Integer Linear Programming Model Research Articles

An iterative method for integrated hump sequencing, train makeup, and classification track assignment in railway shunting yard

In a railway shunting yard, the transformation of inbound trains into properly composed outbound trains is a complex task because it involves decisions of multiple operations processes. This study addresses the integrated optimization of hump sequencing, train makeup, and classification track assignment problem in a railway shunting yard. Several key practical yard operation constraints are considered, including train formulation constraints, hump sequencing constraints, and limitations of the maximum number and capacity of classification tracks. By introducing a new representation of block flow, the integrated problem, which adopts the extended single-stage strategy and the train-to-track policy, is formulated as a unified 0-1 integer linear programming model. The objective of the proposed model is to minimize the weighted-sum of the total dwell time of all railcars and the formulation deviation penalties of all outbound trains. Then, an iterative two-phase decomposition approach is developed to reduce the complexity of solving the integrated problem. The first phase aims to explore all feasible humping sequences using a Branch-and-Bound (B&B) algorithm. Each time a new humping sequence is generated in the first phase, the second phase containing a Branch-and-Price (B&P) algorithm is applied to solve the integrated train makeup and classification track assignment problem with the known humping sequence found in the first phase. In addition, greedy heuristics and lower bounding techniques are designed in both phases to improve computational efficiency. Comprehensive experiments are investigated based on a set of real-life instances. The results show that exact approaches provide optimal solutions, whereas heuristic approaches yield satisfactory solutions within a shorter computation time. Moreover, sensitivity analyses on the number of classification tracks and the effects of different deviation penalties are also performed to gain more managerial insights.

A modified adaptive large neighborhood search algorithm for solving the multi-port continuous berth allocation problem with vessel speed optimization

Despite its fast-growing popularity in maritime transportation, container shipping is still fraught with risks and uncertainties with its complex operating environments. This paper studies the multi-port continuous berth allocation problem with speed optimization (MCBAP). In the MCBAP, vessels visit multiple ports sequentially, and the problem aims at minimizing the sum of vessel sailing cost, waiting cost, delay cost and port handling cost, while satisfying various constraints related to vessel sailing and berthing. A mixed integer linear programming (MILP) model for MCBAP is formulated, and a modified adaptive large neighborhood search (MALNS) algorithm is proposed for solving large-scale MCBAPs. In the MALNS, an efficient initial solution generation strategy is developed, and a series of neighborhood solution generation operators are proposed. Finally, the proposed MILP model and MALNS algorithm are tested on a range of MCBAP instances. The numerical results demonstrate that the MILP model can be solved to optimality with CPLEX, and the MALNS can efficiently solve instances at various scales. In addition, sensitivity analyses on fuel prices and vessel design speeds (the planned maximum speeds) are performed, and management insights have been provided.

Bi-objective optimization for equipment system-of-systems development planning using a novel co-evolutionary algorithm based on NSGA-II and HypE

Previous research on the equipment system-of-systems development planning (ESoSDP) problem has predominantly focused on the portfolio selection of developed equipment (DE) or equipment pending development (EPD), often neglecting the real-world confrontation and practical implementation of DE and EPD. Motivated by this gap, we undertake a novel investigation into the ESoSDP problem, integrating the characteristics of systematization, confrontation, and implementation. To address this, we formulate an integer linear programming model aimed at minimizing total expenditure costs while maximizing operational effectiveness. Specifically, a two-side equipment system-of-systems (ESoS) confrontation network, encompassing both EPD and DE, is designed to evaluate the operational effectiveness of the ESoS. The associated costs are determined by integrating the expenses related to research and development (R&D), procurement, maintenance, and decommissioning activities of both DE and EPD. To address this problem, a co-evolutionary algorithm, named MCEANH, which integrates the NSGA-II and HypE algorithm, is proposed. Within the MCEANH framework, an adaptive crossover-mutation strategy and a knowledge transfer mechanism between NSGA-II and HypE are introduced to enhance its performance. Through a series of comprehensive experiments conducted across nine different solution scales, MCEANH demonstrated superior performance in terms of distribution and convergence when compared to three widely-used multi-objective optimization algorithms, as well as their modified versions incorporating the adaptive crossover-mutation strategy. This study provides essential insights and practical tools for managers of ESOSDP, particularly in light of current trends in systemic confrontation. The research not only contributes to academic discourse but also proposes pragmatic planning schemes for real-world ESoSDP challenges, emphasizing the necessity of integrating real-world confrontation and equipment implementation into ESoSDP.

An integrated optimization strategy for mission planning and control resource allocation for long-endurance unmanned aerial vehicle

PurposeWith burgeoning interest in the low-altitude economy, applications of long-endurance unmanned aerial vehicles (LE-UAVs) have increased in remote logistics distribution. Given LE-UAVs’ advantages of wide coverage, strong versatility and low cost, in addition to logistics distribution, they are widely used in military reconnaissance, communication relay, disaster monitoring and other activities. With limited autonomous intelligence, LE-UAVs require regular periodic and non-periodic control from ground control resources (GCRs) during flights and mission execution. However, the lack of GCRs significantly restricts the applications of LE-UAVs in parallel.Design/methodology/approachWe consider the constraints of GCRs, investigating an integrated optimization problem of multi-LE-UAV mission planning and GCR allocation (Multi-U&G IOP). The problem integrates GCR allocation into traditional multi-UAV cooperative mission planning. The coupling decision of mission planning and GCR allocation enlarges the decision space and adds complexities to the problem’s structure. Through characterizing the problem, this study establishes a mixed integer linear programming (MILP) model for the integrated optimization problem. To solve the problem, we develop a three-stage iterative optimization algorithm combining a hybrid genetic algorithm with local search-variable neighborhood decent, heuristic conflict elimination and post-optimization of GCR allocation.FindingsNumerical experimental results show that our developed algorithm can solve the problem efficiently and exceeds the solution performance of the solver CPLEX. For small-scale instances, our algorithm can obtain optimal solutions in less time than CPLEX. For large-scale instances, our algorithm produces better results in one hour than CPLEX does. Implementing our approach allows efficient coordination of multiple UAVs, enabling faster mission completion with a minimal number of GCRs.Originality/valueDrawing on the interplay between LE-UAVs and GCRs and considering the practical applications of LE-UAVs, we propose the Multi-U&G IOP problem. We formulate this problem as a MILP model aiming to minimize the maximum task completion time (makespan). Furthermore, we present a relaxation model for this problem. To efficiently address the MILP model, we develop a three-stage iterative optimization algorithm. Subsequently, we verify the efficacy of our algorithm through extensive experimentation across various scenarios.

A Data-Driven Optimization Framework for Static Rebalancing Operations in Bike Sharing Systems

Bike sharing systems have been widely deployed in urban cities for first- and last-mile transportation. However, because of the geographical and temporal imbalance of bike demand, bikes need to be reallocated system-wide among stations during the night to maintain a high service level while minimizing demand loss due to stockout or overcapacity. Two technical challenges remain in optimizing the static bike rebalancing operations. One challenge is to accurately predict bike pickup and dropoff demand at each station, considering demand substitution effects and subsequently determining the optimal rebalancing quantity for each station. The other is to efficiently optimize the routing of multiple rebalancing vehicles for large-scale bike sharing systems, considering outlier stations with rebalancing quantities exceeding vehicle capacity. To this end, we propose an end-to-end solution to tackle the aforesaid challenges. Specifically, we first develop deep learning-based predictors that capture the time dependencies of station-level demand, the impact of weather conditions, and the demand substitution effect by nearby stations. Based on the demand rate, a sequential simulation-based demand loss estimator is developed to find the optimal rebalancing quantities that lead to the minimum expected demand loss. Then, a mixed integer linear programming model is formulated to optimize the routing problem of rebalancing vehicles. To address the computational challenge, we propose a data-driven decomposition algorithm to support a multivehicle multivisit rebalancing strategy by decomposing the multivehicle routing problem into smaller and tractable single-vehicle routing problems, which can be solved in parallel. Finally, extensive numerical experiments using real-world data from New York City Citi Bike demonstrate the accuracy of the proposed bike demand predictors, the impact of demand substitution, and the efficiency of the data-driven optimization framework. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Funding: This work was supported by the National Natural Science Foundation of China [Grant 72201222] and the Hong Kong Research Grants Council [Grants CityU 21500220 and CityU 11504322]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0182 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0182 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A model for container stowage planning considering stability constraints

The growth in global trade has led to the deployment of mega-container ships to meet increasing transportation demands. However, the larger vessel sizes pose significant challenges for planners in arranging container loading. This study addresses the container stowage planning problem for near-sea routes, incorporating stability constraints, container types, and technical limitations related to stack weight and stress. We propose a mixed integer linear programming model to describe this problem and generate load plans that minimize total costs associated with re-stowing and bending moments (BM). To solve the model efficiently, we design a dynamic weight and time-varying particle swarm optimization (PSO) algorithm, which combines particle classification and time-varying acceleration constants. The effectiveness of the model and the efficiency of the algorithm are validated through extensive numerical experiments. The results demonstrate that our proposed algorithm outperforms the traditional PSO algorithm in both solution quality and computation time. Additionally, sensitivity analyses are conducted to derive potentially useful managerial insights for supporting shipping liners’ decision-making. This study highlights the benefits of particle classification and time-varying acceleration constants, and reveals the influence of container bay capacity and permitted frame BM.

Profit-oriented balancing of two-sided disassembly lines with resource-dependent task times

PurposeAs a result of environmentally conscious production requirements in the world, the concept of disassembly has been a focus of interest by researchers and practitioners over the last two decades. Disassembly is an important process in circular economy to recover and reuse of parts and materials. End-of-life and large-sized products such as minibuses and trucks may be disassembled on two-sided lines. The ability of using both right and left sides of two-sided lines may increase line efficiency and reduce space requirements across the line. This paper aims to address a two-sided disassembly line balancing problem (TSDLBP), which deals with assigning disassembly tasks, various equipments and assistants to the workstations to maximize total net recovery profit of the line.Design/methodology/approachA detailed explanation of the TSDLBP is first presented in the paper. A new 0–1 integer linear programming model is then proposed for the TSDLBP, aiming at maximizing total net recovery profit from disassembly of products. A set of test problems is generated, and an experimental analysis is conducted to make a comparison between traditional one-sided disassembly lines (TOSDL) and two-sided disassembly lines by means of performance improvement rate.FindingsOptimal results are obtained in 132 (81.48%) out of 162 the TOSDL balancing problems, while 92 (56.79%) out of 162 the TSDLBP using the proposed model. Total net recovery profits are compared on 88 problems for which optimal solutions are obtained in both the TOSDL and the TSDLBP. Results showed that implementing two-sided disassembly lines provides 29.18% increment in total net recovery profit compared to the TOSDL. Furthermore, the effects of different parameter levels on the net recovery profit are analyzed using two-way analysis of variance. According to the results, implementing two-sided disassembly line configuration increases total net recovery profit of the line significantly compared to traditional disassembly line configuration.Originality/valueThe use of disassembly lines has become essential because of increasing consumption that results in a huge number of end-of-life products in the world. Two-sided disassembly lines may be preferred for dismantling large-sized products due to their high disassembly capacity and fewer space requirements. This paper proposes a new mathematical model for disassembly line balancing problem. The proposed model differs from the existing models by means of efficiently assigning limited disassembly resources as well as assigning disassembly tasks to the workstations to maximize total net recovery profit of the production system. The model allows decision-makers to consider several resource limitations when balancing their disassembly lines. The paper also provides a comprehensive experimental study to compare traditional and two-sided disassembly lines by means of profitability of disassembly processes.

A GRASP-based multi-objective approach for the tuna purse seine fishing fleet routing problem

Nowadays, the world’s fishing fleet uses 20% more fuel to catch the same amount of fish compared to 30 years ago. Addressing this negative environmental and economic performance is crucial due to stricter emission regulations, rising fuel costs, and predicted declines in fish biomass and body sizes due to climate change. Investment in more efficient engines, larger ships and better fuel has been the main response, but this is only feasible in the long term at high infrastructure cost. An alternative is to optimize operations such as the routing of a fleet, which is an extremely complex problem due to its dynamic (time-dependent) moving target characteristics. To date, no other scientific work has approached this problem in its full complexity, i.e., as a dynamic vehicle routing problem with multiple time windows and moving targets. In this paper, two bi-objective mixed linear integer programming (MIP) models are presented, one for the static variant and another for the time-dependent variant. The bi-objective approaches allow to trade off the economic (e.g., probability of high catches) and environmental (e.g., fuel consumption) objectives. To overcome the limitations of exact solutions of the MIP models, a greedy randomized adaptive search procedure for the multi-objective problem (MO-GRASP) is proposed. The computational experiments demonstrate the good performance of the MO-GRASP algorithm with clearly different results when the importance of each objective is varied. In addition, computational experiments conducted on historical data prove the feasibility of applying the MO-GRASP algorithm in a real context and explore the benefits of joint planning (collaborative approach) compared to a non-collaborative strategy. Collaborative approaches enable the definition of better routes that may select slightly worse fishing and planting areas (2.9%), but in exchange for a significant reduction in fuel consumption (17.3%) and time at sea (10.1%) compared to non-collaborative strategies. The final experiment examines the importance of the collaborative approach when the number of available drifting fishing aggregation devices (dFADs) per vessel is reduced.

A novel data-driven rolling horizon production planning approach for the plastic industry under the uncertainty of demand and recycling rate

Efficient production planning in the plastic industry requires integrating sustainability practices, particularly the use of recycled plastics. Recycling helps reduce environmental impacts and conserve resources by decreasing the reliance on virgin materials. To develop a responsive plan, dynamic approaches are beneficial to confront fluctuations in uncertain parameters such as recycling rate and market demand. The present study proposes a novel Data-driven Rolling Horizon Planning (DRHP) approach for a sustainable and dynamic production plan in the plastic industry. A dynamic Rolling Horizon (RH) planning framework is formulated as a multi-product, multi-period Mixed Integer Linear Programming (MILP) model for the problem. This model aims to minimize total production cost while taking into account the system’s constraints and sustainability considerations. To deal with uncertainty, the RH-based MILP model is coupled with a rolling Long Short-Term Memory (LSTM) model. The rolling LSTM model leverages historical data of market demand and recycling rate to predict their future values and consequently improve the responsiveness of the production plan. The outperformance of the proposed DRHP approach is demonstrated through an extensive comparison with a robust static counterpart in terms of total production cost and sustainability performance. Results indicate significant reductions, up to 80%, in production costs using the proposed DRHP approach. Furthermore, the effect of rolling duration is investigated concerning production cost, backlog, late-order, and inventory level. Findings highlight the potential of the proposed DRHP approach to mitigate inventory- and late-order-related challenges.

Joint scheduling of hybrid flow-shop with limited automatic guided vehicles: A hierarchical learning-based swarm optimizer

Transportation system in workshop is essential for high-efficient production scheduling. Due to the limited transportation resources, the joint scheduling of production and transportation has emerged as a pivotal issue in modern manufacturing. This paper investigates a joint scheduling of hybrid flow-shop with limited automatic guided vehicles (HFSP-LAGV), which extends the classical hybrid flow-shop scheduling by considering the limited number of the AGVs on the transportation resources. To solve such problem, a mixed integer linear programming (MILP) model is firstly built to formulate HFSP-LAGV. Then, a hierarchical learning-based swarm optimizer (HLSO) is proposed. An encoding and decoding method based on three dispatch rules is proposed. The framework of HLSO comprises a pyramid-based layering strategy, an inter-layer learning and an intra-layer learning. The pyramid-based layering strategy divides the swarm into several layers. In the inter-layer learning, the individuals in higher layers guide the evolution of individuals in lower layers to achieve the exploration of global area. In the intra-layer learning, an offline Q-learning-based local search is designed to implement the self-learning of elite individuals in higher layer to intensify the exploitation of the local area. A Q-learning model that has been pre-trained offline is used to guide the selection of appropriate operator of local search. Experimental results reveal the effectiveness of the designs and the superiority of HLSO over several well-performing methods on solving HFSP-LAGV.

Economically viable reshoring of supply chains under ripple effect

This paper presents stochastic mixed integer linear programming models for economically viable supply chain reshoring under the ripple effect, that is, for the integrated optimization of supply chain reshoring and supply chain viability. The problem objective is to select supply chain facilities for relocation and backup suppliers for recovery supplies to simultaneously minimize cost, maximize service and enhance viability. If the primary supplier is not selected for reshoring, the viability can alternatively be improved by recovery supplies from the backup supplier. The problem is formulated as a multi-portfolio stochastic optimization. The portfolio of suppliers/plants for reshoring is optimized along with an alternative recovery portfolio of backup supplies. This study indicates that reshoring decisions directly affect supply chain viability and have positive impact on both its average and worst-case performance. The more risk-aversive is viable reshoring (the confidence level greater than 0.90), the greater is also improvement of supply chain business-as-usual resilience. If increase of expected customer service is more important than cost reduction, supplier reshoring is preferable to backup sourcing. The findings also indicate that cost- and service-optimal reshoring decisions are getting closer when the government subsidy for reshoring significantly increases such that the reduced reshoring costs and domestic costs do not exceed the achieved reduction of lost sales and backup sourcing. Then both conflicting objectives can be simultaneously achieved, which is a highly desirable solution of the economically viable reshoring of supply chain. Numerical examples based on a real-world industrial supply chain prove practical usefulness of the proposed stochastic approach and its high computational efficiency.

On Interdicting Dense Clusters in a Network

Given a vertex-weighted undirected graph with blocking costs of its vertices and edges, we seek a minimum cost subset of vertices and edges to block such that the weight of any γ-quasi-clique in the interdicted graph is at most some predefined threshold parameter. The value of [Formula: see text] specifies the edge density of cohesive vertex groups of interest in the network. The considered weighted γ-quasi-clique interdiction problem can be viewed as a natural generalization of several variations of the clique blocker problem previously studied in the literature. From the application perspective, this setting is primarily motivated by the problem of disrupting adversarial (“dark”) networks (e.g., social or communication networks), where γ-quasi-cliques represent “tightly knit” groups of adversaries that we aim to dismantle. We first address the theoretical computational complexity of the problem. We then exploit some basic characterization of its feasible solutions to derive a linear integer programming (IP) formulation. This linear IP model can be solved using a lazy-fashioned branch-and-cut scheme. We also propose a combinatorial branch-and-bound algorithm for solving this problem. The computational performance of the developed exact solution schemes is studied using a test bed of randomly generated and real-life networks. Finally, some interesting insights and observations are also provided using a well-known example of a terrorist network. History: Accepted by Russel Bent, Area Editor for Network Optimization: Algorithms & Applications. Funding: The work of S. Butenko was partially supported by the Air Force Office of Scientific Research under Award FA9550-23-1-0300. The work of O. A. Prokopyev was partially supported by the Office of Naval Research under Award ONR N00014-22-1-2678. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0027 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0027 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ . The online appendix is available at https://doi.org/10.1287/ijoc.2023.0027 .

Asynchronous decentralized traffic signal coordinated control in urban road network

AbstractThis study introduces an asynchronous decentralized coordinated signal control (ADCSC) framework for multi‐agent traffic signal control in the urban road network. The controller at each intersection in the network optimizes its signal control decisions based on a prediction of the future traffic demand as an independent agent. The asynchronous framework decouples the entangled interdependence between decision‐making and state prediction among different agents in decentralized coordinated decision‐making problems, enabling agents to proceed with collaborative decision‐making without waiting for other agents’ decisions. Within the proposed ADCSC framework, each controller dynamically optimizes its signal timing strategy with a unique rolling horizon scheme. The scheme's individualized parameters for each controller are determined based on the vehicle travel time between the adjacent intersections, ensuring that controllers can make informed control decisions with accurate arrival flow information from upstream intersections. The signal optimization problem is formulated as a mixed integer linear program model, which adopts a flexible signal scheme without a fixed phase structure and sequence. Simulation results demonstrate that the proposed ADCSC strategy significantly outperforms the benchmark signal coordination methods in terms of average delay, travel speed, stop numbers, and energy consumption. Experimental analysis on computation time validates the applicability of the proposed optimization model for real‐time implementation. Sensitivity analysis on key parameters in the framework is conducted, offering insights for parameter selection in practice. Furthermore, the ADCSC framework is extended to a road network in Qinzhou City, China, with 45 signalized intersections, demonstrating its effectiveness and scalability in the real‐world road network.

Part Feeding and Internal Transportation Decision Making for a Machinery Manufacturer

This research validates a mixed integer linear programming model that minimizes total feeding and acquisition costs by integrating assembly line feeding and vehicle type decisions in a machinery manufacturing company. The results show that the optimal assignment reduces costs by 56% compared with the company's current assignment.

Research on cargo volume prediction and personnel scheduling optimization of sorting center based on BP neural network and improved genetic algorithm

With the development of Internet technology, e-commerce has become the primary choice for consumers to shop, and the huge order transaction volume has generated logistics problems. As an important part of the logistics network, the sorting platform is essential to improve its management efficiency. To enhance the efficiency of the sorting center, the topological relationship diagram is used to show the logistics network, and the cargo volume of the sorting center in the next 30 days and every hour is predicted based on the BP neural network model. Based on the mixed integer linear programming model and the improved genetic algorithm SCAGA model, the total man-days are minimized to ensure that the cargo volume is processed. At the same time, to ensure that the actual hourly efficiency is as balanced as possible, the optimization problem of personnel scheduling is studied. To maintain the efficient operation of the logistics industry, the accurate prediction of the number of goods and the reasonable scheduling of personnel, to provide a set of efficient and economical personnel scheduling programs for the logistics company, thus improving the overall operational efficiency and service quality.

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

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

A mathematical model for potash supply chain management with a strategic logistics perspective

Abstract Accepted by: M. Zied Babai This paper introduces a novel Integer Linear Programming model designed to enhance the efficiency and sustainability of the potash supply chain, a crucial element supporting global agriculture. The developed mathematical optimization model focuses on fleet selection (private/outsource) and incorporates the concept of ‘inter-warehouse collaboration’, which addresses key logistics considerations. Integrating mining, processing, storage and transportation, the model encompasses decision variables like extracted carnallite amount, production, storage levels and shipped potash amount. Illustrated through a case study on the Arab Potash Company in Jordan, the results showcase the model’s proficiency in meeting local and international market demands. The model ensures resilient and sustainable supply chain performance by emphasizing logistics optimization, particularly in fleet selection. The study attains the highest ‘warehouse-to-warehouse’ support for Standard and Granular potash types in the international demand scenario, contributing to efficient production planning and fleet management. In conclusion, the presented mathematical model is a valuable tool for potash industry stakeholders, offering insights for strategic decision-makers involved in production planning and fleet management.

Optimizing Supply Chain Design under Demand Uncertainty with Quantity Discount Policy

In typical business situations, sellers usually offer discount schemes to buyers to increase overall profitability. This study aims to design a supply chain network under uncertainty of demand by integrating an all-unit quantity discount policy. The objective is to maximize the profit of the entire supply chain. The proposed model is formulated as a mixed integer nonlinear programming model, which is subsequently linearized into a mixed integer linear programming model and hence able to obtain a global solution. Numerical examples in the manufacturing supply chain where customer demand follows normal distributions are used to assess the effect of quantity discount policies. Key findings demonstrate that the integration of quantity discount policies significantly reduces total supply chain costs and improves inventory management under demand uncertainty, and decision makers need to decide a balance level between service levels and profits.

Deep reinforcement learning driven trajectory-based meta-heuristic for distributed heterogeneous flexible job shop scheduling problem

As the production environment evolves, distributed manufacturing exhibits heterogeneous characteristics, including diverse machines, workers, and production processes. This paper examines a distributed heterogeneous flexible job shop scheduling problem (DHFJSP) with varying processing times. A mixed integer linear programming (MILP) model of the DHFJSP is formulated with the objective of minimizing the makespan. To solve the DHFJSP, we propose a deep Q network-aided automatic design of a variable neighborhood search algorithm (DQN-VNS). By analyzing schedules, sixty-one types of scheduling features are extracted. These features, along with six shaking strategies, are used as states and actions. A DHFJSP environment simulator is developed to train the deep Q network. The well-trained DQN then generates the shaking procedure for VNS. Additionally, a greedy initialization method is proposed to enhance the quality of the initial solution. Seven efficient critical path-based neighborhood structures with three-vector encoding scheme are introduced to improve local search. Numerical experiments on various scales of instances validate the effectiveness of the MILP model and the DQN-VNS algorithm. The results show that the DQN-VNS algorithm achieves an average relative percentage deviation (ARPD) of 3.2%, which represents an approximately 88.45% reduction compared to the best-performing algorithm among the six compared, with an ARPD of 27.7%. This significant reduction in ARPD highlights the superior stability and performance of the proposed DQN-VNS algorithm.

Augmented ɛ‐constraint‐based matheuristic methodology for Bi‐objective production scheduling problems

AbstractMatheuristic is an optimisation methodology that integrates mathematical approaches and heuristics to address intractable combinatorial optimisation problems, where a common framework is to insert mixed integer linear programming (MILP) models as local search functions for evolutionary algorithms. However, since a mathematical programming formulation only tries to find the solution with the best objective value, matheuristics are rarely adopted to multi‐objective scenarios asking for a set of Pareto optimal solutions, for example, vehicle routing problems and production scheduling problems. In this situation, the ɛ‐constraint, which transforms multi‐objective problems into single‐objective formulations by considering selected objectives as constraints, seems to be a promising approach. First, an augmented ɛ‐constraint‐based matheuristic methodology (ɛ‐MH) is proposed to apply the idea of ɛ‐constraint to embedded MILP models, so that Pareto fronts obtained by meta‐heuristics can be further improved by solving a set of MILP models. Afterwards, four speed‐up strategies are developed to alleviate the computational burden resulting from repeatedly solving mathematical formulations, which also imply preferable scenarios for taking advantages of the ɛ‐MH. Finally, several real‐world bi‐objective scheduling problems are discussed to present potential applications for the proposed methodology.