Gurobi Solver Research Articles

Low-Carbon Optimization Scheduling of Integrated Energy Systems Based on Bilateral Demand Response and Two-Level Stackelberg Game

In the context of low-carbon energy transformation, fully utilizing the integrated demand response (IDR) resources on the load side can improve the operational flexibility and economy of the integrated energy system (IES). However, establishing a reasonable trading mechanism to enhance users’ participation in IDR has become a key issue that IES urgently needs to solve. To this end, this paper first establishes an IES model that includes electricity, heat, and gas. To reduce carbon emissions, a ladder-type carbon trading mechanism is introduced while adding low-carbon technologies such as carbon capture devices and power-to-gas conversion. Secondly, a bilateral IDR mechanism centered on the load aggregator (LA) is proposed, and a multi-agent operation model including IES, LA, and users is established. The IDR subsidy price is dynamically determined through a two-level Stackelberg game model involving IES, LA, and users. Then, through KKT conditions and the Big M method, the two-level game model is turned into an IES-LA game model, which is solved using a combination of the White Shark Optimization method and the Gurobi solver. The final simulation results show that the scheduling model can fully reflect the time value of IDR resources, and the IES cost is decreased by USD 152.22, while LA and user benefits are increased by USD 54.61 and USD 31.85. Meanwhile, the ladder-type carbon trading mechanism and low-carbon technology have effectively achieved low-carbon operation of the system.

Optimizing task assignment and routing operations with a heterogeneous fleet of unmanned aerial vehicles for emergency healthcare services

This paper studies the optimization of task assignment and pickup and delivery operations using a heterogeneous fleet of unmanned aerial vehicles (UAVs). We specifically address the distribution of emergency medical supplies, including medications, vaccines, and essential medical aid, as well as the collection of biological blood samples for testing and analysis. Unique challenges, such as supply shortages, time windows, and geographical considerations, are explicitly taken into account. The problem is first formulated as a mixed-integer linear programming model aimed at maximizing the total profit derived from the execution of a set of emergency healthcare pickup and delivery tasks. An enhanced Q-learning-based adaptive large neighborhood search (QALNS) is proposed for large-scale benchmark instances. QALNS exhibits a superior performance on benchmark instances. It also improves the quality of the solutions on average by 5.49% and 6.86% compared to the Gurobi solver and a state-of-the-art adaptive large neighborhood search algorithm, respectively. Sensitivity analyses are performed on critical factors contributing to the performance of the QALNS algorithm, such as the learning rate and the discount indicator. Finally, we provide managerial insights on the use of the fleet of UAVs and the design of the network.

Code and Data Repository for On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

The goal of this repository to be able to trace and reproduce the results and plots in the referenced article. For the results in this article, we used the general purpose integer programming solver Gurobi version 10.0.0. In addition, you need to have the python packages pandas, numpy, scipy, and pybind11 installed to be able to build the code for computing the weights of the cutting planes from 5.1.1.

Scheduling Splitable Jobs on Identical Parallel Machines to Minimize Makespan using Mixed Integer Linear

The scheduling of parallel machines with and without a job-splitting property, deterministic demand,and sequence-independent setup time with the goal of minimizing makespan is examined in this work.For simultaneous processing by multiple machines, single-stage splitable jobs are broken into random(job) sections. When a job starts to be processed on a machine, an operator has to setup the machinefor an hour. By creating two Mixed Integer Linear Programming models, this work proposes amathematical programming strategy (MILP). A MILP model takes the job-splitting property intoaccount. Another model, however, does not include the job-splitting property. This study investigatesthe performance of the proposed models using Gurobi solver. These programs' numerical calculationsare based on actual problems in the Indonesian city of Bandung's plastics industry. On four identicalparallel injection molding machines, 318 jobs must be finished in 22 periods. The real schedulingmethod is contrasted with these two MILP models. The maximum workload imbalance, the maximumrelative percentage of imbalance, and the makespan of these three scheduling systems are used toevaluate their effectiveness. Without the job-splitting property, MILP can handle the real issue ofscheduling identical parallel machines on injection molding machines to reduce makespan, resultingin a 36% average decrease. The MILP model's job-splitting property can reduce makespan by anadditional 2.40%. The order of relative ranking is MILP with job-splitting property, MILP withoutjob-splitting property, and actual scheduling based on the makespan minimization, workloadimbalance, and relative percentage of imbalance.

Bi-objective Optimization in Role Mining

Role mining is a technique that is used to derive a role-based authorization policy from an existing policy. Given a set of users U , a set of permissions P and a user-permission authorization relation UPA ⊆ U × P , a role mining algorithm seeks to compute a set of roles R , a user-role authorization relation UA ⊆ U × R and a permission-role authorization relation PA ⊆ R × P , such that the composition of UA and PA is close (in some appropriate sense) to UPA . Role mining is therefore a core problem in the specification of role-based authorization policies. Role mining is known to be hard in general and exact solutions are often impossible to obtain, so there exists an extensive literature on variants of the role mining problem that seek to find approximate solutions and algorithms that use heuristics to find reasonable solutions efficiently. In this paper, we first introduce the Generalized Noise Role Mining problem (GNRM) – a generalization of the MinNoise Role Mining problem – which we believe has considerable practical relevance. In particular, GNRM can produce “security-aware” or “availability-aware” solutions. Extending work of Fomin et al., we show that GNRM is fixed parameter tractable, with parameter r + k , where r is the number of roles in the solution and k is the number of discrepancies between UPA and the relation defined by the composition of UA and PA . We further introduce a bi-objective optimization variant of GNRM, where we wish to minimize both r and k subject to upper bounds \(r \le \bar{r} \) and \(k\le \bar{k} \) , where \(\bar{r} \) and \(\bar{k} \) are constants. We show that the Pareto front of this bi-objective optimization problem (BO-GNRM) can be computed in fixed-parameter tractable time with parameter \(\bar{r} +\bar{k} \) . From a practical perspective, a solution to BO-GNRM gives security managers the opportunity to identify a mined policy offering the best trade-off between the number of policy discrepancies and the number of roles. We then report the results of our experimental work using the integer programming solver Gurobi to solve instances of BO-GNRM. Our key findings are that (a) we obtained strong support that Gurobi’s performance is fixed-parameter tractable, (b) our results suggest that our techniques may be useful for role mining in practice, based on our experiments in the context of three well-known real-world authorization policies. We observed that, in many cases, our solver is capable of obtaining optimal solutions when the values of either k or r are small.

Low carbon scheduling method of mine integrated energy system based on two-stage P2G technology

Coal mines derive a large amount of various associated resources that can be used for production and living needs during the mining process. In order to realize the recycling of coal associated resources, a typical architecture of coal mine comprehensive energy system including the recycling of coal derived resources is proposed. Firstly, according to the coal mine resource endowment and its geological characteristics, the corresponding mine integrated energy system is constructed. Then, the refined two-stage P2G technology is combined with the mine integrated energy system to realize the complete consumption of wind and light and the efficient utilization of energy. Finally, the stepped carbon trading mechanism is used to further constrain the carbon emissions and realize the further reduction of carbon emissions in the mining area. The simulation model of the integrated mine energy system is established in MATLAB and the GUROBI solver is called to solve the results. The simulation results show that the method can effectively reduce carbon emissions, realize the complete consumption of wind and light and the efficient use of energy.

Multi-period emergency facility location-routing problems under uncertainty and risk aversion

This paper addresses a multi-period emergency facility location-routing problem, in which the uncertainties in material demands and transportation time, as well as dynamic inventory replenishment and carryover are incorporated in the design of multi-level emergency logistics networks. To measure the risks stemming from uncertain transportation time, mean-CVaR method is used. Then, a risk-averse stochastic programming model for the presented problem is formulated to minimize the total rescue time of the network. Moreover, a genetic algorithm is developed to solve the proposed model. Extensive numerical experiments including the randomly generated instances and a case study on the Wenchuan earthquake in China are conducted to verify the effectiveness of the presented model and algorithm. Experimental results show that the genetic algorithm significantly performs better than the Gurobi solver in terms of both solution quality and solution time.

Last-train timetable synchronization for service compatibility maximization in urban rail transit networks with arrival uncertainties

The last train services of urban rail transit offer all passengers the final chance to reach their destinations. In the context of multimodal transport, considering the random delayed arrivals of late-night vehicles of other transport modes at urban hub stations and their adverse effects on multimodal passengers transferring to urban rail transit, a probabilistic scenario set is established to determine the arrival uncertainties of multimodal passengers. In each scenario, the timetable synchronization problem is formulated as a mixed-integer nonlinear programming model that requires a performance trade-off between destination reachability and the remaining path distance of both non-multimodal and multimodal passengers, integrally termed last train service compatibility. Through linearization techniques, the model is transformed into an equivalent mixed-integer linear programming form, which can be efficiently solved using the Gurobi solver to obtain the optimal last train timetable for each scenario and form an alternative scheme set. An improved probabilistic scenario set-based regret value theory is then developed, in which a novel regret value calculation method is proposed. The scheme with the minimum total weighted opportunity loss in all scenarios is selected as the optimal robust. Real case experiments based on the Chengdu–Chongqing high-speed railway line and Chengdu metro network are conducted to test the performance of our model. The results show that compared with the original timetable, the optimized timetable reduces the number of unreachable passengers by 34.29% and the sum of the average remaining path distance of non-multimodal and multimodal passengers by 57.43% with the help of path planning for all passengers. The proposed approach is proven not only to balance the demands of both reachable and unreachable passengers, but also to significantly improve the robustness of the last train timetable to arrival uncertainties of multimodal passengers and reduce their service inequity.

AGV Scheduling for Optimizing Irregular Air Cargo Containers Handling at Airport Transshipment Centers

Airport transshipment centers play a pivotal role in global logistics networks, enabling the swift and efficient transfer of cargo, which is essential for maintaining supply-chain continuity and reducing delivery times. The handling of irregularly shaped air cargo containers presents new constraints for automated guided vehicles (AGVs), as these shapes can complicate loading and unloading processes, directly impacting overall operational efficiency, turnaround times, and the reliability of cargo handling. This study focuses on optimizing the scheduling of AGVs to enhance cargo-handling efficiency at these hubs, particularly for managing irregular air cargo containers. A mixed-integer linear programming (MILP) model is developed, validated for feasibility with the Gurobi solver, and designed to handle large-scale operations. It incorporates a novel approach by integrating a simulated annealing optimized genetic algorithm (GA). The experimental results demonstrate that the designed algorithm can solve models of considerable size within 8 s, offering superior time efficiency compared to the solver, and an average solution quality improvement of 12.62% over the genetic algorithm, significantly enhancing both the model’s efficiency and scalability. The enhanced AGV scheduling not only boosts operational efficiency but also ensures better integration within the global logistics framework. This research provides a robust foundation for future advancements in logistics technology, offering both theoretical and practical insights into optimizing complex transportation networks.

Adaptive Ant Colony Optimization Algorithm Based on Real-Time Logistics Features for Instant Delivery.

Ant colony optimization (ACO) algorithm is widely used in the instant delivery order scheduling because of its distributed computing capability. However, the order delivery efficiency decreases when different logistics statuses are faced. In order to improve the performance of ACO, an adaptive ACO algorithm based on real-time logistics features (AACO-RTLFs) is proposed. First, features are extracted from the event dimension, spatial dimension, and time dimension of the instant delivery to describe the real-time logistics status. Five key factors are further selected from the above three features to assist in problem modeling and ACO designing. Second, an adaptive instant delivery model is built considering the customer's acceptable delivery time. The acceptable time is calculated by emergency order mark and weather conditions in the event dimension feature. Third, an adaptive ACO algorithm is proposed to obtain the instant delivery order schedules. The parameters of the probability equation in ACO are adjusted according to the extracted key factors. Finally, the Gurobi solver in Python is used to perform numerical experiments on the classical datasets to verify the effectiveness of the instant delivery model. The proposed AACO-RTLF algorithm shows its advantages in instant delivery order scheduling when compared to the other state-of-the-art algorithms.

Modelling reservation strategies for managing peak-hour stranding on an oversaturated metro line

Metro is the main travel model for urban commuters in many metropolises around the world. During peak hours, large numbers of passengers pour into metro stations for rail services, but some are unable to board the trains in time and left stranded on the platform or even queuing outside the stations. The trip reservation (TR) strategy, where passengers preplan their trips and reserve their entry time to the stations. This paper develops an entry reservation strategy (ERS) to optimize the commuter flow during peak hours, and construct a multi-objective passenger flow joint optimization model based on many-to-many passenger demand to minimize the total trip cost of passengers at reservation station and the number of stranded passengers at intermediate stations. The passenger flow optimization problem is formulated as a mixed-integer non-linear programming (MINLP) model. We design an iterative sequential search algorithm combined with the GUROBI solver to obtain the parameters of the optimal ERS and the passenger flow distribution in the metro system after disaggregated reformulation of the complex constraints of the model. We also demonstrate the accuracy and effectiveness of the proposed method with two experiments – an illustrative example and a large-scale case study of Beijing Metro. The results of Beijing Metro experiment show that the joint optimization model with entry reservation strategy (JO-ERS) reduces the number of stranded passengers by 88.46 % compared with the original passenger flow from the AFC.

Quadratic p-Median Problem: A Bender’s Decomposition and a Meta-Heuristic Local-Based Approach

In this paper, the quadratic p-median optimization problem is discussed, where the goal is to connect users to a selected group of facilities (emergency services, telecommunications servers, healthcare facilities) at the lowest possible cost. The problem is aimed at minimizing the cost of connecting these selected facilities. The costs are symmetric, meaning connecting two different points is the same in both directions. This problem extends the traditional p-median problem, a combinatorial problem used in various fields like facility location, network design, transportation, supply chain networks, emergency services, healthcare, and education planning. Surprisingly, the quadratic version has not been thoroughly considered in the literature. The paper highlights the formulation of two mixed-integer quadratic programming models to find optimal solutions to this problem. One model is a classic formulation, and the other is based on set cover theory. Linear versions and Bender’s decomposition formulations for each model are also derived. A Bender’s decomposition is solved using an algorithm that adds constraints during each iteration to improve the solution. Lazy constraints in the Gurobi solver’s branch and cut algorithm are dynamically added whenever a mixed-integer programming solution is found. Additionally, an efficient local search meta-heuristic is proposed that usually finds optimal solutions for tested instances. Challenging instances with up to 60 facilities and 2000 users are successfully solved. Our results show that Bender’s models with lazy constraints are the most effective for Euclidean and random test cases, achieving optimal solutions in less CPU time. The meta-heuristic also finds near-optimal solutions rapidly for these cases.

Co-Evolutionary Algorithm for Two-Stage Hybrid Flow Shop Scheduling Problem with Suspension Shifts

Demand fluctuates in actual production. When manufacturers face demand under their maximum capacity, suspension shifts are crucial for cost reduction and on-time delivery. In this case, suspension shifts are needed to minimize idle time and prevent inventory buildup. Thus, it is essential to integrate suspension shifts with scheduling under an uncertain production environment. This paper addresses the two-stage hybrid flow shop scheduling problem (THFSP) with suspension shifts under uncertain processing times, aiming to minimize the weighted sum of earliness and tardiness. We develop a stochastic integer programming model and validate it using the Gurobi solver. Additionally, we propose a dual-space co-evolutionary biased random key genetic algorithm (DCE-BRKGA) with parallel evolution of solutions and scenarios. Considering decision-makers’ risk preferences, we use both average and pessimistic criteria for fitness evaluation, generating two types of solutions and scenario populations. Testing with 28 datasets, we use the value of the stochastic solution (VSS) and the expected value of perfect information (EVPI) to quantify benefits. Compared to the average scenario, the VSS shows that the proposed algorithm achieves additional value gains of 0.9% to 69.9%. Furthermore, the EVPI indicates that after eliminating uncertainty, the algorithm yields potential improvements of 2.4% to 20.3%. These findings indicate that DCE-BRKGA effectively supports varying decision-making risk preferences, providing robust solutions even without known processing time distributions.

A spatiotemporal optimization method for connected and autonomous vehicle operations in long tunnel constructions

With the advancement of technology, connected and autonomous vehicles (CAVs) can be applied to complex tunnel networks in long tunnel construction to enhance vehicle operation safety and efficiency. This paper proposes an optimization method for CAVs' operation in long tunnel constructions. Firstly, a spatiotemporal coordinated optimization model with decentralized time and hierarchical networks is proposed to minimize the total working time for completing transportation services. The model integrates macro task allocation and micro node control and optimizes the vehicle-space-time relationships of CAVs to prevent conflicts and collisions. Secondly, a heuristic algorithm named Search-Adjustment Genetic Algorithm (SAGA) is developed to solve the problem considering the model's complexity and engineering characteristics. Thirdly, numerical experiments are designed to validate the feasibility and efficiency of the proposed model and algorithm. The results indicate that (1) the proposed model can effectively deconflict CAVs in the road network to ensure safety and obtain a low total working time to fulfill the transportation demand. (2) Compared to the commercial solver Gurobi, the proposed algorithm demonstrates significantly superior solution accuracy and efficiency within an acceptable time limit. (3) The solution ensures the safety and efficiency of CAVs and increases their utilization compared with engineering-oriented methods, resulting in a 50 % reduction in CAV acquisition costs, a 29 % and 85 % reduction in running time and delay respectively, and a reduction in fuel consumption. (4) As the number of transportation services and the complexity of the road network increases, the efficiency gains become more prominent and better adapted to the needs of the actual long tunnel construction project. To sum up, the proposed model and algorithm can ensure the safety and efficiency of providing transportation services in future long tunnel construction. Moreover, it can be adapted for controlling CAVs in road networks such as other construction scenarios and urban road networks.

Design and optimization of zero-carbon integrated energy system for highway service area

Abstract The challenge of achieving dual-carbon targets arises from the issue of high carbon emissions in Chinese highway service areas. To address this, a comprehensive electric-thermal-hydrogen energy system is proposed, consisting of a new energy generation system, heat pumps, electrolyzers, hydrogen fuel cells, energy storage devices, and charging stations. A dispatch model for the integrated energy system is established, with the objective of minimizing the total operating cost while satisfying the constraints of each device and the electric-thermal-hydrogen balance. The model is implemented in MATLAB, and the ideal optimal solution for each decision variable is obtained using the Gurobi solver. The investment and operating costs, as well as the scheduling results for electricity, heat, and cooling, are analyzed. The results indicate that the electrolyzer-fuel cell system can improve the system’s operational stability and flexibility in dispatch optimization. Due to the instability of photovoltaic and wind power devices and the fluctuating nature of loads, energy storage devices are required for peak load shaving in the system. The zero-carbon integrated energy system model can calculate the most economical investment plan based on the load and meteorological conditions of different service areas.

Efficiency-oriented vehicle relocation of shared autonomous electric fleet in station-based car-sharing system

Long waiting delays for users and significant imbalances in vehicle distribution are bothering traditional station-based one-way electric car-sharing system operators. To address the problems above, a “demand forecast-station status judgement-vehicle relocation” multistage dynamic relocation algorithm based on the automatic formation cruising technology was proposed in this study. In stage one, a novel trip demand forecast model based on the long short-term memory network was established to predict users' car-pickup and car-return order volumes at each station. In stage two, a dynamic threshold interval was determined by combining the forecast results with the actual vehicle distribution among stations to evaluate the status of each station. Then vehicle-surplus, vehicle-insufficient, vehicle-normal stations, and the number of surplus or insufficient vehicles for each station were counted. In stage three, setting driving mileage and carbon emission as the optimization objectives, an integer linear programming mathematical model was constructed and the optimal vehicle relocation scheme was obtained by the commercial solver Gurobi. Setting 43 stations and 187 vehicles in Jiading District, Shanghai, China, as a case study, results showed that rapid vehicle rebalancing among stations with minimum carbon emissions could be realized within 15 min and the users’ car-pickup and car-return demands could be fully satisfied without any refusal.

An exact algorithm for simultaneous pickup and delivery problem with split demand and time windows

This study introduces a new variant of the vehicle routing problem (VRP) called the simultaneous pickup and delivery problem with split demand and time windows (SPDP-SDTW). The motivation behind this study stems from real-life urban and rural delivery scenarios, encompassing features such as split demand, simultaneous pickup and delivery, many-to-many pickup and delivery, and time windows. The study thoroughly investigates the properties of the optimal solution for the SPDP-SDTW. Based on these properties, an arc flow model is developed for the SPDP-SDTW. Dantzig Wolfe (DW) decomposition techniques are employed to obtain the master problem and the pricing subproblem. In order to effectively address the SPDP-SDTW, an improved branch and price (I-BP) algorithm is proposed, incorporating a tailored column generation (CG) algorithm, branching strategies, and dual stabilization strategies. The proposed CG algorithm provides a framework that combines the improved adaptive degree heuristic (I-AGH) algorithm and the solver Gurobi. This integration substantially mitigates the computational burden involved in solving the subproblem. Extensive computational experiments conducted on datasets of varying sizes, including small, medium, and large instances, consistently demonstrate that the I-BP algorithm performs the best in both solution quality and computational efficiency when compared to existing exact and heuristic algorithms.

365‐day sectional work zone schedule optimization for road networks considering economies of scale and user cost

AbstractThis study proposes a methodology for deriving the optimal work zone schedule for the annual routine maintenance planning in an infrastructure asset management system considering the (i) economies of scale in work zone costs due to work zone synchronization and (ii) user costs across the road network with traffic assignments. A key aspect of the proposed methodology is the ability to derive in detail optimal work zone schedules of realistic‐scale road networks in 100 m sections for 365 days, which is beneficial in practice. To this end, an optimization model of the work zone schedule is newly formulated as a mixed‐integer programming (MIP) problem, and a novel bilevel solution method for the model utilizing the conventional solver Gurobi Optimizer with MIP algorithms such as root relaxation, dual simplex, barrier methods, and cutting planes is proposed. In application examples, the proposed methodology is applied to two real‐world road networks, which confirms that the optimal work zone schedule can be obtained in 313.7 s for a network with 2640, 100 m, sections and in 168,180 s (46 h and 43 min) for a network with 5038, 100 m, sections.

DivideMerge: A multi-vessel optimization approach for cooperative operation and maintenance scheduling in offshore wind farm

The optimization of maintenance scheduling and routing in offshore wind farms is crucial for the intelligent operation and maintenance of offshore wind energy. It involves determining the optimal timing, assigning vessels, planning routes, and efficiently completing maintenance tasks with maximum efficiency and minimal cost. Traditional branch-and-bound search methods, often used in multi-vessel cooperative maintenance scheduling models, face limitations in finding optimal solutions efficiently, while classic heuristic methods may compromise on global optimality. This paper introduces DivideMerge, a high-performance hybrid algorithm that synergizes heuristic and exact algorithms to address large-scale multi-vessel cooperative maintenance scheduling challenges in offshore wind farms. Initially, a maintenance task constraint decomposition heuristic method is utilized to break down the collaborative optimization scheduling problem into individual vessel scheduling sub-problems, ensuring adherence to the constraints of the original problem. Subsequently, the CP-SAT solver is employed to sequentially solve these sub-problems rapidly. The solutions for individual vessel scheduling are then merged to form a comprehensive solution for the multi-vessel cooperative maintenance scheduling problem. Computational results validate the effectiveness and robustness of DivideMerge, demonstrating a solution speed nearly 1000 times faster than the commercial Gurobi solver, thus offering a significant advancement in the field of offshore wind farm maintenance optimization.

A multi-stage framework for coordinated scheduling of networked microgrids in active distribution systems with hydrogen refueling and charging stations

Due to the increase in electric energy consumption and the significant growth in the number of electric vehicles (EV) at the level of the distribution network, new networks have started using new fuels such as hydrogen to improve environmental indicators and at the same time better efficiency from the excess capacity of renewable resources. In this article, the services that can be provided by hydrogen refueling stations and charging electric vehicles in the optimal performance of microgrids have been investigated. The model proposed in this paper includes a two-stage stochastic framework for scheduling resources in microgrids, especially hydrogen refueling stations and electric vehicle charging. In this model, two main goals of cost minimization and greenhouse gas emissions are considered. In the proposed framework and in the first stage, the service range of microgrids is determined precisely according to the electrical limitations of distribution systems in emergency situations. Then, in the second stage, the problem of energy management in each microgrid will be solved centrally. In this situation, various indicators including the output energy of renewable sources, smart charging of hydrogen and electric vehicle charging stations (EV/FCV) and flexible loads (FL) are evaluated. The final mathematical model is implemented as a multivariate integer multiple linear problem (MILP) using the GUROBI solver in GAMS software. The simulation results on the modified IEEE 118-Bus network show the positive effect of the presence of flexible loads and smart charging strategies by charging stations. Also, the numerical derivation shows that the operating costs of the entire system can be reduced by 4.77% and the use of smart charging strategies can reduce greenhouse gas emissions by 49.13%.