Mixed Integer Linear Programming Model Research Articles

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.

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.

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.

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.

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%.

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.

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.

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.

Optimal long-term planning of CCUS and carbon trading mechanism in offshore-onshore integrated energy system

The offshore-onshore integrated energy system (OOIES) interconnects the offshore oil extraction platforms, offshore wind farms (OWFs), onshore coal/gas-fired power plants and oil distillation factories, which has great potential in CO2 emission reduction. To investigate its long-term CO2 emission reduction and economic benefits in the subsequent operation, an optimal long-term planning (OLTP) model of carbon capture, utilization, and storage (CCUS) and carbon trading mechanism (CTM) in OOIES is proposed. In the model, a CCUS model using the geological reservoir model for CO2 enhanced oil recovery (CO2-EOR) is proposed, a price-based CTM with carbon emission allowance and Chinese certified emission reduction is included, multi-band intervals are used to describe the uncertain prices and OWF outputs, and GlueVaR is introduced to measure the risks caused by uncertainties for risk management. For the solution of the OLTP model, the geological reservoir model with partial differential equations is transformed into mix-integer linear constraints by using the implicit pressure explicit saturation method, the ReLU-based multi-layer perceptron method and the convex envelop relaxation technique. Moreover, by applying multi-band interval possibility degree and GlueVaR's coherence, the original uncertainty optimization model is transformed into a mixed-integer linear programming model. Case study on an actual OOIES on the east bank of the Pearl River estuary demonstrates the effectiveness of the proposed method. Compared to the optimal results without CCUS and CTM, the optimal results of the proposed method reduce the CO2 emission by 13.28 % and the expected value and GlueVaR of total cost by 2.381 % and 22.480 %, respectively.

Blockchain-backed resilient strategies in a stochastic supply chain sourcing and distribution environment under disruption: implications for B2B sector

Purpose With the changing landscape of the globalised business world, business-to-business supply chains face a turbulent ocean of disruptions. Such is the effect that supply chains are disrupted to the point of failure, supply is halted and its adverse effect is seen on the consumer. While previous literature has extensively studied risk and resilience through mathematical modelling, this study aims to envision a novel supply chain model that integrates blockchain to support visibility and recovery resilience strategies. Design/methodology/approach The stochastic bi-objective (cost and shortage utility) optimisation-based mixed-integer linear programming model integrates blockchain through a binary variable, which activates at a particular threshold risk-averse level of the decision-maker. Findings Firstly, visibility is improved, as identified by the average reduction of penalties by 36% over the different scenarios. Secondly, the average sum of shortages over different scenarios is consequently reduced by 36% as the recovery of primary suppliers improves. Thirdly, the feeling of shortage unfairness between distributors is significantly reduced by applying blockchain. Fourthly, unreliable direct suppliers resume their supply due to the availability of timely information through blockchain. Lastly, reliance on backup suppliers is reduced as direct suppliers recover conveniently. Research limitations/implications The findings indicate that blockchain can enhance visibility and recovery even under high-impact disruption conditions. Furthermore, the study introduces a unique metric for measuring visibility, i.e. penalty costs (lower penalty costs indicate higher visibility and vice versa). The study also improves upon shortages and recoveries reported in prior literature by 6%. Finally, blockchain application caters to the literature on shortage unfairness by significantly reducing the feeling of shortage unfairness among distributors. Practical implications This study establishes blockchain as a pro-resilience technology. It advocates that organisations focus on investing in blockchain to enhance their visibility and recovery, as it effectively reduces absolute shortages and feelings of shortage unfairness while improving recovery and visibility. Originality/value To the best of the authors’ knowledge, this is a unique supply chain model study that integrates a technology such as blockchain directly as a binary variable in the model constraint equations while also focusing on resilience strategies, costs, risk aversion and shortage unfairness.

Electric airport ferry vehicle scheduling problem for sustainable operation

This study discusses the challenges and solutions surrounding the scheduling of electric airport ferry vehicles (EAFVs) in the aviation industry. With the rise of electric vehicle technology and the push for sustainability, the adoption of EAFVs presents a significant opportunity for reducing emissions in airport ground transportation. However, scheduling EAFVs involves integrated considerations of operational scheduling and charging scheduling. To address these challenges, this study proposes a mixed-integer nonlinear programming (MINLP) model aimed at minimizing total costs associated with EAFV utilization while adhering to flight time window constraints and ensuring electric power availability. Furthermore, we transform the proposed MINLP model into a solvable mixed-integer linear programming model, facilitating its solution using off-the-shelf commercial solvers. To meet the requirements of the large international airport, we propose a stabilization approach for solving the large-scale instances. Finally, we further analyze the influence of crucial parameters, including the number of EAFVs deployed, the number of flights for servicing, and the charging power of charging facilities. Managerial policies are proposed for the airport managers based on the findings. This research addresses both theoretical and practical aspects, offering valuable policy recommendations for the effective EAFV management and providing a foundation for further exploration in EAFV scheduling. The findings contribute to the advancement of sustainable transportation practices in the aviation industry.

A tri-level stochastic model for operational planning of microgrids with hydrogen refuelling station-integrated energy hubs

Energy hubs are efficient energy systems in which multiple energy carriers are converted, conditioned and stored to supply multiple forms of energy demands such as electricity, gas and heat. On the other hand, the penetration of fuel cell vehicles in transportation sector is increasing. The role of hydrogen refuelling stations is to inject hydrogen into fuel cell vehicles. A hydrogen refuelling station may absorb its required electricity from an energy hub. The operational planning of the microgrids with hydrogen refuelling station-integrated energy hubs has not been addressed before; therefore, the main goal of this research is to develop a hierarchical stochastic framework for operational planning of isolated microgrids with hydrogen refuelling station-integrated energy hubs, considering the uncertainties. In a hierarchical framework, the players are not obliged to submit their models to a central agent, so the privacy of players is preserved; moreover, it is computationally inexpensive. Mixed-integer linear programming models are used for hydrogen refuelling stations and energy hubs, while a mixed-integer quadratic programming model is used for modeling microgrid. CPLEX and GUROBI solvers are respectively used for solving the developed models. SCENRED module is used for scenario reduction. The studied microgrid is a renewable-rich 69-bus radial network. The findings approve the efficiency of the proposed methodology. The impact of batteries and wind generators on the operation of energy hubs has been evaluated.

An Integrated Multi-Product Biodiesel and Bioethanol Supply Chain Model with Torrefaction Under Uncertainty

This study presents an integrated supply chain network model for biodiesel and bioethanol production, incorporating torrefaction under uncertain conditions related to the establishment of new facilities. The proposed mixed-integer linear programming model aims to minimize the total cost of the supply chain while maximizing social objectives such as reducing unemployment. To solve the bi-objective model, a three-stage approach is employed: first, uncertain parameters are defuzzified; second, the augmented epsilon-constraint method is applied to generate a set of efficient Pareto-optimal solutions; and third, robust optimization is used to handle real-world uncertainties, such as disruptions caused by natural disasters and sanctions, ensuring feasibility under different scenarios. The study considers various stages of the supply chain, from feedstock cultivation to processing, transportation, and distribution. A real-life case study in Iran is used to evaluate the effectiveness of the proposed model, highlighting that biodiesel and bioethanol supply chains are interrelated, particularly at the cultivation stage, where each crop impacts the other. In this regard, Kermanshah, Isfahan, Chahar Mahal & Bakhtiari, Khorasan North, Kohgiluyeh & Boyer-Ahmad, and Lorestan are identified as the most suitable provinces for second-generation plant cultivation. Additionally, Azerbaijan East is identified as the best location for a bioethanol refinery, while Tehran and Markazi are the optimal choices for biodiesel refineries. This integrated approach offers a novel solution that prevents impractical overlaps in land use, providing a comprehensive, sustainable, and socially beneficial framework for bioenergy supply chain management.

Integrating Energy-Efficient Train Control in railway Vertical Alignment Optimization: A novel Mixed-Integer Linear Programming approach

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.

Multi-objective grey wolf optimizer based on reinforcement learning for distributed hybrid flowshop scheduling towards mass personalized manufacturing

As an emerging production paradigm, mass personalized manufacturing (MPM) realizes personalized product customization under the premise of ensuring large-scale production. In this paradigm, the rapid switching of the type and quantity of manufacturing tasks increases the difficulty of scheduling. Hence, this paper proposes the distributed hybrid flowshop scheduling problem with an order modularization and tasks assigning method (DHFSP-OMTA), where heterogeneous customer orders are decomposed into standard and personalized production tasks and assigned to different factories. Meantime, towards MPM, a novel mixed integer linear programming model is established to minimize the makespan and total energy consumption simultaneously. Considering the high complexity of DHFSP-OMTA, a multi-objective grey wolf optimizer based on reinforcement learning (MOGWO-RL) is designed. This paper contains the following three improvements. Firstly, the variable tasks splitting method combines two initial heuristic-rule to produce a high-quality population. Secondly, a variable neighborhood search based on reinforcement learning is designed to improve the search quality and jump out of the local optimum. Thirdly, an efficient merging batches method is presented to save transportation energy consumption. The advantages of the proposed algorithm are verified on 18 modified test instances based on the Taillard benchmark with the MPM feature. The results show that MOGWO-RL has the best effectiveness and stability of all comparison algorithms. Therefore, it can be used as a novel method to solve MPM’s scheduling problem.

The green marine waste collector routing optimization with puma selectison-based neighborhood search algorithm

AbstractImproper waste disposal by humans has created significant environmental issues in the marine ecosystem, including endangering aquatic life and accelerating the extinction of certain marine species. Due to the floating nature of the marine debris, the coordinates for collecting activities must be estimated in advance. In this article, GNOME software is used to estimate the coordinates of debris, and then a fleet of several ships is used to collect them. Also, a mixed integer linear programming model is presented for the routing optimization of debris collection fleets. The proposed optimization model formulates the objective function based on numerous factors, including labor cost, rent, and ship insurance, and considers constraints on fuel tank capacity, the time window, and the ship’s cargo capacity. A new hybrid algorithm combining the Puma algorithm and neighborhood search is proposed to address the problem. Metropolis acceptance is used in the simulated annealing algorithm to avoid the local optima and greedy selection. Numerical examples of the marine survey and the port of Rotterdam are used to test the proposed approach, which has been proven effective in several scenarios. Results achieved from the proposed hybrid method demonstrate considerable performance improvement in solving the problem. This approach has decreased total fuel and labor costs by 10–15% compared to conventional methods, with minimized time window violation reaching 25%. These results show a significant reduction in total operational costs with proper scheduling and route planning.

An accelerated Benders decomposition method for distributionally robust sustainable medical waste location and transportation problem

This study addresses the sustainable medical waste location and transportation (SMWLT) problem from the viewpoint of social risk, environmental impact, and economic performance, where model uncertainty includes risk and transportation costs. In practice, it is usually hard to obtain the exact probability distribution of uncertain parameters. To address this challenge, this study first constructs an ambiguity set to model the partial distribution information of uncertain parameters. Based on the constructed ambiguity set, this study develops a new multi-objective distributionally robust chance-constrained (DRCC) model for the SMWLT problem. Subsequently, this study adopts the robust counterpart (RC) approximation method to reformulate the proposed DRCC model as a computationally tractable mixed-integer linear programming (MILP) model. Furthermore, an accelerated Benders decomposition (BD) enhanced by valid inequalities is designed to solve the resulting MILP model, which significantly improves the solution efficiency compared with the classical BD algorithm and CPLEX solver. Finally, a practical case in Chongqing, China, is addressed to illustrate the effectiveness of our DRCC model and the accelerated BD solution method.

Enhancing disaster resilience: optimizing debris management for end-of-life buildings amidst uncertainty

ABSTRACT This study develops a Stochastic Mixed Integer Linear Programming (SMILP) model to optimize the reverse logistics network for debris generated from proactive demolition of end-of-life buildings. Unlike most existing research, this study addresses proactive strategies for the Mitigation stage of disaster management. Our model identifies the optimal number of landfilling areas and sorting facilities, factoring in uncertainties in debris quantity and material quality. It incorporates environmental constraints, such as mandatory sorting processes and recycling thresholds. Multiple scenarios are considered, each with unique parameter values and occurrence probabilities, with the overall objective of minimizing net costs across all scenarios. A realistic case study is used to illustrate the model, demonstrating its capacity to reduce post-disaster recovery costs, improve operational efficiency, and balance financial and environmental considerations. This study offers insights for decision-makers, advocating proactive end-of-life buildings’ management as a disaster preparedness and sustainable practices.

Dynamic scheduling strategy and algorithm for mixed batch scheduling in vacuum freeze-dried fruit processes

Abstract Vacuum freeze-dried fruit processes consisting of heating and holding are modelled as a mixed batch scheduling with the objective of minimizing the makespan. The jobs differ from each other in job family, size, weight and ready time. The batch processing time is determined by the longest job and the total weight of the jobs in the batch. A mixed-integer linear programming model is developed and tested with small-scale examples. Typical batch scheduling strategies are analysed and a machine-based dynamic programming strategy is proposed. The machine-based dynamic scheduling strategy is applied to design improved genetic and particle swarm optimization algorithms, which demonstrate the effectiveness of this strategy. The worst-case ratio of the algorithms using machine dynamic programming strategy are proved. Numerical experiments show that the heuristic algorithm, genetic algorithm, and particle swarm optimization algorithm based on machine dynamic scheduling strategy outperform related algorithms using greedy and job-based dynamic scheduling strategies.

Adaptive large neighbourhood search for the multi-depot arc routing problem with flexible assignment of end depot and different arc types

This article introduces an advanced solution to optimize street sweeping operations by extending a multi-depot arc routing problem. The key enhancement involves flexible end depot assignments, where vehicles start and conclude shifts at designated depots. A notable constraint requires subsequent shifts to begin from the destination depot of the preceding shift. The problem involves servicing highway exclusively during night shifts, while other arc types can be addressed during both day and night. The objective is to identify optimal shifts meeting practical criteria while adhering to constraints like maximum shift duration. To address this, a mixed-integer linear programming (MILP) model is presented. It aims to minimize the number of shifts and total travel time. Given the computational complexity of large instances, an adaptive large neighbourhood search (ALNS) metaheuristic was developed. This approach incorporates specialized operators that address unique attributes such as arc type and depot assignments, ensuring arcs are repositioned based on their type and proximity to depots. This tailored approach provides a distinct advantage over classical ALNS operators, as numerical tests indicate that the specialized operators are more efficient in comparison. The approach is evaluated on larger and a real-world instances, demonstrating notable performance in solution quality and computational efficiency.