The consistency improvement of probabilistic linguistic hesitant fuzzy preference relations and their application in venture capital group decision making

Driven by the entrepreneurial trend of mass entrepreneurship and innovation, venture capital(VC) has been widely concerned and valued by investors. There is no doubt that investment decision plays a critical role in venture capital, however, due to the complexity of the investment environment, it is often difficult for investors to make a definite judgement on an innovative solution. Consequently, to express more accurately the hesitation and ambiguity of investors in the decision-making process, this paper proposes the probabilistic linguistic hesitant fuzzy preference relation(PLHFPR) based on the probabilistic linguistic hesitant fuzzy set(PLHFS). Unlike hesitant fuzzy preference relation (HFPR), PLHFPR not only provides flexible linguistic expression for decision makers, but also gives the occurrence probability of each element in the PLHFPR. Considering that it is difficult for investors to give the exact probability of each element in the PLHFPR, a new probability calculation method is proposed based on the consistency analysis. What's more, the convex consistency index(CCI) is defined to measure the consistency level of the PLHFPR by considering decision maker's risk attitude. For the inconsistent PLHFPR, a weighted nonlinear programming model(WNPM) is constructed to derive an acceptable convex consistent PLHFPR and obtain the PLHFPR priority weight vector. Finally, an example about the venture capital is offered to verify the effectiveness of the proposed method.