPFET, this article discusses the possibility of the VI$\check{s}$ekriterijumsko Kompromisno Rangiranje method under QPFET to handle multicriteria decision-making that enables us to capture 2-D data, considering not only amplitude but also phase angle.

Pythagorean fuzzy set (PFS) is an extension of the intuitionistic fuzzy set. It has a wider space of membership degrees. Thus, it is more capable in expressing and handling the fuzzy information in engineering practice and scientific research. However, PFSs lack a mathematical tool to express stochastic or probability information, rendering it unsuitable for application in many scenarios. In this article, an ordered number pair is used to describe fuzzy information and stochastic information under uncertain environments, namely constrained Pythagorean fuzzy set (CPFS). The CPFS has two components,$\text{CPFS}=(A,P)$, where$A$is the classical PFS, while$P$is a measurement of reliability for$A$. For PFS, CPFS is the first unified description of fuzzy information and probabilistic information, which is a more flexible way to describe knowledge or thinking. Furthermore, the similarity measure of CPFSs is presented, which meets the similarity measure theorems and can better indicate the flexibility of CPFSs. Numerical examples are used to demonstrate that the CPFSs similarity measure is reasonable and effective. The method of similarity measure can be degenerated to the similarity measure of PFSs under specific case and can avoid generating counter-intuitive results. In addition, similarity measure of CPFSs is applied to medical diagnosis and target classification of Iris. These experimental results have proven the practicability and effectiveness of our model.

Dempster-Shafer (D-S) evidence theory has been used in many fields due to the flexibility and effectiveness in modelling uncertainties, which is the extension of classical probability. Uncertainty principle is one of the most important principles in quantum theory, which has been used in many fields. How to set the connection between quantum theory and D-S evidence theory is also an open issue. Hence, the paper proposed the quantum model of mass function to consider the quantum theory and D-S evidence theory. In the proposed quantum method, quantum mass function uses euler formula to represent. The paper also discusses some operations based on the quantum model of the mass function. Moreover, the paper also discusses the relationship between quantum mass function and classical mass function by using some numerical examples. Classical mass function is a special case when there is no interference in quantum mass function. Similar to the other quantum models, this study provides a more wide application in quantum information.

Dempster-Shafer evidence theory has been widely used in many applications due to its advantages with weaker conditions than Bayes probability. How to measure the uncertainty of basic probability assignment (BPA) in Dempster-Shafer evidence theory is an open and essential issue. Tsallis entropy as nonextensive entropy proposed according to multifractals has been used in many fields. In this paper, a new uncertainty measure of BPA is presented based on Tsallis entropy. The key issue is to determine the value of q in Tsallis entropy. In addition, this paper also analyzes the properties of proposed uncertainty measure. Some numerical examples are used to illustrate the efficiency of the proposed method. Finally, the paper also discusses the application of the proposed method in decision-making.