Innovative Applications of O.R.The Airline Container Loading Problem with pickup and delivery
Introduction
In the Airline Container Loading Problem with Pickup and Delivery (ACLPPD), a set of containers and pallets, known as Unit Load Devices (ULD), must be loaded into a compartmentalized cargo aircraft. We consider that pickup and delivery operations occur at different airports during any given trip. The loading task is illustrated in Fig. 1. We propose an exact solution approach that relies on a mixed integer linear program to find the optimal ULD assignment.
Air cargo represents 10 percent of the world trade volume, but its value is in excess of $6.4 trillion per annum, which is approximately 35 percent of the world trade value (IATA, 2013a). Thus, air cargo transportation plays a highly significant economic role. Optimizing loading assignment on board is critical to airlines for several reasons. First, correct loading conditions safety. Inappropriate loading can cause significant damage, and place the aircraft, the freight or even the crew at risk. Therefore, this paper models a wide set of constraints for operators to consider daily. The proposed model applies to all aircraft and loads, and complies with international standards. Considering the same constraints as Limbourg, Schyns, and Laporte (2012), we adapt such constraints to the case of a sequence of routes, called legs, while considering the additional case of hazardous products and oversized ULDs. Second, optimal loading has a positive impact on aerodynamics, thus resulting in less fuel consumption, i.e., reduced cost and environmental impact. This issue is crucial for airlines, affected by rising oil prices and increased pressure to reduce carbon dioxide emissions. This paper analyzes fuel and handling operations in order to minimize costs. The management of these first two requirements is done through a proper distribution of the ULD weights within the aircraft. This part is a Weight & Balance Problem. The third reason optimal loading is important for airlines is that managing operations on the ground is challenging, especially when the trip includes several legs with P&D operations. Reducing the number of handling operations reduces time, which in turn reduces labor costs per flight. Such reduction also allows shorter turnaround time, i.e., the time that elapses from the moment the plane arrives to the moment it leaves again, thus reducing airport fees. Time saved could be used for other valuable operations. Optimizing loading plans is also crucial and constitutes another reason to consider this problem. Indeed, loadmasters must build plans within an extremely short time, and doing so manually requires significant time. On the other hand, with an interactive computerized efficient tool, loadmasters would be able to consider different alternatives and select the best solution with respect to their experience and the real conditions faced on the ground.
In this context, the problem no longer consists merely, as in Limbourg et al. (2012), in positioning ULDs to reach a proper equilibrium, but also in defining the unloading and loading operation sequence at airports. Because there is only one path between any ULD and the exit door, this path must be free to unload ULDs. The task is to minimize, at each airport, the number of ULDs in transit to be unloaded in order to have access to the ULDs reaching their delivery point. The same question arises when pickup occurs. The problem is even more complex when several doors can be used, as occurs occasionally. The cost of these handling operations is the second element of our proposed objective function. It is important to notice that we face two conflicting objectives: optimizing board assignments for fuel and for ground operations. Our contribution is to propose an exact approach to solve simultaneously both the Weight & Balance Problem over a multi-leg trip, and the sequencing problems associated to pickups and deliveries. We resort to a mixed integer linear program where the objective is to minimize both costs.
Currently, this extremely complex problem (NP-hard) is still essentially solved manually based on best practices. Because load planners have extremely short time windows to choose assignments, they focus mainly on finding a feasible and reasonable solution. As a rule, they do not incorporate P&D operations in the planning process. A common method for managing several legs is, indeed, to plan each leg independently. Accordingly, almost the entire cargo can be unloaded at intermediate airports, and the ULDs that have not reached final destinations are reloaded subsequently, which is the worst possible scenario for ground operations. We show, based on of our first results from real data provided by industrial partners, that our approach allows significant savings.
The remainder of this paper is organized as follows. Section 2 outlines the problem and the assumptions involved. Related literature and contributions are presented in Section 3. Section 4 describes the problem in more detail, and provides the proposed model’s mathematical formulation. Section 5 provides information on the theoretical complexity of the problem, whereas Section 6 illustrates the performance of the approach through numerical results. Finally, some conclusions are drawn.
Section snippets
Problem summary and assumptions
ACLPPD can be informally summarized as:
min Fuel and loading operations costs on the entire trip (global optimization) s.t. Pickup and delivery sequences are feasible Customer demand is satisfied (each ULD is loaded) Each ULD fits in an aircraft position A position accepts only one ULD Some positions overlap and cannot be used simultaneously Longitudinal stability is within certified limits (ZFW, TOW, LW) Lateral stability is within certified limits Weight per position is below certified limit Combined
Related literature and contributions
This problem is an Assignment Problem (AP) that is referred in the literature as belonging to the family of Weight & Balance Problems. Over the past years, more attention has been paid to the problem that precedes ACLPPD by considering how to optimize freight loading within ULDs (Chan, Bhagwat, Kumar, Tiwari, Lam, 2006, Li, Tao, Wang, 2009, Paquay, Schyns, Limbourg, 2014, Tang, 2011, Tang, Chang, 2010, Wu, 2010, Yan, Shih, Shiao, 2008) independently of aircrafts. The scientific literature on
Main parameters and variables
Our model is built on three main sets of parameters. The first is the set of legs that are the different parts of a trip that separates two successive airports. This model considers two legs. However, generalization to more legs is simple. A trip composed of two legs is a common case for long-range flights, whereas the consideration of more legs essentially complicates notation. The second main set of parameters is the set of ULDs. For each ULD, we know its type (IATA code), weight wi, and
Complexity
Let us now provide insight into the complexity of the problem. The first term of the objective function is to consider balanced loading and fuel consumption. We show below that the problem defined by this first part of the objective function and for one leg is already NP-hard.
Definiton We define ACLPPDD as the decision version of ACLPPD that asks whether the objective function (7) can reach a null value. Lemma 5.1 ACLPPDD is in NP. Proof Because the problem is expressed as a mixed integer linear problem, inputting the
Implementation and results
Our mathematical model was tested on a set of real-world instances provided by TNT Airways, a wholly owned subsidiary of TNT Express. Their main activity is to provide TNT Express with an airfreight network that connects daily all TNT Express locations throughout the world, and more specifically, in Europe. TNT Express is one of the leading delivery integrators in Europe. The model was implemented in Java and relies on the IBM ILOG CPLEX 12 library (default parameters). Thanks to a graphical
Conclusions
In this paper, we analyzed ACLPPD, which is a crucial problem encountered daily by airlines. We considered trips of several legs at the end of which P&D operations might occur. We proposed a new mixed integer linear model.
Our contributions are multiple. First, the model is based on international standards and is valid for most commercial operators. We integrated, and adapted to the multi-leg context, a large set of the constraints they encounter. Most operators should be able to use this
Acknowledgements
This work was initiated as part of a research project with TNT Airways. Special thanks are due to E. Meyer, M. Clety, H. Marchal, and J.M. Urbani. We would also like to thank T. Kleyntssens for his preliminary work on some constraints. Virginie Lurkin is supported by “Fonds National de la Recherche Scientifique” (FNRS). This work was also partially funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (grant P7/36). We are also grateful to three
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