The stochastic $p$-hub center problem with service-level constraints

Abstract

Small package delivery companies offer services where packages are guaranteed to be delivered within a given time-frame. With variability in travel time, the configuration on the hub-and-spoke delivery network is vital in ensuring a high probability of meeting the service-level guarantee. We present the stochastic $p$-hub center problem with chance constraints, which we use to model the service-level guarantees. We discuss analytical results, propose solution heuristics, and present the results from computational experiments.

Introduction

Customers looking for delivery services are increasingly requiring delivery companies to offer fast and reliable service, as well as guarantees on when deliveries will be made. These guarantees are characterized by the next-day or second-day delivery services offered by companies such as UPS and FedEx. In UPS’ case, for example, these services represented $6.8 billion in revenue in 2006 (a 6.2% increase from 2005) from delivery of 1.27 million packages a day, accounting for 22% of its domestic revenues [1].

With such large volumes of packages transported between many different origin–destination points during a single year, it is paramount that the delivery networks operate efficiently and reliably so as to be able to meet the service guarantees. The configuration of the network thus has an important role in this regard [2], [3]. Package delivery companies operate complex hub-and-spoke networks, which have many sorting hub centers and many more local service centers. The delivery network configuration will determine the company's ability to meet service guarantees, or determine what service guarantees it can offer. The $p$-hub center problem ($p$HCP), one of many different hub location models, can be used as an aid in designing such time-sensitive delivery services.

The $p$HCP simultaneously finds the optimal location of $p$ hubs and the assignment of non-hub nodes to the $p$ hubs so as to minimize the longest origin–destination path in the network. The solution to the $p$HCP essentially provides an upper bound on the delivery times in the hub-and-spoke network, or put differently, the $p$HCP returns the minimum time period needed to guarantee that a package from any origin to any destination will be delivered within that time-frame. This minimum delivery time value can be used to design time-constrained service offerings to customers, or to ascertain if certain current service guarantees are feasible.

Considering the solution of the $p$HCP from a strategic point of view, once a service guarantee has been established, a company can then evaluate alternative delivery transportation modes for those origin–destination pairs that are not tightly constrained by the time service guarantee. For example, the company may choose to use a slower but lower cost transportation option to service a particular origin–destination pair and still meet the service guarantee, or perhaps use a combination of different transportation modes to service the demand flows [4]. The company could also consider differential pricing strategies to account for customers’ service demands and the company's operational constraints.

With variability in the transportation time from the origin to the destination, there is the possibility that a package may not be delivered on time. A failure in on-time delivery may result in a company having to compensate an unhappy customer. For example, FedEx and UPS refund the delivery service charges in the event of a service failure [5], [6]. While this remunerative value is known, it is more difficult to quantify the potential cost of lost goodwill, and thus a company may insist on a minimum service level instead.

In this paper, we introduce the stochastic $p$-hub center problem (S$p$HCP), which employs a chance-constrained formulation [7], [8] to model the minimum service-level requirement. This model accounts for the variability in travel times when designing the hub network so that the maximum travel time through the network is minimized. Our work is the first to incorporate stochastic travel times in hub location models. We know of only one other published article on stochastic hub location problems, which is that by Marianov and Serra [9]. In their paper, the authors focus on stochasticity at the hub nodes by representing the hub airports as $M/D/c$ queues and limiting via chance constraints the number of airplanes that can be in a queue at the hub airports.

In the next section, we provide an overview of the hub location problem, and review the literature related to the problem presented in this paper. A general formulation for the S$p$HCP is given in Section 3, and we also present a mixed-integer linear programming formulation for the S$p$HCP under the assumption that the travel time on each link is normally distributed, and that the travel time on a link is independent of that on any other link. Because of the size of the mixed-integer linear programming problem, CPLEX is only able to solve problems with small number of nodes (less than 10 nodes). In Section 4, we discuss some insights obtained from analyzing the optimal solutions of the S$p$HCP, and then propose several heuristics for solving the S$p$HCP. Results from our computational experiments are provided in Section 5. Concluding remarks and future research directions arising from this work are discussed in Section 6.