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Due to the complexity of the real problems and the limitations of expert's knowledge, it is often difficult for decision makers to give the exact decision information. How to accurately and adequately express the evaluation information of decision makers becomes the primary problem of multi-attribute decision making(MADM). Therefore, a large number of fuzzy sets(such as linguistic hesitant fuzzy sets(LHFS), probabilistic hesitant fuzzy sets(PHFS), dual hesitant fuzzy sets(DHFS)) that express decision makers' judgment information have been widely used in multi-attribute decision making. Inspired by these fuzzy sets, this paper proposes a new set called probabilistic linguistic dual hesitant fuzzy sets(PLDHFS). Based on the PLDHFS, the probabilistic linguistic dual hesitant fuzzy preference relation(PLDHFPR) is further given. In order to obtain a reasonable ranking weight vector, the equivalence consistency between the PLDHFPR and the interval-valued fuzzy preference relation(IVFPR) is proposed, and we present the objective programming model(OPM) based on the multiplicative consistency of the IVFPR. Moreover, we establish a consistency improvement model for inconsistent PLDHFPR. Finally, an example of venture capital is analyzed to demonstrate the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]