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mixed-integer linear programming for computing optimal experimental designs

JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2025年 234卷

作者： Harman, Radoslav Rosa, Samuel Comenius Univ Bratislava Slovakia

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

关键词： Optimal design Exact design mixed-integer linear programming A-optimality G-optimality

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A multi-echelon globalized agro-industrial supply chain under conditions of uncertainty: A two-stage fuzzy-possibilistic mixed-integer linear programming model

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EXPERT SYSTEMS WITH APPLICATIONS 2025年 270卷

作者： Garrido, Alexander Cardenas-Barron, Leopoldo Eduardo Buitrago, Oscar Yecid Alvarez-Pomar, Lindsay Univ Rosario Sch Engn Sci & Technol Bogota 111711 Colombia Tecnol Monterrey Sch Engn & Sci Dept Ind & Syst Engn E Garza Sada 2501 Sur Monterrey 64849 Nuevo Leon Mexico Univ Mil Nueva Granada Ind Engn Dept Bogota Colombia Univ Distrital Francisco Jose de Caldas Ind Engn Dept Bogota 110111 Colombia

Supply chain (SC) design problem is probably one of the most critical challenges in the production/operations management field, with lasting effects on customer service and cost measurements. Managers are therefore obliged to find the best possible SC configuration that provide the highest level of responsiveness and operational efficiency, which subsequently improves customer satisfaction and fosters loyalty. In that vein, this research proposes a general approach to determine a resilient strategic design-i.e., able to rapidly adjust to market changes and disruptions, offering a competitive edge in an increasingly globalized economy-of a singular multilayered and multi-product globalized agro-industrial SC with direct shipping in conditions of uncertainty. In order to obtain the shortest distance to move and store products from the supplier echelon-or crops-to the last echelon in the SC-or customers-, a two-stage fuzzy-possibilistic mixed-integer linear programming (FP-MILP) model is formulated and numerically evaluated through different restrictive conditions. The proposed FP-MILP model-based on the blending-problem approach-incorporates deterministic and fuzzy datasets in the objective function and constraints, efficiency calculations in crops and feedstocks, as well as different thermal floors and cultivation geographic areas. To illustrate the application of the FP-MILP model, several experiment scenarios are performed on a real four-tier globalized agro-industrial SC, and the results obtained are analyzed and compared with each other.

关键词： Supply chain resilient design Fuzzy possibilistic programming Agro-industrial supply chain Fuzzy mathematical programming mixed-integer linear programming

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Reactive tabu search and mixed-integer linear programming for multi-day assignment, scheduling, and routing problems of specialised education and home-care services

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INTERNATIONAL JOURNAL OF PRODUCTION RESEARCH 2025年第5期63卷 1779-1802页

作者： Saleh, Mira Bou Chariete, Abderrahim Schwartz, Leo Grunder, Olivier El Hassani, Amir Hajjam Univ Lorraine LORIA F-54000 Nancy France Univ Technol Belfort Montbeliard SINERGIES F-90000 Belfort France

In this paper, we address the Multi-Day Assignment, Scheduling, and Routing Problem for Specialized Education and Home Care Services (SEHCS-MASRP), which involves heterogeneous employees and missions, posing a complex optimisation challenge. To tackle this, we propose a novel mixed-integer linear programming (MILP) model that considers employee qualifications, service requirements, scheduling constraints, routing decisions, and multiple objectives across the planning horizon. Additionally, we develop two metaheuristic approaches: a Reactive Tabu Search (RTS) algorithm incorporating either a Probabilistic Greedy Heuristic (PGH) or a Greedy Randomized Adaptive Search Procedure (GRASP) for initial solutions and a tailored genetic algorithm (GA). The three approaches aim to minimise wasted and overtime hours, total travel distances, and the number of assignments with an unsatisfied specialty while balancing wasted hours, overtime hours, and travel distances among the employees. Gurobi uses the proposed MILP model to find the optimal solutions, which are then compared with RTS and GA results across various instance sizes based on real-life SEHCS scenarios. Experimental results demonstrate the efficiency of MILP, RTS, and GA. MILP achieves proven optimal solutions for smaller to large instances. For huge instances, RTS generates high-quality solutions within reasonable computing times, outperforming GA performance. Notably, RTS consistently finds solutions within 5% of optimality for most instances.

关键词： mixed-integer linear programming reactive tabu search genetic algorithm assignment problems specialised education and home care services multi-day planning horizon

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Integrating Energy-Efficient Train Control in railway Vertical Alignment Optimization: A novel mixed-integer linear programming approach

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TRANSPORTATION RESEARCH PART C-EMERGING TECHNOLOGIES 2025年 171卷

作者： Sun, Yichen Ni, Shaoquan Chen, Dingjun He, Qing Xu, Shuangting Gao, Yan Chen, Tao Southwest Jiaotong Univ Sch Transportat & Logist Chengdu 611756 Sichuan Peoples R China Southwest Jiaotong Univ Natl United Engn Lab Integrated & Intelligent Tran Chengdu 611756 Sichuan Peoples R China Southwest Jiaotong Univ Natl Engn Lab Integrated Transportat Big Data Appl Chengdu 611756 Sichuan Peoples R China Southwest Jiaotong Univ Sch Civil Engn Chengdu 611756 Sichuan Peoples R China Southwest Jiaotong Univ Key Lab High Speed Railway Engn Minist Educ Chengdu 611756 Sichuan Peoples R China

Incorporating train control into the railway design process enables a practical and comprehensive evaluation of the lifecycle utility of a track profile. This paper proposes a novel integrated approach, termed EETC-VAO, which combines railway track Vertical Alignment Optimization (VAO) and Energy-Efficient Train Control (EETC). Initially formulated as a mixed- integer Nonlinear programming (MINLP) problem, EETC-VAO aims to meet various geometric constraints and simultaneously minimize construction costs, traction energy consumption, and section running times in both directions. The model is subsequently reformulated into an equivalent mixed-integer linear programming (MILP) model using linearization methods and is further enhanced with valid inequalities, logic cuts, and a warm start algorithm with random velocity generation. The model has been extensively tested across a variety of case studies and train types, from synthetic small-scale scenarios to challenging real-world cases spanning from 3 to 71.2 km. Our findings demonstrate that operational costs can be significantly reduced with only marginal increases in construction costs. The integrated approach achieves reductions in total lifecycle costs of up to 40%, revealing a critical trade-off between construction and operational expenses. Notably, our results also indicate that lower construction costs do not inherently conflict with reduced operational costs, emphasizing the critical importance of integrating the train control scheme into the VAO problem.

关键词： Railway Vertical Alignment Optimization Energy-Efficient Train Control mixed-integer linear programming Optimization Lifecycle cost

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mixed-integer linear programming Approaches for Tree Partitioning of Power Networks

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IEEE Transactions on Control of Network Systems 2025年

作者： Lan, Leon Zocca, Alessandro Amsterdam University College Amsterdam Netherlands California Institute of Technology California United States

In transmission networks, power flows and network topology are deeply intertwined due to power flow physics. Recent literature shows that a specific more hierarchical network structure can effectively inhibit the propagation of line failures across the entire system. In particular, a novel approach named tree partitioning has been proposed, which seeks to bolster the robustness of power networks through strategic alterations in network topology, accomplished via targeted line switching actions. Several tree partitioning problem formulations have been proposed by considering different objectives, among which power flow disruption and network congestion. Furthermore, various heuristic methods based on a two-stage and recursive approach have been proposed. The present work provides a general framework for tree partitioning problems based on mixed-integer linear programming (MILP). In particular, we present a novel MILP formulation to optimally solve tree partitioning problems and also propose a two-stage heuristic based on MILP. We perform extensive numerical experiments to solve two tree partitioning problem variants, demonstrating the excellent performance of our solution methods. Lastly, through exhaustive cascading failure simulations, we compare the effectiveness of various tree partitioning strategies and show that, on average, they can achieve a substantial reduction in lost load compared to the original topologies. © 2014 IEEE.

关键词： mixed-integer linear programming

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A mixed-integer linear programming Model for Addressing Efficient Flexible Flow Shop Scheduling Problem with Automatic Guided Vehicles Consideration

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APPLIED SCIENCES-BASEL 2025年第6期15卷 3133-3133页

作者： Wang, Dekun Wu, Hongxu Zheng, Wengang Zhao, Yuhao Tian, Guangdong Wang, Wenjie Chen, Dong Zhengzhou Univ Sch Mech & Power Engn Zhengzhou 450001 Peoples R China Beijing Univ Civil Engn & Architecture Sch Machinery & Vehicles Beijing 100044 Peoples R China

With the development of Industry 4.0, discrete manufacturing systems are accelerating their transformation toward flexibility and intelligence to meet the market demand for various products and small-batch production. The flexible flow shop (FFS) paradigm enhances production flexibility, but existing studies often address FFS scheduling and automated guided vehicle (AGV) path planning separately, resulting in resource competition conflicts, such as equipment idle time and AGV congestion, which prolong the manufacturing cycle time and reduce system energy efficiency. To solve this problem, this study proposes an integrated production-transportation scheduling framework (FFSP-AGV). By using the adjacent sequence modeling idea, a mixed-integer linear programming (MILP) model is established, which takes into account the constraints of the production process and AGV transportation task conflicts with the aim of minimizing the makespan and improving overall operational efficiency. Systematic evaluations are carried out on multiple test instances of different scales using the CPLEX solver. The results show that, for small-scale instances (job count <= 10), the MILP model can generate optimal scheduling solutions within a practical computation time (several minutes). Moreover, it is found that there is a significant marginal diminishing effect between AGV quantity and makespan reduction. Once the number of AGVs exceeds 60% of the parallel equipment capacity, their incremental contribution to cycle time reduction becomes much smaller. However, the computational complexity of the model increases exponentially with the number of jobs, making it slightly impractical for large-scale problems (job count > 20). This research highlights the importance of integrated production-transportation scheduling for reducing manufacturing cycle time and reveals a threshold effect in AGV resource allocation, providing a theoretical basis for collaborative optimization in smart factories.

关键词： integrated scheduling flexible flow shop scheduling makespan AGV scheduling mixed-integer linear programming exact algorithm

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A Timer and mixed integer linear programming Load Shedding Scheme for Resilient DC Microgrids

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IEEE ACCESS 2025年 13卷 6632-6642页

作者： Babagana, Abdulrahman Jimoh, Isah A. Seferi, Yljon Burt, Graeme Univ Strathclyde Inst Energy & Environm Glasgow G1 1XW Scotland

Islanded DC microgrids are vulnerable to voltage instability caused by excessive power demand, which can adversely impact downstream consumers and disrupt overall microgrid operation. Existing load-shedding techniques face limitations such as over-shedding due to fixed voltage thresholds and time delays, predetermined load-shedding actions that fail to account for disturbance magnitude, and delayed stabilization caused by sequential load-shedding steps. To address these challenges, this paper proposes a novel load-shedding strategy for islanded DC microgrids that integrates a short-timer mechanism with mixed integer linear programming (MILP) optimization. The proposed approach reduces reliance on communication systems and achieves optimal load-shedding decisions using local voltage measurements. Simulation results on a DC microgrid model adapted from the IEEE 37-bus network demonstrate the effectiveness of the proposed scheme. The scheme results in a 20% reduction in unnecessary load shedding, an 18% improvement in voltage stabilization (measured as the final voltage after a disturbance), and a 25% decrease in response time compared to conventional methods. The results show that the proposed strategy ensures that the DC bus voltage remains above the critical threshold of 720 V, enhancing system reliability by minimizing voltage transients, reducing regulation time, and maintaining power balance. These improvements highlight the potential of the proposed scheme to support robust, secure, and resilient DC microgrid operation.

关键词： DC microgrid distributed energy resources load shedding mixed-integer linear programming optimization resilience DC microgrid distributed energy resources load shedding mixed-integer linear programming optimization resilience

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APOLLO-MILP: AN ALTERNATING PREDICTIONCORRECTION NEURAL SOLVING FRAMEWORK FOR mixed-integer linear programming

arXiv

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arXiv 2025年

作者： Liu, Haoyang Wang, Jie Geng, Zijie Li, Xijun Zong, Yuxuan Zhu, Fangzhou Hao, Jian Ye Wu, Feng MoE Key Laboratory of Brain-inspired Intelligent Perception and Cognition University of Science and Technology of China China Noah’s Ark Lab Huawei Technologies Hong Kong Tianjin University China Shanghai Jiao Tong University China

Leveraging machine learning (ML) to predict an initial solution for mixed-integer linear programming (MILP) has gained considerable popularity in recent years. These methods predict a solution and fix a subset of variables to reduce the problem dimension. Then, they solve the reduced problem to obtain the final solutions. However, directly fixing variable values can lead to low-quality solutions or even infeasible reduced problems if the predicted solution is not accurate enough. To address this challenge, we propose an Alternating prediction-correction neural solving framework (Apollo-MILP) that can identify and select accurate and reliable predicted values to fix. In each iteration, Apollo-MILP conducts a prediction step for the unfixed variables, followed by a correction step to obtain an improved solution (called reference solution) through a trust-region search. By incorporating the predicted and reference solutions, we introduce a novel Uncertainty-based Error upper BOund (UEBO) to evaluate the uncertainty of the predicted values and fix those with high confidence. A notable feature of Apollo-MILP is the superior ability for problem reduction while preserving optimality, leading to high-quality final solutions. Experiments on commonly used benchmarks demonstrate that our proposed Apollo-MILP significantly outperforms other ML-based approaches in terms of solution quality, achieving over a 50% reduction in the solution gap. © 2025, CC BY.

关键词： mixed-integer linear programming

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Optimal Operation of a Tablet Pressing Machine Using Deep-Neural-Network-Embedded mixed-integer linear programming

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INVENTIONS 2025年第2期10卷 29-29页

作者： Li, Jialong Wu, Lan Qin, Yuang Zhi, Haojun Univ Technol Sch Elect Engn Henan Zhengzhou 450007 Peoples R China Univ Liverpool Sch Elect Engn Elect & Comp Sci Liverpool L69 3GJ England

This paper presents a deep neural network (DNN)-embedded mixed-integer linear programming (MILP) model for fault prediction and production optimization in tablet pressing machines. The DNN predicts the probability of failures during the tablet pressing process by analyzing key operational parameters such as pressure, temperature, humidity, speed, vibration, and number of maintenance cycles. The MILP model optimizes the temperature and humidity settings, production schedules, and maintenance planning to maximize total profit while minimizing penalties for fault pressing, energy consumption, and maintenance costs. To integrate DNN into the MILP framework, Big-M constraints are applied to linearize the Rectified linear Unit (ReLU) activation functions, ensuring solvability and global optimality of the optimization problem. A case study using the Kaggle dataset demonstrates the model's ability to dynamically adjust production and maintenance schedules, enhancing profitability and resource utilization under fluctuating electricity prices. Sensitivity analyses further highlight the model's robustness to variations in maintenance and energy costs, striking an effective balance between cost efficiency and production quality, which makes it a promising solution for intelligent scheduling and optimization in complex manufacturing environments.

关键词： deep neural network mixed-integer linear programming fault prediction tablet pressing machines maintenance scheduling manufacturing systems

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mixed-integer linear programming Formulation for Short-term Scheduling of Cascaded Hydroelectric Plants with Pumped-storage Units

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ELECTRIC POWER COMPONENTS AND SYSTEMS 2013年第15期41卷 1456-1468页

作者： Jia, Jiangtao Xi An Jiao Tong Univ Syst Engn Inst Xian 710049 Peoples R China

In this article, a short-term scheduling model for cascaded hydroelectric chain plants with pumped-storage units is discussed, which considers in detail the various factors of the hydro system and units. A mixed-integer linear programming formulation is established. The main contribution of the article is that a systematic method is established such that the complex multivariable production functions of hydro units can be approximated by a piecewise linear function, and the characteristics of pumped-storage units are described by introducing some ancillary integer variables. Numerical testing results show that the mixed-integer linear programming formulation given in this article is efficient and effective.

关键词： cascaded hydro systems short-term scheduling non-linear production function pumped-storage units mixed-integer linear programming

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