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Athlete training optimization is an important part of sports management, and the scientific management program is of practical significance for athletes to improve their sports level. This paper collects sports training data for cluster analysis, establishes the optimization function of the distance from the data point to the center of the class, solves it, and determines the consistency matrix to judge the data category. Establish the column generation optimization model, get the linear programming model based on the matrix change column generation, in the process of transforming the model to solve the recursive matrix recursive relationship formula and solve the objective function. According to the solution of the optimization function to adjust the athletes’ sports training program. After the test, it was found that the pre-and post-test scores of the three tests of the experimental class were improved by 1.03 seconds, 1.05 meters, and 1.33 meters, respectively, and were significantly ahead of the scores of the control group. According to the satisfaction analysis, 76.67% of students were satisfied with the training effect. It indicates that the new training mode improved by the algorithm has good application effects.

In this paper, in the process of solving the mathematical model of sports facility optimization, the integer linear programming issue of choosing a sports facility is solved using a branch delimitation approach paired with a recursive matrix-based column-generating method. As part of the process of creating a scientific evaluation system for physical education, a hierarchical gray correlation-based evaluation model for the quality of physical education in colleges and universities is built by combining the hierarchical analysis weight-solving method and the solution method of the gray correlation degree. After the analysis, 81.1% of the teachers and students were satisfied after physical education facilities were optimized with the physical education teaching facilities. The gray correlation coefficient of physical education hardware facilities and software is 0.7, and the gray correlation coefficient of physical education effect is 0.18.

• A public transit accessibility under heat exposure model is studied with time-dependent consideration. • A column generation method to urban rail-bus-walk service network integration is devised. • Experiments in Beijing show the impact of heat exposure on public transit accessibility. The evolution of urban transportation systems is increasingly driven by the integration of rail, bus, and walking pathways, forming a cohesive service network. This urban rail-bus-walk service network integration is pivotal in addressing urban challenges and ensuring seamless mobility. Nevertheless, the effectiveness of such integrated systems depends on their accessibility and the willingness of individuals to take them as their primary transportation mode. In the context of climate change, extreme heat events pose an increasing threat to passenger comfort and health, which in turn, affects individuals' decisions to use public transit. Additionally, walking to transit stops or waiting for transportation becomes more challenging during such extreme heat conditions. This paper tackles the challenge of comprehensively addressing the evolving issues related to public transit accessibility in response to rising temperatures, an area of research that has received limited attention until now. This paper introduced an approach aimed at optimizing the urban rail-bus-walk service network with a focus on both accessibility and heat exposure considerations. By calculating the transportation accessibility level under varying temperature conditions and identifying its vulnerability to high temperatures, this paper presents a methodology to enhance the transit service. Such improvement involves not only adjustments to the integration network but also considering of cooling facilities, such as shaded routes and shelters. Column generation is applied to solve this problem and the optimization result demonstrates improvement of accessibility under the extreme heat condition. [ABSTRACT FROM AUTHOR]