Integer Programming Model Research Articles

Variable neighborhood genetic algorithm for multi-order multi-bin open packing optimization

In the rapidly evolving e-commerce landscape, efficient packaging and logistics reduce costs and enhance customer satisfaction. This study addresses the problem of dynamic bin size optimization in e-commerce logistics by proposing a series of intelligent algorithms. Considering real-world constraints such as item separation requirements, a Mixed Integer Programming Model for Multi-Order Multi-Box Open-Dimension Rectangular Packing (MOMB-ODRPP) is formulated. The Stacked Clustering Algorithm (SCA) series, One-Dimensional Fixed Stacked Clustering Algorithm (ODF-SCA), Two-Dimensional Fixed Stacked Clustering Algorithm (TDF-SCA), and Variable Neighborhood Descent Spatial Ordering Algorithm (VND-SOA) series are employed to solve the MOMB-ODRPP model and improve order packing rates and optimize bin sizes. Computational experiments using real-world data from JD’s e-commerce operations reveal that the TDF-SCA algorithm series outperforms the ODF-SCA series by approximately 5% in Case 4. In contrast, the VND-SOA-S1 and VND-SOA-S2 algorithms achieve improvements of 0.83% and 0.76%, respectively, over the TDF-SCA-P2 algorithm in Cases 4 and 11. The comparative analysis highlights the practical implications of bin size optimization, with Case 11 providing a more viable option for standardizing bin sizes in e-commerce logistics.

Fleet Repositioning, Flag Switching, Transportation Scheduling, and Speed Optimization for Tanker Shipping Firms

In response to the European Union (EU)’s sanctions on Russian oil products, tanker shipping firms may adopt two strategies to reoptimize their shipping networks. The first strategy is to switch the flag states of tankers that are not eligible to operate on certain routes. The second strategy is to reposition tankers based on their flag states, i.e., moving those tankers that are eligible from other groups to specified routes. To help tanker shipping firms minimize the total operating cost during the planning horizon in the context of EU oil sanctions, including costs of fleet repositioning, flag switching, and fuel, this study investigates an integrated problem of fleet repositioning, flag switching, transportation scheduling, and speed optimization considering the dynamic relationships among fuel consumption, speed, and load. By formulating the problem as a nonlinear integer programming model and applying various linearization techniques to convert the nonlinear model into a linear optimization model solvable by off-the-shelf linear optimization solvers, this study demonstrates the practical application potential of the proposed model, with the longest solution time of less than two hours for a numerical instance with seven routes. Furthermore, through sensitivity analyses on important factors including unit fuel prices, crude oil transportation demand, and the tanker repositioning cost, this study provides managerial insights into the operations management of tanker shipping firms.

Time-dependent hydrogen fuel cell vehicle routing problem with drones and variable drone speeds

This paper proposes a novel problem called the time-dependent hydrogen fuel cell vehicle routing problem with drones and variable drone speeds (TDHFCVRP-D-VDS). In this problem, a fleet of hydrogen fuel cell vehicles (HFCVs) are equipped with multiple unmanned aerial vehicles (UAVs) to perform delivery and pickup services within the customers’ time windows. The UAV is capable of performing pickup operations after the delivery service, as long as it does not exceed its energy capacity. The HFCV can launch and retrieve the UAV multiple times as needed throughout the routing process. We establish a mixed integer programming model to simultaneously minimize the total cost and makespan, and verify its accuracy by Gurobi. To tackle larger-scale instances, we propose a non-dominated sorting genetic algorithm III with intelligent selection (NSGA-III-IS). The initial population is generated using four distinct approaches aimed at enhancing both diversity and quality. Considering the customers’ time windows, we factor in the temporal-spatial distances between customers when generating the initial population. Our approach employs a two-phase algorithm for developing initial solutions. In the first phase, the algorithm focuses on generating UAV trips, while in the second phase, it creates joint delivery routes for both the HFCV and the UAV. To enhance the optimization process, we have developed four strategies for optimizing UAV speed. These strategies dynamically adjust the UAV’s speed in response to the customers’ time windows and the HFCVs’ arrival times. Additionally, we have integrated an intelligent selection mechanism to optimize the execution probabilities of both general and problem-specific operators. The experimental results demonstrate the following: (1) The proposed NSGA-III-IS outperforms other variant algorithms and two benchmark algorithms; (2) UAVs significantly benefit from variable flight speeds, resulting in reduced costs and improved efficiency; (3) The HFCV with UAV joint delivery pattern is superior for reducing carbon emissions compared to other joint delivery patterns; (4) Longer customer time windows and optimized UAV speed strategies are effective in reducing the total cost, makespan, and total UAV hover waiting time; (5) Finally, a method that combines multi-attribute decision-making with principal component analysis is utilized to aid decision-makers in selecting satisfactory solutions.

A bidirectional branch-and-price algorithm with Pulse procedure for the Electric Vehicle Routing Problem with flexible deliveries

Delivery flexibility offers customers the convenience of receiving their orders at a location and time that best suits their schedule. Moreover, it enables delivery companies to better manage their operations and decrease costs. Nonetheless, this flexibility brings added complexities to route optimization. This study focuses on the Electric Vehicle Routing Problem with Flexible Deliveries, which becomes more challenging when the fleet comprises electric vehicles with limited autonomy. To solve this problem, we develop a novel column generation method embedded within a bidirectional branch-and-price algorithm enhanced with Pulse procedure. We adopt an enhanced bidirectional algorithm to solve the pricing subproblem to overcome common weaknesses of the classical labeling algorithms such as label storing. The BP is improved with state-of-the-art acceleration techniques including (i) a bidirectional search mechanism in which both forward and backward labels are created, (ii) a method to prevent solving the pricing sub-problem at each iteration, and (iii) an integer programming model that generates upper bounds. We present a new data set including instances with up to 120 customers and an extensive computational study evaluating the performance of the proposed BP algorithm. Our results show that the proposed algorithm can effectively solve instances with up to 60 customers, 237 delivery locations, and 13 recharging stations.

A hybrid multi-criteria decision-making model for waste facilities location considering system resilience

The location of waste treatment facilities is a crucial strategic decision in urban management. Most existing studies focus on mathematical modeling and algorithm solving for multi-objective models aimed at minimizing cost or pollution. They lack consideration for the resilience of decision solutions in the face of uncertain and unpredictable disruptive incidents. This paper aims to bridge the research gap by proposing a top-down hybrid multi-criteria decision-making model that incorporates cost, pollution, social discontent, and system resilience simultaneously. The proposed model integrates various methods such as integer programming modeling, simulation modeling, and catastrophe analysis. It addresses the challenge that disruptive incidents cannot be precisely expressed. Additionally, it quantitatively analyzes the factors that impact system resilience. Finally, the proposed model is validated by applying it to a waste management network based on the real case of Shenzhen city. The results demonstrate notable differences in the resilience indexes among the solutions in the Pareto-optimal set, which confirms the significance of system resilience from another aspect. Further catastrophe analysis reveals several valuable insights for urban waste management. This paper offers a new perspective to cope with various risks related to waste management and provides a method reference and decision support for the construction of urban resilience.

Multi-objective optimization for assigning reviewers to proposals based on social networks

Most of the literature on reviewer assignment problem (RAP) considers fitness or efficiency as the evaluation criterion of an assignment, e.g., the degree of matching between reviewers and proposals. In this study, we consider fairness of an assignment and establish two multi-objective linear integer programming models for proposal assignment. We introduce a definition of fairness based on the closeness of social relationships between scholars. The closeness of social relationships is quantified based on social networks among reviewers and serves as part of objective functions. An improved space-partitioning algorithm is designed to solve the multi-objective models with integer objectives. When the problem size is large, two meta-heuristic algorithms are applied. Experiments on real and randomly generated cases are provided to illustrate the effectiveness of the proposed models and algorithms.

Reliable truck-drone routing with dynamic synchronization: A high-dimensional network programming approach

Within the context of burgeoning multimodal transportation systems, the emerging integration of drones and trucks is shaping the landscape of last-mile delivery. In this service-centric industry, uncertainty in travel time challenges the stability of delivery systems, highlighting the need for a more reliable and effective methodology. To tackle this challenge, we propose a reliable truck-drone collaborative routing problem enabling dynamic synchronization between trucks and drones in this paper. An integer programming model based on a space–time-state network has been developed to minimize the total travel time as well as their associated variabilities. Furthermore, to overcome the computational challenges for large-scale scenarios, a multi-dimensional-based hierarchical decomposition has been developed to identify a high-quality solution effectively. Numerical experimental results based on an illustrative network and a real-world network demonstrate the effectiveness of the introduced algorithm. Our proposed methodology advances operational planning for truck-drone collaborative delivery by appropriately navigating unavoidable variations of travel time, shedding light on the broad development prospects of the multimodal transportation system.

Optimization of transition from open-pit to underground mining considering environmental costs

The problem of optimizing the transition from open-pit to underground mining remains a challenge within the field of mining engineering, with far-reaching implications for sustainable development. Several researchers have tackled this transition problem, primarily focusing on maximizing project profitability while grappling with the complexities associated with solving it for real large-scale ore deposits. This study presents a mixed integer programming model that accounts for costs associated with preventing, mitigating, or compensating for adverse environmental impacts on net present value (NPV). Then, a Python-based code was developed using the Divide and Conquer algorithm, with the goal of reducing various operational modes and boosting solver speed to overcome the limitations encountered in the existing literature, particularly when dealing with large-scale deposits. The developed solution aims to transform a block model with varying dimensions into a selective mining unit with uniform dimensions, align the pushback's size and location with the mine's depth, length, and width, convert the matrix into a functional format, and clear previous calculations and results upon achieving the maximum NPV in the new phase. The study also evaluated the environmental impact on the transition problem using this methodology, applying it to the Sungun large-scale copper deposit. The results favor a sequential combined mining approach, yielding an NPV of 7.51 billion USD at a transition depth of 887.5 m with environmental costs included, and 7.61 billion USD at a depth of 950 m without environmental cost considerations.

Prediction-based decomposition optimisation for multi-portfolio supply chain resilience strategies under disruption risks

ABSTRACT This paper focuses on the design of supply chain (SC) risk mitigation and recovery strategies during long-term disruptions caused by the COVID-19 outbreak, which affect both suppliers and plants. Consequently, concurrent disruptions in supply and production are observed, which vary in duration and result in time-varying reductions in supply and production capacity. To cope with long-term disruptions, a modified multi-portfolio approach that integrates simulation and predictions is proposed to develop efficient mitigation and recovery plans. This approach involves selecting primary and recovery supply and production portfolios concurrently. To achieve this objective, time-dependent mixed integer programming (MIP) models that incorporate preparedness and recovery measures are developed to optimise SC operations. A prediction-based decomposition optimisation method is proposed to solve MIP problems and coordinate supply and production portfolios under disruptions and uncertainties. Furthermore, a heuristic approach is established to provide a comprehensive solution process. Finally, computational experiments and comparative analysis are conducted on a real-life case study. The results demonstrate that the proposed modelling and optimisation methods can effectively address disruptions and improve SC resilience. In addition, the developed models and approaches have the potential to serve as decision-making tools in SC management during disruptions.

A bi-level optimization framework for the power-side virtual power plant participating in day-ahead wholesale market as a price-maker considering uncertainty

This paper develops a novel bilevel decision-making framework for a power-side virtual power plant (VPP) comprising refined power to gas (P2G), wherein the upper-level is the cost minimization objective of VPP, while the lower-level is social cost minimization objective of wholesale electricity market. Meanwhile, wind power uncertainty is handled by distributionally robust optimization method. By utilizing Karush-Kuhn Tucker optimality condition and strong duality theory, the bilevel model is interpreted as a tractable single-level mixed integer programming model. Simulation results show that: relative to the models with removing uncertainty and refined P2G device, 1) the proposed approach realizes outstanding economic benefit by obtaining an increase with a rate by 7.47 % and 105.09 % in total actual profit, respectively. Regarding environmental benefit, CO2 emission volume experiences a 0.89 % decline and 0.90 % increase, and the CO2 intensity of the proposed strategy is the lowest with the value of 193. 25kg/MWh. 2) The proposed model can achieve the economic and low-carbon dispatching of the VPP. Wind power curtailment in the proposed model witnesses a trivial increase and reduced by 53.19 %, respectively. 3) The model exhibits merits due to its trade-off between robustness and computational complexity.

Large-scale dynamic surgical scheduling under uncertainty by hierarchical reinforcement learning

Dynamic surgical scheduling within a workday is a complicated decision-making process. The critical challenge is that the actual duration of surgery and the arrival process of emergency patients are uncertain and unknown in advance. In this work, we propose a two-level dynamic scheduling framework based on hierarchical reinforcement learning to solve dynamic surgical scheduling problems considering both elective and emergency patients. Specifically, with the realisation of uncertainty, the upper-level agent (UA) dynamically decides whether to trigger rescheduling to optimise the workday total cost. The lower-level agent (LA) aims at obtaining subscheduling solutions when rescheduling is triggered. The subproblem at the LA can be formulated as a mixed integer programming model, which can be generalised to unrelated parallel machine scheduling with machine eligibility restrictions and sequence- and machine-dependent setup times. Such problems can be solved in small-scale cases and suffers the combinatorial explosion in large scale cases. To address this issue, we propose a heuristic method that is built upon deep reinforcement learning to obtain high-quality solutions. We conduct extensive simulation experiments with real data to test the effective of our framework. The results for different scenarios show that our proposed framework outperforms existing methods in terms of overall performance and has strong generalisation ability.

Optimal deployment of private 5G multi-access edge computing systems at smart factories: Using hybrid crow search algorithm

In smart factories, an increasing number of mobile intelligent devices are deployed to meet the growing demands for flexible manufacturing. These devices, equipped with various sensors, synchronize a substantial amount of data with cloud servers for real-time monitoring and control. Fifth generation mobile networks (5G) combined with multi-access edge computing (MEC) provide the capabilities for multiple connections, large-scale data transmission, and efficient response times, becoming the mainstream network architectures for the needs. Additionally, with the emerging trend of energy-saving, recent works have addressed energy consumption as a significant cost factor. However, most previous studies tended to focus solely on cloud servers, neglecting energy consumption at edge computing and underestimating the impact of heterogeneous mobile device. As the scope of smart factory networks continues to expand, the challenges become increasingly critical. This study presents a private 5G MEC system architecture tailored for smart panel factories. The concerned problem is formulated as an integer programming model, which determines deployment locations and quantities for MEC servers and 5G small cells, including picocells and femtocells, so as to minimize the overall deployment costs while meeting the constraints of connectivity, capacity, latency, coverage, serviceability, and energy consumption. Since the edge server deployment problem is NP-hard, this study further proposes a hybrid metaheuristic algorithm, CSAVNS, which combines the strengths of crow search algorithm (CSA) and variable neighborhood search (VNS). In the global search phase of CSA, the VNS local search is introduced to enhance the algorithm's capabilities. Experimental analysis demonstrates that the proposed CSAVNS outperforms other algorithms in terms of solution solving capabilities.

A Novel Rolling Horizon Based Solution Framework for Scheduling Airplane Maintenance

Maintenance, Repair and Overhaul (MRO) activities on aircraft and systems in aviation is a service sector that relies heavily on skilled workforce. The output of MRO activities is basically bringing the system reliability values, which decrease because of the use of aircraft and systems in certain flight times and landing and take-off numbers, to the default levels determined during the design phase. MRO companies are accountable to the civil aviation authorities who directly authorize them to ensure the required levels of reliability of their products. Airline companies request MRO services at the most convenient time and cost. However, the maintenance of an aircraft is a process that can take up to five weeks, includes plenty of jobs some of which may have stochastic durations, and many over-costed qualified technicians spend thousands of man-hours. In addition, each MRO company is involved in the maintenance of several airplanes arriving at different time intervals. In the study such a problem faced by an MRO company is addressed. The company’s aim is to schedule several incoming airplane maintenance projects. A framework that employs an integer programming (IP) model working on a rolling horizon (RH) setting is used.

Integrated optimization of transport capacity and train timetabling with time-dependent passenger demand

When facing a surge in passenger demand especially during holidays, dispatchers have to adjust train operation organization for busy high-speed railway corridors. This paper is devoted to optimizing transport capacity considering the capacity pool and flexible train composition with time-dependent passenger demand to generate new timetables for both regular and additional trains. A bi-objective mixed integer programming model is formulated to minimize the waiting time of passengers and operation costs for capacity utilization. A local search technique is hybridized into an elitist heuristic algorithm based on NSGA-II to generate high-quality solutions. Two sets of numerical experiments are carried out based on a small-scale example and real-world data from the Wuhan-Guangzhou high-speed railway corridor to verify the effectiveness of the proposed methodology. The results show that (1) when considering time-dependent passenger demand, the satisfaction rate exceeds 90%; (2) when optimizing transport capacity, satisfaction is guaranteed while bringing reduction in operation costs.

Optimal Routing and Scheduling of Flag State Control Officers in Maritime Transportation

Maritime transportation plays a pivotal role in the global merchandise trade. To improve maritime safety and protect the environment, every state must effectively control ships flying its flag, which is called flag state control (FSC). However, the existing FSC system is so inefficient that it cannot perform its intended function. In this study, we adopt an optimization method to tackle this problem by constructing an integer programming (IP) model to solve the FSC officer routing and scheduling problem, which aims to maximize the total weight of inspected ships with limited budget and human resources. Then we prove that the IP model can be reformulated into a partially relaxed IP model with the guarantee of the result optimality. Finally, we perform a case study using the Hong Kong port as an example. The results show that our model can be solved to optimality within one second at different scales of the problem, with the ship number ranging from 20 to 1000. Furthermore, our study can be extended by considering the arrangement of working timetables with finer granularity and the fatigue level of personnel.

Logistics vehicle routing optimisation with synchronised transfer

This paper presents a synchronized transfer strategy in which operators can schedule logistics vehicles among employees and arrange the transfer of commodities. The transfer time consists of the vehicle waiting time and commodity transition time. By adopting the strategy, logistics vehicles can visit fewer communities, resulting in savings in transportation cost under ensuring punctual delivery. An integer programming model based on a space-time-state network is proposed to describe the complex transfer process. To simplify the model, time window and vehicle capacity constraints are embedded into the network. The alternating direction method of multipliers (ADMM) is designed to solve the model. In the ADMM-based solution framework, the original model can be converted into a series of shortest-path searching subproblems and iteratively solved using a dynamic programming (DP) algorithm. Our computational results show that the proposed model and algorithm are efficient and competitive.

A mathematical model and solution algorithms for optimising system-level configurations of reconfigurable single part flow lines

This study addresses a system-level configuration selection problem for reconfigurable single part flow lines (R-SPFLs) with parallel identical flexible machines at each stage. The problem is to determine the number of stages, the number of machines of a certain type at each stage and the assignment of operations to each stage in order to satisfy the demand of a part type. Precedence relations among the operations and space limitations are also considered. For the objective of minimising the sum of machine purchase and operation processing costs, a nonlinear integer programming model is developed that gives the optimal solutions. Then, due to the problem's complexity, a basic variable neighbourhood search algorithm is proposed that constructs an initial solution using a greedy-type heuristic and improves it using a shaking and a local search improvement methods. Moreover, the basic algorithm is extended to a general one that uses a variable neighbourhood descent method. Computational results show that the two algorithms outperform the previous one significantly in both solution quality and computation times, and the general algorithm improves the basic one significantly. Finally, the applicability of the variable neighbourhood search algorithms is shown by reporting a brief case study.

Autonomous data detection and inspection with a fleet of UAVs

Consider an area of interest A, where a set of nsites lie. Two kinds of information can be captured from each site: light and heavy information. A fleet of m homogeneous UAVs, each one equipped with a battery B, is available at a common depot, where the flight mission of each UAV starts and finishes. The problem we consider focuses on a single flight of the fleet of UAVs and aims at collecting their light information from all sites (that can be retrieved, not necessarily passing over each site, but simply “close” to it). At the same time, the fleet will have to select a limited number of sites from which to collect their heavy information. Flying among sites and acquiring information from them (both light and heavy) has a battery cost. On the other hand, a profit is associated with the action of acquiring heavy information from a site. We refer to the extraction of light and heavy information from a site as to weakly or strongly cover the site. The aim of the problem consists of retrieving light information from all sites while maximizing the overall profit, keeping the battery consumption of each UAV within B. In this paper, we model this real-life situation as a new combinatorial optimization problem that we call m3DIP, for which we provide a mixed integer programming model. Given the high degree of complexity of the problem, in this way we are not able to provide a solution in a reasonable time. To address larger instances we propose a matheuristic in which we exploit a path-based algorithm filled with only a subset of feasible cycles (paths) provided by different heuristics. The output indicates which path to select and the set of nodes to be strongly and weakly covered by each trip. We compare our matheuristic with the results obtained by every single heuristic on a large set of instances, showing that the matheuristic strongly outperforms them. An interesting insight is that even paths provided by a heuristic with very bad performances can be useful if combined with paths provided by other heuristics and if the coverage decisions are reoptimized by the matheuristic. We also show the benefit of adding fictitious additional points that UAVs can visit to weakly cover a subset of sites, without actually visiting none of them.

Uncertainty Analysis of Aircraft Center of Gravity Deviation and Passenger Seat Allocation Optimization

The traditional method of allocating passenger seats based on compartments does not effectively manage an aircraft’s center of gravity (CG), resulting in a notable divergence from the desired target CG (TCG). In this work, the Boeing B737-800 aircraft was employed as a case study, and row-based and compartment-based integer programming models for passenger allocation were examined and constructed with the aim of addressing the current situation. The accuracy of CG control was evaluated by comparing the row-based and compartment-based allocation techniques, taking into account different bodyweights and numbers of passengers. The key contribution of this research is to broaden the range of the mobilizable set for the aviation weight and balance (AWB) model, resulting in a significant reduction in the range of deviations in the center of gravity outcomes by a factor of around 6 to 16. The effectiveness of the row-based allocation approach and the impact of passenger weight randomness on the deviation of an airplane’s CG were also investigated in this study. The Monte Carlo method was utilized to quantify the uncertainty associated with passenger weight, resulting in the generation of the posterior distribution of the aircraft’s center of gravity (CG) deviation. The outcome of the row-based model test is the determination of the range of passenger numbers that can be effectively allocated under different TCG conditions.

Logic-based Benders decomposition for order acceptance and scheduling on heterogeneous factories with carbon caps

We study an integrated order acceptance and scheduling problem for heterogeneous factories with carbon caps, order-dependent processing speeds, and setup times. Three joint decisions including order selection, order assignment and order sequencing are made to maximize profits, i.e., total revenue minus total production and tardiness cost. Firstly, we develop a mixed integer programming model to simultaneously optimize the three decisions and propose several dominance rules to enhance the model. Secondly, due to the particular structure of the problem, we propose a logic-based Benders decomposition (LBBD) method that decomposes the complicated original problem into a master problem and a number of subproblems. The master problem aims to determine order acceptance and order assignment, and each subproblem is for determining order sequencing in each factory. A branch and bound algorithm is then designed to quickly solve the subproblems. The branch and check framework is implemented to accelerate the solving process of the LBBD. Finally, numerical experiments verify that the proposed dominance rules play a promising role in enhancing the model, and the results of the algorithm comparison experiments show the significant advantages of proposed LBBD algorithm combined with the branch and check framework. Sensitivity experiments reveal that carbon cap serves as a major constraint on order acceptance.