The selection of venture capital investment projects is one of the most important decision-making activities for venture capitalists. Due to the complexity of investment market and the limited cognition of people, most of the venture capital investment decision problems are highly uncertain and the venture capitalists are often bounded rational under uncertainty. To address such problems, this article presents an approach based on regret theory to probabilistic hesitant fuzzy multiple attribute decision-making. Firstly, when the information on the occurrence probabilities of all the elements in the probabilistic hesitant fuzzy element (P.H.F.E.) is unknown or partially known, two different mathematical programming models based on water-filling theory and the maximum entropy principle are provided to handle these complex situations. Secondly, to capture the psychological behaviours of venture capitalists, the regret theory is utilised to solve the problem of selection of venture capital investment projects. Finally, comparative analysis with the existing approaches is conducted to demonstrate the feasibility and applicability of the proposed method.

Electric power industry has been undergoing enormous transformations. Therefore, it is necessary to improve the security of electric power system and the decision capacity in the emergency process. As a complicated system, electric power system is affected by many factors, the reasoning of which can be regarded as a process that combines intuition and cognition. Uncertainty characterizes human cognitive and reasoning processes. Several extensions of fuzzy cognitive map (FCM) model have been suggested to handle multifarious sources of uncertainty. Nevertheless, the uncertainty from human doubt may arise in the assignment of membership degrees, which is neglected in current FCMs. To deal with this problem, a novel approach based on hesitant fuzzy sets (HFSs) and FCMs, called hesitant fuzzy cognitive maps (HFCMs), is presented in this paper. The proposed method, which possesses the features of tackling hesitancy explicitly during the experts' assessments, is demonstrated by an example on the analysis of risk factors affecting electric power system. It can provide a better simulation of the inherent uncertainty in real problems through the hesitancy represented in experts' knowledge, and a what-if analysis for describing hesitant fuzzy scenario is well developed with the HFCM model.

The Belt and Road (B&R) Initiative receives enthusiastic response, the aim of which is to develop cooperative partnerships with countries along the routes and build a community of common destiny. So far, Chinese companies have invested in many different countries along the B&R. Generally, the investment decision making problems are characterized by high risk and uncertainty. Then how to make an appropriate investment decision will be a thorny issue. In this paper, probabilistic hesitant fuzzy set (PHFS) is used for handling uncertainty in multiple attribute decision making (MADM), and the criteria importance through intercriteria correlation (CRITIC) approach is extended to obtain attribute weights, no matter whether the weight information is incompletely known or not. Considering that the existing probabilistic hesitant fuzzy distance measures fail to meet the condition of distance measure, a new distance between PHFSs is proposed and applied to investment decision making for countries along the B&R. In the last, comparative analyses are performed to illustrate the advantages of the presented approach.

Appropriate classification of medical insurance fraud events can not only be effective in preventing and combating fraud, but also greatly improve the utilization of medical resources. Due to the uncertainty inherent in medical insurance fraud, identifying and classifying the fraud are non-trivial tasks. In addition, the selection of classification radius by traditional methods is often highly subjective. To this end, a case-based reasoning (CBR) approach in probabilistic hesitant fuzzy environment and its application to classifying the severity of medical insurance fraud events are investigated in this article. At first, the probabilistic hesitant fuzzy element (PHFE) is regarded as a discrete probability distribution, and its distribution function is defined. On this basis, a distribution discrepancy degree is proposed to make up for the shortage of existing measures between PHFEs. Then, a probabilistic hesitant fuzzy decision-making method based on CBR is proposed, which considers both decision data and the expert’s own knowledge and experience. Finally, the proposed method is used to classify the severity of medical insurance fraud events, and the rationality and superiority of the method are verified by comparative analysis. First published online 15 July 2025