Optimizing consolidation processes in hubs: The hub-arrival-departure problem

Highlights

• We address a novel consolidation problem in hubs.

• Equivalence among two (on first sight completely different) objectives is proven.

• Exact and heuristic solution procedures are presented.

• A simulation study explores the robustness of different objectives.

Abstract

To realize economies of scale in transportation, hubs are important entities in today’s distribution networks. Hubs, such as cross docks for trucks, central ports for container vessels, transshipment yards for freight trains, and air hubs for airplanes, allow to consolidate less-than-vehicle-load shipments and to improve the utilization of transport capacities. In this context, we consider a generic optimization problem to streamline hub processes. We assign vehicles consolidating their shipments at a central hub each an arrival time slot where a vehicle is unloaded and a later departure time slot where it is loaded with the goods it takes back to its designated territory. Since the processing of goods inside a hub terminal takes some time, we want to synchronize the exchange of goods among vehicles such that three different consolidation-oriented objectives are optimized. For instance, we aim to minimize the number of goods missing their dedicated departures after a fixed consolidation time for goods inside the hub. We prove important structural properties, provide a comprehensive analysis of computational complexity, and derive suited solution procedures. Our problem occurs in the real world at the air hub of postal service provider DHL in Leipzig, where freight airplanes from all over the world exchange goods and need to be assigned landing and departure slots on the runway. Our problem, however, is generic, and we sketch some extensions in order to adapt it to further hubs.

Introduction

Ever since 1955 when Delta Air Lines pioneered the hub-and-spoke system at its hub in Atlanta (USA) for the airline industry (Guo, Weidinger, & Boysen, 2019), hub terminals have played a crucial role in many modern distribution networks. Nowadays they exist for any kind of freight vehicles and also for vehicles moving people, which are, however, not address by this paper. Beyond the airline industry, there are, for instance, cross docks (Van Belle, Valckenaers, & Cattrysse, 2012) and parcel distribution centers (Fedtke & Boysen, 2017) for trucks, container ports for vessels and inland barges (Steenken, Voß, & Stahlbock, 2004), as well as railway yards for freight trains (Boysen, Fliedner, Jaehn, & Pesch, 2013). Hubs are intermediate nodes in distribution networks where less-than-vehicle-load shipments can be consolidated to full-vehicle loads. In this way, transport capacities are better utilized and economies of scale in transportation can be realized.

On the negative side, each stop at a hub slows down the distribution process, jeopardizes on-time deliveries, and the double handling of goods, unloaded from one vehicle, consolidated at the hub, and loaded onto another vehicle, costs time and money. To avoid these negative consequences, a close synchronization among inbound vehicles delivering goods to the hub, efficient inner-hub processes, and outbound vehicles departing with the consolidated shipments is essential. To support this task and to assist managers of real-world hub terminals with their daily work, a rich body of scientific literature for each specific type of hub terminal has accumulated.

When aiming at real-world decision support, naturally, the specifics of each vehicle and terminal type have to be considered. When assigning gates to aircraft, for instance, security reasons forbid wing tip overlap of two big aircraft assigned to neighboring gates (Dorndorf, Jaehn, & Pesch, 2008), and a decision support tool to be applied in a real-world airport has to consider this restriction. In spite of the undisputed peculiarities of each hub type, however, a second stream of research has established over the years in this area. Since the general aims and problem structures when synchronizing inbound and outbound flows at hubs are always very similar, it might also be a fruitful idea to rather take a generalist’s perspective and to focus on generic models suited for general hub processes. This stream of research aims to extract the basic structural properties of synchronization problems in hubs in order to derive basic algorithmic concepts and building block models that are adaptable and expandable to specific hub types in a second step. Some papers of this research stream are, for instance, provided by Boysen, Emde, Stephan, & Weiß (2015), Briskorn, Fliedner, & Tschöke (2020), Guo et al. (2019), O’Kelly (2010) and Yu & Egbelu (2008). In the tradition of this second stream of hub research, this paper investigates a generic synchronization problem, which we call the hub-arrival-departure (HAD) problem. HAD can briefly be characterized as follows.

We are given a set of vehicles each servicing two territories before and after visiting the hub (e.g., an origin and the same or a different destination depot for a local region). That is, a vehicle arrives at the hub from its origin territory and unloads its goods posted there but dedicated to other territories during an arrival slot. Then, the vehicle waits for some time in a holding lot until other vehicles from other origin territories have arrived and, finally, the vehicle is called to the service point of the hub once again during its departure slot. Here, the vehicle receives its dedicated goods having arrived from other territories and departs back toward its destination territory. Thus, our decision task is to assign each vehicle an arrival slot and a later departure slot. Processing goods in a hub takes a consolidation time, which is required to sort, prepare, and rearrange goods for their departure. In this decision context, we consider three alternative objectives: (i) For a given number of equidistant slots for processing vehicles and a constant consolidation time, we want to minimize the number of shipments missing their departure vehicle. (ii) We minimize the makespan of vehicle processing, i.e., the number of slots required for arrivals and departures of vehicles, so that at least a predefined number of shipments reach their departure vehicles on time given a constant consolidation time. (iii) For a given number of slots and an allowed number of goods missing their departure, we want to maximize the minimum consolidation time. Note that the latter objective mainly aims at robust solutions to protect against unforeseen consolidation delays within a hub. This issue is discussed in further detail in Section 6.

Example: Consider a hub processing four vehicles 1, 2, 3, and 4. The exchange of goods among these vehicles is given by the handover graph in Fig. 1(a), where nodes represent vehicles and arcs stand for goods delivered by the vehicle of the tail node dedicated to the vehicle of the head node. We assume that there is exactly one shipment to be delivered if we have an arc from one node to an other. Figs. 1(b) to (d) depict three alternative solutions, where each node represents one of the equidistant slots and its assigned vehicle, with the leftmost (rightmost) slot being the earliest (latest) slot. In each solution, each vehicle receives two slots, where the arrival (departure) slot is marked light (dark) gray and empty slots remain empty. In solution (b), for instance, vehicle 3 unloads the goods brought to the hub in its arrival slot 3 and loads the goods it receives from other vehicles in departure slot 7. Solution (b) is optimal with regard to objective (i) assuming that we have eight slots available and a given consolidation time of four slots it takes to prepare goods after arrival for departure. Our aim is to minimize the number of delayed shipments. The only shipment missing its departure is the one delivered on vehicle 2 dedicated to vehicle 1. A single time unit is not sufficient to prepare the goods for departure. Solution (c) is optimal for objective (ii), where a makespan of nine time slots is required, so that all shipments reach their departure in time, given a minimum consolidation time of four slots. Finally, solution (d) allows for a maximum consolidation time of three, if we have eight given time slots and all shipments are required to timely reach their departure vehicle. This solution is optimal for objective (iii).

This paper is dedicated to the analysis of our HAD problem. We investigate its computational complexity and prove essential structural properties. For instance, we show that optimal solutions of objective (ii) can easily be transferred into optimal solutions of objective (iii) and vice versa. Furthermore, we provide suited solution procedures and test the abilities of our three objectives to produce robust solution in case the actual arrival time of vehicles and consolidation times of goods are bound to unforeseen disturbance. HAD is a generic problem that is relevant whenever vehicles, connecting a hub with a specific depot, travel back and forth fixed territories. This setting can generally occur in all different types of hub nodes and is especially applied by postal service providers (Fedtke & Boysen, 2017). In the following, we briefly sketch a real-world application where our problem setting is directly relevant.

The air hub in Leipzig (Germany) is one of three major hubs in postal service provider DHL’s worldwide express freight network. Here, about 60 airplanes, arriving from all over the world, need to land and be processed in a short time-frame of a few hours during the night, so that express postal packages can rapidly be consolidated among aircraft heading back to their respective territory. Thus, each airplane, servicing a specific territory, is to be assigned a landing and a later departure time slot, assuming a single runway operation at the airport. Moreover, since DHL operates a standardized fleet of airplanes, sequence-dependent setup times due to the vortex caused by differently sized planes are a non-issue (Boysen & Fliedner, 2011) and the equidistant landing slots of our HAD are no shortcoming. The packages, once unloaded from the aircraft and transported toward the terminal building, are isolated and automatically sorted by destination by a large sortation conveyor and then reloaded onto the respective aircraft (Fedtke & Boysen, 2017). For this standardized consolidation process, a fixed consolidation time is also not far from reality, so that our decision problem (already in its elementary form) is pretty close to applicability in a real-world hub. As a generic problem, however, other hubs may require different adaptions, which we sketch in Section 7.

The remainder of the paper is structured as follows. Section 2 reviews the relevant literature. A detailed problem definition is given in Section 3. Some elementary structural properties and computational complexity are investigated in Section 4. Exact and heuristic solution approaches are presented in Section 5. Our computational study (see Section 6) explores the performance of our solution approaches and investigates the robustness of our objectives. Finally, Section 7 concludes the paper and discusses some possible extensions of our generic problem.