Mixed Integer Linear Programming Research Articles

A reformulation approach to resolving the inconsistency of max–min equations on fuzzy algebra

Modelling with fuzzy relations in approximate reasoning is obstructed sometimes by the inconsistency of obtained fuzzy relational equations. This paper tackles the inconsistency resolving problem for a finite system of max–min equations by modifying only the right-hand side vector as slightly as possible with respect to the sum of absolute deviations. It is demonstrated that this problem may be reformulated equivalently as a polynomial-sized mixed integer linear programming problem. Although such a reformulation results in a problem of much larger size than its original compact form, it may be solved to optimality on instances of moderate size or even large size by an off-the-shelf solver for mixed integer linear programming and in some sense does not require a tailored solving method.

Land layout optimisation for virtual land sales in the metaverse: two-dimensional assortment problem

With the growing popularity of the metaverse, the virtual real estate in the metaverse has generated immense investment enthusiasm. Considering the virtual land sale mode allowing the potential buyers to request the purchase of rectangular land parcels of a specified size, this study investigates how to lay out the requested land parcels for minimising the size of the required rectangular open land, which can be formulated completely as the two-dimensional assortment problem (2DAP). Due to the strong NP-hardness of the 2DAP, an effective and efficient heuristic solution approach named binary adjoining algorithm (BAA) is presented for tackling the 2DAPs in large scales. The conducted computational experiments show that the BAA can outperform the state-of-the-art piecewise-linearisation mixed integer linear programming model as well as four existing advanced metaheuristic techniques designed for the 2DAP, in both solution quality and computational time, on the small-size instances. The superiority of the BAA over either foregoing reference method on the large-size instances with up to 60 requested land parcels is also demonstrated.

Integrating Energy Generation and Demand in the Design and Operation Optimization of Energy Communities

The optimization of the energy system serving users’ aggregations at urban level, such as Energy Communities, is commonly addressed by optimizing separately the set of energy conversion and storage systems from the scheduling of energy demand. Conversely, this paper proposes an integrated approach to include the demand side in the design and operation optimization of the energy system of an Energy Community. The goal is to evaluate the economic, energetic, and environmental benefits when users with different demands are aggregated, and different degrees of flexibility of their electricity demand are considered. The optimization is based on a Mixed-Integer Linear Programming approach and is solved multiple times by varying (i) the share of each type of user (residential, commercial, and office), (ii) the allowed variation of the hourly electricity demand, and (iii) the maximum permitted CO2 emissions. Results show that an hourly flexibility of up to 50% in electricity demand reduces the overall system cost and the amount of energy withdrawn from the grid by up to 25% and 31%, respectively, compared to a non-flexible system. Moreover, the aggregation of users whose demands match well with electricity generation from renewable sources can reduce CO2 emissions by up to 30%.

From features to benefits: a data-driven approach for the economic design of adaptive quality control

The current economic design of control charts assumes specific quality distributions, limits parameter choices, and over-relies on historical samples, hindering companies from determining the most economical parameters accurately. Leveraging industrial big data, we propose a data-driven, mixed-integer linear programming model for the economic design of adaptive control charts. Control limits are dynamically designed as a function of features to minimise quality costs. Considering the trade-off between false alarms and penalty costs, we develop three models: a basic model incorporating big data, a model with cost-penalised features, and a model that uses regularisation to manage overfitting. We simulate the model using new performance measures. Our findings demonstrate the economic value of adaptive control limits strategies incorporating feature data compared to benchmarks. We expanded the model to an endogenous sample size and sampling interval framework, further demonstrating the superiority of our approach. We undertook a case study using real-world data from a casting company and revealed that employing our approach culminates in a 24.6% reduction in costs relative to the company's existing quality control protocols. Our approach enables manufacturers to make strategic decisions about quality control by operationalising big data, thereby proving advantageous in reducing quality costs.

Impact of accelerated review policy on portfolio planning of vaccine companies

BackgroundWith the introduction of the accelerated drug review policy in China, the clinical research and development time and the review and approval time of drugs have been shortened accordingly. Especially under the influence of the COVID-19 pandemic, the vaccine formulations released through the accelerated review policy are springing up, and the question of how the accelerated review policy affects the investment portfolio of vaccine enterprises has also attracted more and more attention.Aims and methodsThe article uses mixed-integer linear programming to develop a new model on portfolio planning for vaccine companies based on the accelerated review policy context. The model is constructed using the Gurobi extension class of .NET, and the investment decision is made and simulated by the Gurobi solver to investigate the portfolio planning decision of a vaccine company maximizing the net present value of its vaccine production portfolio with the increase of available capital over a 20-year time horizon.ResultsThe NPV under the accelerated review policy is significantly higher than the net present value under the standard review policy when the available capital exceeds RMB 900 million. And the difference between the two of them peaks at RMB 1.87 billion when the available capital is RMB 1.9 billion; break-even occurs about 1.3 years earlier in the accelerated review policy than in the standard review; and when the available capital is the same, firms in the accelerated review policy choose to produce four products earlier and make the decision to invest in facility construction earlier; scenarios in the accelerated review policy are not as sensitive to changes in model parameters as they are in the standard review.ConclusionThe accelerated review policy is effective in providing incentives for commercialisation. The results of this study will provide an effective reference for vaccine companies to make scientific portfolio planning under the accelerated review policy.

Metro train rescheduling approach during the transitional period

ABSTRACT Metro systems will inevitably be affected by train delays. Train delays during the peak to off-peak transitional period potentially lead to passengers being stranded. To address this problem, this study develops a train rescheduling model that comprehensively considers train delays, service cancellations, and the passenger service quality. The rescheduling strategy of adding temporary train services is particularly considered based on operational characteristics during the transitional period. Next, the model is transformed into a mixed-integer linear programming (MILP) model using linearization methods. To improve computational efficiency, a two-stage approach is proposed. Finally, numerical examples based on practical data from the Beijing Metro Yizhuang line are conducted to verify the effectiveness and efficiency of the proposed approach. The computational results show that the model proposed in this study can reduce the number of stranded passengers during the transitional period, and the two-stage approach can obtain high-quality feasible solutions in a short time.

Autonomous Truck Scheduling and Platooning Considering Cargo Consolidation

Thanks to advancements in automated driving technology, autonomous trucks (ATs) can form platoons with minimal inter-vehicle distances on highways, significantly reducing air drag and fuel consumption for fleets. Given the dispersed distribution and small quantities of cargo, fleet operators should manage ATs to enable cargo consolidation during platooning. In this way, fleet operators can enhance operational efficiency and reduce fuel consumption. This study addresses the AT scheduling and platooning problem considering cargo consolidation. The problem is the scheduling of ATs to transport cargo while consolidating cargo and forming platoons between two terminals, all while minimizing operational costs. A mixed-integer linear programming (MILP) model is formulated for the proposed problem. In addition, we conduct extensive numerical experiments to evaluate the proposed model. The results show that Gurobi can solve instances with different sizes to optimality or near-optimality. Impact analysis is also conducted to explore the influences of several factors, such as maximal platoon size and the load capacity of AT, on the system performance and to provide managerial insights.

Mixed‐Integer Motion Planning for Nonholonomic Robots Under Visible Light Communication Constraints

ABSTRACTThis work is concerned with the problem of motion planning for a group of nonholonomic robots under Visible Light Communication (VLC) connectivity requirements. Recently, multi‐robot systems operating with VLC networks have been demonstrated as an attractive solution for inspection tasks in pipelines and underground facilities. However, there are still no established methodologies to automate their operation. We address this problem with an optimization framework based on Mixed‐Integer Linear Programming (MILP) to design a motion planner that safely coordinates the robot team while jointly addressing the nonlinearity of the robots' dynamics, the directed line‐of‐sight requirement for VLC connections, and the connectivity maintenance of the ensuing directed networks. We propose a novel method to encode the nonlinear dynamics of nonholonomic robots into the MILP formulation and demonstrate through comparative trials that, although the resulting optimization search space is more restricted, the approach still outperforms mixed‐integer nonlinear programming in terms of optimization time and solution quality. The efficacy of the proposal is also demonstrated through realistic simulations of the Turtlebot3 robot platform.

Planning of distributed energy storage with the coordination of transmission and distribution systems considering extreme weather

As the penetration level of renewable energy is continuously growing, it is essential for transmission and distribution system operators to collaborate on optimizing the siting and sizing of distributed energy storage to enhance the operational flexibility and economic efficiency. Given the frequent occurrence of extreme weather in recent years, the planning should also account for such factors. Hence, a planning method of distributed energy storage with the coordination of transmission and distribution systems considering extreme weather is proposed. Firstly, a Gaussian mixture model-based chance constraint is established to describe the uncertainty of wind and solar power, ensuring high confidence that the bus voltage of the distribution system is within a safe range. Secondly, aiming to maximize the social welfare, a bi-level planning model for distributed energy storage is developed. The upper-level addresses the siting and sizing issues of distributed energy storage, while the lower-level characterizes the day-ahead clearing problem of power market. By leveraging Karush-Kuhn-Tucker (KKT) conditions and linearization techniques, the bi-level model is transformed into a single-level mixed integer linear programming model that is easier to solve. Finally, numerical analysis is conducted on a modified IEEE 24-node system combined with two IEEE 33-node systems. The case study verifies the effectiveness of the proposed model.

Integrated Energy Management in Small-Scale Smart Grids Considering the Emergency Load Conditions: A Combined Battery Energy Storage, Solar PV, and Power-to-Hydrogen System

This study introduces an advanced Mixed-Integer Linear Programming model tailored for comprehensive electrical and thermal energy management in small-scale smart grids, addressing emergency load shedding and overload situations. The model integrates combined heat and power sources, capable of simultaneous electricity and heat generation, alongside a mobile photovoltaic battery storage system, a wind resource, a thermal storage tank, and demand response programs (DRPs) for both electrical and thermal demands. Power-to-hydrogen systems are also incorporated to efficiently convert electrical energy into heat, enhancing network synergies. Utilizing the robust Gurobi solver, the model aims to minimize operating, fuel, and maintenance costs while mitigating environmental impact. Simulation results under various scenarios demonstrate the model’s superior performance. Compared to conventional evolutionary methods like particle swarm optimization, non-dominated sorting genetic algorithm III, and biogeography-based optimization, the proposed model exhibits remarkable improvements, outperforming them by 11.4%, 5.6%, and 11.6%, respectively. This study emphasizes the advantages of employing DRP and heat tank equations to balance electrical and thermal energy relationships, reduce heat losses, and enable the integration of larger photovoltaic systems to meet thermal constraints, thus broadening the problem’s feasible solution space.

Integrated optimization for sizing, placement, and energy management of hybrid energy storage systems in renewable power systems

Power systems reliant on renewable energy sources (RES) encounter supply-demand imbalances and stability challenges due to their inherent uncertainties. Hybrid energy storage systems (HESS) have emerged as a flexible and cost-effective solution to address these issues. This paper proposes an integrated optimization method for the capacity, location, and energy management of a HESS in RES-based power systems. The method optimizes battery energy storage system (BESS), electrolyzer (EL), fuel cell (FC), and hydrogen storage tank (HST) to minimize total costs, including power purchase, curtailment compensation, operation and maintenance (O&M), and investment costs, alongside voltage profile deviation index (VPDI) and active power loss index (APLI). A novel hybrid framework is proposed, utilizing mixed-integer linear programming (MILP) optimization at the master level and non-dominated sorting genetic algorithm II (NSGA-II) optimization at the slave level. This framework ensures that MILP provides robust solutions, while NSGA-II derives balanced optimal solutions from the optimal Pareto front. The proposed method was validated using real seasonal data from Korea and tested in IEEE 33 and IEEE 69 bus systems against various benchmark cases. Results indicate significant improvements, with VPDI enhanced by 16.00 % and 10.11 %, APLI by 15.64 % and 12.15 %, and total costs reduced by 58.67 % and 50.56 %, respectively. The sensitivity analysis demonstrated the robustness of the proposed method, showing zero sensitivity to iterations and positive correlations with increased levels of RES. Further verification against other optimization algorithms showed that the proposed method excels in cost efficiency, voltage stability, and line loss reduction, providing highly reliable results.

Optimal operation of hydrogen-based multi-energy microgrid integrating water network and transportation sector

To address the rising demand for hydrogen energy and its reliance with the water sector, this study presents an optimal scheduling framework for a multi-energy microgrid (MEMG) that integrates electric, thermal, water, and hydrogen energy networks. To this end, a mixed-integer linear programming (MILP) model is formulated to minimize both operational costs and emissions. A bi-variate piecewise McCormick envelope technique is utilized to manage the non-linear constraints associated with the water network. The model also incorporates the transportation sector, including electric and hydrogen vehicles (EVs, HVs), with vehicle-to-grid (V2G) technology, and models their associated uncertainties using Monte Carlo simulation (MCS). Additionally, the sale of oxygen as a by-product of the hydrogenation process is also considered. The case study shows significant economic and environmental benefits, with a 29.66% cost reduction and 22.26% emissions decrease from water network integration. Oxygen sales further reduce costs by 14.19%, and V2G technology contributes an additional 2.35% cost and 6.01% emissions reduction. The proposed linear approximation method achieved superior performance, with a root mean square error (RMSE) of 0.72 and a relative error of 2.132%.

Learning to Cut via Hierarchical Sequence/Set Model for Efficient Mixed-Integer Programming.

Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), which formulate many important real-world applications. Cut selection heavily depends on (P1) which cuts to prefer and (P2) how many cuts to select. Although modern MILP solvers tackle (P1)-(P2) by human-designed heuristics, machine learning carries the potential to learn more effective heuristics. However, many existing learning-based methods learn which cuts to prefer, neglecting the importance of learning how many cuts to select. Moreover, we observe that (P3) what order of selected cuts to prefer significantly impacts the efficiency of MILP solvers as well. To address these challenges, we propose a novel hierarchical sequence/set model (HEM) to learn cut selection policies. Specifically, HEM is a bi-level model: (1) a higher-level module that learns how many cuts to select, (2) and a lower-level module-that formulates the cut selection as a sequence/set to sequence learning problem-to learn policies selecting an ordered subset with the cardinality determined by the higher-level module. To the best of our knowledge, HEM is the first data-driven methodology that well tackles (P1)-(P3) simultaneously. Experiments demonstrate that HEM significantly improves the efficiency of solving MILPs on eleven challenging MILP benchmarks, including two Huawei's real problems.

A modified adaptive large neighborhood search algorithm for solving the multi-port continuous berth allocation problem with vessel speed optimization

Despite its fast-growing popularity in maritime transportation, container shipping is still fraught with risks and uncertainties with its complex operating environments. This paper studies the multi-port continuous berth allocation problem with speed optimization (MCBAP). In the MCBAP, vessels visit multiple ports sequentially, and the problem aims at minimizing the sum of vessel sailing cost, waiting cost, delay cost and port handling cost, while satisfying various constraints related to vessel sailing and berthing. A mixed integer linear programming (MILP) model for MCBAP is formulated, and a modified adaptive large neighborhood search (MALNS) algorithm is proposed for solving large-scale MCBAPs. In the MALNS, an efficient initial solution generation strategy is developed, and a series of neighborhood solution generation operators are proposed. Finally, the proposed MILP model and MALNS algorithm are tested on a range of MCBAP instances. The numerical results demonstrate that the MILP model can be solved to optimality with CPLEX, and the MALNS can efficiently solve instances at various scales. In addition, sensitivity analyses on fuel prices and vessel design speeds (the planned maximum speeds) are performed, and management insights have been provided.

The location routing problem with time windows and load-dependent travel times for cargo bikes

Last-mile delivery with traditional delivery trucks is ecologically unfriendly and leads to high road utilization. Thus, cities seek for different delivery options to solve these problems. One promising option is the use of cargo bikes in last-mile delivery. These bikes are typically released at micro hubs, which are small containers or facilities located at advantageous places in the city center. Since the bike’s travel speed depends on its remaining load and the street gradient, placing the hubs at valleys might cause additional work for rides. Therefore, the following question arises: How high is the impact of load-dependent travel times on micro hubs’ cost-optimal placements?To answer this question, we introduce the location routing problem with time windows and load-dependent travel times. We formulate the problem as a mixed-integer linear program and introduce an adaptive large neighborhood search with a problem-specific procedure for micro hub placements and problem-specific operators to solve larger instances. In a numerical study, we find that load-dependent travel times significantly influence the location of hubs, following that hubs with a higher elevation are preferably used. Moreover, customers are served from hubs with a similar elevation. This would not be the case if load-dependent travel times are ignored, resulting in an increase in costs by up to 2.7% or, instead, to up to 26% infeasible solutions as time windows are not adhered to.

Product tracing or component tracing? Blockchain adoption in a two-echelon supply chain management

Consumer awareness of product authenticity and carbon footprint tracing are among the most remarkable reasons for adopting blockchain in the supply chain in today’s world. However, the research literature has not yet examined specific ways to adopt blockchain in the supply chain. This study aims to develop a decision support tool to deal with the adoption of blockchain technology to design a two-echelon supply chain. In this regard, four specific cases for integrating supply chain and blockchain are developed based on types of tracing and block generation authority. In product tracing, green products are investigated throughout the supply chain, while in component tracing, green products are examined between the components of the supply chain. As it is necessary to record and verify the supply chain information by authorities in the blockchain network, in this work, such authorities are taken into account for both links and members of the supply chain. As far as we know, this is the first attempt to classify the various methods of adopting blockchain in Green Supply Chain Management (GSCM) and propose mathematical optimization models related to them. In this line, four Mixed-Integer Linear Programming (MILP) models with the aim of minimizing the costs related to the physical supply chain and blockchain deployment are developed for the integration of the supply chain with blockchain technology. They are treated by the Branch and Efficiency (B&E) algorithm and Simultaneous Data Envelopment Analysis (SDEA) model considering common (cost and service) and innovative (blockchain) criteria. The results showed that link-based and component tracing models are cost-effective. In addition, the cost objective function of green product tracing is more sensitive to the number of blocks than that of component tracing. Eventually, the study provides great opportunities for decision-makers and managers to understand how to adopt blockchain in terms of supply chain network characteristics, cost, transparency, and service.

Analysis of the inter-city shared mobility system based on autonomous electric vehicles

While autonomous electric vehicles (AEVs) have been preliminarily implemented in shared travel services from both technical and theoretical perspectives, challenges such as dispersed fleets, limited service quality, and travel safety concerns have confined existing research to intra-city shared mobility, with few studies focusing on inter-city shared mobility for AEVs. With China’s metropolitan regions expanding and intercity travel rapidly increasing, examining the use of shared autonomous electric vehicles (SAEVs) for intercity travel is essential for future transportation systems. To address this issue, this study models the inter-city shared mobility problem utilizing mixed integer linear programming. The problem is defined on a directed graph and addressed using a variable neighborhood search algorithm. Numerical experiments are designed based on the actual shared mobility data between Yinchuan and Shizuishan, Ningxia Province, China. From the perspective of the operating platform, the potential influencing factors of the inter-city shared mobility system are analyzed, including passenger travel demand, fleet size, vehicle range, and the number of charging stations. The results demonstrate that increasing temporal and spatial imbalance in passenger travel demand and excessive fleet size will lead to lower total revenue of the operation platform. Conversely, within certain thresholds, increasing the number of charging stations, vehicle range, and fleet size will increase the total revenue. Continuing to increase the number of charging stations and vehicle range has no significant effect on total revenue and service rate but will lead to a waste of basic resources. The study is expected to provide a solution for building an inter-city shared mobility system based on SAEVs. By fully understanding the system’s limitations and advantages, we aim to provide a reference template for constructing future inter-city shared mobility systems.

The Dial-a-Ride problem with meeting points: A problem formulation for shared demand–responsive transit

In this paper, a formulation for the Dial-a-Ride Problem with Meeting Points (DARPmp) is introduced. The problem consists of defining routes that satisfy trip requests between pick-up and drop-off points while complying with time window, ride time, vehicle load, and route duration constraints. A set of meeting points is defined, and passengers may be asked to use these meeting points as alternative pickup or drop-off points if this results in routes with lower costs. Incorporating meeting points into the DARP is achieved by formulating a mixed-integer linear program. Two preprocessing steps and three valid inequalities are introduced, which improve the computational performance when solving the DARPmp to global optimality. Two versions of the Tabu Search metaheuristic are proposed to approximate the optimal solution in large-scale networks due to the NP-hardness of DARPmp. Performing numerical experiments with benchmark instances, this study demonstrates the benefits of DARPmp compared to DARP in terms of reducing vehicle running costs.

Fault-tolerant hierarchical energy management system for an electrical power system on more-electric aircraft

The concept of More-Electric Aircraft (MEA) has the potential to improve the environmental, economic, and reliability performance in the energy and transportation sectors. To achieve this potential, it has become a tendency to develop complex architectures of Electrical Power Systems (EPSs) for MEA to supply increasing electrical power demands. Moreover, a reliable and intelligent Energy Management System (EMS) is critical to coordinate the various EPS subsystems to ensure safe and efficient flight, following the real-time EPS operating requirements and safety criteria, while reducing the operating costs for all flight stages. This paper presents a fault-tolerant hierarchical EMS, for an innovative multi-converter-based aircraft EPS, to configure the system, ensure power distribution, and manage energy storage in multiple faulty scenarios over different time scales. There are two levels in this EMS: The High Level (HL) is based on Model Predictive Control (MPC), formulated by Mixed-Integer-Linear-Programming (MILP), to optimise the long-term EPS performance while considering future predictions; The Low Level (LL) adopts deterministic rules to cope with load changes and fault occurrences over the short term, during the HL sample intervals, with a faster clock. In particular, the LL controller contains four modes: to either cooperate with the HL online MPC or to operate independently, in either EPS normal or faulty conditions. The proposed EMS is evaluated in two cases, firstly considering load deviations in a normal operating scenario, and then considering behaviour in fault scenarios. The results indicate that the proposed EMS successfully reduces the EPS operational costs while ensuring quick responses to dynamic changes with either EPS component faults or EMS internal faults.

Bi-objective dynamic tugboat scheduling with speed optimization under stochastic and time-varying service demands

With the growing emphasis on green shipping to reduce the environmental impact of maritime transportation, optimizing fuel consumption with maintaining high service quality has become critical in port operations. Ports are essential nodes in global supply chains, where tugboats play a pivotal role in the safe and efficient maneuvering of ships within constrained environments. However, existing literature lacks approaches that address tugboat scheduling under realistic operational conditions. To fill the research gap, this is the first work to propose the bi-objective dynamic tugboat scheduling problem that optimizes speed under stochastic and time-varying demands, aiming to minimize fuel consumption and manage service punctuality across a heterogeneous fleet. For the first time, we develop an extended Markov decision process framework that integrates both reactive task assignments and proactive waiting decisions, considering the dual objectives. Subsequently, an initial schedule for known requests is established using a mixed-integer linear programming model, and an anticipatory approximate dynamic programming method dynamically incorporates emerging demands through task assignments and waiting plans. This approach is further enhanced by an improved rollout algorithm to anticipate future scenarios and make decisions efficiently. Applied to the Singapore port, our methodology achieves a 12.8% reduction in the total sail cost compared to the tugboat company’s scheduling practices, resulting in significant daily savings. The results with benchmarking against three methods demonstrate improvements in cost efficiency and service punctuality, meanwhile, extensive sensitivity analysis provides managerial insights for operational practice.