Probabilistic linguistic term sets (PLTSs), which assign different weights to various linguistic terms, offer an efficient framework for expressing preferences. Meanwhile, the TODIM method, grounded in prospect theory, is adept at incorporating the cognitive behaviors of decision-makers into the decision-making process. In this paper, we extend the TODIM method to solve multi-attribute group decision making (MAGDM) problems with PLTSs. At first, we extend the Frank operators to PLTSs and propose the probabilistic linguistic Frank weighted averaging (PLFWA) operator based on adjusted rules of PLTSs. Further, some desirable properties of them are studied. Meanwhile, we present an innovative distance measure, deeply anchored in linguistic scale functions, designed to overcome the shortcomings of current distance metrics. What’s more, the combined weights for attributes can be obtained by the criteria importance though intercriteria correlation (CRITIC) method and the best-worst method (BWM) and the steps of the extended TODIM method for PLTSs are proposed. Finally, a numerical example for the purchase selection of electric vehicles is given, and some sensitivity and comparative analysis are used to illustrate the effectiveness and rationality of this new method.

Proppants are one of the key materials for hydraulic fracturing, whose main role is to support fractures and create a channel through which oil and gas can flow. The nature of proppants is the most talked about feature besides their cost, for example, their sphericity, turbidity, particle size, or strength. The porosity, permeability, and fracture conductivity of proppants in fractures are also the main indicators to measure the performance of them. These indicators are usually obtained through physical experiments. However, experimental results often differ depending on the experimental scheme. Different stacking methods of proppant particles lead to this phenomenon. The nature of proppant particles in fractures varies with the way they accumulate. This paper will start with the microscopic arrangement of proppant particles. Considering the randomness and certainty of three-dimensional particle stacking and arrangement, the Monte Carlo stochastic method and an optimization model were used to conduct three-dimensional computer simulation of proppant particles. This lays an important foundation for revealing the randomness and regularity of the micro arrangement of proppant particles.

The key performance parameters of proppant are mainly the crushing rate and fracture conductivity, which are usually evaluated using physical experimental methods. However, the testing method for fracture conductivity has limitations, such as its long time-consumption, high testing costs, instability, and even the presence of large errors in testing results under the same conditions. The purpose of this paper is to propose a calculation method that can replace physical experiments. Firstly, we analyze the random and deterministic phenomena in the contact relationship between proppant particles from a microscopic perspective. Subsequently, we develop a physical model of the microscopic arrangement of these particles, enabling us to conduct further computer simulations of their microscopic configuration. Secondly, we conduct a microscopic mechanical analysis of the contact between proppant particles and between particles and boundaries and establish a corresponding mathematical model. Then, utilizing the simulation and mechanical analysis results of the proppant, we calculate the crushing rate. Considering the crushing rate of proppant, we improve the Kozeny–Carmen equation to determine the fracture permeability, and subsequently calculate the fracture conductivity. Finally, the calculated results are compared with the experimental results. The results show that the calculated values for the proppant crushing rate and fracture conductivity matched well with experimental data, and that the model’s calculation values were more accurate. As the number of simulations increased, the accuracy of the calculation results became higher. Research shows that the fracture conductivity is influenced by factors such as the particle size, microstructure, and crushing rate. Numerical calculation methods can replace physical experiments and provide theoretical support for engineering applications of hydraulic fracturing proppant materials.

Probabilistic linguistic term sets (PLTSs) offer a superior means of capturing decision-makers’ evaluations, accurately and reliably reflecting their opinions. It account for both the inherent ambiguity of linguistic terms and the uncertainty associated with those terms. In this article, we elaborate on the application of the Frank operator within the context of probabilistic linguistic multiple attribute group decision making (MAGDM), leveraging its adjusted rules. In addition, we introduce a novel distance measure, rooted in linguistic scale functions, to address the limitations of existing distance metrics. Based on this new distance measure, we develop an extended TODIM method specific to the PLTSs environment.Concurrently, we enhance the rationality of attribute weighting by integrating the criteria importance though intercrieria correlation(CRITIC) method and the best-worst method (BWM) within the PLTSs framework. To demonstrate the practical application of our proposed approach, the purchase selection of electric vehicles was taken as an example to show the procedure of the proposed method. We outline the step-by-step procedure and conduct sensitivity and comparative analyses to validate the effectiveness and rationality of our methodology.