The discrete time window assignment vehicle routing problem

Highlights

• We provide a time window assignment model in a VRP setting with demand uncertainty.

• We develop a state-of-the-art exact branch-price-and-cut method.

• We develop column generation heuristics for good solutions in large instances.

• Numerical experiments show that using multiple demand scenarios is better than one.

Abstract

In this paper we introduce the discrete time window assignment vehicle routing problem (DTWAVRP) that can be viewed as a two-stage stochastic optimization problem. Given a set of customers that must be visited on the same day regularly within some period of time, the first-stage decisions are to assign to each customer a time window from a set of candidate time windows before demand is known. In the second stage, when demand is revealed for each day of the time period, vehicle routes satisfying vehicle capacity and the assigned time windows are constructed. The objective of the DTWAVRP is to minimize the expected total transportation cost. To solve this problem, we develop an exact branch-price-and-cut algorithm and derive from it five column generation heuristics that allow to solve larger instances than those solved by the exact algorithm. We illustrate the performance of these algorithms by means of computational experiments performed on randomly generated instances.

Introduction

In distribution networks, it is common for a supplier and a customer to agree on a time window in which a delivery will be made. This time window is often used repeatedly within some period of time in which multiple deliveries are made at regular intervals. At the moment of choosing a time window for a customer, its demand is usually unknown and may fluctuate for different deliveries. When the demands of all customers become known for a given day, a vehicle routing problem with time windows (VRPTW) must be solved to construct a delivery schedule within the agreed time windows.

The time window assignment vehicle routing problem (TWAVRP) was recently introduced by Spliet and Gabor (2014). This problem can be viewed as a two-stage stochastic optimization problem (see Birge and Louveaux, 1997). Given a set of customers to be visited on the same day regularly during some period of time (e.g., a season), the TWAVRP consists of assigning in the first stage a time window to each customer before customer demand is known. In the second stage, when demand is revealed for each day of the time period, vehicle routes that respect the assigned time windows are constructed. The assigned time windows have a predetermined width and can start at any time within an exogenous time window that can be customer-dependent. The objective of the TWAVRP is to minimize the expected total transportation cost over the whole time period that is equivalent to minimizing the expected cost for a single day.

In this paper, we study the discrete TWAVRP (DTWAVRP) that differs from the TWAVRP by considering for each customer a finite set of candidate time windows from which one has to be selected. For example, a customer might divide the day in blocks of 2 hours commencing on the hour and require one of these blocks to be the assigned time window. We have encountered such time window assignment problems (discrete or not) while collaborating with Dutch retail chains, and believe they are common in this industry. Here, the retailers (customers) are heavily dependent on the time window to be fixed in advance and kept for some time. For instance, a retailer might receive all its deliveries on the same day of the week and more or less the same hour of the day for an entire year. This is crucial for many operational purposes like inventory management and the scheduling of personnel. Considering a discrete set of time windows is often more practical for the retailers, especially to ease the personnel scheduling process which must take into account various regulations. Furthermore, it can give them the opportunity to express preferences for the time windows. Maximizing the satisfaction of these preferences might be taken into account as a secondary objective during the optimization process, an option that is not considered in this paper.

As is common for two-stage stochastic optimization problems, we suggest to approximate the probability distributions of the customer demands by a finite set of possible scenarios of demand realizations. In this framework, the DTWAVRP is clearly NP-hard as, in the case of one scenario and one candidate time window per customer, it reduces to the VRPTW. When it involves several scenarios, the DTWAVRP corresponds to solving several VRPTWs (one per scenario) that are linked together by the choice of the time windows. The VRPTW is a well-studied problem for which many exact and heuristic algorithms have been developed (see, e.g., the surveys of Baldacci, Mingozzi, Roberti, 2012, Bräysy, Gendreau, 2005a, Bräysy, Gendreau, 2005b, Kallehauge, Larsen, Madsen, Solomon, 2005).

We believe that in the scientific literature, the problem of assigning time windows before knowing demand has been largely overlooked so far. It was only tackled recently by Spliet and Gabor (2014) who designed an exact branch-price-and-cut algorithm for the TWAVRP that can solve instances with up to 25 customers and three scenarios. In their solution approach, the pricing problem is modeled as an elementary shortest path problem with linear node costs (induced because the set of time windows is not discrete) and capacity and time window constraints. Due to the complexity of this problem route relaxation is required, but only basic route relaxation can be utilized efficiently: allowing all cyclic routes and eliminating 2-cycles. For the DTWAVRP we also develop a branch-price-and-cut algorithm, but in this case we are able to employ the more sophisticated ng-route relaxation introduced by Baldacci, Mingozzi, and Roberti (2011). The complexity of the pricing problem of Spliet and Gabor (2014) also prohibits the use of certain types of valid inequalities as these add to the complexity. For instance subset row inequalities, introduced by Jepsen, Petersen, Spoorendonk, and Pisinger (2008), are not used in the algorithm for the TWAVRP, whereas they are used in our algorithm for the DTWAVRP. Furthermore, as opposed to the algorithm for the TWAVRP, we add a heuristic pricing algorithm to speed up computations and develop several column generation heuristics.

Introduced by Groër, Golden, and Wasil (2009), the consistent vehicle routing problem (ConVRP) shares some similarities with the DTWAVRP when considering demand scenarios. In this deterministic problem, each customer must be visited on different days of a given horizon (not all customers must be serviced each day) following a consistent schedule, that is, the arrival times at a customer from one day to another cannot differ by more than a limited amount of time. Moreover, it is required that each customer is always visited by the same driver. Groër et al. (2009) found optimal solutions to ConVRP instances involving up to 12 customers and three scenarios using a commercial mixed integer programming solver. They reported computation times of up to several days. Furthermore, they developed a local search heuristic to solve instances with over 3700 customers.

Jabali, Leus, van Woensel, and de Kok (2013) considered another related problem that involves the assignment of time windows in a vehicle routing problem with stochastic travel times and deterministic demands. They developed a tabu search algorithm for solving it. Also, Agatz, Campbell, Fleischmann, and Savelsbergh (2011) studied a stochastic problem faced by e-tailers providing home delivery that consists of selecting which time slots to offer per zip code for making deliveries. They developed a local search heuristic.

The main contributions of this paper are as follows. First, we propose a new problem, the DTWAVRP. Second, we develop a state-of-the-art exact branch-price-and-cut algorithm to solve it and report computational results obtained on randomly generated instances to evaluate the effectiveness of some of its components. Third, from this exact algorithm, we derive several column generation heuristics that can find good solutions to the DTWAVRP in limited computation times. This allows to find solutions to instances that are larger than those solved by the exact algorithm. We report computational results to compare the performance of five such algorithms. Finally, we also provide additional results to assess the benefits of using several demand scenarios to assign time windows to the customers instead of using a single one defined by the average demands as it is usually done in practice.

In the next section, we define the DTWAVRP using demand scenarios and we present an integer programming formulation for it. In Section 3, we introduce the proposed branch-price-and-cut algorithm. In Section 4, we describe the five column generation heuristics. The results of our computational experiments are reported in Section 5. Finally, conclusions are drawn in Section 6.