Branch-and-Cut for the split delivery vehicle routing problem with time windows

The split delivery vehicle routing problem with time windows (SDVRPTW) is a notoriously hard combinatorial optimization problem. First, it is hard to find a useful compact mixed-integer programming (MIP) formulation for the SDVRPTW. Standard modeling approaches either suffer from inherent symmetries (mixed-integer programs with a vehicle index) or cannot exactly capture all aspects of feasibility. Because of the possibility to visit customers more than once, the standard mechanisms to propagate load and time along the routes fail. Second, the lack of useful formulations has rendered any direct MIP-based approach impossible. Up to now, the most effective exact algorithms for the SDVRPTW have been branch-and-price-and-cut approaches using path-based formulations. In this paper, we propose a new and tailored branch-and-cut algorithm to solve the SDVRPTW. It is based on a new, relaxed compact model, in which some integer solutions are infeasible for the SDVRPTW. We use known and introduce some new classes of valid inequalities to cut off such infeasible solutions. One new class is path-matching constraints that generalize infeasible-path constraints. However, even with the valid inequalities, some integer solutions to the new compact formulation remain to be tested for feasibility. For a given integer solution, we build a generally sparse subnetwork of the original instance. On this subnetwork, all time-window-feasible routes can be enumerated, and a path-based residual problem then solved to decide on the selection of routes, the delivery quantities, and thereby the overall feasibility. All infeasible solutions need to be cut off. For this reason, we derive some strengthened feasibility cuts exploiting the fact that solutions often decompose into clusters. Computational experiments show that the new approach is able to prove optimality for several previously unsolved instances from the literature

The Split Delivery Vehicle Routing Problem with Time Windows and Customer Inconvenience Constraints

In classical routing problems, each customer is visited exactly once. By contrast, when allowing split deliveries, customers may be served through multiple visits. This potentially results in substantial savings in travel costs. Even if split deliveries are beneficial to the transport company, several visits may be undesirable on the customer side: at each visit the customer has to interrupt his primary activities and handle the goods receipt. The contribution of the present paper consists in a thorough analysis of the possibilities and limitations of split delivery distribution strategies. To this end, we investigate two different types of measures for limiting customer inconvenience (a maximum number of visits and the temporal synchronization of deliveries) and evaluate the impact of these measures on carrier efficiency by means of different objective functions (comprising variable routing costs, costs related to route durations, fixed fleet costs). We consider the vehicle routing problem with time windows in which split deliveries are allowed (SDVRPTW) and define the corresponding generalization that takes into account customer inconvenience constraints (SDVRPTW-IC). We design an extended branch-and-cut algorithm to solve the SDVRPTW-IC and report on experimental results showing the impact of customer inconvenience constraints. We finally draw useful insights for logistics managers on the basis of the experimental analysis carried out

A Branch-Price-and-Cut Algorithm for the Capacitated Multiple Vehicle Traveling Purchaser Problem with Unitary Demand

The multiple vehicle traveling purchaser problem (MVTPP) consists of simultaneously selecting suppliers and routing a fleet of homogeneous vehicles to purchase different products at the selected suppliers so that all product demands are fulfilled and traveling and purchasing costs are minimized. We consider variants of the MVTPP in which the capacity of the vehicles can become binding and the demand for each product is one unit. Corresponding solution algorithms from the literature are either branch-and-cut or branch-and-price algorithms, where in the latter case the route-generation subproblem is solved on an expanded graph by applying standard dynamic-programming techniques. Our branch-price-and-cut algorithm employs a novel labeling algorithm that works directly on the original network and postpones the purchasing decisions until the route has been completely defined. Moreover, we define a new branching rule generally applicable in case of unitary product demands, introduce a new family of valid inequalities to apply when suppliers can be visited at most once, and show how product incompatibilities can be handled without considering additional resources in the pricing problem. In comprehensive computational experiments with standard benchmark sets we prove that the new branch-price-and-cut approach is highly competitive

The Team Orienteering Problem with Overlaps : An Application in Cash Logistics

The team orienteering problem (TOP) aims at finding a set of routes subject to maximum route duration constraints that maximize the total collected profit from a set of customers. Motivated by a real-life automated teller machine cash replenishment problem that seeks for routes maximizing the number of bank account holders having access to cash withdrawal, we investigate a generalization of the TOP that we call the team orienteering problem with overlaps (TOPO). For this problem, the sum of individual profits may overestimate the real profit. We present exact solution methods based on column generation and a metaheuristic based on large neighborhood search to solve the TOPO. An extensive computational analysis shows that the proposed solution methods can efficiently solve synthetic and real-life TOPO instances. Moreover, the proposed methods are competitive with the best algorithms from the literature for the TOP. In particular, the exact methods can find the optimal solution of 371 of the 387 benchmark TOP instances, 33 of which are closed for the first time

The split delivery capacitated team orienteering problem

In this article, we study the capacitated team orienteering problem where split deliveries are allowed. A set of potential customers is given, each associated with a demand and a profit. The set of customers to be served by a fleet of capacitated vehicles has to be identified in such a way that the profit collected is maximized, while satisfying constraints on the maximum time duration of each route and the vehicle capacity constraints. When split deliveries are allowed, each customer may be served by more than one vehicle. We show that the profit collected by allowing split deliveries may be as large as twice the profit collected under the constraint that each customer has to be served by one vehicle at most. We then present a branch-and-price exact algorithm and a hybrid heuristic. We show the effectiveness of the proposed approaches on benchmark instances and on a new set of instances that allow us to computationally evaluate the impact of split deliveries

Comparison of policies in dynamic routing problems

We consider a company that has to satisfy customers' pick-up requests arriving over time every day. The overall objective of the company is to serve as many requests as possible at a minimum operational cost. When organizing its business the company has to fix some features of the service that may affect both service quality and operational costs. Some of these features concern the time a request is taken into account to plan its service, the associated deadline and the way requests are managed when the system is overloaded. In this paper we analyse several policies that can be implemented by the management of a carrier company in a multi-period context. For example, a company might reject all the requests that cannot be feasibly scheduled or accept all the requests and rely on a backup service in order to serve requests that are difficult to handle. Another interesting issue considered in this paper is the impact of collaborative service where two or more carrier companies, with their own customers, decide to share customers in order to optimize the overall costs. We set up a general framework to allow comparison of alternative service policies. Extensive computational results evaluating the number of lost requests and the distance travelled provide interesting insights

Branch-and-Price-and-Cut for the Active-Passive Vehicle-Routing Problem

This paper presents a branch-And-price-And-cut algorithm for the exact solution of the active-passive vehicle-routing problem (APVRP). The APVRP covers a range of logistics applications where pickup-And-delivery requests necessitate a joint operation of active vehicles (e.g., trucks) and passive vehicles (e.g., loading devices such as containers or swap bodies). The objective is to minimize aweighted sum of the total distance traveled, the total completion time of the routes, and the number of unserved requests. To this end, the problem supports a flexible coupling and decoupling of active and passive vehicles at customer locations. Accordingly, the operations of the vehicles have to be synchronized carefully in the planning. The contribution of the paper is twofold: First, we present an exact branch-And-price-And-cut algorithm for this class of routing problems with synchronization constraints. To our knowledge, this algorithm is the first such approach that considers explicitly the temporal interdependencies between active and passive vehicles. The algorithm is based on a nontrivial network representation that models the logical relationships between the different transport tasks necessary to fulfill a request as well as the synchronization of the movements of active and passive vehicles. Second, we contribute to the development of branch-And-price methods in general, in that we solve, for the first time, an ng-path relaxation of a pricing problem with linear vertex costs by means of a bidirectional labeling algorithm. Computational experiments show that the proposed algorithm delivers improved bounds and solutions for a number of APVRP benchmark instances. It is able to solve instances with up to 76 tasks, four active, and eight passive vehicles to optimality within two hours of CPU time

A mathematical programming algorithm for planning and scheduling an Earth observing SAR constellation

We consider the planning and scheduling problem of SAR constellations for the observation of the Earth. We define a simplified problem which only takes into account the most binding constraints of the real problem. We present a Lagrangean relaxation algorithm that provides upper bounds and we show that it can also be used to guide an existing heuristic algorithm to find better solutions to the real problem

The capacitated team orienteering problem with incomplete service

In this paper we study the capacitated version of the Team Orienteering Problem (TOP), that is the Capacitated TOP (CTOP) and the impact of relaxing the assumption that a customer, if served, must be completely served. We prove that the profit collected by the CTOP with Incomplete Service (CTOP-IS) may be as large as twice the profit collected by the CTOP. A computational study is also performed to evaluate the average increase of the profit due to allowing incomplete service. The results show that the increase of the profit strongly depends on the specific instance. On the tested instances the profit increase ranges between 0% and 50%. We complete the computational study with the increase of the profit of the CTOP due to split deliveries, that is multiple visits to the same customer, and to split deliveries combined with incomplete service