Mixed Integer Linear Programming Research Articles

Machine learning augmented branch and bound for mixed integer linear programming

Machine learning augmented branch and bound for mixed integer linear programming

A Mixed Integer Linear Programming Model for Optimal Sizing of Hydrogen Refueling Station Powered Using a PV-Grid System

A Mixed Integer Linear Programming Model for Optimal Sizing of Hydrogen Refueling Station Powered Using a PV-Grid System

Nature-Based Secondary Resource Recovery under Climate Change Uncertainty: A Robust Multi-Objective Optimisation Methodology

The management of high-volume (HV) waste poses a persistent challenge in sustainable materials management and represents an untapped opportunity in circular economy models. This study proposes a conceptual decision-making framework to operationalise a novel circular economy strategy for HV waste, involving temporary storage to facilitate nature-based secondary resource recovery. Using an illustrative case study of a candidate HV waste (legacy mining waste), we apply a robust multi-objective spatial optimisation approach at a national scale, employing an exact solution approach. Our methodology integrates mixed-integer linear programming to evaluate the economic viability, social benefits, and impacts of climate change uncertainties on nature-based solutions (NbS) implementation across diverse scenarios. The results demonstrate that NbS can enhance economic feasibility by incorporating carbon sequestration and employment benefits while demonstrating resilience against climate change projections to ensure long-term sustainability. The findings suggest that although NbS can improve the circular economy of HV nationally, it is essential to assess additional ecosystem services and address multiple uncertainties for effective macro-level sustainability assessment of HV management. This study offers a robust decision-making framework for policymakers and stakeholders to plan and implement nature-based circular economy strategies for HV waste streams at a national level while effectively managing long-term planning uncertainties.

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

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.

Task and motion planning using mixed integer linear programming for solving fetch-and-carry tasks by a mobile manipulator

This manuscript describes a TAsk and Motion Planning (TAMP) method for mobile manipulators. We focus on fetch-and-carry tasks, and we aim to simultaneously generate both a sequence of actions and a motion sequence for each action. The number of factors to be considered in solving such a problem, and the interactions among them are complex due to the multifaceted characteristics of mobile manipulation. As a result, straightforwardly performing simultaneous TAMP may require a significant amount of processing time. Therefore, we reduce the complexity of the problem by formulating fetch-and-carry planning for a Mixed Integer Linear Programming (MILP) problem. The proposed method can obtain the sequence of actions and the movement of the mobile manipulator in less than one second in many cases. The effectiveness of the proposed method is verified in environments in which delivery objects and obstacles are placed in various patterns, using a robot with an omnidirectional mobile platform and a serial link manipulator.

Balancing Staff Finishing Times vs. Minimizing Total Travel Distance in Home Healthcare Scheduling

Cost reduction and staff retention are important optimization objectives in home healthcare (HHC) systems. Home healthcare operators need to balance their objectives by optimizing resource use, service delivery and profits. Minimizing total travel distances to control costs is a common routing problem objective while minimizing total finishing time differences is a scheduling objective whose purpose is to enhance staff satisfaction. To optimize routing and scheduling, we propose mixed integer linear programming with a bi-objective function, which is a subset of the vehicle routing problem with time windows (VRPTWs). VRPTWs is a known NP-hard problem, and optimal solutions are very hard to obtain in practice. Metaheuristics offer an alternative solution to this type of problem. Our metaheuristic uses the simulated annealing algorithm and weighted sum approach to convert the problems to single-objective problems and is equipped with operators including swapping, moving, path exchange and ruin and recreate. The results show, firstly, that the algorithm can effectively find the Pareto front, and secondly, that minimizing total finishing time differences to balance the number of jobs per caretaker is an efficient way to tackle HHC scheduling. A statistical test shows that the algorithm can obtain the Pareto front with a lower number of weighted sum problems.

Investigation of a model predictive control (MPC) strategy for seasonal thermochemical energy storage systems in district heating networks

This research investigates the integration of model predictive control (MPC) with seasonal thermochemical energy storage systems (STES) within district heating networks, focusing on the Nottingham district heating as a case study. The primary aim is to develop an MPC strategy that utilizes machine learning models for accurate heat load forecasting. This strategy optimizes the charging and discharging cycles of thermochemical energy storage systems to mitigate the mismatch between heating energy supply and demand by storing surplus heat during summer and utilizing it during winter. We employed and validated machine learning models, including support vector regression (SVR), regression trees, and long short-term memory (LSTM) networks, using historical heat load and meteorological data. A validated numerical model of the thermochemical energy storage system (TCES) was integrated into the MPC framework, formulated as a mixed-integer linear program to optimize the STES's operations. The performance of the MPC strategy was benchmarked against a rule-based control approach under varying supply capacities to evaluate scalability and robustness. Our findings reveal that each machine learning model achieved comparable performance, with CVRMSE values within the 9–11% range. The LSTM model, in particular, provided accurate multi-step forecasts essential for the MPC framework. Incorporating these models into the MPC strategy allowed for precise heat demand predictions, enhancing the management of energy storage and distribution. Results confirmed that MPC effectively shifts energy seasonally, reduces reliance on auxiliary heating during winter, and minimizes waste heat. The MPC strategy outperformed the rule-based control by storing a significantly higher percentage of waste heat and meeting a greater portion of the additional heat demand that was not covered by the auxiliary heat supply. The system demonstrated effective performance under varying supply capacities, with the MPC strategy efficiently utilizing stored heat to meet demand at 80% supply capacity, achieving a waste heat reduction to 4% and meeting most of the heat demand. However, performance declined at 60% capacity, indicating the need for careful consideration of supply capacities in system design. This study highlights the potential of integrating machine learning models with MPC to enhance the performance and adaptability of district heating systems with STES, minimizing waste heat and efficiently meeting energy demands.

Benchmark of mixed-integer linear programming formulations for district heating network design

Optimal routing and investment decisions are key design criteria to reduce the high investment costs of district heating systems. However, these optimization problems have prohibitively high computational costs for large districts. Four different mixed-integer linear optimization frameworks are benchmarked in this study in order to compare their computational scaling. The frameworks exhibit significant differences in solving times for synthetic benchmarks and real-world urban districts of up to 9587 potential edges. The new open-source framework topotherm, developed for this work, exhibits the best computational performance when only one time step is optimized. The comparison between the models Résimont, DHmin, DHNx, and topotherm shows two main trends. First, fewer integer variables do not necessarily translate to lower solving times, and second, using redundant binary variables, which introduce symmetries into the constraints, leads to higher solving times. None of the considered optimization frameworks is able to solve the largest benchmark problems for five time steps within the allowed time limit and tolerance. These findings highlight the challenges of and pressing need to develop efficient models for simultaneous optimization of district heating network topology, pipe sizing, and operation.

A novel and cost-effective model for cloud energy storage based on a dual storage setup in active distribution grid with resilience consideration

The rising occurrences of natural disasters, terrorist actions, and cyber-attacks that result in extensive, long-lasting, and expensive disruptions have necessitated a shift in focus towards the resilience of electrical grids for network operators. However, the task of designing a resilient network remains complex and costly. One potential solution to bolster resilience is the deployment of battery energy storage devices on the consumer side, known as distributed energy systems (DES). Despite its effectiveness, the high construction costs and lengthy payback period associated with investing in energy storage devices have led consumers to exhibit reluctance in adopting them. Cloud energy storage (CES) is an innovative and cost-effective solution to address those challenges. In the CES platform, investors install storage facilities in the network which can be rented by consumers to fulfill their needs and they become holders of the virtual batteries. By adopting this approach, consumers are relieved from the burden of maintenance, repair, and installation. While a single CES facility offers reduced costs and increased comfort for consumers, it compromises the resilience of the grid when compared to the Distributed Energy Storage (DES) mechanism. In order to bridge this gap, this paper proposes a dual CES model which serves as an intermediate solution between DES and single CES. The dual CES model strikes a balance between the resilience of the grid and cost-effectiveness. It provides a higher level of resilience compared to a single CES and a lower level compared to DES. Additionally, the costs associated with the dual CES model fall between that of a single CES and DES. This model not only increases the profit margin for investors but also enhances the overall comfort and well-being of consumers compared to the single CES. To validate the proposed model, a Mixed Integer Linear Programming (MILP) problem is formulated and simulated on the IEEE 33 bus network. Three cases are considered: DES, single CES, and dual CES. The results indicate that the dual CES reduces consumers' costs by 28 %, losses by 27 %, unsupplied load costs by 45 %, and return on investment by 33 %. Moreover, it increases the stability margin time by 2 intervals and improves robustness by 47 %.

Digital Twin-Driven Dynamic Scheduling of Flexible Manufacturing System in the Context of Smart Factory Producing Brass Accessories

This study aims to deal with a dynamic scheduling problem of a real Flow Shop follow-up of an assembly process considering the specific constraints and requirements of a brass accessories manufacturing (BAM) company. We basically set up a Digital Twin-driven dynamic scheduling approach for Industry 4.0 by addressing uncertainties of machine availability. In fact, the setup of this Digital Twin (DT) is the outcome of the combination of both optimization and simulation. For the first step, we have elaborated a Mixed Integer Linear Programming (MILP) scheduling model by taking into account the dedicated requisites of our case study. Concerning simulation, a 3D simulation platform of a workshop producing brass accessories controlled by a Cyber-Physical Production System (CPPS) has been developed, in our previous work, including constraints and stochastic aspects. The simulation constraints are difficult or impossible to be modeled in the MILP model. These designs are integrated with the real workshop to construct the Digital Twin. The proposed tool enables the rescheduling of production orders based on machine disturbances and unpredictability. Validation scenarios have been designed and conducted to highlight the efficacy of the Digital Twin approach utilizing the brass accessories case study. We are confident that this represents the initial endeavor addressing the optimization of production rescheduling in a flow shop follow-up of a mixed assembly system.

Energy-aware production lot-sizing and parallel machine scheduling with the product-specific machining tools and power requirements

This study addresses a multi-product lot-sizing and scheduling problem with sequence-dependent setup times, considering that the machining operations cause energy consumption. The production facility comprises identical parallel machines under which the production of each product requires a certain set of tools. The energy requirement of production depends on the product-specific machining tools. The problem deals with determining the minimum cost lot-sizing and scheduling plan considering the energy capacity of the production facility. We formulate the problem as a mixed integer linear programming model by introducing energy consumption-related costs and constraints. We perform a case study on CNC milling and turning workshops. Further, we propose an heuristic approach combining a decomposition-based Simulated Annealing heuristic and Fix&Optimise algorithms to handle larger-sized problem instances. The computational performance of the proposed heuristic approach is evaluated against the proposed mixed integer linear programming model on a numerical study. Our numerical experiments reveal that the proposed heuristic approach is capable of providing cost-efficient solutions without compromising time efficiency.

An integrated approach for an aircraft routing and fuel tankering problem

In the highly competitive airline industry, it is crucial for airlines to minimise operating costs and strengthen their competitiveness. Fuel costs constitute a substantial portion of airline operating costs and are directly related to airlines' profits. In this research, we support the airlines' fuel management by presenting a model that integrates aircraft routing and fuel tankering decisions. The effectiveness of fuel tankering can be enhanced by considering the aircraft routing decisions together, leading to a significant cost reduction for airlines. The integrated model is developed as a mixed-integer linear programming model. The symmetry-breaking methods and decomposition-based heuristic algorithm are also proposed to alleviate the computational burden. A set of computational results illustrates the significant cost-saving effects of the proposed model and the heuristic algorithm.

Stochastic transitions of a mixed-integer linear programming model for the construction supply chain: chance-constrained programming and two-stage programming

Stochastic transitions of a mixed-integer linear programming model for the construction supply chain: chance-constrained programming and two-stage programming

Modeling cost-optimal fuel choices for truck, ship, and airplane fleets: The impact of sustainability commitments

For the next 25 years, numerous transport operators have introduced sustainability commitments to mitigate carbon emissions. However, static goals to reduce fossil fuel usage and adopt renewable fuel technologies pose economic risk. This study analyzes the costs of strategic fleet replacement through a mixed integer linear program. Comparing three Norwegian transport operators, we determine cost-optimal investments in fossil and renewable fuel technologies for two approaches: One integrating and the other neglecting sustainability commitments. Our findings reveal that the truck operator’s sustainability commitments incur minimal additional costs, with battery-electric trucks proving cost-effectiveness early. In contrast, ship and airplane operators exhibit significant differences in fleet replacement decisions, resulting in additional costs ranging from +6% to +31% for ships and +4% to +11% for airplanes, across fuel cost and carbon price scenarios. Norwegian carbon price regulations fall short in incentivizing a cost-competitive technology transition for ships and airplanes. Additional policies are needed to encourage gradual fleet replacement.

CCUS source-sink matching model based on sink well placement optimization

Realizing the dual-carbon target is a major strategic plan made by the Party Central Committee and a solemn commitment to address climate change of global significance. Source-sink matching technology in carbon capture, utilization, and storage (CCUS) technology is crucial to achieving the dual-carbon target. However, most previous studies constructed source-sink matching models based on sink-fixed sequestration potential, making it difficult to accurately assess the potential of sequestration sites and thus affect the source-sink matching scheme. Therefore, a dynamic storage potential assessment model has been established which can quickly assess the injection capacity of sinks under different well deployment schemes and then a source-sink matching model based on the overall optimization of sink deployment schemes has been constructed by combining the dynamic storage potential assessment model. The model considers the source matching problem as a mixed-integer linear programming (MILP) problem and obtains the computational results of the dynamic storage potential assessment model to realize the correct matching and simultaneous optimization of the source-sink pipeline network layout and the sink-well deployment scheme. With the proposed model, source-sink matching cases for national-level and project-level have been analyzed. It has been shown that the dynamic storage potential assessment model and the combined method improve the accuracy of the source-sink matching model in the form of MILP while minimizing the total cost and additional computations.

A Vehicle-to-Grid planning framework incorporating electric vehicle user equilibrium and distribution network flexibility enhancement

The rapid surge in electric vehicle (EV) adoption, coupled with advancements in charging technologies, emphasizes the critical necessity for expanding EV recharging infrastructure. Simultaneously, the Distribution Network (DN) encounters escalating challenges in meeting charging demand during peak traffic periods. Consequently, there is a mounting demand for the deployment of innovative Vehicle-to-Grid (V2G) technologies to augment the DN’s flexibility in power dispatch and alleviate travel costs for EV users. Hence, this paper proposes an EV-user-equilibrium-(UE)-constrained V2G planning framework that enhances flexibility in the DN. The framework aims to ascertain the optimal placement and capacity of EV charging stations (EVCSs) and V2G charging piles within the Transportation Network (TN). It takes into account the equilibrium condition stemming from competitive EV charging and routing behaviors alongside the optimal expansion of DN energy resources to accommodate the electricity supplied by the V2G piles. This study commences by analyzing EV drivers’ travel decisions, considering the influence of charging and V2G pile locations and sizes. Subsequently, we tackle the Traffic Assignment Problem with User Equilibrium (TAP-UE) model to characterize the steady-state traffic flow distribution of EVs. Following this, we formulate the optimization model for the Coordinated Power and Transportation Network (CPTN), which encompasses the optimal expansion of DN facilities and traffic flow regulation under UE conditions. To mitigate the computational complexity associated with the V2G planning model, we introduce a series of linearization methods to obtain a manageable Mixed-Integer Linear Programming (MILP) solution. Finally, to validate the efficacy of our proposed planning framework, we apply it to two test systems, including a real-world case study. Through these case studies, we explore the necessity and potential benefits of V2G technologies.

Research on Train Loading and Unloading Mode and Scheduling Optimization in Automated Container Terminals

In some automated container terminals, railway lines have been implemented into the port, saving container transfer time. However, the equipment scheduling level of the railway yard needs to be improved for managers. In the equipment scheduling of loading and unloading containers for railway trains, the operation modes “full unloading and full loading” and “synchronous loading and unloading” are often adopted. Due to the long length of the railway yard and the line of one train, there are two ways to arrange loading and unloading tasks for automated rail-mounted gantry cranes (ARMGs): one is to pre-assign tasks for ARMGs, and the other is to not pre-assign tasks for ARMGs. To investigate the efficacy of these different operation modes and methods of assigning tasks, this study formulated three mixed-integer linear programming (MILP) models with the goal of minimizing the ARMG task completion time. An adaptive large neighborhood search algorithm was used to tackle the scheduling problem. The scheduling effects of different operation modes and methods for assignment tasks were compared in terms of their calculation time and the completion time of ARMG tasks. Notably, the findings reveal that, with an increase in the number of tasks, the “pre-assign” task arrangement had a limited effect on the completion time of the ARMG tasks, made the calculation time shorter, and reduced the complexity of the problem. From the perspective of the completion time of ARMG tasks, the time under the “synchronous loading and unloading” operation mode was less than that of the “full unloading and full loading” operation mode. Therefore, it is recommended that the managers of the railway yard in an automated container terminal adopt the “synchronous loading and unloading” operation mode but determine the task assignment method according to decision time requirements. In addition, when the number of tasks is large, to decrease the time to complete ARMG tasks, the manager can adopt the “non-pre-assign” task distribution method.

Carbon neutral energy systems: Optimal integration of energy systems with CO2 abatement pathways

AbstractReducing emissions requires transitioning towards decarbonized systems through avoiding, processing, or offsetting. Decisions on system design are associated with high costs which can be reduced at the planning stage through optimization. The temporal variations in power demand and renewable energy supply significantly impact the design of a low‐emissions energy system. Effective decision‐making must consider such impact in a comprehensive framework that accounts for the potential synergies between different options. This work presents a mixed integer linear programming model that considers the impacts of energy supply and demand dynamics to optimize the design and operation of an integrated energy system while adhering to a set emissions limit. The model integrates renewable power with CO2 capture, utilization, and sequestration by considering H2 production and storage. The case study showed including negative emissions technologies and CO2 capture and processing with renewable energy allows achieving net zero emissions power.

THE MODIFIED MEAN ABSOLUTE DEVIATION INTEGER LINEAR PROGRAMMING MODEL FOR GLOBAL PORTFOLIO ASSET ALLOCATION

This study develops a practical yet robust Mean Absolute Deviation (MAD) Mixed-Integer Linear Programming (MILP) model for a global investment portfolio tailored to the constraints faced by Malaysian retail investors, such as transactional fees and foreign currency exchange spreads, to provide more accurate estimations of portfolio returns. The MAD ILP model is modified to address these factors, aiming to enhance utility and efficiency. The study utilizes a Maybank 12-month fixed deposit cash account and ten selected Exchange-Traded Funds (ETFs) from iShares, Vanguard, and State Street Global Advisors for global portfolio construction. Forecasting techniques were utilized to generate expected returns for the ETFs and foreign currency exchange spreads. Based on the MAD model proposed by Konno and Wijayanayake (2001), the study introduces three modifications: incorporating transaction cost elements specific to Malaysian investors, including foreign currency exchange spreads in return calculations, and transforming the model from mixed-integer to pure integer model to accommodate minimum transaction lot constraint. Verification and validation exercises demonstrate the model’s ability to replicate real-world portfolio construction and reliable simulations. The model offers a practical tool for self-ascribed investors aiming to optimize global investment portfolios while considering unique constraints and transaction costs.

Short-term load distribution model for giant cascade serial diversion-type hydropower stations

With the development of complex cascade serial diversion-type hydropower plants (CSDHP) in Southwest China, it has become imperative to study short-term scheduling challenges that are distinctive to this type of hydropower facility. Unlike conventional cascade hydropower stations, which have large reservoirs to regulate the outflow between stations, the CSDHP system consists of several diversion-type hydropower plants connected in series, requiring strict matching of turbine discharge between upstream and downstream facilities. In addition, these plants impose complex operation constraints, such as multiple units sharing one tunnel, which greatly increases the difficulty of short-term scheduling. This paper proposes a short-term optimal scheduling model for CSDHP systems aimed at achieving efficient load distribution among units while minimizing water consumption. The model takes into account the strict matching of turbine discharge and the complex constraints of multiple units sharing one tunnel. Subsequently, the model is transformed into a mixed integer linear programming problem using specified linearization techniques, with complexity further addressed via an iterative resolution method. A case study of a cascade illustrates the model’s efficacy in facilitating effective unit load distribution under complex hydraulic and electrical constraints, which substantially reduces the system’s water consumption by up to 6.1% during the dry season and 1.8% during the flood season, and reducing the number of crossing forbidden zones by 50.0% and 52.0%, respectively. The result shows the models’ potential to enhance day-ahead scheduling for the CSDHP system.