Mixed Integer Linear Programming Model Research Articles

Electrification of transportation: A hybrid Benders/SDDP algorithm for optimal charging station trading

This paper examines the electrification of transportation as a response to environmental challenges caused by fossil fuels, exploring the potential of battery electric vehicles and hydrogen fuel cell vehicles as alternative solutions. However, a significant barrier to their widespread adoption is the limited availability of charging infrastructure. Therefore, this study proposes the development of comprehensive charging stations capable of accommodating both battery and hydrogen vehicles to address this challenge. The energy is purchased from the day-ahead and intraday auction-based electricity markets, where the electricity price is subject to uncertainty. Therefore, a two-stage stochastic programming model is formulated while the price scenarios are generated utilizing a k-means clustering algorithm. Given the complexity of the proposed model, an efficient solution approach is developed through the hybridization of the Benders decomposition algorithm and stochastic dual dynamic programming. In the Benders master problem, day-ahead bidding variables are determined, whereas the Benders sub-problem addresses intraday bidding and charging station scheduling variables, employing stochastic dual dynamic programming to tackle its intractability. Additionally, we transform the mixed integer linear program model of the second stage problem into a linear program, confirming its validity through KKT conditions. Our model provides practical insights for making informed decisions in electricity markets based on sequential auctions. While the bidding curves submitted to the day-ahead market remain unaffected by scenarios, those submitted to the intra-day market show dependence on fluctuations in day-ahead market prices.

A reinforcement learning-based multiobjective heuristic algorithm for multiple-truck routing problems with heterogeneous drones

As a result of new technological developments, drone usage has become common in last-mile logistics activities. Owing to the limited flight range of drones, drones may service customers by using trucks as hubs. Although the truck routing problem with drones (TRP-D) has been studied frequently, studies that have addressed this problem in a multiobjective manner are rare. In this study, multiobjective TRP-D is discussed. In this problem, multiple trucks are considered. Each truck is equipped with multiple heterogeneous drones. A multiobjective mixed-integer linear programming (MILP) model is proposed. Since the problem is NP-hard, a reinforcement learning (RL)-based multiobjective heuristic algorithm is proposed to obtain solutions in large-scale cases. A three-phase heuristic feasible-solution-generation algorithm is proposed. The variable neighbourhood descent (VND) algorithm with RL is proposed as a local search algorithm for the multiobjective optimization problem. The results of the proposed heuristic algorithm are compared with the Pareto optimal solutions for small-sized test problems via the augmented ε-constraint (AUGMECON) method and the CPLEX solver. The proposed algorithm is also compared with state-of-the-art multiobjective heuristic algorithms for large-scale problems. According to the results, the proposed algorithm is an efficient multiobjective heuristic algorithm for the addressed problem.

Applying the Simulated Annealing Algorithm to the Set Orienteering Problem with Mandatory Visits

This study addresses the set orienteering problem with mandatory visits (SOPMV), a variant of the team orienteering problem (SOP). In SOPMV, certain critical sets must be visited. The study began by formulating the mathematical model for SOPMV. To tackle the challenge of obtaining a feasible route within time constraints using the original MILP approach, a two-stage mixed-integer linear programming (MILP) model is proposed. Subsequently, a simulated annealing (SA) algorithm and a dynamic programming method were employed to identify the optimal route. The proposed SA algorithm was used to solve the SOP and was compared to other algorithms, demonstrating its effectiveness. The SA was then applied to solve the SOPMV problem. The results indicate that the solutions obtained using SA are superior and more efficient compared to those derived from the original MILP and the two-stage MILP. Additionally, the results reveal that the solution quality deteriorates as the ratio of the set of mandatory visits increases or the maximum allowable travel time decreases. This study represents the first attempt to integrate mandatory visits into SOP, thereby establishing a new research direction in this area. The potential impact of this research is significant, as it introduces new possibilities for addressing complex combinatorial optimization problems.

Optimal cost predictive BMS considering greywater recycling, responsive HVAC, and energy storage

A crucial aspect of a sustainable city is ensuring that energy and water supplies can adequately meet urban demand. With the scarcity of natural resources and growing electricity and water demand, it becomes increasingly important for consumers to manage their resource usage more efficiently. This paper proposes a novel perspective of demand-side management coordination strategy for a building's water-energy nexus to enhance the resilience and efficiency of the overall electricity-water-heating system. The model is formulated to optimally coordinates the onsite greywater recycling system, heating, ventilation, air conditioning (HVAC) loads, and distributed energy generation systems with a bidirectional grid connection in a residential building. All subsystems are controlled by a model predictive controller (MPC) receiving real-time time of use (ToU) pricing from electricity and water utilities. The presented mixed integer linear programming model is verified to meet the customers' demands while reducing the operational costs. Results are compared with benchmark systems lacking the water recycling or energy storage system showing 8.3 % operational cost reduction while reducing potable water consumption by 21.5 %. The effect of increased MPC control horizon is also studied showing reduction in cost with increased horizon. Detailed analysis of the proposed framework computational burden and effect of prediction errors is performed to prove the MPC adaptability and robustness. Testing under increased room size and different user preferences further validate the efficacy of the proposed scheme in reducing the operational costs.

Dynamic airport gate assignment with improved Shuffled Frog-Leaping Algorithm and triangle membership function

The rapid development of the air transportation industry has increased air traffic, posing challenges to the task of airport gate assignment (AGA) for flights. Most past studies have solved the AGA problem (AGAP) using deterministic models, which are incapable of dealing with uncertainty and dynamic conditions at airports. Thus, this research employs fuzzy theory and proposes a triangular membership function to handle flight uncertainty in the AGAP. In addition, an improved metaheuristic, termed the improved Shuffled Frog-Leaping Algorithm (ISFLA), is proposed to circumvent the computationally intractable problems commonly faced by exact approaches when handling large instances. In this research, the AGAP is first formulated as a stochastic Mixed-Integer Linear Programming (MILP) model, with stochastic flight lateness and earliness considered. The objective of this model is to minimize the total cost, which consists of three sub-costs: passenger walking distances, non-preferred gate (NPG) assignments for planes, and fuzzy idle times of gates. These three sub-costs correspond to the major concerns of passengers, airlines, and airports, respectively. The cooperation between the ISFLA and the triangular membership function demonstrates their capability to effectively handle big AGAP instances. Furthermore, the experimental results show that the ISFLA outperforms the standard SFLA, Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Firefly Algorithm (FA).

Collaboration and resource sharing in the multidepot time-dependent vehicle routing problem with time windows

Concerns about energy conservation and emission reduction have highlighted the importance of environmentally sound logistics networks in urban areas. These networks are deeply intertwined with urban traffic systems, where variations in transit speeds can significantly increase the energy consumption and carbon emissions of delivery vehicles, compromising the environmental sustainability of urban deliveries. To address this, we propose a multidepot time-dependent vehicle routing problem with time windows, enhancing route planning flexibility and resource configuration. Our approach begins with a route spatiotemporal decomposition method to estimate vehicle travel times and emissions based on varying vehicle speeds. We then develop a multiobjective mixed integer linear programming model that aims to minimize total operating costs, the number of vehicles, and carbon dioxide emissions. A hybrid heuristic algorithm combining spectral clustering, multiobjective ant colony optimization, and variable neighborhood search is proposed to solve the model. This algorithm incorporates collaboration and resource sharing strategies, a pheromone initialization mechanism, a novel heuristic operator that accounts for time dependency, and a self-adaptive update mechanism, enhancing both solution quality and algorithm convergence. We compare the performance of our algorithm with that of the CPLEX solver, multiobjective ant colony optimization, non-dominated sorting genetic algorithm-Ⅲ, and multiobjective particle swarm optimization. The results demonstrate the superior convergence, uniformity, and spread of our proposed algorithm. Furthermore, we apply our model and algorithm to a real-world case in Chongqing, China, analyzing optimized results for different time intervals and vehicle speeds. This study offers robust methodologies for theoretically and practically addressing the multidepot time-dependent vehicle routing problem with time windows, contributing to the development of economical, efficient, collaborative, and sustainable urban logistics networks.

Maximizing virtual power plant profit: A two-level optimization model for energy market participation

Managing dispersed generation via virtual power plants (VPPs) is crucial for maximizing profits in electricity markets. This paper presents a model aimed at maximizing VPP profit through participation in the energy market. The proposed model addresses grid and security constraints of units using deterministic programming, formulated as an equilibrium-constrained, two-level mathematical optimization model. The first level focuses on maximizing VPP profit, while the second optimizes social welfare. Applying duality theory transforms this two-level model into a mixed-integer linear programming model, further refined using Karush–Kuhn–Tucker (KKT) optimality conditions. Given the inherent conflict in these objectives, a novel algorithm employing water flow dynamics is proposed for solving the model. To enhance method performance, the Pareto criterion and fuzzy decision-making are incorporated. Model tests are conducted on a standard 24-bus IEEE grid, demonstrating its efficiency. For the single-objective problem without line congestion, the solving time was 12 s. Introducing line congestion increased the profit by 13.4 %, from $40,413.21 to $45,837.32. In the two-objective problem without congestion, the profit ranged between $36,928.72 and $42,813.28, and emissions ranged from 275.21 to 2,916.32 pounds. With congestion, the profit range increased by a maximum of 8.7 %, and emissions were reduced by up to 4.6 %.

Pickup and Delivery Vehicle Routing Problem with Hybrid Missions and Third-Party Vehicle Rentals: A Case Study of Animal Feed Industry

The vehicle routing problem for small and medium-sized animal feed manufacturers is a complex challenge because each transport route can vary significantly between delivering products to customers and picking up raw materials to return to the factory. Currently, businesses lack well-planned management for utilizing a limited number of trucks. As a result, they often incur additional costs by relying on third-party vehicle rentals. This research aims to reduce transportation costs for small and medium-sized animal feed manufacturers by solving a formulated mixed-integer linear programming (MILP) model, using experimental data that accurately reflects the actual problems faced. The MILP model results help reduce transportation costs by up to 18.5% compared to current practices by optimizing route sequences and incorporating the use of third-party vehicle rentals. This also serves as a guideline for reducing transportation costs in other similar industries.

Virtualized network functions resource allocation in network functions virtualization using mathematical programming

Network Functions Virtualization (NFV) revolutionizes network services by eliminating the need for dedicated hardware. This virtualization enables flexible and efficient deployment of various network functions like proxies, firewalls, and load balancers. Providing the service requested by the user in the network is done by a sequence of virtual network functions, which are known as service functions chain. One of the main challenges in the development of network functions virtualization architecture is the allocation of resources to the requested network services in network infrastructures, this challenge is called network function virtualization resource allocation problem. Therefore, this paper addresses the resource allocation problem in Network Functions Virtualization (NFV) architectures using mathematical programming techniques. A multi-objective mixed-integer linear programming (MILP) model is proposed to optimize resource allocation for virtual network functions (VNFs). The model incorporates constraints related to node and link resource capacities, as well as delay requirements. The objective functions focus on maximizing network throughput, minimizing node resource costs (CPU cores and memory), reducing capital and operational expenses, and ensuring efficient execution time. These constraints and objective functions are formally defined by mathematical functions. The proposed mathematical model is implemented and solved using the Cplex solver. To evaluate the effectiveness of the proposed mathematical model, various network topologies were evaluated under different parameters. These parameters included the length of Service Function Chains (SFCs), the number and length of flows, node resource capacities, the number of nodes and VNFs. The experimental results demonstrated the model’s ability to efficiently allocate resources to VNFs across these different scenarios.

Preemptive Software Project Scheduling Considering Personality Traits

The success of software projects is inseparable from the effective scheduling of multi-skilled human resources. In software development, employees’ personality traits often affect the performance of their work. Additionally, software developers are usually multi-tasking, which makes software development tasks preemptive. In view of this, we investigate the preemptive software project scheduling problem considering personality traits (PSPSP-PT), which aims at assigning employees with different skills and personality traits to suitable tasks and determining the start time of each task with the consideration of preemption, multi-skill, and precedence relations such that the total cost of the project is minimized. We formulate a mixed-integer linear programming model for the PSPSP-PT and design a dual priority rules-based heuristic (DPRH). Based on a benchmark dataset that is generated using a full factorial design, computational experiments are conducted to analyze the performance of our DPRH. Comparative analysis with existing algorithms reveals that our DPRH is competitive in terms of efficiency and effectiveness.

A novel bi-objective model for sustainable and efficient airport logistics management: A case study on Copenhagen airport

Due to the growing volume of air transportation and the scarcity of airport resources, aviation planners are paying more attention to the efficiency of airport resource management. Optimizing the assignment and scheduling of ground facilities plays an important role in reducing ground congestion, flight delays, and passenger dissatisfaction. To prevent any collisions among aircraft on the apron, taxi network, and runways, safety considerations should be strictly followed by traffic controllers. Minimizing taxi time is another matter of importance in airport logistics management that helps mitigate environmental concerns (e.g., fuel consumption, tire abrasion, and noise pollution). In this study, a bi-objective mixed-integer linear programming model is developed to optimize the ground resource scheduling problem, taking into account the challenges of gate assignment, taxi planning, and runway scheduling. The model is developed in a way that includes safety and sustainability considerations. The validation of the proposed model is investigated through several examples based on Copenhagen airport using the augmented ε-constraint method. Since the simultaneous scheduling of multiple resources is an NP-hard problem, two multi-objective metaheuristic algorithms are employed to deal with the large-scale problems. The performance of the algorithms is then compared in terms of coverage and convergence rates. Additionally, the results are discussed in detail to provide managerial implications for airport authorities.

Optimizing Drone Delivery Paths from Shared Bases: A Location-Routing Problem with Realistic Energy Constraints

Recently, Amazon patented fulfillment centers for drones on a large scale in densely populated areas. A network of such shared centers can be used for landing and launching drones as an alternative to the traditional private bases in near future. This paper studies how a user of such a shared delivery infrastructure can optimally determine the required bases among the existing bases to perform her delivery tasks. To this end, a location-routing problem is studied where the problem determines the drone launching centers and their routes to deliver parcels. A realistic energy function is used that incorporates the effect of load weight for calculating energy consumption in all flight phases including take-off, level flight, hovering, and landing. Although the energy consumption is nonlinear, the problem is formulated as a mixed-integer linear programming model and strengthened by valid inequalities and a pre-processing algorithm that enables us to solve instances with up to 100 customers using off-the-shelf optimization solvers. Moreover, a heuristic method is presented to solve instances with up to 200 customers. Results indicate the importance of incorporating the load-dependent energy formula in contrast to using fixed flight duration constraints for drones. The value of allowing multiple visits on each trip versus only simple back-and-forth trips is also assessed. The advantage of using an integrated location-routing approach is shown over the sequential approach in which the locations of bases are decided first and the routes are constructed next. Moreover, to manage future demand uncertainty, the model is extended for the case that a number of demand scenarios are given.

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.

A new mathematical model and meta-heuristic algorithm for order batching, depot selection, and assignment problem with multiple depots and pickers

The ‘order picking problem’ involves the efficient and organized retrieval of items from shelves to fulfill customer orders. In this study, we consider a new configuration of warehouse system with multiple pickers and depots (m-pickers & n-depots) for manually operated ‘picker-to-parts’ warehouses. The efficiency of the process is measured based on two metrics: i- total order picking distance of each picker ii- total number of pickers assigned to each of depots. The handled problem is called as ‘Order Batching, Depot Selection and Assignment Problem with Multiple Depots and Multiple Pickers (OBDSAPMDMP)11OBDSAPMDMP = Order Batching, Depot Selection, and Assignment Problem with Multiple Depots and Multiple Pickers.’. To solve this complex problem, a new bi-objective Mixed-Integer Linear Programming (MILP) formulation for small-sized problems and a meta-heuristic called ‘Dependent Harmony Search (DHS)22DHS = Dependent Harmony Search.’ for large-sized problems are proposed. The performance of DHS algorithm is evaluated by comparing the optimal results attained by MILP model. For the problem size of 10 orders, the average gap (%) in distance between the solution of DHS and MILP is 4.22%, although in some experiments DHS can find the optimal solution in a very short time. Also, in related analysis, it is seen that constructing multiple depots instead of one left-most located depot decreases total order picking distance by 7.11% on average.

Addressing staffing challenges through improved planning: Demand-driven course schedule planning and instructor assignment in higher education

This paper presents a novel decision support system (DSS) to address the University Course Timetabling Problem (UCTP). The solution decomposes the NP-complete UCTP into two sub-problems, allowing a structured approach to addressing the complexities inherent in the UCTP process. A mixed integer linear programming (MILP) model is proposed to integrate academic year course schedule planning and instructor assignment, accommodating various constraints to meet student demands. The model optimizes the number of course sections and strategically schedules instructors, aiming to reduce the number of new and distinct courses assigned to them. Historical data from an academic department encompassing multiple disciplines, including Computer Information Systems, Business Management, and Business Analytics, at a large public university in the U.S. is used to develop the model, and the results are compared with the actual course schedule and instructor assignment. The results demonstrate that the proposed DSS would result in a 14 % reduction in the number of course sections offered, translating to approximately $130,000 in annual savings. Additionally, it could significantly reduce the number of new courses assigned to instructors by up to 81 % and the number of distinct course sections assigned to them by 29 %.

A distribution-free-based approach for stochastic food closed-loop supply chain

ABSTRACT Resource scarcity has driven growing interest in circular economy (CE). Closed-loop supply chain (CLSC) with returnable transport items (RTIs) in the food industry is an important component of CE. However, existing works on food CLSC with RTIs have not simultaneously considered the perishability, facility location, and uncertain demand under limited information. Therefore, this work addresses a new food CLSC optimisation problem. We first propose a non-linear chance-constrained programming model. It is then transformed into a mixed-integer linear programming model via using the distribution-free (DF) method and sample average approximation (SAA) method, respectively. An illustrative example reveals that the DF method needs only 10.50% of the computation time of the SAA method. To address large-scale problems, an improved Lagrangian relaxation (LR) method is developed. To address the computational challenge in large-scale problems, an improved Lagrangian relaxation (LR) algorithm is developed. Results show that CPLEX achieves a gap of 75.57%, while the LR surpasses it by finding near-optimal solutions with a gap of 1.22%, using only 31.82% of the computation time required by CPLEX. For this work, the main insights are summarised: (1) extending product shelf life can reduce the total cost; and (2) to alleviate uncertain demand and production risks, production capacity and product inventory capacity can be appropriately expanded, but excessive investment may not improve returns.

A constraint programming approach for resource-constrained flexible assembly flow shop scheduling problem with batch direct delivery

Production and distribution are both crucial components of supply chains. Integrated production and distribution scheduling (IPDS) in the context of flexible assembly flow shop scheduling and batch delivery problems is often overlooked. A realistic problem inspired by the production and distribution processes of a dishwasher factory can be modeled as a resource-constrained flexible assembly flow shop scheduling problem with batch direct delivery (RCFAFSP-BDD). In this problem, order requirements are decomposed into several production tasks processed at different stages of the workshop, and are then delivered in batches via a third-party logistics provider to regional distributors at various locations. Auxiliary resource restrictions, hierarchical coupling constraints, machine eligibility restrictions and sequence-dependent setup times, are incorporated into the problem as operational constraints. To the best of our knowledge, this is the first attempt to solve this problem. This work formulates a mixed-integer linear programming (MIP) model to minimize total costs, including tardiness, inventory, and delivery costs. Given the problem’s strong NP-hard nature, the focus is on developing an efficient solution approach using constraint programming (CP). A CP model is proposed and enhanced with multiple redundant constraints. To reduce runtime, two branching strategies are designed for the CP model. Numerical experiments with varying instance scales reveal that the proposed CP model outperforms the MIP model in accuracy and efficiency within the given time limit. The redundant constraints and search strategy can reduce CP model runtime by up to 263.83%. Compared to manual scheduling at the studied factory, the CP model can cut costs by up to 26.59% for real data, offering viable alternatives for factory planners.

Model Predictive Control of a Stand-Alone Hybrid Battery-Hydrogen Energy System: A Case Study of the PHOEBUS Energy System

Model predictive control is a promising approach to robustly control complex energy systems, such as hybrid battery-hydrogen energy storage systems that enable seasonal storage of renewable energies. However, deriving a mathematical model of the energy system suitable for model predictive control is difficult due to the unique characteristics of each energy system component. This work introduces mixed integer linear programming models to describe the nonlinear multidimensional operational behavior of components using piecewise linear functions. Furthermore, this paper develops a new approach for deriving a strategy for seasonal storage of renewable energies using cost factors in the objective function of the optimization problem while considering degradation effects. An experimentally validated simulation model of the PHOEBUS Energy System is utilized to compare the performance of two model predictive controllers with a hysteresis band controller such as utilized for the real-world system. Furthermore, the sensitivity of the model predictive controller to the prediction horizon length and the temporal resolution is investigated. The prediction horizon was found to have the highest impact on the performance of the model predictive controller. The best-performing model predictive controller with a 14-day prediction horizon and perfect foresight increased the total energy stored at the end of the year by 18.9% while decreasing the degradation of the electrolyzer and the fuel cell.

Optimal planning and management of the energy–water–carbon nexus in hybrid AC/DC microgrids for sustainable development of remote communities

This paper presents a two-stage stochastic mixed-integer nonlinear programming model, which is linearized to a mixed-integer linear programming (MILP) form, for optimizing the energy–water–carbon (EWC) nexus in remote AC/DC microgrids aimed at sustainable community development. The model optimizes the selection, location, and operation of diesel generators, voltage source converters, photovoltaic systems, wind turbines, and battery energy storage systems while managing CO2 emissions. It accurately models the AC/DC microgrid’s operation, determining optimal voltages, currents, and converter states (inverter or rectifier). Uncertainties in power demand, water consumption, solar irradiation, and wind speed are modeled using stochastic scenarios generated via k-means clustering. The MILP model is implemented in GAMS and solved with the commercial solver CPLEX, ensuring global optimization. Results show that integrating distributed energy resources, compared to a case without these elements, reduces CO2 emissions of the diesel generators and the operational costs of the EWC nexus, highlighting the proposed approach’s environmental and economic benefits. These findings underline the importance of incorporating distributed energy resources in the sustainable development of microgrids.

Typical daily scenario extraction method based on key features to promote building renewable energy system optimization efficiency

Given its intrinsic volatility and randomness, the effective design of a building renewable energy system (BRES) necessitates long-period, short-step boundaries. Nevertheless, this poses computational burdens. In this context, time-series aggregation (TSA) emerges as a valuable approach, which involves obtaining a set of typical scenarios to substitute original time series. Numerous studies have focused on mathematical algorithms to minimize the deviation between typical scenarios and original time series. However, the premise of enhancing model-solving efficiency by extracting typical scenarios is to ensure the optimality of planning. Henceforth, identifying and quantifying key features in the original time series that significantly influence the planning, and further extracting need to be addressed. This paper proposes a typical day extraction method based on key features, named the feature-based method. Key features are summarized through an extensive literature review and then quantitatively expressed through different indicator models. Finally, a feature-based mixed integer linear programming (MILP) model is established for extraction. To evaluate the effectiveness of the feature-based method, it is applied to design a BRES and compared with traditional TSA methods such as Averaging Method (AM) and k-means. Results indicate that the feature-based method can effectively retain key features, thereby achieving more efficient and accurate designs.