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mixed-integer linear programming for project scheduling under various resource constraints

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2024年第1期319卷 79-88页

作者： Klein, Nicklas Gnagi, Mario Trautmann, Norbert Univ Bern Dept Business Adm Engehaldenstr 4 CH-3012 Bern Switzerland

Project scheduling is an important management task in many companies across different industries. Generally, projects require resources, such as personnel or funds, whose availabilities are limited, giving rise to the challenging problem of resource-constrained project scheduling. In this paper, we consider the scheduling of a project consisting of precedence-related activities that require time and two types of resources for execution: storage resources representing, e.g., the project budget;and renewable resources representing, e.g., personnel or equipment. Storage resources are consumed by activities at their start or produced upon their completion, while renewable resources are allocated to activities at their start and released upon their completion. The resource-constrained project scheduling problem with consumption and production of resources (RCPSP-CPR) consists of determining a minimum-makespan schedule such that all precedence relations are respected, the demand for each renewable resource never exceeds its capacity, and the stock level of each storage resource never falls below a prescribed minimum. Due to the consideration of storage resources, the feasibility variant of this problem is NP-complete. We propose a novel compact mixed-integer linear programming (MILP) model based on a novel type of sequencing variables. These variables enable us to identify which activities are processed in parallel and whether a sequencing of activities is necessary to respect the resource capacities. Our computational results indicate that our novel model significantly outperforms state-of-the-art MILP models for all considered scarcity settings of the storage resources. Additionally, our results indicate a superior performance for instances of the well-known resource-constrained project scheduling problem (RCPSP).

关键词： Project scheduling mixed-integer linear programming Resource-constrained project scheduling Consumption and production of resources

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mixed-integer linear programming for computing optimal experimental designs

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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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mixed-integer linear programming, constraint programming and column generation approaches for operating room planning under block strategy

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APPLIED MATHEMATICAL MODELLING 2022年 105卷 438-453页

作者： Ghandehari, Najmeh Kianfar, Kamran Univ Isfahan Fac Adm Sci & Econ Dept Management Esfahan *** Iran Univ Isfahan Fac Engn Esfahan *** Iran

The planning of operating rooms under block strategy is addressed in this study. The decisions are about opening the operating rooms and assigning specialties and surgeons to blocks at the tactical level, and sequencing the surgeries at the operational level. This problem aims to minimize the costs of opening the operating rooms and their overtime, the waiting costs of patients, and the surgeons' idle costs. We propose two mixed integer linear programming models, a constraint programming (CP) model and a constraint programming-based column generation (CPCG) method for handling the problem. The performance of the models is evaluated by random test instances. The results indicate that CP and CPCG models are more efficient than the linear programming models in terms of computational time, and the number of variables and constraints. The proposed method CPCG generates optimal solutions for problem instances of up to 30 surgeries in less than 4 min. The CP model finds the optimal solutions in about one minute but proving the optimality of the found solutions is time-consuming in some instances. The maximum optimality gap for the proposed two-step linear programming model is 2%, while its run time is less than 20 s. A sensitivity analysis is done on the main parameters of the problem like objectives' weights, opening cost of ORs, unit waiting cost of patients, and the maximum available time in surgery blocks. Among the three objectives, the unit waiting cost of patients has the most sensitivity to variations of the objective function weights.(c) 2022 Elsevier Inc. All rights reserved.

关键词： Operating room planning Block strategy Master surgery schedule mixed-integer linear programming Constraint programming Column generation

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mixed-integer linear programming models for the type-II resource-constrained assembly line balancing problem

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ASSEMBLY AUTOMATION 2022年第5期42卷 585-594页

作者： Michels, Adalberto Sato Costa, Alysson M. Univ Melbourne Sch Math & Stat Melbourne Vic Australia

Purpose Resource-constrained assembly lines are widely found in industries that manufacture complex products. In such lines, tasks may require specific resources to be processed. Therefore, decisions on which tasks and resources will be assigned to each station must be made. When the number of available stations is fixed, the problem's main goal becomes the minimisation of cycle time (type-II version). This paper aims to explore this variant of the problem that lacks investigation in the literature. Design/methodology/approach In this paper, the authors propose mixed-integer linear programming (MILP) models to minimise cycle time in resource-constrained assembly lines, given a limited number of stations and resources. Dedicated and alternative resource types for tasks are considered in different scenarios. Findings Besides, past modelling decisions and assumptions are questioned. The authors discuss how they were leading to suboptimal solutions and offer a rectification. Practical implications The proposed models and data set fulfil more practical concerns by taking into account characteristics found in real-world assembly lines. Originality/value The proposed MILP models are applied to an existing data set, results are compared against a constraint programming model, and new optimal solutions are obtained. Moreover, a data set extension is proposed due to the simplicity of the current one and instances up to 70 tasks are optimally solved.

关键词： Assembly line balancing Resource constraints Type-II problem mixed-integer linear programming

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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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Accelerating operation optimization of complex chemical processes: A novel framework integrating artificial neural network and mixed-integer linear programming

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CHEMICAL ENGINEERING JOURNAL 2024年 481卷

作者： Zhou, Jianzhao Shi, Tao Ren, Jingzheng He, Chang Hong Kong Polytech Univ Dept Ind & Syst Engn Hong Kong Peoples R China Hong Kong Polytech Univ Res Ctr Resources Engn Carbon Neutral Kowloon Hong Kong Peoples R China Hong Kong Polytech Univ Res Inst Adv Mfg Dept Ind & Syst Engn Hung HomKowloon Hong Kong Peoples R China Sun Yat Sen Univ Guangdong Engn Ctr Petrochem Energy Conservat Sch Mat Sci & Engn Guangzhou 510275 Peoples R China

We present an automated framework that integrates rectified linear unit activated artificial neural network (ReLU-ANN) and mixed-integer linear programming (MILP) to enable efficient operational-level optimization of complex chemical processes. Initially, data is generated through rigorous simulations to pre-train surrogate models based on ReLU-ANN (classification and regression), and subsequently, MILP is employed for optimization by linearly formulating these models. This novel framework efficiently handles complex convergence constraints through a classification neural network which will be used for high-throughput screening data for regression, while simultaneously implementing an 'optimizing while learning' strategy. By iteratively updating the neural network based on optimization feedback, our approach streamlines the optimization process and ensure the feasibility of optimum solution. To demonstrate the versatility and robustness of our proposed framework, we examine three representative chemical processes: extractive distillation, organic Rankine cycle, and methanol synthesis. Our results reveal the framework's potential in enhancing optimization effect while concurrently reducing computational time, surpassing the capabilities of typical optimization algorithms. As for the three processes, optimization effectiveness improved by 10.11%, 28.69%, and 5.45%, respectively, while execution time were reduced by 71.71%, 54.49%, and 59.38%. This notable enhancement in optimization efficiency stems from a substantial reduction in costly while ineffective objective function evaluations. By seamless integration of ReLU-ANN and MILP, our proposed framework holds promise for improving the optimization of complex chemical processes, yielding superior results within significantly reduced timeframes compared to traditional approaches.

关键词： Distillation Organic Rankine cycle Methanol synthesis Operational level optimization Artificial neural network 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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More mixed-integer linear programming models for solving three-stage remanufacturing system scheduling problem

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COMPUTERS & INDUSTRIAL ENGINEERING 2024年 194卷

作者： Wang, Wenjie Tian, Guangdong Luo, Mengqi Zhang, Honghao Yuan, Gang Niu, Kejia Zhengzhou Univ Sch Mech & Power Engn Zhengzhou 450001 Peoples R China Beijing Univ Civil Engn & Architecture Sch Machinery & Vehicles Beijing 100044 Peoples R China Shandong Univ Sch Mech Engn Jinan 250061 Peoples R China Southeast Univ Sch Mech Engn Nanjing 212000 Peoples R China Henan Univ Sci & Technol Sch Appl Engn Sanmenxia 472199 Peoples R China

This paper aims to deal with the three-stage remanufacturing system scheduling problem (3T-RSSP) that minimizes the maximum completion time i.e., makespan. To this end, inspired by distinct modeling ideas, four mixedinteger linear programming (MILP) models are formulated. The modeling ideas used in this paper are sequencebased idea, position-based idea, time-based idea, and adjacent sequence-based idea. Numerical experiments are executed on a set of small-scaled, medium-scaled and large-scaled instances to test the performance of these formulated MILP models in terms of size complexity and computation complexity. Experimental results demonstrate that the sequence-based MILP model (Model 1) is the most efficient one while the time-based MILP model (Model 3) is worst one. The Wilcoxon signed-rank test is also conducted to evaluate the four models. Model 1 is further compared with different optimization solvers, heuristic and some state-of-art heuristics. Comparison results verify the effectiveness and good portability of the MILP model. Moreover, some key parameters contained in 3T-RSSP including end-of-life (EOL) products count, reprocessing lines count, disassembly workstations/reassembly workstations count are also evaluated. Notably, it is concluded that the increase of disassembly workstations/reassembly workstations count can improve the three-stage remanufacturing system's flexibility and contribute to the enhancement of production efficiency.

关键词： Three-stage remanufacturing system scheduling Makespan mixed-integer linear programming Modeling and optimization

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