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.
