Small-scale Instances Research Articles

Delivery routing for a mixed fleet of conventional and electric vehicles with road restrictions

The enforcement of stringent regulations capping carbon emissions has prompted city logistics enterprises to substitute electric vehicles (EVs) for conventional vehicles (CVs). For city logistics enterprises with a mixed fleet of CVs and EVs, this study investigates the delivery routing problem with road restrictions (DRPRR) for accessible time windows and bearing weight. We formulate the DRPRR as a mixed integer programming model to minimise the total operations cost. The model consists of the set-up cost of CVs and EVs, the diesel cost of CVs, the electricity cost of EVs, the carbon tax cost, and the penalty cost of vehicles waiting for roads to be accessible. To effectively solve the model, an adaptive large neighbourhood search (ALNS) algorithm is developed that consists of tailored destroy and repair operators with two alternative solution acceptance criteria. Numerical experiments are conducted to validate the effectiveness of the proposed model and ALNS algorithm. In small-scale instances, the ALNS with the Metropolis criterion finds the best solutions for 41 of 48 instances while maintaining a deviation of less than 2.5% from CPLEX in the remaining 7 instances, and its running time is significantly shorter than CPLEX. In large-scale instances, the ALNS with the Metropolis criterion has stronger solving ability and better stability than benchmark algorithms (i.e. GA-LS, LNS, and ALNS with a threshold acceptance criterion). We also address a real-world case and conduct a sensitivity analysis to provide useful managerial insights. Specifically, implementing road restrictions on accessible time windows and the carbon tax policy simultaneously is more appropriate from the comprehensive perspective of market activity and carbon emissions.

A Material Allocation Model for Public Health Emergency under a Multimodal Transportation Network by Considering the Demand Priority and Psychological Pain

In a public health emergency, residents urgently require a large number of rescue materials for treatment or protection. These rescue materials are usually located far from the emergency area. The government must organize rescue materials transportation by selecting suitable transport modes. Thus, we propose a material allocation model for public health emergencies under a multimodal transportation network to determine the best rescue material supply route. In this model, we set the demand priorities according to the emergency degrees to decide the transportation sequence. Meanwhile, we introduce the psychological pain cost brought by the rescue material shortage into the proposed model to trade off the priority and fairness of demand. Having compared it to the research literature, this is the first study that considers multiple categories of materials, absolute pain costs, relative pain costs and demand priority under multimodal transportation. The research problem is formulated into an integer programming model, and we develop a modified genetic algorithm to solve it. A set of numerical examples are conducted to test the performance of the proposed algorithm, and to investigate features and applications of the proposed model. The results indicate that the modified genetic algorithm performs better in the calculation examples at different scales. For small-scale instances, the algorithm produces consistent results with Gurobi. As the instance size increases, Gurobi fails to find the optimal solution within 1800 s, while this algorithm is able to find the optimal solution within an acceptable time frame. Additionally, when dealing with large-scale instances, the algorithm exhibits a significant advantage in terms of runtime. Sensitivity analysis of key factors indicate that (1) Adjusting the relative pain cost coefficient can make the best trade-off between fairness, economy and timeliness; (2) Compared with a single mode of transport, multimodal transport can reduce the psychological pain cost and the logistics cost; (3) Improving the loading and unloading capacity of nodes can reduce the delivery time of materials and the psychological pain cost of residents, but the influence of other factors and cost-effectiveness need to be considered.

An improved deep Q-learning algorithm for a trade-off between energy consumption and productivity in batch scheduling

The single-batch machine, commonly found in industrial manufacturing, can concurrently process a group of jobs in variable-speed batches, leading to fluctuating levels of both energy consumption and processing time. Identifying an optimal balance between total energy consumption and makespan presents a challenge due to the intricate nature of their relationship in real-world scenarios. This study introduces a mixed-integer programming model designed to determine optimal machine operating states for diverse workloads, meeting customer requirements while concurrently achieving energy savings for sustainability. An energy-aware batch scheduling deep Q-learning network (EBSDQN) framework has been created, encompassing sequencing policies, batching rules, and speed adjustment policies. This framework is explicitly crafted to tackle the NP-hard problem, a challenge that commonly perplexes commercial solvers like Gurobi when seeking optimal solutions within a 360-second time frame for small-scale instances. The EBSDQN design is fortified with efficient action decoding policies, refined reward assessments, exploitative neighborhood rules, and a streamlined buffer training process. This approach not only conserves training time but also adapts dynamically in accordance with the principles of the Markov Decision Process (MDP) and deep neural network. The developed algorithm exhibits impressive robustness, reaching convergence after 600 episodes of training. In separate comparisons with a commercial solver, two single-objective algorithms, and two multi-objective algorithms, our algorithm consistently demonstrates superior overall performance across the same instances. In the worst-case analysis, further examination delves into the influences of job features. In summary, this study underscores the high sensitivity of both TEC and Cmax to job processing time distribution, suggesting the prioritization of TEC over Cmax in optimization for sustainable planning and operations. Furthermore, this research has the potential to accelerate the integration of artificial intelligence into the manufacturing sector.

An efficient solution space exploring and descent method for packing equal spheres in a sphere

The problem of packing equal spheres in a spherical container is a classic global optimization problem, which has attracted enormous studies in academia and found various applications in industry. This problem is computationally very challenging, and many efforts focus on small-scale instances with the number of spherical items less than 200 in the literature. In this work, we propose an efficient local search heuristic algorithm named solution space exploring and descent for solving this problem, which can quantify the solution’s quality to determine the number of exploring actions and quickly discover a high-quality configuration. Besides, we propose an adaptive neighbor item maintenance method to speed up the convergence of the continuous optimization process and reduce the time consumption. Computational experiments on a large number of benchmark instances with 5≤n≤400 spherical items show that our algorithm significantly outperforms the state-of-the-art algorithm. Specifically, our algorithm improves 274 best-known results and matches 84 best-known results out of the 396 well-known benchmark instances.

Modeling and optimization of the hybrid flow shop scheduling problem with sequence-dependent setup times

The hybrid flow shop scheduling problem (HFSP) is an extension of the classic flow shop scheduling problem and widely exists in real industrial production systems. In real production, sequence-dependent setup times (SDST) are very important and cannot be neglected. Therefore, this study focuses HFSP with SDST (HFSP-SDST) to minimize the makespan. To solve this problem, a mixed-integer linear programming (MILP) model to obtain the optimal solutions for small-scale instances is proposed. Given the NP-hard characteristics of HFSP-SDST, an improved artificial bee colony (IABC) algorithm is developed to efficiently solve large-sized instances. In IABC, permutation encoding is used and a hybrid representation that combines forward decoding and backward decoding methods is designed. To search for the solution space that is not included in the encoding and decoding, a problem-specific local search strategy is developed to enlarge the solution space. Experiments are conducted to evaluate the effectiveness of the MILP model and IABC. The results indicate that the proposed MILP model can find the optimal solutions for small-scale instances. The proposed IABC performs much better than the existing algorithms and improves 61 current best solutions of benchmark instances.

Optimal Solution for the Integrated Model of the Problems in Capacitated Vehicle Routing and Vehicle Scheduling at the Multi-door Depot

To have an efficient Supply Chain (SC), the coordination and integration of the activities in the SC are mandatory. Routing vehicles is an optimization problem in the SC. When the routed vehicles return to the depot and the doors at the depot are busy, the returned vehicles must wait to unload the accumulated shipments. Therefore, properly scheduling these vehicles to those doors at the depot to minimize the waiting time is an optimization problem in the SC. Therefore, in this study, routing vehicles to collect the shipments from suppliers and scheduling vehicles to doors at the depot, based on a first-come-first-serve basis, are simultaneously solved. Hence, the objective of this integrated vehicle routing and scheduling problem (VR&SP) is to minimize the total cost which contains the following components: vehicle travelling cost between suppliers, loading cost at the suppliers, vehicle waiting cost, unloading cost at the depot and vehicle operations cost. A Mixed Integer Quadratic Programming (MIQP) model is developed to solve the integrated VR&SP. The Branch and Bound algorithm method is employed to obtain the exact optimal solution to this MIQP using LINGO optimization software. The compatibility of the developed MIQP model is verified by the randomly generated small-scale instances of VR&SP. Therefore, it can be concluded that, this model solves the vehicle routing to suppliers and vehicle scheduling to doors at the depot simultaneously. Since VR&SP is a NP-hard problem, heuristics or meta-heuristic methods are recommended to solve the large-scale instances of VR&SP. Moreover, it can be recommended for further studies to amend this model by incorporating additional constraints to make it more applicable to real-world scenarios.

A multi-day waste collection and transportation problem with selective collection and split delivery

This paper addresses a multi-day waste collection and transportation problem with selective collection and split delivery (MDWCTP-SCSD). Rather than emptying waste according to a fixed schedule, garbage trucks only visit community waste collection sites whose waste level reaches a predetermined threshold, reducing collection costs but increasing overflow risk. Collection sites likely to overflow the next day if unvisited today are also selected for preventive collection. Split delivery is permitted. The MDWCTP-SCSD is formulated as a mixed integer linear programming model. To solve this NP-hard problem, a heuristic combining variable neighborhood search and simulated annealing is proposed. In addition to classical operators, three new operators are developed to handle split delivery. Experiments demonstrate the proposed algorithm is competitive in solving two split delivery vehicle routing benchmark sets; outperforms CPLEX in solving small-scale instances within an hour; and quickly solves large-scale instances with high quality. Noteworthy findings indicate fleet costs and size can be reduced by approximately 50% if only visiting sites whose waste level reaches 80%.

A branch-price-and-cut algorithm for the local container drayage problem with controllable vehicle interference

This paper investigates a local container drayage problem with controllable vehicle interference (LCDP&CVI) under the tractor-and-trailer separation mode in which a tractor can be coupled or decoupled with a trailer at a customer location or a terminal. The container drayage requests proposed by customers should be fulfilled by a set of vehicle fleets, and each vehicle fleet comprises of a certain number of tractors and trailers that should be matched to each other. The controllable vehicle interference (CVI) means that the routes of any tractors from different vehicle fleets are independent of each other, implying that the interference between tractors is restricted only to the tractors from the same vehicle fleets. A mixed-integer linear programming (MILP) model is proposed for the LCDP&CVI. To exactly solve the problem, we develop a branch-price-and-cut algorithm based on the reformation of the MILP model, including a master problem and a pricing sub-problem. The pricing sub-problem that deals with the temporal synchronization constraints is formulated as the multi-vehicle shortest path problem with interference. By adding the tractor to the temporal-spatial network, a new representation of the pricing sub-problem is constructed, which enables us to solve the sub-problem by the multi-label method. In addition, the symmetrical elimination and dominance rules are incorporated into the multi-label method to enhance its effectiveness. Extensive numerical experiments are conducted to validate the proposed model and solution algorithm. For small-scale instances, the proposed solution algorithm is able to solve instances with up to 24 requests to optimality within two hours, which outperforms significantly the commercial MILP solver in terms of efficiency and effectiveness. For large-scale instances, the proposed algorithm can yield competitive solutions that are superior to the heuristic algorithms reported in the literature. Our experimental results also show that the controllable vehicle mechanism can strike a good balance between computational efficiency and operational costs by meticulously designing the structure of vehicle fleets.

Towards efficient airline disruption recovery with reinforcement learning

Disruptions to airline schedules precipitate flight delays/cancellations and significant losses for airline operations. The goal of the integrated airline recovery problem is to develop an operational tool that provides the airline with an instant and cost-effective solution concerning aircraft, crew members and passengers in face of the emerging disruptions. In this paper, we formulate a decision recommendation framework which incorporates various recovery decisions including aircraft and crew rerouting, passenger reaccommodation, departure holding, flight cancellation and cruise speed control. Given the computational hardness of solving the mixed-integer nonlinear programming (MINP) model by the commercial solver (e.g., CPLEX), we establish a novel solution framework by incorporating Deep Reinforcement Learning (DRL) to the Variable Neighborhood Search (VNS) algorithm with well-designed neighborhood structures and state evaluator. We utilize Proximal Policy Optimization (PPO) to train the stochastic policy exploited to select neighborhood operations given the current state throughout the Markov Decision Process (MDP). Experimental results show that the objective value generated by our approach is within a 1.5% gap with respect to the optimal/close-to-optimal objective of the CPLEX solver for the small-scale instances, with significant improvement regarding runtime. The pre-trained DRL agent can leverage features/weights obtained from the training process to accelerate the arrival of objective convergence and further improve solution quality, which exhibits the potential of achieving Transfer Learning (TL). Given the inherent intractability of the problem on practical size instances, we propose a method to control the size of the DRL agent’s action space to allow for efficient training process. We believe our study contributes to the efforts of airlines in seeking efficient and cost-effective recovery solutions.

A simulated annealing with variable neighborhood descent approach for the heterogeneous fleet vehicle routing problem with multiple forward/reverse cross-docks

With a greater awareness of the challenges regarding environmental, societal, political, and economic factors, where reverse logistics has become a significant part of supply chain networks, this paper presents an integrated forward and reverse logistics network, named the Heterogeneous Fleet Vehicle Routing Problem with Multiple Forward/Reverse Cross-Docks (HF-VRPMFRCD). We consider a heterogeneous fleet of vehicles with different loading capacities and transportation costs. We also consider multiple cross-docks with two different operations: forward and reverse processes. The former focuses on delivering the demand from suppliers to customers, while the latter aims at returning unsold products from customers to suppliers. We propose a Simulated Annealing with Variable Neighborhood Descent (SAVND) algorithm for solving HF-VRPMFRCD, where Variable Neighborhood Descent (VND) is a local search heuristic embedded in the framework of Simulated Annealing (SA). SAVND outperforms the state-of-the-art algorithm in solving the Heterogeneous Fleet Vehicle Routing Problem with Multiple Cross-Docks (HF-VRPMCD), where the VND heuristic significantly improves the quality of solutions. For HF-VRPMFRCD benchmark instances, SAVND provides optimal solutions for small-scale instances and better solutions than those of the GUROBI solver for remaining larger instances. Lastly, we present and discuss the benefits of integrating the forward and reverse processes.

Optimising the flight turnaround schedules: An improved sliding time windows approach based on MILP and CP models

This paper addresses the scheduling of turnaround operations for flights in civil airports. A series of complex issues, including the heterogeneity of flights and operators, resource constraints on turnaround operations, temporal restrictions and alternative service patterns of turnaround operations, multi-functional operators, multi-trip services and multi-to-one services, are jointedly considered. We propose two formulations for the problem: A mixed-integer linear programming (MILP) model, and a constraint programming (CP) model. For comparison purpose, we also present simplified versions of the models in which the multi-to-one services are prohibited. The uncertainty are incorporated by modifying the models based on robust optimisation principles. The numerical experiments are conducted using instances generated based on real flight data at Beijing Capital International Airport. It is shown that under current computer configurations, small-scale instances can be solved to the optimum by CPLEX within an acceptable period of time using the MILP model, while moderate-scale instances can be tackled using the CP formulation. To cope with large-scale instances, an improved sliding time window method is developed, and evaluated with different parameter combinations, showing that practical turnaround schedule would best be updated two to three times per hour. Besides, after a case study in both static and stochastic scenarios, we illustrate that multi-to-one services should be specifically considered and highlighted, for their existence could significantly improve the operator utilisation rate and reduce flight delay, especially when the uncertainties are taken into account.

Solving a new mathematical model for the integrated cockpit crew pairing and rostering problem by meta-heuristic algorithms under the COVID-19 pandemic

This study examines the problem of crew pairing and rostering under COVID-19 conditions. To keep the cockpit crew members safe against COVID-19, the problem is investigated in such a way that each cockpit crew member spends less daily sit time period in the airports and returns to his/her home base at the end of the day to take a rest. This study aims to introduce a Mixed-Integer Linear Programming (MILP) formulation for the problem. In order to solve the problem, three meta-heuristic algorithms including Genetic Algorithm (GA), Firefly Algorithm (FA), and Particle Swarm Optimization (PSO) are applied based on a new chromosome representation. The proposed algorithms could obtain solutions with the least possible number of cockpit crew members to cover the existing flights by considering some rules and regulation related to employing cockpits. Moreover, the findings indicate that the algorithms can provide solutions near the optimal solutions (1.94%, 2.49%, and 2.43% gaps for the GA, FA, and PSO on average, respectively) for the small-scale instances extracted from the data sets. Additionally, the proposed GA can find lower-cost solutions in comparison to the FA and PSO in approximately similar CPU time for problem instances with different sizes.

Multi-objective multi-fidelity optimisation for position-constrained human-robot collaborative disassembly planning

ABSTRACT Human-robot collaborative disassembly lines are widely used by remanufacturing companies to disassemble end-of-life (EOL) products. When disassembling large-sized EOL products, each workstation on a disassembly line is generally divided into multiple operating positions, so that different operators can disassemble the same product at their respective positions at the same time, thereby greatly improving efficiency. This paper focuses on a position-constrained human-robot collaborative disassembly planning (PC-HRCDP) problem for the above-mentioned lines, including three subproblems of disassembly sequence planning, disassembly line balancing and robot path planning. A multi-objective mixed integer programming model for PC-HRCDP is developed to solve small-scale instances. Furthermore, a multi-objective multi-fidelity optimisation (MO-MFO) algorithm is proposed to solve large-scale instances. Comprehensive experiments are conducted based on 10 problem instances generated in this study. Experimental results show that the proposed MO-MFO is better than a high-fidelity optimisation algorithm in terms of running time. In addition, benefiting from the strategy of MO-MFO to allocate the limited high-fidelity computational budget to solutions in the two stages of multi-objective optimisation and optimal sampling, MO-MFO is significantly better than the existing representative multi-fidelity optimisation algorithms in terms of the hyper-volume and the inverted generational distance.

Integrated production and transportation scheduling problem under nonlinear cost structures

We consider an integrated production and transportation scheduling problem arising in several make-to-order settings, where customer orders have release times and pre-specified delivery time windows. These orders are first processed in a plant and, thereafter, delivered to their customer sites by transporters (e.g., freight trains, ships, or air flights) with fixed departure times and nonlinear transportation cost functions. If the processing of an order is completed but not immediately delivered, it will be stored temporarily, incurring inventory holding costs. The objective is finding an integrated schedule for production and transportation such that the total cost of inventory holding and transportation is minimized. In this study, we consider the following two cases: one where partial delivery is not allowed, and the other where partial delivery is allowed. For each case, we propose an exact algorithm to find optimal solutions for small-scale instances, and a heuristic algorithm to find near-optimal solutions for large-scale instances. For the randomly generated test instances, the computational results reveal that the exact algorithms significantly outperform a commercial optimization solver in terms of the computational times and the number of instances solved within a specified time limit, and the heuristic algorithms are capable of generating near-optimal solutions within a reasonable computational time. Finally, we incorporate considerations of sequence-dependent setup time into the problem and apply the approaches for designing the algorithms to address the more general problem.

Stability measure for prefab balancing in prefabrication construction supply chain management

Effective management of the prefabrication construction supply chain management (PCSCM) requires careful consideration of rational resource allocation and Just-in-Time (JIT) delivery. The transportation and storage of materials can be a significant expense in the construction industry, making it essential to prioritize the efficient allocation of resources. When resource allocation and delivery schedules are not properly managed, it can result in additional costs and project delays. Current practices mainly depend on the foreman’s experience, leading to unpunctual delivery and insufficient distribution. Punctual delivery involves collaboration with many stakeholders, including main contractor, logistics companies, foreman, etc. This paper proposes a resource allocation model with a penalty mechanism to comprehensively consider the benefit of the main contractor under PCSCM, aiming to minimize costs and strategically allocate resources. Buffer space, as a crucial safety place, is beneficial for balancing the stability of PCSCM and controlling the cost among stakeholders. This research takes into account several practical considerations, including limitations on storage and buffer space, penalty of unpunctual delivery, and the availability of multiple transportation routes. Especially, research seldom includes the penalty mechanism with a soft time window related to the demand and supply of resources based on the storage limitation. The numerical analysis demonstrates that the result of the proposed PSO is similar to CPLEX in small-scale instances. The results show that implementing a penalty also significantly saves the CPU around 38% for the main contractor compared with the model without penalty mechanism in a practical case. These contributions allow stakeholders to optimize resource allocation and project timelines while minimizing waste through a rational penalty mechanism.

An artificial immune differential evolution algorithm for scheduling a distributed heterogeneous flexible flowshop

This paper studies a distributed heterogeneous flexible flowshop problem (DHFFP) with sequence-dependent setup and transport times, where each job has a release time and each production stage has multiple unrelated parallel machines. An integer linear programming (ILP) model and a new artificial immune differential evolution (AIDE) algorithm are proposed aiming to minimize the makespan and total tardiness simultaneously. Firstly, the AIDE employs a two-dimensional vector encoding scheme including the information of job permutation and factory allocation for solution representation. In addition, a dynamic decoding scheme is designed to construct a feasible schedule. Secondly, three distributed Nawaz–Enscore–Ham (NEH) based constructive heuristics are presented to provide the initial antibody population. Thirdly, a discrete differential evolution algorithm is introduced to mutate the cloned antibodies. Further, clonal suppression is applied by eliminating the antibodies with lower stimulation values. Simulated annealing (SA) based local search is incorporated with four problem-specific neighborhood structures to enhance the algorithm. Finally, the optimal values of the algorithmic parameters are determined by the orthogonal test. Some numerical experiments of the ILP model on small-scale instances are conducted to show the effectiveness of the proposed AIDE algorithm. The AIDE algorithm is also compared with several existing algorithms for different scale instances. The results demonstrate that the AIDE algorithm generates average relative percentage deviation of 12.51% and 7.99% respectively for small & medium-scale instances and large-scale instances, which is the best-performing among all algorithms.

Open vehicle routing problem with moving shipments at the cross-docking terminal

In this study, one of the internal operations at the Cross-docking Terminal (CDT), moving shipments (MS) from receiving doors to shipping doors of CDT is integrated with Open Vehicle Routing Problem (OVRP) and the problem is indicated here as OVRPCD-MS. As an additional feature, asymmetric distance between any two customers is assigned by incorporating a characteristic of one-way routes between cities in real-life transportation. The objective is to minimize the total transportation cost which incurs travelling cost between customers, service cost at customer points, service cost at the receiving and shipping doors of CDT, cost of moving shipments inside the CDT and finally the cost of hiring fleets of vehicles. To solve the OVRPCD-MS problem, a Mixed Integer Linear Programming (MILP) model is developed. The programming models are implemented in LINGO (version 18) optimization software. Branch and Bound algorithm is employed to solve ten small-scale instances generated randomly. The applicability of the proposed MILP model is observed. The required fleets of vehicles to be hired and run time to reach the optimal solution are determined. The study revealed that the average run time is exponential for small-scale instances. Thus, it can be concluded that this proposed model can be used for last time planning for small-scale instances. Also, the combinatorial nature of the vehicle routing problem makes OVRPCD-MS as NP-hard. Therefore, this study recommends that heuristic or meta-heuristic methods are more appropriate for the large-scale instances of OVRPCD-MS to reach near-optimum solutions.

The time-dependent electric vehicle routing problem with drone and synchronized mobile battery swapping

Logistics enterprises are moving towards high efficiency and green. As an emerging technology, unmanned aerial vehicle (UAV, also known as drone) is very useful to cope with flexible and fast delivery. Electric vehicles (EVs), as new energy vehicles, are the technical support for green logistics development. This paper considers the last-mile delivery system in which a fleet of homogeneous EVs, equipped with a UAV each, serve a set of customers, and meanwhile battery swapping vehicles (BSVs) offer mobile battery swapping service for EVs. A mixed integer programming (MIP) model is formulated to minimize the sum of depreciation cost, driver wages and energy consumption cost of EVs, UAVs, and BSVs, by considering the nonlinear energy consumption of each vehicle, time-dependent speeds of EVs and BSVs and monetary time value on depreciation cost. A large neighborhood search algorithm based on Q-learning (LNS-QL) is designed in which Q-learning is used to select a combination of tailored destroy and repair operators to solve three sub-problems, (1) the UAV multi-visit problem, (2) the EV and UAV synchronized scheduling problem, (3) the EV and BSV synchronized mobile battery swapping problem. Besides, this paper also designs Lagrangian relaxation algorithm for comparison with LNS-QL. Extensive experiments verify the correctness of the model by testing small-scale instances. Further analysis demonstrates the accuracy and stability of LNS-QL on medium and large-scale instances as well as computational time and verifies the effectiveness of the mobile battery swapping mode in the cooperative delivery system of EVs and UAVs. Actual cases analyze the characteristics of EV delivery from the cost and emissions perspectives by comparing with FV delivery.

Neural TSP Solver with Progressive Distillation

Travelling salesman problem (TSP) is NP-Hard with exponential search space. Recently, the adoption of encoder-decoder models as neural TSP solvers has emerged as an attractive topic because they can instantly obtain near-optimal results for small-scale instances. Nevertheless, their training efficiency and solution quality degrade dramatically when dealing with large-scale problems. To address the issue, we propose a novel progressive distillation framework, by adopting curriculum learning to train TSP samples in increasing order of their problem size and progressively distilling high-level knowledge from small models to large models via a distillation loss. In other words, the trained small models are used as the teacher network to guide action selection when training large models. To accelerate training speed, we also propose a Delaunary-graph based action mask and a new attention-based decoder to reduce decoding cost. Experimental results show that our approach establishes clear advantages over existing encoder-decoder models in terms of training effectiveness and solution quality. In addition, we validate its usefulness as an initial solution generator for the state-of-the-art TSP solvers, whose probability of obtaining the optimal solution can be further improved in such a hybrid manner.

Mathematical Optimization of Carbon Storage and Transport Problem for Carbon Capture, Use, and Storage Chain

The greenhouse effect caused by carbon dioxide (CO2) emissions has forced the shipping industry to actively reduce the amount of CO2 emissions emitted directly into the atmosphere over the past few years. Carbon capture, utilization, and storage (CCUS) is one of the main technological methods for reducing the amount of CO2 emissions emitted directly into the atmosphere. CO2 transport, i.e., shipping CO2 to permanent or temporary storage sites, is a critical intermediate step in the CCUS chain. This study formulates a mixed-integer programming model for a carbon storage and transport problem in the CCUS chain to optimally determine ship allocation, ship departure scheduling, and CO2 storage and transport. Taking advantage of the structure of the problem, we transform the mixed-integer programming model into a simpler model that can be computed efficiently. To evaluate the performance of the simpler model, numerous computational experiments are conducted. The results show that all small-scale instances (each with 10 power plants) and medium-scale instances (each with 30 power plants) can be solved optimality by Gurobi within 14.33 s. For large-scale instances with 60 and 65 power plants, feasible solutions with average gap values of 0.06% and 6.93% can be obtained by Gurobi within one hour, which indicates that the proposed methodology can be efficiently applied to practical problems. In addition, important parameters, including the unit fuel price, the time-charter cost, and the ship sailing speed, are examined in sensitivity analyses to investigate the impacts of these factors on operations decisions. In summary, a lower fuel price, a lower charter cost, or a higher ship sailing speed can increase the profit of the CCUS chain.