Jiayu Shen

Vocational undergraduate programs increasingly expect students to use probability and statistics to make decisions in workplace-like settings, yet instructors often lack practical tools for evaluating open-ended modeling work consistently and instructionally. This study developed and applied a course-embedded analytic rubric for statistical and probabilistic modeling in a required mathematics course. The rubric addressed six dimensions: contextual problem understanding, assumptions and data quality, model formulation, quantitative reasoning, validation and uncertainty, and vocational communication. Two authentic tasks, a manufacturing quality-sampling decision and a service waiting-time evaluation, were implemented, and 203 anonymized student submissions were scored. Two trained raters scored all submissions, with a third rater scoring a subsample to examine rater effects. Evidence from inter-rater agreement, many-facet Rasch measurement, and correlations with an independent transfer-oriented assessment evaluated scoring quality and educational usefulness. Results indicated good overall scoring consistency, ordered category functioning, small rater severity differences, and moderate-to-strong relations between rubric scores and transfer-oriented performance. Students performed more strongly on problem understanding, model formulation, and quantitative reasoning than on validation, uncertainty, and vocational communication. Beyond demonstrating scoring quality in a vocational setting, the study conceptualizes statistical modeling competence as a course-embedded, multidimensional, formative performance construct. This shows that rubric-based assessment can operationalize modeling not only as technical execution, but also as the coordinated integration of problem framing, quantitative reasoning, validation, and vocational communication. The findings suggest that a course-embedded rubric can serve as both a scoring tool and a practical framework for feedback, instructional adjustment, and curriculum improvement in vocational undergraduate mathematics.

This paper analyzes the transportation issue involving multiple objectives and items with fixed costs amid uncertainty, which aims to increase net profit while minimizing carbon emissions, to determine an optimal product shipping strategy. This paper introduces the use of uncertain theory to address the transportation dilemma, considering various challenges such as potential uncertainties during the actual transport process. It involves defining variables such as supply, demand and the rate of product sampling qualification as uncertain factors, constructing mathematical models, and deriving the corresponding model as well as the respective equivalent form by means of uncertainty theory. A linear weighted method is adopted to reflect the significance of each objective as identified by policymakers and suggest a sparrow optimization algorithm combined with butterfly search for numerical experiments to discover the optimal solution. This demonstrates the practicality and effectiveness of the proposed models.

Due to social and psychological factors, the COVID-19 pandemic has impacted international trade, dampened consumption globally, and resulted in conservative investment and spending. To stimulate economic recovery while promoting the establishment of a positive consumption awareness among people, enterprises endeavor to enhance competitiveness and expand market share through various means, such as advertising and discounting. With more attention paid to environmentally friendly products, there are greater challenges encountered by green supply chain management. In this study, a green supply chain network problem is analyzed, involving a manufacturer and a retailer. In addition to the construction of centralized and decentralized decision models, two cost-sharing contracts are adopted to reduce promotion costs for the retailer and improve the level of greenery in products. With the help of game theory, equilibrium decisions can be made by solving the models. According to the results of numerical experiment, cost-sharing contracts can effectively improve the level of greenery in products and the profitability of the supply chain, despite the inability to achieve a win–win situation.

In order to improve the teaching quality and efficiency of vocational education, it is necessary to cultivate students' core competencies. Mathematics is a fundamental course in higher vocational education teaching, which is of great significance for cultivating students' logical thinking ability and enhancing their future scientific and technological research and innovation work. With the continuous reform and development of education and teaching, the vocational mathematics should focus on improving students' core literacy, and through mathematical thinking and literacy training, cultivate students' ability to solve practical problems. This article is based on the current situation of mathematics curriculum teaching in vocational colleges. By elaborating on the characteristics and functions of mathematical core literacy, analyzing the importance of core literacy, and proposing teaching strategies based on improving core literacy, we aim to cultivate talents who adapt to the new era through reasonable educational model.

The uncertain production-inventory problem with deteriorating items is investigated and an optimal control model is developed in the present paper. The uncertain production-inventory problem is perturbed by an uncertain canonical process. Based on uncertainty theory, an optimistic-value optimal-based control model is established. The present study aims to find the optimistic value of revenue at a certain confidence level. The uncertainty theory is used to obtain the equation of optimality. Using the Hamilton–Jacobi–Bellman principle, a nonlinear partial differential equation that has to be satisfied by a value function is obtained. Assuming a specific form of the solution, backsubstituting the partial differential equation to find functions of time is conducted, and the functions are then used to solve the partial differential equation. Numerical experiments with different demand functions are used to assess the feasibility of this model and this method.

This paper constructs an uncertain mathematical model for the fixed charge transportation problem in which several kinds of items are transported from different sources to various destinations via different methods. In addition, after‐sale service such as product return and exchange that help to raise the customer’s satisfaction is considered during the trade to meet the reality. Meanwhile, some indeterministic factors may occur during the transportation process; probability theory is not the best option due to the lack of adequate historical data. In the light of this statement, some parameters like supplies, demands, return rate, and exchange rate are all defined as uncertain variables to formulate the uncertain programming model. After that, the equivalent deterministic models are derived with the help of uncertainty theory. Finally, some numerical experiments are implemented via three algorithms; the optimal values are displayed and compared to show the application of this problem. The study of this paper provides an applicable method for the decision maker.