High demand and volume fluctuations, constrained transportation capacity, and weak network effects compel retailers to rely on external logistic service providers (LSPs) instead of operating their own transportation fleet. This outsourcing process requires retailers to propose tours to LSPs that then carry out the store deliveries. LSPs typically bill their service using a tariff with a zone structure representing geographical positions. The transportation costs depend on the zones visited and the delivery volume, which is subject to volume discounts. Further, LSPs apply detour limits to prevent excessive travel distances. Zone tariffs are a standard pricing scheme to ease transportation planning in the industry. Despite its high practical relevance, the literature in this field is limited, and a general model and solution approach are lacking. Our work closes this gap by providing the first comprehensive model for the Capacitated Vehicle Routing Problem with a Zone Tariff (C-VRP-ZT) and by developing a generally applicable exact solution method. The developed Branch-and-Check (BAC) framework includes valid inequalities and multiple acceleration techniques. We prove the computational efficiency of our approach using benchmark instances and derive managerial insights. We analyze different tariff structures concerning central characteristics such as the zone layout and order consolidation. Our results show how the tariff characteristics impact the cost of retailers and the margins of LSPs. We further solve a real-world application in retailing and analyze the cost/revenue split between a retailer and an LSP. These results provide insights into the threshold when a retailer should outsource its transportation process.
The strength of a graph was introduced by Cunningham (1985) as the minimum ratio of the number of edges that could be removed from the graph to the number (minus one) of the connected components created by such a removal. The role of the strength of a graph as a connectivity and resilience measure is further highlighted by its connection to the spanning tree packing number of a graph, namely, the maximum number of edge-disjoint spanning trees that can be contained in a graph. In this work, we investigate the question of determining a subset of a given graph's vertices of maximum cardinality that induces a subgraph of at least the prescribed strength. We show that the stated problem is polynomially solvable, and present the corresponding algorithm. In addition, we propose a mathematical programming-based cutting plane method for computing graph strength, which can be integrated into larger mathematical programming models. Numerical experiments on a diverse array of real-world graphs illustrate the computational properties of the proposed algorithms.
We consider an original dynamic dial-a-ride service designed for sparsely populated areas. The service relies on vehicles capable of switching between road and an existing abandoned rail network. It defines a Dial-A-Ride Problem (DARP) with rail scheduling constraints. In the DARP, a set of users must be picked up and dropped off at desired locations, while adhering to time windows and maximum travel time constraints. In the dynamic context, the system has to evaluate the acceptance of new customers and their integration into vehicle routes. Rail scheduling highly complicates the problem by creating interdependencies between vehicles. We develop a solution method combining the Adaptive Large Neighborhood Search framework, a set-covering approach, and multiple anticipatory scenarios generated with fictitious requests. We evaluate the effectiveness of our approach with realistic instances generated from a specific abandoned railway in France.
In this article, we motivate and define variants of perfect matching under precedence constraints where a perfect matching is built incrementally and precedence constraints ensure that an edge may only be added to the matching if the edge's predecessor vertices have already been covered. We study the complexity of the problem and particularly consider -canonical precedence constraints where only edges that are "close" to the current matching may be added to the matching. For the & Nscr; & Pscr; -hard perfect matching under 1-canonical precedence constraints, we identify polynomial-time solvable cases.
This article studies a vehicle routing problem involving a fleet of heavy-duty vehicles and pickup-and-delivery requests for crude items that are both heavy and high-temperature. The objective is to route the fleet in such a way that maximizes resource efficiency and operational efficiency while simultaneously avoiding thermal overload of any vehicle in the fleet. This is achieved by considering the curb weight of the vehicles and the weight of the items being transported when optimizing vehicle routing. Additionally, transshipments of items en route are considered. To maintain the fleet's mechanical preservation, thermal overload of the vehicles must be avoided. Therefore, temperatures of items and vehicles, as well as their interdependence, are considered. We introduce the vehicle temperature predictor, which is based on Newton's law of cooling and allows us to estimate the vehicle temperature en route. However, considering the vehicle temperature presents two challenges. First, solution feasibility is often jeopardized by vehicle temperature constraints. Second, solutions are produced that are heat-efficient but not resource-efficient. Our metaheuristic solution framework addresses both challenges by means of a large neighborhood search using adapted, novel, and feature-based heuristics. We provide valuable insights into resource-efficient and heat-efficient routing policies by experimenting with test instances that mimic real-world data from a partnered steel plant. Additionally, we present a mixed integer nonlinear programming formulation and demonstrate the effectiveness of our proposed metaheuristic solution approach by obtaining optimal solutions for small test instances.
The gradual growth of road infrastructure worldwide has heightened the importance of sustainable roadside management for preserving adjacent ecosystems and maintaining critical ecosystem services. Although the Arc Routing Problem (ARP) presents a promising framework for optimising operations carried out along an arc, its application to roadside management remains unexplored despite its advantages over traditional routing problems. This study presents a comprehensive systematic literature review to analyse how ARPs have been applied across various real‐world scenarios and to identify relevant features for roadside management applications. The presented analysis is conducted in two stages. First, a bibliometric analysis introduces the context of the ARP scientific domain. Then, a deeper analysis is based on a detailed examination of model characteristics, network configurations, and sustainability challenges across different application domains. The findings reveal a predominance of theoretical work with few real‐life applications but highlight significant opportunities for adapting existing ARP frameworks to roadside management, particularly in integrating sustainability criteria and local context considerations. However, crucial gaps in current ARP applications are identified, notably the limited incorporation of sustainable and regenerative practices in model formulation. This review provides a structured decision support framework for researchers and practitioners, outlining specific directions for developing ARP models tailored to sustainable roadside management while emphasising the need to balance operational efficiency with sustainable objectives.
In this paper, we introduce new Semidefinite Programming (SDP) dual bounding procedures and two Branch-and-bound algorithms based on them, BBLAG and BBLAG+, for solving the -Cluster Problem, KCP. The algorithms compute Lagrangian relaxation SDP bounds by relaxing the positive semidefiniteness constraint and attaching conveniently defined matrices of Lagrangian multipliers to it. Operating under a more standard approach found in the literature, BBLAG only requires the matrix of Lagrangian multipliers to be positive semidefinite; symmetry is not required. BBLAG+, on the other hand, enforces both conditions and, as a result, solves a reformulation of the Lagrangian Dual problem without the need of projecting the matrix of Lagrangian multipliers onto the cone of symmetric positive semidefinite matrices. The positive impact of avoiding the projection step is tremendous, as it is the most CPU time-consuming operation involved in the computation of our SDP bounds, by means of Lagrangian Relaxation. Because of that, BBLAG+ is four times faster than BBLAG. Additionally, BBLAG+ attained quite competitive results when compared to the existing KCP algorithms from the literature, BiqCrunch and BiqBin, that also rely on SDP bounds. In practice, the SDP bounds computed by BBLAG+ neither dominate nor are dominated by the SDP bounds computed by its competitors. For the densest subgraph problem, one of the five KCP variants tested here, the best results were provided by BiqCrunch followed by BiqBin. For four other sets of KCP instances, BBLAG+ usually provides better results than BiqCrunch and BiqBin, for a wide range of values of and graph densities. In summary, BBLAG+ seems to be a robust approach for a wide range of KCP instances.
Medical on-demand couriers play an important role in urban medical services. They transport medical supplies, tests, blood, or organs between different locations in the city within a short amount of time. Transportation orders are issued spontaneously and differ in their priority and time restrictions. Since all orders must be served, the goal is to minimize the priority-dependent delay penalty of all orders. To this end, we propose an anticipatory re-optimization approach by integrating a parametric Cost Function Approximation (CFA) into a Large Neighborhood Search (LNS). Whenever a new order is placed, the LNS iteratively optimizes the current route plans. The CFA addresses the tradeoff between minimizing delays for pending orders and the fleet's flexibility to react to future orders. To this end, it is used both to evaluate decision candidates (i.e., new route plans) and to assess insertion options within the LNS repair operator. We demonstrate the advantages of our approach in comprehensive computational experiments and provide insights into the method and problem.
The vehicle routing problem with time windows is a well-researched problem in literature. We study the 2-index flow formulation for the problem and propose a relatively little-used approach of polar duality/local cuts to compute new general valid inequalities for the problem. Our method of applying polar duality is quite distinct from and complimentary to an earlier attempt of applying the same idea to this problem and produces significantly better results on instances with tight time windows. On almost all 25-customer Solomon instances with tight time windows, our approach is capable of producing very strong lower bounds that are close to 100% of the optimal solution within reasonable computing time. On larger instances too the lower bounds are significantly better than those reported in the literature. We also present a new version of the previously proposed k-path inequalities that are easy to compute as well as effective. These inequalities also lead to a substantial improvement in lower bounds and solution times for some classes of instances. Computational tests performed on benchmark instances indicate significant improvement in computing time and decrease in the number of nodes in the branch-and-bound tree as compared to extant methods that employ the flow formulation for the problem.
Nearest, Farthest, and Cheapest Insertion are three well-known polynomial-time approximation algorithms for the Traveling Salesman Problem (TSP). This paper aims to report on a fourth insertion algorithm, called Largest Insertion, from both a theoretical and an experimental viewpoint. On the theoretical side, the worst-case performance of the algorithm is studied: In particular, it is shown that there exist instances for which the value of the solution computed by Largest Insertion approaches three times the optimum in the Euclidean plane. On the experimental side, the outcome of computational tests is reported: When compared with the other three insertion algorithms, Largest Insertion scores second on random Euclidean instances and first on large random graphical instances.
This work considers branch-price-and-cut algorithms for variants of the vehicle-routing problem in which subset-row inequalities (SRIs) are used to strengthen the linear relaxation. SRIs often help to substantially reduce the size of the branch-and-bound search tree. However, their use is computationally costly because SRIs modify the structure of the respective column-generation subproblem, which is a shortest-path problem with resource constraints (SPPRC). Each active SRI requires the addition of a resource to the labeling algorithm that is invoked for solving the SPPRC in every iteration. In the context of time-window constraints, the concept of unreachable customers has been used for preprocessing (time-window reduction, arc elimination, precedence identification) as well as for improving the dominance between labels in the elementary SPPRC and its relaxations. We show that the identification of unreachable customers can also help to improve the dominance due to a modified comparison of SRI-related resources. Computational experiments with a fully fledged branch-price-and-cut algorithm for the (standard and electric) vehicle-routing problem with time windows demonstrate the effectiveness of the approach: Overall computation times decrease; for some difficult instances, they may even be cut in half, while the required modifications of a computer implementation for combining SRIs with unreachable customers are minor.
We introduce the nested -center problem, a multi-period variant of the well-known -center problem. Using the nesting concept allows us to obtain solutions consistent over the considered time horizon, that is, facilities that are opened in a given time period stay open for subsequent periods. This consistency is important in real-life applications, as closing (and potentially later re-opening) of facilities between time periods can be undesirable. We consider two versions of our problem, with the difference being the objective function. The first version considers the sum of the absolute regrets (of nesting) over all time periods, and the second version considers minimizing the maximum relative regret over the time periods. We present three mixed-integer programming formulations for the version with an absolute regret objective and two formulations for the version with a relative regret objective. For all the formulations, we present valid inequalities. Based on the formulations and the valid inequalities, we develop branch-and-bound/branch-and-cut solution algorithms. These algorithms include a preprocessing procedure that exploits the nesting property and starting heuristics and primal heuristics. We conducted a computational study on instances from the literature for the -center problem, which we adapted to our problems. We also analyze the effect of nesting on the solution cost and the number of open facilities.
The use of underground freight transportation (UFT) is gaining attention because of its ability to quickly move freight to locations in urban areas while reducing road traffic and the need for delivery drivers. Since packages are transported through the tunnels by electric motors, the use of tunnels is also environmentally friendly. We examine the use of tunnels to transport individual orders, motivated by the last-mile delivery of goods from e-commerce providers. The use of UFT for last-mile delivery requires more complex network planning than for direct lines that have previously been considered for networks connecting large cities. We introduce a new network design problem based on this delivery model and transform the problem into a fixed-charge multicommodity flow problem with additional constraints. We show that this problem, the -UFT, is NP-hard, and provide an exact solution method for solving large-scale instances. Our Benders decomposition-based solution approach exploits the combinatorial sub-structures of the problem in a cutting planes fashion. We provide computational results for real urban environments to build a set of insights into the structure of such networks and evaluate the benefits of such systems. We estimate the costs for implementing UFT systems and break them down into a per-package cost. Our estimates indicate at least a 40% cost savings from using a UFT over traditional delivery models. This indicates that UFT systems for last-mile delivery are a promising area for future research.
The split delivery vehicle routing problem with time windows (SDVRPTW) extends the well-known VRPTW by the option of satisfying individual customer demands with more than one vehicle. The SDVRPTW is relatively well studied from an exact viewpoint, but efficient heuristics that are able to provide near-optimal solutions within short runtimes have not been proposed in the literature. In this article, we design a heuristic for the SDVRPTW, called granular tabu search with balanced customer splitting (GTS-BCS), that is based on the principle of a priori customer splitting introduced by Chen, International Transactions in Operational Research, 24, 27-41, 2017. The main idea of this approach is to split customers into subcustomers with an a priori splitting rule and then use a solver for the unsplit counterpart of the problem to solve the resulting split-instance. We use theoretical properties of optimal SDVRPTW solutions and numerical experiments to find well-balanced splitting rules that find a good tradeoff between the flexibility needed to explore a large number of promising splits of the original demand and keeping the number of generated subcustomers as low as possible. We observe that starting the search from a so-called no-split solution, that is, a solution in which customer splits are only possible for customers whose demand exceeds the vehicle capacity, has strong positive effects on both solution quality and runtimes. The final configuration of our GTS-BCS is able to compute near-optimal solutions on the common SDVRPTW benchmark sets within comparatively short runtimes. We also investigate whether the proposed splitting rules can be transferred to the standard SDVRP. To this end, we use the same a priori customer splitting framework as in GTS-BCS and integrate the hybrid genetic search of Vidal, Computers and Operations Research, 140, 105643, 2022 for solving the unsplit counterpart of the SDVRP - the CVRP. Numerical experiments show that the resulting algorithm is nearly competitive to the state-of-the-art dedicated methods for the SDVRP.
This article studies a network interdiction problem where the success of the first player's attempt to remove arcs is based on decision- dependent success probabilities, and the second player (the follower) responds by pushing maximum flow. In this multi-stage model, previous interdictions affect future interdictions' success probabilities. The decision-dependent structure requires an objective function that includes a conditional probability that is nonlinear, which we reformulate into a mixed integer program using probability flow networks over all possible scenarios. However, this reformulation involves a large number of scenarios. To reduce the scenario space, we present a scenario clustering approach that exploits the problem structure while preserving optimality. We also provide a scenario clustering heuristic for further reductions and faster solution times where optimality is not guaranteed. We illustrate our method by solving instances on various physical grid and supply chain networks. We also investigate how changing decision-dependent success probabilities over time can influence interdiction policies over these networks.
For many routing applications, it is not only necessary to minimize total costs but also to allocate them to individual customers. In this context, the allocation according to the Shapley value is a well-known method highly regarded for its fulfillment of major fairness criteria. However, its computational complexity restricts its applicability, especially for NP-hard underlying optimization problems like routing problems, since the exact computability of the Shapley values is no longer possible within a reasonable time. In this study, we propose a general Machine Learning-based Shapley Value Approximator (MLSVA) and apply it to routing problems based on the exploitation of routing-specific problem structures as features, enabling real-time approximations. Within an extensive numerical study, we show that our MLSVA outperforms current state-of-the-art approximation results for the Traveling Salesman Problem (TSP) and, at the same time, achieves very good results for the Capacitated Vehicle Routing Problem (CVRP), for which it is the first efficient method that quickly delivers high-quality approximations. On average, we approximate Shapley values with a mean absolute percentage error of 2.4% for the TSP and 3.5% for the CVRP. Further, the MLSVA achieves strong approximation results even when trained on biased labels. This shows the scalability of the MLSVA and allows high-quality approximations even for large routing problems. Finally, we demonstrate the generalizability of our approach by successfully approximating Shapley values for items in a variant of the bin packing problem.
The Split Delivery Vehicle Routing Problem with Two-dimensional Loading Constraints (2L-SDVRP) integrates vehicle routing, split delivery, and two-dimensional packing constraints. In the 2L-SDVRP, customers can be served by multiple vehicles, and their demands consist of different two-dimensional rectangular items that must be packed in the vehicles' bases. The problem involves determining the least-cost routes that satisfy all customer demands while ensuring the feasible packing of items in each vehicle. We present tailored branch-and-cut (BC) methods for solving the 2L-SDVRP. One of the methods is based on an effective, relaxed two-index vehicle flow formulation that is newly introduced in this paper. To evaluate the performance of the BC methods, computational experiments were conducted using both benchmark instances and new realistic instances inspired by cases from Brazilian logistics companies. The results indicate the superior performance of the method based on the two-index formulation, which obtained optimal solutions for 14 more instances than the other approach on the benchmark instances. This method also performed better on newly created instances, improving solutions by 5.6% on average.
The Vehicle Routing Problem with Synchronization (VRPSync) aims to minimise the total routing costs while considering synchronization requirements that must be fulfilled between tasks of different routes. These synchronization requirements are especially relevant when it is necessary to have tasks being performed by vehicles within given temporal offsets, a frequent requirement in applications where multiple vehicles, crews, materials, or other resources are involved in certain operations. Although several works in the literature have addressed this problem, mainly the deterministic version has been tackled so far. This paper presents a robust optimization approach for the VRPSync, taking into consideration the uncertainty in vehicle travel times between customers. This work builds on existing approaches in the literature to develop mathematical models for the Robust VRPSync, as well as a branch-and-cut algorithm to solve more difficult problem instances. A set of computational experiments is also devised and presented to obtain insights regarding key performance parameters of the mathematical models and the solution algorithm. The results suggest that solution strategies where certain standard problem constraints are only introduced if a candidate solution violates any of those constraints provide more consistent improvements than approaches that rely on tailor-made cutting planes, added through separation routines. Furthermore, the analysis of the Price of Robustness indicators shows that the adoption of robust solutions can have a significant increase in the total costs, however, this increase quickly plateaus as budgets of uncertainty increase.
We propose a variant of the shortest path problem where the order in which vertices occur in the path is subject to precedence constraints. Precedence constraints are defined in terms of vertex pairs (a,b)$$ \left(a,b\right) $$ which indicate that a vertex a$$ a $$ is the predecessor of a vertex b$$ b $$. A feasible (not necessarily simple) path may visit a vertex only upon having covered all its predecessors. The problem generalizes the graphic TSP Path, which makes it APX-hard. We propose a dynamic program and identify input classes for which the dynamic program yields an optimal solution in polynomial time. We also explore the limits of efficient solvability by proving that the problem remains hard even when significantly restricting the structure of the graph or the structure of the precedence constraints: Surprisingly, the problem remains hard even when restricted to spiders.
This article proposes an implicit enumeration approach to solve the maximum ratio s-plex and the maximum ratio s-defective clique problems. The approach is inspired by the classical Bron-Kerbosch algorithm for enumerating all maximal cliques in a graph, which is extended to enumerating structures that are hereditary on induced subgraphs. Such structures include s-plexes and s-defective cliques, among many others. The performance of the proposed approach is compared with that of the methods based on mixed integer linear programming (MILP), binary search, and Newton's iteration through numerical experiments on randomly generated and real-life network instances.
