Logic-based Benders Decomposition Research Articles

A new branch-and-Benders-cut algorithm for the time-dependent vehicle routing problem

Daily traffic congestion poses significant challenges for companies operating in urban areas. By considering predicted travel times throughout the day, route planning systems can improve delivery schedules, thereby reducing costs associated with delays and congestion. Although the time-dependent vehicle routing problem presents a more realistic model for city logistics, it also introduces significant computational challenges due to the size of the network of the associated models. This paper presents the first exact method using logic-based Benders decomposition for the time-dependent vehicle routing problem. Two problem formulations based on two, and three-index vehicle routing models are proposed and solved using a branch-and-cut procedure. Moreover, a logic-based branch-and-Benders-cut algorithm is developed. Computational experiments on real-world instances demonstrate that the proposed algorithm achieves better solutions, especially when applied to the three-index formulation. Typical optimality gaps are lower than 5% even in the bigger instances. Moreover, the paper evaluates various design choices for the algorithm and their impact on the solutions of the problem. Finally, managerial insights highlight the importance of solving time-dependent vehicle routing problem by demonstrating that congestion adds significant delays and costs to transportation systems.

Logic-based benders decomposition algorithm for robust parallel drone scheduling problem considering uncertain travel times for drones

The integration of trucks and drones in last-mile delivery has introduced new capabilities to the transportation industry. These two vehicles simultaneously offer unique features, which have improved performance and efficiency in the process of delivering products. This paper investigates a robust parallel drone scheduling traveling salesman problem with supporting drone, where drone travel times are uncertain and products gradually arrive at a depot over time. In this problem, a truck, a supporting drone, and service drones are located in the depot to deliver products with the goal of minimizing the total completion time. A mathematical model is proposed which is improved using the earliest release dates rule, followed by the development of an exact logic-based benders decomposition algorithm to solve the problem. In this algorithm, customers are initially assigned to the service drones or the truck in a master problem, and subsequent auxiliary problems are addressed utilizing the earliest release dates rule and a dynamic programming algorithm. Finally, various cuts are enhanced through strengthening techniques and sequentially added into the master problem. Numerical experiments demonstrate the efficiency of the improved mathematical model and the proposed algorithm. Furthermore, sensitivity analysis has provided several managerial recommendations for enhancing the delivery system performance.

A two-echelon multi-trip vehicle routing problem with synchronization for an integrated water- and land-based transportation system

This study focuses on two-echelon synchronized logistics problems in the context of integrated water- and land-based transportation (IWLT) systems. The aim is to meet the increasing demand in city logistics as a result of the growth in transport activities, including parcel delivery, food delivery, and waste collection. We propose two models, a novel mixed integer linear joint model, and a logic-based Benders’ decomposition (LBBD) model, for a two-echelon problem under realistic settings such as multi-trips, time windows, and synchronization at the satellites with no storage and limited resource capacities. The objective is to optimize transfers and satellite assignments, thereby reducing overall logistics costs for street vehicles and vessels. Computational experiments demonstrate that the LBBD model is more robust in terms of solution quality and solution time on average while the added value of the LBBD is more evident when solving large-scale instances with 100 customers, reducing the overall costs by 10.6% on average and significantly reducing the fleet costs on both networks. Furthermore, we assess the effect of changing cost parameters and satellite locations in the proposed IWLT system–analyzing system behavior and suggesting potential improvements–and evaluate several system alternatives in city logistics–consisting of different transportation network designs (single- and two-echelon), vehicle types, and operational constraints. On average, the proposed two-echelon IWLT system reduces the number of kilometers traveled by vehicles at street level by ranging from 20% to 30% compared to a typical single-echelon service design that relies solely on trucks.

Optimal allocation and route design for station-based drone inspection of large-scale facilities

The utilization of drones to conduct inspections on industrial electricity facilities, including large-sized wind turbines and power transmission towers, has recently received significant attention, mainly due to its potential to enhance inspection efficiency and save maintenance costs. Motivated by the advantages of drones for facility inspection, we present a novel station-based drone inspection problem (SDIP) for large-scale facilities. The objective of SDIP is to determine the locations of multiple homogeneous automatic battery swap stations (ABSSs) equipped with drones, assign facility inspection tasks to the ABSSs with operation duration constraints, and design drone inspection routes with battery capacity constraints, such that minimize the sum of fixed ABSS costs and drone travel costs. The SDIP can be regarded as a variant of the location-routing problem, which is NP-hard and difficult to solve optimally. To obtain the optimal solution of SDIP efficiently, we firstly formulate this problem into an arc based formulation and a route based formulation, and then develop a logic-based Benders decomposition (LBBD) algorithm to solve it. The SDIP is decomposed into a master problem (MP) and a set of subproblems (SPs). The MP is solved by a branch-and-cut (BC) procedure. Once a feasible integer solution is found, the linear relaxation of SPs are solved by a stabilized column generation to generate Benders cuts. If the cost of all the SPs’ optimal LP solutions plus the cost of the MP’s solution is less that current best cost, the SPs are exactly solved by a Branch-and-Price (BP) algorithm to generate the logic cuts. The numerical results on five scales of randomly generated instances validate the effectiveness of the LBBD algorithm. Specifically, the LBBD can solve all small- and middle-sized instances, and seven out of ten large-sized instances in 1000 s. Furthermore, we conduct a sensitivity analysis by varying the attributes of ABSSs and drones, and provide valuable managerial insights for large-scale facility inspection.

The picker routing problem in mixed-shelves, multi-block warehouses

Unlike conventional warehouses, where the inventory of an item is concentrated on a single shelf, the inventory in mixed-shelves warehouses is broken down into units and dispersed throughout the warehouse. As a result, the locations from where items are to be retrieved must be chosen when designing the picker tour. The rare research on the picker routing problem (PRP) in mixed-shelves warehouses focuses on (and heavily exploits the properties of) 1-block warehouses, which are warehouses with only two cross aisles. Here, we tackle the more general PRP in mixed-shelves, multi-block warehouses. To solve the problem, we propose a logic-based Benders decomposition method whereby the master problem selects the locations from where each item is retrieved and the subproblem designs the associated picker tour. We introduce tailored optimality cuts and prove their validity. In addition, we propose a set of various techniques to enhance the performance of the logic-based Benders decomposition, including a lower bounding function and valid inequalities. We show the efficiency and effectiveness of the proposed method through extensive computational experiments carried out on both standard instances from the literature (for the 1-block setting) and newly generated instances (for our case). We also leverage our algorithms to generate managerial insights into the benefits of multi-block layouts and mixed-shelves policies in warehousing.

Integrated optimization of multi-machine scheduling and berth allocation in automated ore terminals

As water transportation undergoes a new technological revolution, automated terminal technology has become a key issue in reducing costs and improving efficiency. However, the operation theory limitation makes it problematic for machines to cooperate effectively and operate profitably, which is notably salient at ore terminals with their unique structure. This study focuses on the integrated scheduling of a new automated ore terminal. It comprises three main processes: (a) berth allocation, (b) splitter scheduling, and (c) feeder scheduling. We attempt to tackle this issue by developing a mixed-integer linear programming (MILP) model to minimize the vessels’ maximum departure time. Considering the complexity of the MILP model, we propose a logic-based Benders decomposition algorithm to solve the efficiency problem. The results indicate that the algorithm can solve realistically sized problem instances. The main factors affecting the computation time are the congestion level, total cargo types, and vessel service time. The integrated scheduling methods can effectively solve the actual challenges faced by terminals and provide suggestions and references for terminal expansion planning.

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.

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.

The flexible airport bus and last-mile ride-sharing problem: Math-heuristic and metaheuristic approaches

Airport buses play a crucial role in addressing the last-mile problem of air travel, especially in cities and countries lacking inner-city rail transit systems. Nevertheless, airport buses are currently witnessing a decline in ridership due to drawbacks such as long departure intervals, inflexible stops, and considerable distances between stops. Consequently, delivering high-quality airport bus services has become a pressing concern for public transport operators. Motivated by new flexible buses and ride-sharing services, this paper explores a flexible airport bus service that integrates ride-sharing services for passengers traveling from bus stops to their destinations. This problem entails integrated decisions involving bus stop selection, passenger assignment to drop-off bus stops, as well as bus and ride-sharing routing. Accordingly, this problem presents more challenges in decision-making than traditional flexible bus or ride-sharing routing problems. We first develop an arc-based mixed-integer linear programming model. Subsequently, we design a double decomposition math-heuristic algorithm that builds upon logic-based Benders decomposition and column generation algorithms to obtain a near-optimal solution within practical computation time limits for practical-scale instances. Additionally, we implement an adaptive large neighborhood search algorithm to evaluate the solution quality of this math-heuristic algorithm and to solve large-scale instances. To validate the effectiveness of both the model and the algorithms, we conduct numerical experiments using instances derived from Shenzhen airport bus lines. The experimental results demonstrate that the flexible service mode offers significant advantages in reducing both passenger ride time and vehicle mileage over traditional airport bus or taxi modes.

Logic-based Benders decomposition for bi-objective parallel machine selection and job scheduling with release dates and resource consumption

This work addresses a new bi-objective parallel machine selection and job scheduling problem with release dates and resource consumption. It consists in optimally selecting subcontractors (machines) from a set of geographically dispersed locations and scheduling the orders (jobs) to the selected subcontractors for processing while meeting the order release dates and resource consumption restrictions, so as to simultaneously minimize the maximum completion time, i.e., the makespan, and the total cost including machine usage cost and resource consumption cost. The problem is first formulated into a bi-objective mixed-integer linear program based on linear ordering (LO-MILP), and then valid inequalities are explored based on property analysis. To solve it, an ɛ-constraint method based on LO-MILP (ɛ-LO-MILP) is first proposed. To more efficiently solve it, we also develop a tailored logic-based Benders decomposition combined with ɛ-constraint method (ɛ-LBBD) where a novel method to obtain a tight lower bound of the identical parallel machine with machine-dependent release dates and some problem-specific cuts are proposed. Numerical experiments on an illustrative example are conducted to show the applicability of the model and algorithm and intuitively reveal the trade-off between production efficiency and cost. Experimental results on 200 instances with up to 100 orders demonstrate that ɛ-LBBD can reduce the computation time by about 28.92% compared to ɛ-LO-MILP and yields better Pareto solutions than ɛ-LO-MILP and the well-known non-dominated sorted genetic algorithm II do.

Optimal Establishments of Massive Testing Programs to Combat COVID-19: A Perspective of Parallel-Machine Scheduling-Location (ScheLoc) Problem

Massive testing to identify COVID-19-infected people is crucial in combating COVID-19. However, from the perspective of facility location problems, many current massive testing programs are not properly set, leading to unreasonable travelling distances, long makespan, unbalanced workload, and long queues. This article proposes a decision framework for developing massive testing programs. Specifically, a biobjective parallel-testing-site Scheduling-location (ScheLoc) model is formulated, simultaneously minimizing the makespan and total travelling distance. The former can help reduce the time length of potential virus spread, and the latter can help alleviate the risk of virus spread and traveler inconvenience. To solve the proposed biobjective ScheLoc problem, in addition to the standard <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm {\epsilon }$</tex-math></inline-formula> -constraint method, we further develop two novel strategies. The first one iteratively solves simpler approximate MIP models (IMIP). The second innovatively extends the classical logic-based Benders decomposition approach to solve biobjective problems (B-LBBD). A Hong Kong-based case study shows that the proposed decision framework can significantly reduce the makespan and travelling distance (with a mean of 13 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> and 5.1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> , respectively) and enhance workload balancing. Besides, the developed solution methods, especially the B-LBBD, outperform the adapted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm {\epsilon }$</tex-math></inline-formula> -constraint method in various aspects.

A Decomposition Method for the Group-Based Quay Crane Scheduling Problem

This study addresses the quay crane scheduling problem (QCSP), which involves scheduling a fixed number of quay cranes to load and unload containers from ships in a maritime container terminal. The objective is to minimize the completion time while adhering to precedence, safety margin, and noncrossing constraints. Efficient scheduling of quay cranes plays a crucial role in reducing the time vessels spend at terminals. To solve the QCSP, we explore different schedule directions for the quay cranes. Specifically, we consider three directions: unidirectional, where the quay cranes maintain a consistent movement direction from upper to lower bays or vice versa after initial repositioning; bidirectional, allowing the cranes to change direction once during operations; and multidirectional, permitting freely changing movement direction during operations. For the bidirectional QCSP, we propose a new compact mathematical formulation. To obtain valid lower bounds on the optimal completion time, we derive various relaxations of this new formulation based on the different schedule directions. Our solution framework employs logic-based Benders decomposition, decomposing the problem into an assignment master problem and operation-sequence slave subproblems. Extensive computational experiments using benchmark instances from existing literature and newly generated instances validate the efficiency and effectiveness of the lower bounds and the exact solution approach. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms–Discrete. Funding: This work was supported by the Major Program of the National Natural Science Foundation of China [Grants 72192830 and 72192831], the National Natural Science Foundation of China [Grant 72102034], and the 111 Project [Grant B16009]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0298

Multi-trucks-and-drones cooperative pickup and delivery problem

This study aims to propose a decision methodology on scheduling trucks and drones for truck-and-drone cooperative delivery and pickup system. A fleet contains multiple truck groups; each truck group is a truck with carrying multiple drones. The fleet serves a set of dispersed customers who have the requirements of pickup and delivery services as well as their due time for service. A mixed-integer linear programming (MILP) model is formulated in this study for routing the trucks and drones in the fleet so that each customer’s pickup or delivery requirements could be served by either a truck or a drone before their required due time. For solving the MILP model efficiently, this study designs a novel hybrid algorithm by combining the column generation and the logic-based Benders decomposition. Based on the main frame of column generation algorithm, the hybrid algorithm uses logic-based Benders decomposition to solve the pricing problem, and dynamic programming to solve subproblems of logic-based Benders decomposition for the purpose of accelerating the whole algorithm’s solving process. Numerical experiments are also conducted on the context of the Hangzhou city so as to validate the efficiency of the proposed hybrid algorithm. Some managerial implications are also derived on the basis of some sensitivity analysis. The proposed methodology, i.e., the MILP model and the novel hybrid algorithm, is potentially useful for platform operators who run the truck-and-drone based urban delivery systems.

Logic-based Benders decomposition for order acceptance and scheduling in distributed manufacturing

This paper studies an order acceptance and scheduling problem in distributed manufacturing to minimize the total costs consisting of rejection cost, production cost, transportation cost and tardiness cost. Two mixed-integer programming models are formulated, which are further improved by the proposed enhancement techniques. A logic-based Benders decomposition (LBBD) method and a branch and check search framework are developed in an attempt to realize the optimal and intelligent decisions of order acceptance, order assignment and scheduling. The proposed LBBD method following the principle of “divide and conquer” divides the original problem into determining the master problem of order acceptance and assignment, and determining the subproblem of order sequencing in each factory. A dynamic programming algorithm is also proposed to efficiently solve the subproblems. Extensive computational experiments are conducted, and the results demonstrate the effectiveness and efficiency of the enhancement techniques for the formulations, as well as the LBBD method. Besides, the value of distributed manufacturing is verified and the sensitivity of key cost factors is presented.

Distributed appointment assignment and scheduling under uncertainty

We investigate a stochastic distributed appointment assignment and scheduling problem, which consists of assigning appointments to distributed service units and determining service sequences at each service unit. In particular, the service time duration and release time uncertainties are well-considered. The solution to this generic problem finds interesting applications in distributed production systems, healthcare systems, and post-disaster operations. We formulate the problem as a two-stage stochastic program to minimise the total transportation cost and expected makespan, idle time or overtime, and apply the sample average approximation method to make the problem tractable. We then develop a stochastic logic-based Benders decomposition method, decomposing the problem into a master problem and a subproblem. The master problem determines the appointment assignment variables, and the subproblem handles the sequence and service start time variables. Benders optimality cuts are generated from the subproblem's solution and added to the master problem. The developed stochastic logic-based method is advantageous since it can manage many scenarios in parallel. We further consider each appointment's due date, minimise the weighted earliness and tardiness, and adjust the developed method to solve this variant. Experiments on random instances demonstrate the excellent performance of the proposed model and methods.

A logic-based Benders decomposition solution approach for two covering problems that consider the underlying transportation

We investigate two maximal covering location problems with capacity restrictions, minimum workload, and transportation. The problems are inspired by a waste collection problem in which large waste containers are scattered throughout the municipality, and the residents bring their waste to these containers. We take the residents’ preferences into account when allocating them to locations. When a container is full, a vehicle transports an empty container from the disposal facility (depot) to that location and replaces it. We propose a mixed-integer linear programming formulation for the problems in which vehicles can carry one or two containers, and apply a logic-based Benders decomposition approach for the latter. Here, the sub problem is a multi-period minimum weight perfect matching problem. We show that our logic-based Benders decomposition approach outperforms the direct formulation in terms of solution quality and speed. We further show that transportation of two containers at a time reduces the distance to be driven by 29.5% on average, without compromising the covering level. Furthermore, we analyze the effect of imposing a minimum workload as well as the effect of changing the focus between transportation and covering.

Logic-based benders decomposition for wildfire suppression

In this paper we consider a network interdiction problem where the follower is characterised by a spreading phenomenon. As a case study we consider in detail a wildfire suppression problem. The landscape is modelled as a directed graph, with nodes representing regions of the landscape, and arcs representing adjacency relationships. The fire spread is modelled using the minimum travel time principle. The goal is to locate fire suppression resources in a landscape in order to minimise the total area burned. As previous exact algorithms in this space are intractable on reasonable instances, the literature focuses on heuristics. In this paper we propose a new exact algorithm utilising logic-based Benders decomposition. We benchmark the new approach against mixed-integer programming and metaheuristic approaches. Our approach is able to quickly solve instances from the literature. We are also able to solve larger instances to optimality in a reasonable amount of time.

A logic-based Benders decomposition algorithm for a repair crew routing and drone scheduling problem after a natural disaster

Natural disasters may lead to the isolation of rural areas due to the destruction of roads. Therefore, as a direct consequence of the impossibility of distribution of relief goods, more damages are expected in the aftermath of disasters in these areas. In this situation, we assume that damaged roads must be repaired immediately by a single repair crew and based on a specific route to prevent further damages and facilitate access to ground relief. There exists usually some stored medical equipment in isolated areas so that it can be consumed until destroyed roads become accessible again. In order to reflect this issue in this research, deadlines are considered for making isolated areas accessible. These deadlines are flexible and can be updated whenever a drone visits an isolated area and provide the area with limited amounts of air relief items. An integer linear programming is developed to formulate the repair crew routing problem and drone scheduling in an integrative manner. The developed model is complex and fails to provide the optimal solution even for instances including more than six demand areas and six damaged roads within one hour when the deadline is tight. Therefore, we developed a logic-based Benders decomposition algorithm that is able to find the optimal solution of instances including sixteen nodes in less than three quarters. Using a set of randomly generated instances, we compare the performance of the proposed solution approaches and provide a set of sensitivity analyses based on significant parameters of the problem.

Computational Evaluation of Cut-Strengthening Techniques in Logic-Based Benders’ Decomposition

Cut-strengthening techniques have a significant impact on the computational effectiveness of the logic-based Benders’ decomposition (LBBD) scheme. While there have been numerous cut-strengthening techniques proposed, very little is understood about which techniques achieve the best computational performance for the LBBD scheme. This is typically due to implementations of LBBD being problem specific, and thus, no systematic study of cut-strengthening techniques for both feasibility and optimality cuts has been performed. This paper aims to provide guidance for future researchers with the presentation of an extensive computational study of five cut-strengthening techniques that are applied to three different problem types. The computational study involving 3000 problem instances shows that cut-strengthening techniques that generate irreducible cuts outperform the greedy algorithm and the use of no cut strengthening. It is shown that cut strengthening is a necessary part of the LBBD scheme, and depth-first binary search and deletion filter are the most effective cut-strengthening techniques.

Logic-Based Benders Decomposition in Answer Set Programming for Chronic Outpatients Scheduling

AbstractIn answer set programming (ASP), the user can define declaratively a problem and solve it with efficient solvers; practical applications of ASP are countless and several constraint problems have been successfully solved with ASP. On the other hand, solution time usually grows in a superlinear way (often, exponential) with respect to the size of the instance, which is impractical for large instances. A widely used approach is to split the optimization problem into subproblems (SPs) that are solved in sequence, some committing to the values assigned by others, and reconstructing a valid assignment for the whole problem by juxtaposing the solutions of the single SPs. On the one hand, this approach is much faster due to the superlinear behavior; on the other hand, it does not provide any guarantee of optimality: committing to the assignment of one SP can rule out the optimal solution from the search space. In other research areas, logic-Based Benders decomposition (LBBD) proved effective; in LBBD, the problem is decomposed into a master problem (MP) and one or several SPs. The solution of the MP is passed to the SPs that can possibly fail. In case of failure, a no-good is returned to the MP that is solved again with the addition of the new constraint. The solution process is iterated until a valid solution is obtained for all the SPs or the MP is proven infeasible. The obtained solution is provably optimal under very mild conditions. In this paper, we apply for the first time LBBD to ASP, exploiting an application in health care as case study. Experimental results show the effectiveness of the approach. We believe that the availability of LBBD can further increase the practical applicability of ASP technologies.