Trapezoidal interval type-2 fuzzy soft stochastic set and its application in stochastic multi-criteria decision-making

Trapezoidal interval type-2 fuzzy soft set (TrIT2FSS) (Zhang and Zhang, Appl Math Model 37:4948–4971, 2013) is a delicious generalization of soft set theory where the performance of the alternatives over some parameters is assigned by means of trapezoidal interval type-2 fuzzy numbers (TrIT2FNs). Zhang and Zhang (2013) proposed an approach to multi-criteria group decision-making (MCGDM) problems under trapezoidal interval type-2 fuzzy soft set. However, due to the increasing complexity of decision-making problems, the performance of the alternatives may be determined randomly with respect to some possible states related to the decision-making problems which are known as stochastic multi-criteria decision-making (SMCDM) problems. To handle such type of SMCDM problems based on trapezoidal interval type-2 fuzzy soft set, no research exists till now. To fill up this lacuna, here we have formulated a new generalization of TrIT2FSS to solve the MCDM problems under stochastic environment. First, we have introduced the notion of trapezoidal interval type-2 fuzzy soft stochastic set (TrIT2FSSS). Second, we deal with the obligatory definition of expected trapezoidal interval type-2 fuzzy soft set (ETrIT2FSS). Finally, we have proposed a methodology for handling the trapezoidal interval type-2 fuzzy soft stochastic set (TrIT2FSSS) based stochastic MCDM problems. Additionally, in this model we have assumed that the weights of each of the parameters are partially known. We have proposed a process to obtain the weights of the parameters using signed distance measurement. An application of our proposed approach regarding the selection of the best company for investment of a bank has also been given to describe the feasibility and effectiveness of our proposed approach.