The modern decision-making environment is complex and dynamic. Global supply chain networks are increasingly exposed to unsystematic risks. Therefore, decision-makers need greater flexibility and precision to better manage uncertain market changes and complex international environments. To construct an accurate multi-criteria group decision-making (MCGDM) model, it is necessary to select appropriate evaluation criteria and identify alternative ranking methods based on specific decision problems. To develop a suitable MCGDM model for dynamic environments, this paper develops a new MCGDM model based on probabilistic interval-valued intuitionistic hesitant fuzzy sets (PIVIHFSs), regret theory, and the extended ELimination Et Choice Translating Reality (ELECTRE) III method. Firstly, this paper proposes two new aggregation operators, including the generalized probabilistic interval-valued intuitionistic hesitant fuzzy weighted averaging (GPIVIHFWA) operator and the generalized probabilistic interval-valued intuitionistic hesitant fuzzy weighted geometric (GPIVIHFWG) operator. To incorporate the decision-maker (DM)'s regret aversion, a bidirectional projection measure is proposed to calculate the advantages and disadvantages between two probabilistic interval-valued intuitionistic hesitant fuzzy elements (PIVIHFEs). The regret values of PIVIHFEs are determined using the bidirectional projection measure instead of utility values in the regret-rejoice function. Then, this paper constructs an extended ELECTRE III method and establishes a decision-making model based on the Borda rule for ranking and selecting the best alternatives. Finally, the effectiveness and robustness of the proposed model are verified through a numerical example, and the results are discussed through sensitivity analysis and comparative analysis.

Aiming at the common problems of quality evaluation method, this paper first establishes a fuzzy software quality evaluation model according to the relationship of software quality subcharacteristics and indicators; furthermore, considering the uncertainty and individual deviations of expert judgment results, this paper corrects and tests the consistency of the incomplete information sorting given by the experts and obtains an integration sorting of gathering different expert opinions through the idea of circling modification; at last, this paper proposes the weighted mutation rate which is used to measure the development balance degree and determines weights of evaluation indicators via weighted mutation rate correction incompletion G1 method, which avoids the problem of integration of subjective and objective weights.

In the realm of management decision-making, the selection of green suppliers has long been a complex issue. Companies must take a holistic approach, evaluating potential suppliers based on their capabilities, economic viability, and environmental impact. The decision-making process, fraught with intricacies and uncertainties, urgently demands the development of a scientifically sound and efficient method for guidance. Since the concept of Fermatean fuzzy sets (FFSs) was proposed, it has been proved to be an effective tool for solving multi-attribute decision-making (MADM) problems in complicated realistic situations. And the Power Bonferroni mean (PBM) operator, combining the strengths of the power average (PA) and Bonferroni mean (BM), excels in considering attribute interactions for a thorough evaluation. To ensure a comprehensive and sufficient evaluation framework for supplier selection, this paper introduces innovative aggregation operators that extend the PBM and integrate probabilistic information into Fermatean hesitant fuzzy sets (FHFSs) and Fermatean probabilistic hesitant fuzzy sets (FPHFSs). It successively proposes the Fermatean hesitant fuzzy power Bonferroni mean (FHFPBM), Fermatean hesitant fuzzy weighted power Bonferroni mean (FHFWPBM), and Fermatean hesitant fuzzy probabilistic weighted power Bonferroni mean (FHFPWPBM) operators, examining their key properties like idempotency, boundedness, and permutation invariance. By further integrating PBM with probabilistic information into FPHFSs, three new Fermatean probabilistic hesitant fuzzy power Bonferroni aggregation operators are developed: the Fermatean probabilistic hesitant fuzzy power Bonferroni mean (FPHFPBM), Fermatean probabilistic hesitant fuzzy weighted power Bonferroni mean (FPHFWPBM), and Fermatean probabilistic hesitant fuzzy probabilistic weighted power Bonferroni mean (FPHFPWPBM). Subsequently, a MADM method based on these operators is constructed. Finally, a numerical example concerning the selection of green suppliers is presented to demonstrate the applicability and effectiveness of this method using the FPHFPWPBM operator.

It is difficult to describe the hesitation and uncertainty of experts by single-valued information, and the differences in the importance of attributes are often ignored during the decision-making process. This paper introduces the probability and interval values into Fermatean hesitant fuzzy set (FHFS) and creatively proposes the probabilistic interval-valued Fermatean hesitant fuzzy set (PIVFHFS) to deal with information loss. This new fuzzy set allows decision makers to use interval-valued information with probability to express their quantitative evaluation, which broadens the range of information expression, effectively reflects the important degree of different membership degrees, and can describe uncertain information more completely and accurately. Under the probabilistic interval-valued Fermatean hesitant fuzzy environment, several new aggregation operators based on Hamacher operation are proposed, including the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher weighted averaging (PIVFHFHWA) operator and geometric (PIVFHFHWG) operator, and their basic properties and particular forms are studied. Then, considering the general correlation between different attributes, this paper defines the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher Choquet integral averaging (PIVFHFHCIA) operator and geometric (PIVFHFHCIG) operator and discusses related properties. Finally, a multi-attribute decision-making (MADM) method is presented and applied to the decision-making problem of reducing carbon emissions of manufacturers in the supply chain. The stability and feasibility of this method are demonstrated by sensitivity analysis and comparative analysis. The proposed new operators can not only consider the correlation between various factors but also express the preference information of decision makers more effectively by using probability, thus avoiding information loss in decision-making progress to some extent.

Inspired by the GARCH-MIDAS model, we revisit the relationship between Chinese futures and macroeconomic factors. We introduce the level of the macroeconomic variables into the GARCH-MIDAS model in order to test the impact of the macroeconomic level on the variance of futures’ return volatility. Based on the empirical results, we find the level of macroeconomic variables has a significant impact on the volatility of Chinese futures’ return. The influence of the macroeconomic level factor on the futures’ return volatility is statistically significant.