The Flexible Periodic Vehicle Routing Problem

Highlights

• A Mathematical model for the Flexible Periodic Vehicle Routing Problem (FPVRP) is proposed.

• A Load-based formulation is developed to solve the FPVRP.

• A set of several inequalities are proposed to reinforce the load-based FPVRP formulation.

• A Load-based formulation for the Periodic Vehicle Routing Problem (PVRP) and a load-based formulation for the Flexible Periodic Vehicle Routing Problem with InventoryConstraints (FPVRP-IC) are proposed for a fair comparison in the computational experience.

• A computational experience was carried out in order to show that FPVRP formulation outperforms PVRP and FPVRP-IC formulations in terms of solution quality.

Abstract

This paper introduces the Flexible Periodic Vehicle Routing Problem (FPVRP) where a carrier has to establish a distribution plan to serve his customers over a planning horizon. Each customer has a total demand that must be served within the horizon and a limit on the maximum quantity that can be delivered at each visit. A fleet of homogeneous capacitated vehicles is available to perform the services and the objective is to minimize the total routing cost. The FPVRP can be seen as a generalization of the Periodic Vehicle Routing Problem (PVRP) which instead has fixed service frequencies and schedules and where the quantity delivered at each visit is fixed. Moreover, the FPVRP shares some common characteristics with the Inventory Routing Problem (IRP) where inventory levels are considered at each time period and, typically, an inventory cost is involved in the objective function. We present a worst-case analysis which shows the advantages of the FPVRP with respect to both PVRP and IRP. Moreover, we propose a mathematical formulation for the problem, together with some valid inequalities. Computational results show that adding flexibility improves meaningfully the routing costs in comparison with both PVRP and IRP.

Introduction

Vehicle Routing Problems (VRPs) (Toth and Vigo, 2014) deal with the definition of a distribution plan where a fleet of vehicles is used to deliver (or pick-up) goods to a set of customers. These problems involve a single-period plan, meaning that the delivery operations start at the beginning of the period and should terminate within the end of it when all customers are served. However, several real–world applications deal with periodic delivery operations over a given time horizon. For example, periodic delivery service may be found in delivery of groceries or collection of waste. We refer to Francis et al. (2008) for a more exhaustive list of periodic service applications. The Periodic Vehicle Routing Problem (PVRP) (Campbell and Wilson, 2014) is a vehicle routing problem where the service to the customers has to be provided over multiple periods. It is assumed that customers must be served with a certain frequency according to a given schedule and they should receive a fixed quantity at each visit. The problem is to choose the visit schedule for each customer and to organize vehicle routes. Another broad class of routing problems dealing with multi-period plans is the class of Inventory Routing Problems (IRPs) (Bertazzi, Speranza, 2012, Bertazzi, Speranza, 2013, Coelho, Cordeau, Laporte, 2013). IRPs differ from PVRPs in that they do not fix the service frequency nor the quantity delivered to each customer at each visit. Instead, each customer has a predefined rate of goods consumption per period, and the inventory level is calculated at each time period. The distribution plan has to be such that each customer must be able to satisfy his consumption rate at each time period.

In this paper, we present a new problem which deals with periodic demand without resorting to inventories. We call this problem the Flexible Periodic Vehicle Routing Problem (FPVRP). In the FPVRP, customers are assigned to a total demand that has to be satisfied within the end of the planning horizon. The quantity delivered at each visit should not exceed the customer capacity, which is typically lower than the total demand. Thus, multiple visits to each customer are performed over the planning horizon. The FPVRP can be seen as an extension of the PVRP where no fixed frequency schedule is set. Instead, if the customer capacity corresponds to the ratio between the total demand and the frequency established in the PVRP, then the FPVRP becomes a generalization of the PVRP where each customer must be visited a number of times which is at least equal to the PVRP frequency. Moreover, the quantity delivered at each visit has to be defined. This clearly increases the flexibility with respect to the PVRP setting and may produce cost savings.

The FPVRP is also related to the IRP, where no fixed frequency is set for the customer visits and the delivered quantity is a decision variable. However, contrary to the IRP, no inventory level is considered in the FPVRP. Again, this gives the FPVRP a higher flexibility with respect to the IRP which may result in cost savings.

The main contributions of the paper are the following. We introduce the FPVRP, which is a new and challenging problem dealing with flexibility in periodic delivery operations. We show that, theoretically, the FPVRP can produce arbitrarily large improvements with respect to both the PVRP and the IRP when considering the routing cost. Indeed, such improvements come at the expenses of introducing additional decisions to the problem related to defining when the customers should be visited and how much should be delivered at each visit. This clearly increases the complexity of the problem, as will become evident in the computational results. We then propose a mathematical formulation for the FPVRP based on commodity flow variables, which we call load-based formulation. This choice is consistent with recent works on other VRP variants for which load-based formulations have been proposed (Archetti, Bianchessi, Irnich, Speranza, 2014, Letchford, Salazar-González, 2015). We also present a load-based formulation for the PVRP and for the IRP, which we use for the computational experiments, as they are quite effective and they do not require the use of sophisticated techniques (like branch-and-cut or column generation) for their implementation.

The paper is organized as follows. In Section 2, a review of the related literature is presented. Then, Section 3 provides a formal definition of the FPVRP, the PVRP and the IRP. The worst case ratio between the optimal values of the PVRP and the FPVRP and between the optimal values of the IRP and the FPVRP are analyzed, respectively, in Section 4. Load-based mathematical programming formulations for the FPVRP, the IRP, and for the PVRP are proposed in Section 5. In order to strengthen the formulation and improve the computing times, several families of inequalities are introduced for the FPVRP in Section 6. Computational experiments and the analysis of numerical results are shown in Section 7. Finally, some concluding remarks and future work are given in Section 8.