Expected hesitant VaR for tail decision making under probabilistic hesitant fuzzy environment

Abstract

Generally, to reasonably make decision, all evaluation information should be aggregated, and thus, the ranking and the optimal alternative can be obtained. However, in some extreme cases, the decision maker (DM) can only focus on the tail information such as the big-loss or big-gain values and wants to ask the simple question “How bad can a thing become?” or “How good can a thing become?” To address this type of decision-making issue, this paper introduces the definition of value at risk (VaR), which is a famous term in the financial field, and the probabilistic hesitant fuzzy element (PHFE), which is a general hesitant fuzzy element (HFE) and has recently become a popular topic. Then, the hesitant VaR (HVaR) is defined, and its mathematical presentation is provided to measure the tail information of the PHFEs. It is found that the tail information calculated by the HVaR is segmentary, and only the boundary value is used. Therefore, this paper further develops the expected HVaR (EHVaR) to improve the HVaR, which can describe the entire tail information. Two simple examples are provided to show and compare the proposed HVaR and EHVaR. To apply the EHVaR into a group decision making that focuses on the tail information, this paper proposes a dynamic programming model to calculate the weights of the DMs based on the principle that the more accurate PHFE should be given a bigger weight. Then, the tail group decision making steps based on the EHVaR are presented. Finally, this paper provides an example of selecting the optimal stock for four newly listed stocks in China to demonstrate the effectiveness of the proposed approaches.