Mixed-integer Programming Model Research Articles

Aerial drone fleet deployment optimization with endogenous battery replacements for direct delivery of time-sensitive products

Aerial drones offer a distinct potential to reduce the delivery time and energy consumption for the delivery of time-sensitive and small products. However, there is still a need in the relevant industry to understand the performance of drone-based delivery under different business needs and drone operating conditions. We studied a drone deployment optimization problem for direct delivery of time-sensitive products with release dates to customers maintaining a specified time window. This paper presents a new mixed-integer programming model, new valid inequalities, a new greedy heuristic algorithm, and a Genetic algorithm to help business owners optimally schedule and route their drone fleet minimizing the required fleet size, the required number of additional batteries, and total energy consumption. A realistic feature of the optimization method is that instead of replacing the drone battery after each return to the depot, it keeps track of the remaining energy in the drone battery and decides on battery replacements accounting for the drone routing and the user-specified minimum required battery energy. Numerical results based on real data from drone flight tests and prepared food delivery industry provide insights into the effect of different practical drone operating parameters on the required fleet size, the required number of battery replacements, and energy consumption. Results demonstrate that the proposed heuristic algorithm substantially outperforms the accelerated CPLEX in runtime while sacrificing the solution quality by a small amount. Additionally, results show that using a mixed fleet of hexacopter and quadcopter drones reduces the total energy consumption by 48.52% compared to using a homogeneous fleet of only hexacopters.

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.

Combinatorial optimization methods for yarn dyeing planning

AbstractManaging yarn dyeing processes is one of the most challenging problems in the textile industry due to its computational complexity. This process combines characteristics of multidimensional knapsack, bin packing, and unrelated parallel machine scheduling problems. Multiple customer orders need to be combined as batches and assigned to different shifts of a limited number of machines. However, several practical factors such as physical attributes of customer orders, dyeing machine eligibility conditions like flotte, color type, chemical recipe, and volume capacity of dye make this problem significantly unique. Furthermore, alongside its economic aspects, minimizing the waste of natural resources during the machine changeover and energy are sustainability concerns of the problem. The contradictory nature of these two makes the planning problem multi-objective, which adds another complexity for planners. Hence, in this paper, we first propose a novel mathematical model for this scientifically highly challenging yet very practical problem from the textile industry. Then we propose Adaptive Large Neighbourhood Search (ALNS) algorithms to solve industrial-size instances of the problem. Our computational results show that the proposed algorithm provides near-optimal solutions in very short computational times. This paper provides significant contributions to flexible manufacturing research, including a mixed-integer programming model for a novel industrial problem, providing an effective and efficient adaptive large neighborhood search algorithm for delivering high-quality solutions quickly, and addressing the inefficiencies of manual scheduling in textile companies; reducing a time-consuming planning task from hours to minutes.

Uncertain programming model for designing multi-objective reverse logistics networks

Given the contradiction between the rapid growth of products and the modest recovery rate of end-of-life products, there is a pressing need to understand the societal significance of establishing a reverse logistics network for end-of-life products. This research constructs an open-loop five-tier reverse logistic network model encompassing customers, centres for collection, disassembly and inspection, remanufacturing, and disposal. A multi-objective mixed-integer nonlinear programming model under uncertainty has been developed. Unlike previous research, this model accounts for surrounding residents' disutility of facilities while simultaneously minimising economic costs and environmental impact. Besides, uncertainty theory is introduced in solving the proposed model. More specifically, the formulated model converts all uncertain variables into uncertain distributions by implementing the uncertain multi-objective programming method. Furthermore, a customised non-dominated sorting genetic algorithm III (NSGA-III) is proposed and is employed for the first time to address facility selection and recycling volume distribution within the network. The model is then validated using a real-life case study focusing on end-of-life vehicles in Changchun, China. This research could assist decision-makers in both governmental and private sectors in achieving a balanced approach to the interests of the economy, environment, and local communities comprehensively when designing reverse supply chains.

An Improved Q-Learning Algorithm for Optimizing Sustainable Remanufacturing Systems

In our modern society, there has been a noticeable increase in pollution due to the trend of post-use handling of items. This necessitates the adoption of recycling and remanufacturing processes, advocating for sustainable resource management. This paper aims to address the issue of disassembly line balancing. Existing disassembly methods largely rely on manual labor, raising concerns regarding safety and sustainability. This paper proposes a human–machine collaborative disassembly approach to enhance safety and optimize resource utilization, aligning with sustainable development goals. A mixed-integer programming model is established, considering various disassembly techniques for hazardous and delicate parts, with the objective of minimizing the total disassembly time. The CPLEX solver is employed to enhance model accuracy. An improvement is made to the Q-learning algorithm in reinforcement learning to tackle the bilateral disassembly line balancing problem in human–machine collaboration. This approach outperforms CPLEX in both solution efficiency and quality, especially for large-scale problems. A comparative analysis with the original Q-learning algorithm and SARSA algorithm validates the superiority of the proposed algorithm in terms of convergence speed and solution quality.

Optimization methods for plastics management supply chain design

AbstractThis article introduces three mixed integer programming (MIP) models to address a network design problem for mixed plastic waste (MPW) supply chains. By tracking waste compositions throughout the supply chain, the models optimize the technologies needed to process MPW. The three models adopt different approaches to preserve composition information in the supply chain. We also remark on how to improve solution times with additional constraints, and how the models can be easily modified to handle larger‐scale problems. The proposed models provide an approach for examining emerging MPW recycling technologies that may be more sensitive to input composition, as well as determining the extent to which advanced sorting is useful.

A robust mixed-integer binary programming model for operating theater scheduling to the patient and the surgeon under uncertainty in an open-heart Surgery Department

In hospitals, the surgical ward is both a cost and revenue center. In this ward, hospitals face challenges such as increasing demand, limited resources, and rising costs. Consequently, the decisions made have an implications effect on the hospital's performance. Therefore, in this paper a robust mixed-integer binary programming model is proposed with three objectives of maximizing the efficiency of available resources, minimizing the patients waiting time, and minimizing surgery costs that are formulated utilizing the augmented epsilon constraint approach. This model allocates the operating room to the patient and the surgeon and then obtains the required bed capacity inside the downstream units for stand-alone cardiac hospitals. This model includes different preferences for hospital, surgeon, and patient: waiting time, patient cancellations, tardiness, uncertainties in surgery durations, the patient operation start times, the overtime per working day, time windows, SICU beds, planning horizon, and the idle times of the surgeons, operating theater, and working day. The proposed model is solved using robust optimization to deal with stochastic. The proposed model is formulated on the stochastic programming method proposed by Bertsimas and Sim. In the proposed model, a rolling horizon method is used to reschedule the program after cancellation. The computational results illustrate that the rolling horizon method reduces waiting time and increases throughput. The results illustrate that the benefit obtained from the introduced model has improvements in reducing the surgery costs, and patient waiting time, and increasing the efficiency of available resources. This study has been performed at Shahid Rajaei Heart Hospital in Iran.

A Study on Disrupted Flight Recovery Based on Logic-Based Benders Decomposition Method

Aiming at the disrupted flight recovery problem, this paper established a mixed-integer programming model based on the resource assignment model to minimize the recovery cost. To deal with the large-scale examples, the Logic-Based Benders decomposition algorithm is designed to divide the problem into a master problem and sub-problems. The algorithm uses MIP in the master problem to determine flight cancellations and aircraft replacements. In the sub-problems, MIP or CP is used to determine the departure time of delayed flights. Later, incorporating sectional constraints into the main problem and iterating until an optimal solution is obtained. Furthermore, the added cutting plane constraint in the iterations of the Benders decomposition algorithm are strengthened to eliminate more inferior solutions. By comparing the results of CPLEX, the Logic-Based Benders decomposition algorithm, and the enhanced Benders decomposition algorithm, it is verified that the improved Benders decomposition algorithm can solve large-scale examples more efficiently with a faster time and fewer iterations.

Planning of electric vehicle charging stations: An integrated deep learning and queueing theory approach

This study presents a hybrid solution for the charging station location-capacity problem. The proposed approach simultaneously determines the location and capacity of charging stations (i.e., number of charging piles), and assigns piles to electric vehicles based on their level of charge. The problem is formulated as a bi-objective mixed-integer nonlinear programming model to minimize the total cost of establishing charging stations together with the average customers’ waiting time. The proposed solution combines queueing theory with mathematical modelling to estimate the average waiting time. A deep learning algorithm is then developed to enhance the precision of waiting time estimation. Another contribution is involving a deep neural network model in improving NSGA-II algorithm. Numerical experiments are conducted in Halifax, Canada to assess the performance of the proposed framework. The results demonstrate the strong predictive performance of the deep learning algorithm and highlight the limitations of traditional queueing models in estimating waiting times in charging stations (i.e., 99.8% improvement in computation time, as well as accuracy improvement of time estimations from 13% to 1.6% deviation). Several valuable insights are obtained to improve the operational performance of charging stations such as achieving a significant (i.e., 61.5%) drop in the average waiting time across the network by a modest (i.e., 29.2%) increase in the initial investments. Also, it reveals that the variability of service rate significantly impacts the average waiting time (i.e., a 50% increase in the variability of service rate causes a substantial 950.56% surge in the average waiting time). The findings underscore the need to control service rate fluctuations to reduce wait times and boost driver satisfaction. The improved NSGA-II algorithm shows 12.77% improvement in the Pareto front solutions. Finally, the proposed prioritization strategy based on the charging level of vehicles could reduce the average waiting time and cost compared to the first-come-first-served strategy.

A short-term wind-hydrothermal operational framework in the presence of pumped-hydro storage

Short-term operation is one of the most significant aspects of power systems operation and planning and it mainly includes resource scheduling including renewable and non-renewable energy sources. The mentioned problem is modeled as an optimization problem with one or more objectives depending upon the priorities of the decision maker (system operator) and several equality and inequality constraints. In this respect, this paper proposes a short-term operational model for the hydrothermal power systems in the presence of wind power generation and pumped-hydro storage (PHS) units. The problem is formulated as a single-objective optimization problem that aims to minimize the total operating cost including the fuel cost of thermal units. The problem has been comprehensively investigated through five case studies where the first case study is used for the sake of validation. Then, the other case studies assessed the impacts of wind power generation, PHS unit, and uncertainty on the problem. The five case studies are as follows: Case 1: Short-term hydrothermal (STH) scheduling which aims to minimize the total cost. Case 2: Deterministic short-term hydrothermal-wind (STHTW) scheduling which aims to minimize the total cost. Case 3: Stochastic wind-hydro-thermal scheduling that seeks to minimize the total cost taking into consideration the uncertainty of the wind power generation. Case 4: Deterministic wind-hydro-thermal scheduling in the presence of the PHS unit with the total cost minimization as the objective function. Case 5: Stochastic wind-hydro-thermal scheduling in the presence of the PHS unit that is used to evaluate the impact of uncertainty of wind power generation with the total cost minimization as the objective function. It is noteworthy that the first three case studies have been formulated using a non-linear programming (NLP) model and solved using the CONOPT solver in GAMS. Case studies 4 and 5 have been presented using a mixed-integer non-linear programming (MINLP) model and solved using the DICOPT solver in GAMS. A sensitivity analysis has also been carried out to assess the system's performance with different loading conditions. In this respect, the total cost and PHS operation with different amounts of load demand were evaluated and the results have been presented and discussed.

A data-driven approach to urban charging facility expansion based on bi-level optimization: A case study in a Chinese city

This paper addresses the optimization of urban charging facilities expansion in response to the growing charging demand and proposes a data-driven bi-level optimization framework. The proposed framework combines a mixed-integer programming model for siting and capacity planning, an improved sparrow search algorithm (ISSA) for hyperparameter optimization of the CNN-BiLSTM model, and the application of Benders decomposition with Lagrangian relaxation using Gurobi for solving the model. The proposed approach is empirically evaluated through a case study conducted in an administrative area of a Chinese city, using comprehensive charging records and other relevant information. The results demonstrate that the ISSA consistently outperforms other algorithms, showing an average improvement of 23.97 % in root mean square error, 19.25 % in average absolute percentage error, and 1.50 % in the determination coefficient. The modified Diebold-Mariano test confirms the significant advantage of ISSA. Based on the model analysis, the study recommends constructing 20 new charging stations in the case city, with more than half equipped with fast chargers. This strategy ensures that 95 % of the area can find a charging station within 5 min while also ensuring the operator's profitability. The research findings provide effective support for investment decisions related to charging facilities.

Battery swapping, vehicle rebalancing, and staff routing for electric scooter sharing systems

Electric scooter (e-scooter) sharing systems provide on-demand electric scooter rental services. E-scooters are equipped with swappable batteries managed by staff who visit each scooter and replace depleted batteries with charged ones. The e-scooters in this system are free-floating; they can be located anywhere without having to be returned to designated stations. Due to this characteristic of the scattered location of e-scooters, operation decisions with associated staff routing are more challenging than station-based services. Thus, it is critical to implement the efficient management of e-scooter redistribution and charging decisions with proper staff routing to successfully prepare for user demand within a limited operation time while minimizing the total operating cost. To this end, we introduce a battery swapping and vehicle rebalancing problem with staff routing for e-scooter sharing systems, formulated as a mixed integer programming (MIP) model. To derive solutions efficiently for a large-scale instance in practice, we propose a clustered iterative construction approach where the problem is decomposed into two phases. The first phase clusters regions using an approximation of intra-region and inter-region operation costs with the minimum spanning tree approach. The second phase efficiently derives multiple candidate regional sequences by our partial permutation procedure and following the sequences, iteratively solves a significantly reduced size of the problem to construct operation assignments. Our numerical experiments on the generated instances and real-world instances demonstrate that the proposed two-phase algorithm shows significantly superior performance in practically large scale instances than the benchmarks without clustering or the iterative procedures of our algorithm.

Matheuristic for the lot-sizing and scheduling problem in integrated pulp and paper production

Vertical pulp and paper production is challenging from a process point of view. Managers must deal with floating bottlenecks, intermediate storage levels, and by-product production to control the whole process while reducing unexpected downtimes. Thus, this paper aims to address the integrated lot sizing and scheduling problem considering continuous digester production, multiple paper machines, and a chemical recovery line to treat by-products. The aim is to minimize the total production cost to meet customer demands, considering all productive resources and encouraging steam production (which can be used in power generation). Production planning should define the sizes of production lots, the sequence of paper types produced in each machine, and the digester working speed throughout the planning horizon. Furthermore, it should indicate the rate of by-product treatment at each stage of the recovery line and ensure the minimum and maximum storage limits. Due to the difficulty of exactly solving the mixed integer programming model representing this problem for real-world instances, mainly with planning horizons of over two weeks, constructive and improvement heuristics are proposed in this work. Different heuristic combinations are tested on hundreds of instances generated from data collected from the industry. Comparisons are made with a commercial Mixed-Integer and Linear Programming solver and a hybrid metaheuristic. The results show that combining the greedy constructive heuristic with the new variation of a fix-and-optimize improvement method delivers the best performance in both solution quality and computational time and effectively solves realistic size problems in practice. The proposed method achieved 69.41% of the best solutions for the generated set and 55.40% and 64.00% for the literature set for 1 and 2 machines, respectively, compared with the best solution method from the literature and a commercial solver.

Deep learning enhanced mixed integer optimization: Learning to reduce model dimensionality

This work introduces a framework to address the computational complexity inherent in Mixed-Integer Programming (MIP) models by harnessing the potential of deep learning. By employing deep learning, we construct problem-specific heuristics that identify and exploit common structures across MIP instances. We train deep learning models to estimate complicating binary variables for target MIP problem instances. The resulting reduced MIP models are solved using standard off-the-shelf solvers. We present an algorithm for generating synthetic data enhancing the robustness and generalizability of our models across diverse MIP instances. We compare the effectiveness of (a) feed-forward neural networks (ANN) and (b) convolutional neural networks (CNN). To enhance the framework's performance, we employ Bayesian optimization for hyperparameter tuning, aiming to maximize the occurrence of global optimum solutions. We apply this framework to a flow-based facility location allocation MIP formulation that describes long-term investment planning and medium-term tactical scheduling in a personalized medicine supply chain.

Matheuristics for mixed-model assembly line balancing problem with fuzzy stochastic processing time

Our work aims to investigate methods for solving the mixed-model assembly line balancing problem (MALBP) under uncertainty with the objective of minimizing the number of workstations. Specifically, we model task processing time as fuzzy stochastic variables (FRVs) due to the inherent uncertainties and variations in the manufacturing environment. Additionally, we introduce a ranking method for FRVs and propose a mathematical model to address MALBP. The recently developed Red Fox Optimization (RFO) algorithm is also discretized for the first time to support solving this problem. Finally, matheuristic algorithms combine a metaheuristic such as the popular Genetic Algorithm (GA), Particle Swarm Optimization (PSO), or the Discretized Red Fox Optimization (DRFO) algorithm with the Mixed-Integer Programming (MIP) model to generate the best solution in a reasonable time. Our comparative results demonstrate that the GA-MIP combination outperforms the others in both objective value and computational time.

Joint optimization of train timetable and passing facility: A capacity allocation strategy

To addreess the challenge of overcrowding in the urban rail transit system, this paper proposes a novel joint optimization strategy that optimizes the timetable and the passing facility’s working status (PFWS) simultaneously according to the passenger demands. To this end, a mixed-integer nonlinear programming (MINLP) model is formulated to jointly minimize overcrowding on trains and platforms, as well as the passenger travel time. The passing facility setting and the passenger flow activities depiction are particularly taken into account by the rigorous constraints. Subsequently, we design a simulation-based hybrid variable neighborhood search (SHVNS) algorithm to efficiently generate near-optimal solution. This algorithm incorporates a variable neighborhood search (VNS) algorithm, a randomly reset strategy, and a simulation-based algorithm. The VNS algorithm includes five neighborhood operators to enhance search efficiency. Finally, various experiments with the real-world data from Chongqing rail transit line 6 in China are tested to prove the applicability and the performance of our proposed model and algorithm. The computational results show that the joint optimization strategy outperforms a current operation strategy in terms of the overcrowded situation on trains and platforms, often by more than 60%, and some management implications and theoretical support can be obtained for the urban rail transit operator.

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.

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.

Optimal location of base stations for cellular mobile network considering changes in users locations

In the past few decades, the demand for cellular networks has grown tremendously due to the growing demand for data. However, the vast majority of the users are dynamic since they use their smart phones and tablets to connect to these networks when they move to one location to another to attend sport games or other events for example. The location of these events might not cover the large demand. In this paper, we address the classical problem of locating base stations for a mobile cellular network to serve mobile users in a given geographical area considering the users' movements within the network. We developed a mixed integer programming model to provide the optimal location of base stations at different time periods with the network's minimum total cost (i.e., installation and operation of base stations). This paper considers the COST-231-Walfisch-Ikegami propagation model for the path loss calculation with a non-line-of-site situation. To demonstrate the capability of the proposed model, it is experimented with a real-world problem on the Red Sea coast of Saudi Arabia. The model was capable of finding the optimal base station locations with minimum installation and operational costs considering the capacity and quality of service constraints.

A multi-objective model for selecting response strategies of primary and secondary project risks under interval-valued fuzzy uncertainty

Risks are uncertain events that can affect criteria such as project cost, quality, and completion time and ultimately lead to project failure. That is why the project risk management process, which is one of the most fundamental parts of project management, must be done properly. In this paper, a new multi-objective fuzzy model is proposed to respond appropriately to the primary and secondary project risks. The contributions, such as risk interdependency, considering the project with multi-mode activities, resource considerations, as well as attention to interval-valued fuzzy uncertainties are regarded simultaneously for the first time. The purpose of the proposed multi-objective mathematical model is to minimize cost, quality reduction, and project completion time. Then, a new two-stage uncertain solution approach is introduced in this paper. In this solution approach, using the equivalent method in the first step and then using the extended multi-choice goal programming method in the second step, for both the lower and upper limits of the membership functions for interval-valued fuzzy numbers, the computational processes are performed. To show the application and effectiveness of the proposed model, a case study adapted from the literature is given. Finally, sensitivity analysis is conducted to analyze the behavior of the developed mixed-integer programming model. The analysis of obtained results indicates that the utilization of the proposed model leads to better and more appropriate solutions.