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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.

Purpose: To incorporate four‐dimensional computed tomography (4DCT)‐based ventilation imaging into intensity‐modulated radiation therapy (IMRT) treatment planning for pulmonary functional avoidance. Methods and Materials: Nineteen locally advanced lung cancer patients are retrospectively studied. 4DCT images are employed to create ventilation maps for each patient via a density‐change‐based algorithm with mass correction. The regional ventilation is directly incorporated into the mathematical formulation of a direct aperture optimization model in IMRT treatment planning to achieve functional avoidance and a voxel‐based treatment plan. The proposed functional avoidance planning and voxel‐based planning are compared to the conventional treatment planning approach purely based on the anatomy of patients. Paired sample t‐tests are conducted to see whether dosimetric differences among the three approaches are significant. Results: Similar planning target volume (PTV) coverage is achieved by anatomical, functional avoidance, and voxel‐based approaches. The voxel‐based treatment planning performs better than both functional avoidance and anatomical planning to the lung. For a total lung, the average volume reductions in a functional avoidance plan from an anatomical plan, a voxel‐based plan from an anatomical plan, and a voxel‐based plan from a functional avoidance plan are 7.0%, 16.8%, and 10.6%, respectively for V40; and 0.4%, 6.4%, and 6.0%, respectively for mean Lung Dose (MLD). For a functional lung, the reductions are 8.8%, 17.2%, and 9.2%, respectively, for fV40; and 1.1%, 6.2%, and 5.2%, respectively, for functional mean lung dose (fMLD). These reductions are obtained without significantly increasing doses to other organs‐at‐risk. All the pairwise treatment planning comparisons for both total lung and functional lung are statistically significant (p‐value <α=0.05$< \alpha =0.05$) except for the functional avoidance plan with the anatomical plan pair in which the p‐value >α=0.05$> \alpha =0.05$. From these results, we can conclude that voxel‐based treatment planning outperforms both anatomical and functional‐avoidance planning. Conclusions: We propose a treatment planning framework that directly utilizes functional images and compares voxel‐based treatment planning with functional avoidance and anatomical treatment planning. [ABSTRACT FROM AUTHOR]

• 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]

The conventional method of PM2.5 prediction requires high-precision instruments to obtain data on the concentration of pollutants, resulting in a high prediction costs. In this work, we attempt to use image data to estimate PM2.5 concentration. The concentration of atmospheric PM2.5 is closely linked to the image’s dark channel intensity, contrast, and color difference of HSI. The increase in atmospheric PM2.5 concentration leads to a decrease in the non-sky area dark channel intensity, image contrast, and HSI spatial color difference. In this paper, a Column-Generation PM2.5 prediction model based on image mixture kernel was proposed by analyzing the relationship between PM2.5 and image features. First, the sampling period was taken as 1 h, and 8:00–17:00 was taken as the sampling range daily. The scene images were recorded in different weather conditions, and five image features were extracted, including contrast, dark channel intensity, and HSI color difference. Secondly, the image data has the characteristics of large sample size and uneven distribution, and the prediction model consists of a single kernel function, which makes it difficult to meet the prediction accuracy requirement. Therefore, the linear kernel function, polynomial kernel function, and Gauss kernel function were chosen to construct a composite model according to the concept of kernel structure from simple to complex. Then each kernel's Gram matrix was calculated based on training samples, and all gram matrices were placed into a mixture kernel matrix. Using the column generation algorithm and mixture kernel matrix, the prediction model was developed and the parameters of the model were solved. Finally, simulation experiments were performed; the results show that the prediction model based on the image mixture kernel of Column-Generation PM2.5 can meet the prediction accuracy requirements. The model has higher prediction accuracy and better model stability in comparison with the single-kernel prediction model. A computational complexity analysis shows that the prediction model based on the image mixture kernel of column-generation PM2.5 has no significant increase in computational complexity in comparison with the one-kernel prediction model.

Multi-microgrid (MMG) system is a new method that concurrently incorporates different types of distributed energy resources, energy storage systems and demand responses to provide reliable and independent electricity for the community. However, MMG system faces the problems of management, real-time economic operations and controls. Therefore, this study proposes an energy management system (EMS) that turns an infinite number of MMGs into a coherence and efficient system, where each MMG can achieve its goals and perspectives. The proposed EMS employs a cooperative game to achieve efficient coordination and operations of the MMG system and also ensures a fair energy cost allocation among members in the coalition. This study considers the energy cost allocation problem when the number of members in the coalition grows exponentially. The energy cost allocation problem is solved using a column generation algorithm. The proposed model includes energy storage systems, demand loads, real-time electricity prices and renewable energy. The estimate of the daily operating cost of the MMG using a proposed deep convolutional neural network (CNN) is analyzed in this study. An optimal scheduling policy to optimize the total daily operating cost of MMG is also proposed. Besides, other existing optimal scheduling policies, such as approximate dynamic programming (ADP), model prediction control (MPC), and greedy policy are considered for the comparison. To evaluate the effectiveness of the proposed model, the real-time electricity prices of the electric reliability council of Texas are used. Simulation results show that each MMG can achieve energy cost savings through a coalition of MMG. Moreover, the proposed optimal policy method achieves MG's daily operating cost reduction up to 87.86% as compared to 79.52% for the MPC method, 73.94% for the greedy policy method and 79.42% for ADP method.

A cutting stock algorithm based on four-block layout was proposed to solve the two-dimensional plate shearring and punching of circular parts. With this layout, a sheet was divided into four blocks. Only the strips with the same direction and length were arranged in each block, and several rows of the same circular parts were arranged in the strips. A four block layout method for generating circular parts on a single sheet by constructing a layout algorithm. Firstly, the layout of circular parts in the strips was determined. Secondly, recursive algorithm was constructed to generate the layout of the strips in blocks. Finally, implicit enumeration algorithm was used to determine the optimal four-block partition. The column generation algorithm was used to call the above layout algorithm to generate a number of different layouts. Based on the simplex principle, a group of layouts were selected to form the cutting plan, and the decimal solution was rounded. The results showed that the plate utilization ratio of the algorithm in the study is 0. 49%, 0. 32%, 6. 04% and 1. 50% higher than those of the four literature algorithms, respectively, and the calculation time can meet the needs of practical applications. [ABSTRACT FROM AUTHOR]

Combined cooling, heating, and power (CCHP) microgrids are a special form of a microgrid that is attracting increasing attention. This study contributes to the goal of minimising the operation cost of CCHP microgrids by proposing a hierarchical two‐stage robust optimisation dispatch model for multiple CCHP microgrid systems. The uncertainties associated with wind power output, electric power, heating, and cooling loads, and transmission line failures are considered in the proposed model. Moreover, the electricity purchasing and selling prices of each microgrid are independently determined. The proposed model applies the outputs of fuel cells, energy storage devices, and gas turbines, the distribution factor of waste heat, and the power transmission between the microgrids and an external grid as control variables. The optimised dispatch problem is solved using McCormick envelopes relaxation and a novel column and constraint generation algorithm that provides enhanced optimisation performance by implementing co‐evolutionary theory. In this way, the microgrid system is divided into several sections, and each section is represented as an individual min–max–min problem. The rationality and validity of the proposed model and the superiority of the solution performance of the improved algorithm are verified through simulation case studies involving a system composed of four CCHP microgrids. [ABSTRACT FROM AUTHOR]

It is an effective way to regard the electric vehicles as the demand response for reducing the negative impact of large‐scale introduction on the power system. Aiming at the microgrid with demand response, the adaptive uncertainty sets‐based two‐stage robust optimisation method is established in this study. The coordination of micro‐gas turbine, energy storage, and demand response etc. are considered in the economic dispatch model. To effectively consider the uncertain variable contained in the microgrid, the concept of adaptive uncertainty sets is proposed in this study. The uncertainty sets are achieved by the long short‐term memory network and modified fuzzy information granulation. To handle the adaptive uncertainty sets‐based robust optimisation model, the column and constraint generation algorithm and strong duality theory are introduced to decompose the model into a master problem and a subproblem with mixed‐integer linear structure. To verify the performance of the proposed adaptive uncertainty sets‐based two‐stage robust optimisation method, measured data from a plateau city of China are introduced in the simulation test. The simulation results demonstrate the effectiveness of the model and solution strategy. [ABSTRACT FROM AUTHOR]

The problem of disrupted flight recovery is a very complicated real-time network optimization problem, which belongs to NP-hard problem. Considering the situation caused by the shortage of aircraft resources, the closure of airports and the unplanned maintenance, this paper adopts multiple recovery options, such as flight delays, flight cancellations, flight swaps and so on. Based on Dantzig-Wolfe decomposition principle, the mathematical optimization models of master problem and sub-problem are established respectively. And column generation algorithm is introduced to solve this large integer programming problem. In the solving process, first, the initial feasible routes are constructed and the master problem is solved with CPLEX software based on the lines. Secondly, in view of the characteristics of the problem, some properties are proposed, then the sub-problem is solved by using the improved "label-setting algorithm", and in each iteration, multiple columns with reduced cost are added to reduce the number of iterations, then the branch and bound algorithm is used to handle the non-integer solutions. Finally, the correctness and efficiency of the method are verified by testing the practical instances of various scales, and the test results are analyzed and summarized. [ABSTRACT FROM AUTHOR]

Green scheduling is an important means to achieve sustainable industrial development and enhance the green efficiency of enterprises. Given the characteristics of a modern crude oil supply system, the shuttle tanker fleet green scheduling problem (FGSP) is discussed considering a carbon tax and variable tanker speed factor. To minimize the green operating cost (i.e., sum of general operating cost and carbon tax) of the tanker fleet, an integer programming model for shuttle tanker fleet green scheduling (FGSM) is established. The FGSM optimizes the number and sizes of tankers, the number and positions of floating production storage and offloading units (FPSO) at which to berth and the scheduling plan (i.e., berthing order and sailing speed) of each tanker in the fleet. Based on the column generation algorithm, a shuttle tanker fleet green scheduling algorithm is designed to solve the above model accurately. The experimental results show that considering the speed factor, the green operating cost of an example fleet decreases. For different carbon tax rates, speed optimization is an effective way to reduce the green operating cost of the fleet. The above results also show that the FGSM and algorithm can effectively solve the FGSP, improve the operation level and efficiency, and reduce the green operating cost of oil companies. • A model is established to optimize fleet design and scheduling plan. • The model aims to minimize the green operating cost of the fleet. • The model considers the impact of carbon tax and vessel speed factor. • A column generation algorithm is designed to solve the model accurately. [ABSTRACT FROM AUTHOR]

The inverse max+sum spanning tree problem is considered by modifying the sum-cost vector under weighted l1 norm. On an undirected network G(V, E, c, w), a cost c(e) and a weight w(e) are prescribed for each edge e∈E. The max+sum spanning tree (MSST) problem is to find a spanning tree T∗ which makes the combined weight maxe∈Tw(e)+∑e∈Tc(e) as small as possible. It can be solved in O(mlogn) time, where m=|E| and n=|V|. Whereas, in an inverse MSST problem, T0 is a given spanning tree of G, which is not an optimal MSST. The sum-cost vector c is to be modified to c¯ so that T0 becomes an optimal MSST of the new network G(V,E,c¯,w). The goal is to minimize the modification cost ‖c¯-c‖1 incurred by adjusting the sum-cost vector under weighted l1 norm. The inverse MSST problem under weighted l1 norm can be formulated as a linear programming problem with exponential number of constraints. Then a column generation algorithm is proposed to solve its dual problem. It is extremely interesting that in each iteration the choice of entering variable can be determined by solving an MSST problem with new sum-cost c~. Computational experiments show that the column generation algorithm can find the optimal solutions of the inverse problem although the number of iterations as well as the running time grow fast with the increase in the size of the instance. [ABSTRACT FROM AUTHOR]

With the growing traffic of containerized shipping worldwide, container liners have seen increasing cooperation. Slot co-chartering has drawn wide attention as a way of cooperation within liner alliances. However, liners face the issue of finding proper measures to optimize alliance operations. In response to this, this paper comprehensively considers shipping capacity dispatch for containerized shipping companies within liner alliances. First, we explained the general rules of liner shipping dispatch and analyzed the interfering factors. Second, we created a liner shipping capacity dispatch model without consideration of slot co-chartering costs and a liner shipping capacity dispatch model with consideration of slot co-chartering costs and proposed a new column generation algorithm to solve both problems. Third, we tested our algorithm in a real case of a large Chinese containerized shipping company, which belongs to Ocean Alliance, and an optimal ship dispatch strategy on relevant routes and a decision on optimal slot chartering can be reached. Computational results indicate that the proposed column generation algorithm exhibits effective and efficient performance for large-scale slot co-chartering in liner shipping. [ABSTRACT FROM AUTHOR]

The exact solution and heuristic solution have their own strengths and weaknesses on solving the Vehicle Routing Problems with Time Windows (VRPTW). This paper proposes a hybrid Column Generation Algorithm with Metaheuristic Optimization (CGAMO) to overcome their weaknesses. Firstly, a Modified Labelling Algorithm (MLA) in the sub-problem of path searching is analysed. And a search strategy in CGAMO based on the demand of sub-problem is proposed to improve the searching efficiency. While putting the paths found in the sub-problem into the main problems of CGAMO, the iterations may fall into endless loops. To avoid this problem and keep the main problems in a reasonable size, two conditions on saving the old paths in the main problem are used. These conditions enlarge the number of constraints considered in the iterations to strengthen the limits of dual variables. Through analysing the sub-problem, we can find many useless paths that have no effect on the objective function. Secondly, in order to reduce the number of useless paths and improve the efficiency, this paper proposes a heuristic optimization strategy of CGAMO for dual variables. It is supposed to accelerate the solving speed from the view of on the dual problem. Finally, extensive experiments show that CGAMO achieves a better performance than other state-of-the-art methods on solving VRPTW. The comparative experiments also present the parameters sensitivity analysis, including the different effects of MLA in the different path selection strategies, the characteristics and the applicable scopes of the two pathkeeping conditions in the main problem. First published online: 25 Oct 2013

The exact solution and heuristic solution have their own strengths and weaknesses on solving the Vehicle Routing Problems with Time Windows (VRPTW). This paper proposes a hybrid Column Generation Algorithm with Metaheuristic Optimization (CGAMO) to overcome their weaknesses. Firstly, a Modified Labelling Algorithm (MLA) in the sub-problem of path searching is analysed. And a search strategy in CGAMO based on the demand of sub-problem is proposed to improve the searching efficiency. While putting the paths found in the sub-problem into the main problems of CGAMO, the iterations may fall into endless loops. To avoid this problem and keep the main problems in a reasonable size, two conditions on saving the old paths in the main problem are used. These conditions enlarge the number of constraints considered in the iterations to strengthen the limits of dual variables. Through analysing the sub-problem, we can find many useless paths that have no effect on the objective function. Secondly, in order to reduce the number of useless paths and improve the efficiency, this paper proposes a heuristic optimization strategy of CGAMO for dual variables. It is supposed to accelerate the solving speed from the view of on the dual problem. Finally, extensive experiments show that CGAMO achieves a better performance than other state-of-the-art methods on solving VRPTW. The comparative experiments also present the parameters sensitivity analysis, including the different effects of MLA in the different path selection strategies, the characteristics and the applicable scopes of the two path-keeping conditions in the main problem. [ABSTRACT FROM PUBLISHER]

Airlines' expensive resources, especially aircraft, are to be optimally scheduled to cover flights of timetables. However, the irregular flight, due to bad weather, mechanical fault and so on, is inevitable. Moreover, flight delays become more severe with the rapid development of the air transport industry in China and have huge irregular flight cost. In order to alleviate flight delays impact on the flight plan, we present a double objective multi-commodity network flow model of flight delay propagation-based aircraft scheduling and minimize the total delay propagation and airline operation cost as the optimization objective. Branch-and-price solution and column generation algorithm are used to solve the problem. Computational results obtained by using data from a major domestic airline show that our approach can reduce delay propagation significantly, thus improving on-time performance and robustness of aircraft scheduling, and decreasing the total cost simultaneously. [ABSTRACT FROM AUTHOR]

The paper presents a copula-based extension of Conditional Value-at-Risk and its application to portfolio optimization. Copula-based conditional value-at-risk (CCVaR) is a scalar risk measure for multivariate risks modeled by multivariate random variables. It is assumed that the univariate risk components are perfect substitutes, i.e., they are expressed in the same units. CCVaR is a quantile risk measure that allows one to emphasize the consequences of more pessimistic scenarios. By changing the level of a quantile, the measure permits to parameterize prudent attitudes toward risk ranging from the extreme risk aversion to the risk neutrality. In terms of definition, CCVaR is slightly different from popular and well-researched CVaR. Nevertheless, this small difference allows one to efficiently solve CCVaR portfolio optimization problems based on the full information carried by a multivariate random variable by employing column generation algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes a bi-level programming model to solve the design problem for bus lane distribution in multi-modal transport networks. The upper level model aims at minimizing the average travel time of travelers, as well as minimizing the difference of passengers’ comfort among all the bus lines by optimizing bus frequencies. The lower level model is a multi-modal transport network equilibrium model for the joint modal split/traffic assignment problem. The column generation algorithm, the branch-and-bound algorithm and the method of successive averages are comprehensively applied in this paper for the solution of the bi-level model. A simple numerical test and an empirical test based on Dalian economic zone are employed to validate the proposed model. The results show that the bi-level model performs well with regard to the objective of reducing travel time costs for all travelers and balancing transit service level among all bus lines. [ABSTRACT FROM AUTHOR]

The Column Generation Algorithm (CGA) is commonly cited in the bibliography as alternative to solve large scale optimization problems. This article deals with the description of the GC algorithm in the context of the student assignation to the public schools. Some of the wea�nesses and shortcomings encountered in the implementation and application to a real problem are discussed and some strategies to sort it out are presented. This work pointed out how the CGA could be used within a general Branch and Price procedure to solve problems with additional constrains.

We propose a new dimensionality reduction method called maximum margin projection (MMP), which aims to project data samples into the most discriminative subspace, where clusters are most well-separated. Specifically, MMP projects input patterns onto the normal of the maximum margin separating hyperplanes. As a result, MMP only depends on the geometry of the optimal decision boundary and not on the distribution of those data points lying further away from this boundary. Technically, MMP is formulated as an integer programming problem and we propose a column generation algorithm to solve it. Moreover, through a combination of theoretical results and empirical observations we show that the computation time needed for MMP can be treated as linear in the dataset size. Experimental results on both toy and real-world datasets demonstrate the effectiveness of MMP. [ABSTRACT FROM AUTHOR]

Crew scheduling problem is one of the hardest and most comprehensive problems encountered in airline planning. In crew scheduling problem, it is aimed to find the minimum costly set of pairings in that each flight leg is covered at least by one crew pairing. In this study, a column generation approach that is commonly used in crew scheduling literature in which variables are dynamically generated, is used to solve the problem. The master problem is formulated as a set covering problem while the subproblem is formulated as a shortest path problem. Initial pairings which are sufficient to obtain a feasible solution, are produced using a linear programming model. The master problem, sub-problem and the model used to generate initial pairings are encoded by GAMS optimization program in an integrated manner and this integrated model is solved iteratively. The algorithm is applied to a private airline company's crew scheduling problem using real data and optimal crew schedules are obtained. [ABSTRACT FROM AUTHOR]

Abstract: A variety of analytical and simulation-based models and algorithms have been developed for the dynamic user equilibrium (DUE) traffic assignment problem. This paper aims to develop a theoretically sound simulation-based DUE model and its solution algorithm, with particular emphasis on obtaining solutions that satisfy the DUE conditions. The DUE problem is reformulated, via a gap function, as a nonlinear minimization problem (NMP). The NMP is then solved by a column generation-based optimization procedure which embeds (i) a simulation-based dynamic network loading model to capture traffic dynamics and determine experienced path travel costs for a given path flow pattern and (ii) a path-swapping descent direction method to solve the restricted NMP defined by a subset of feasible paths. The descent direction method circumvents the need to compute the gradient of the objective function in finding search directions, or to determine suitable step sizes, which is especially valuable for large-scale simulation-based DUE applications. Computational results for both small and large real road networks confirm that the proposed formulation and solution algorithm are effective in obtaining near-optimal solutions to the DUE problem. [Copyright &y& Elsevier]

With the popularity and development of mobile Internet, new transportation network services such as customized buses are expected to become a new way of popular transportation in crowded metropolises. In this paper, we propose a real-time customized bus routes design model. We divide the proposed model into two-stage problem to solve. The first stage problem is the vehicle route problem with time window (VRPTW), and the second stage problem is bilateral matching problem. For the first stage, the column generation algorithm is used to solve the problem. For the second stage, we solve it by the improved H-R bilateral matching algorithm. Finally, data on customized bus in Tianjin city of China are used to verify the accuracy of the real-time customized bus routes design model. The optimized results show that the average attendance rate has reached 70.8%, and the service rate of passenger has reached 88.5%. The results have proved that the real-time customized bus routes design model has a practical applicability in operation. [ABSTRACT FROM AUTHOR]