In real life, with the trend of outsourcing logistics activities, choosing a third-party logistics (3PL) provider has become an inevitable choice for shippers. One of the most difficult decisions logistics consumers are facing the selecting the 3PL provider that best meets their needs. Decision making (DM) is an important in dealing with such situations because it allows them to make reliable decisions in a short period of time, as incorrect decisions can result in huge financial losses. In this regard, this article provides a new multi criteria group decision making method (MCGDM) under Pythagorean fuzzy rough (PyFR) set. A series of new PyFR Einstein weighted averaging aggregation operators and their basic aspects are described in depth. To evaluate the weights of decision experts and criteria weights we established the PyFR entropy measure. Further, using multiple aggregation methods based on PyFR information, a novel algorithm is offered to solve issues with ambiguous or insufficient data to obtain reliable and preferable results. First, decision-experts use PyFR sets to represent their evaluation information on alternatives based on the criteria. Then, apply all these proposed PyFR Einstein aggregation lists to rank all alternatives and find the best optimal result. Finally, to demonstrate the feasibility of the proposed PyFR decision system, a real example of choosing a 3PL is given.

Finding the best transportation project and logistic service provider is one for the most important aspects of the development of a country. This task becomes more complicated from time to time as different criteria are involved. Hence, this paper proposes an approach to the linguistic three-way decision-making (TWDs) problem for selecting sustainable transportation investments and logistic service providers with unknown criteria and expert weight information. To this end, we first propose a new tool, the Pythagorean double hierarchy linguistic term sets (PyDHLTSs), which is a combination of first hierarchy linguistic term sets and second hierarchy linguistic term sets which can describe uncertainty and fuzziness more flexibly in decision-making (DM) problems. In addition, we propose some aggregation operators and basic operational laws for PyDHLTSs. A new decision-making technique for PyDHLTSs based on decision-theoretic rough sets (DTRSs) is proposed in the three-way decisions. Next, the conditional probability is computed using grey relational analysis in a PyDHLTSs environment, which improves decision-making. The loss function is computed by using the proposed aggregation operator, and the decision's results are determined by the minimum-loss principle. Finally, a real-world case study of a transportation project and logistic service provider is considered to demonstrate the efficiency of the proposed methods.