Generally, in real decision-making, all the pieces of information are used to find the optimal alternatives. However, in many cases, the decision-makers (DMs) only want “how good/bad a thing can become.” One possibility is to classify the alternatives based on minimum (tail) information instead of using all the data to select the optimal options. By considering the opportunity, we first introduce the value at risk (VaR), which is used in the financial field, and the probabilistic interval-valued hesitant fuzzy set (PIVHFS), which is the generalization of the probabilistic hesitant fuzzy set (PHFS). Second, deemed value at risk (DVaR) and reckoned value at risk (RVaR) are proposed to measure the tail information under the probabilistic interval-valued hesitant fuzzy (PIVHF) environment. We proved that RVaR is more suitable than DVaR to differentiate the PIVHFEs with example. After that, a novel complete group decision-making model with PIVHFS is put forward. This study aims to determine the most appropriate alternative using only tail information under the PIVHF environment. Finally, the proposed methods’ practicality and effectiveness are tested using a stock selection example by selecting the ideal stock for four recently enrolled stocks in China. By using the novel group decision-making model under the environment of PIVHFS, we see that the best stock is E4 when the distributors focus on the criteria against 10% certainty degree and E1 is the best against the degree of 20%, 30%, 40% and 50% using the DVaR method. On the other hand when RVaR method is used then the best alternative is E4 and the worst is E2 against the different certainty degrees. Furthermore, a comparative analysis with the existing process is presented under the PHF environment to illustrate the effectiveness of the presented approaches.
Interval-Valued Probabilistic Hesitant Fuzzy Set Based Muirhead Mean for Multi-Attribute Group Decision-Making
