Network Cargo Capacity Management
We consider the problem faced by an airline that is flying both passengers and cargo over a network of locations on a fixed periodic schedule. Bookings for many classes of cargo shipments between origin-destination pairs in this network are made in advance, but the weight and volume of aircraft capacity available for cargo as well as the exact weight and volume of each shipment are not known at the time of booking. The problem is to control cargo accept/reject decisions so as to maximize
... d profits while ensuring effective dispatch of accepted shipments through the network. This network stochastic dynamic control problem has very high computational complexity. We propose a linear programming and stochastic simulation-based computational method for learning approximate control policies and discuss their structural properties. The proposed method is flexible and can utilize historical booking data as well as decisions generated by default control policies. manuscript no. with a process called virtual nesting which maps a multiple-leg booking request onto one of several virtual classes for each leg involved. The request is accepted if and only if the virtual classes on each leg are open. This control mechanism was first implemented by American Airlines around 1983. The second type of control is based on the idea of bid-prices introduced by Smith and Penn (1988) and Simpson (1989). In this type of control, a booking request is accepted if and only if its revenue exceeds the sum of threshold prices -bid prices -of the resources necessary to satisfy it. There is a wide range of computational approaches to finding "good" bid prices. Talluri and van Ryzin (1998) have studied theoretical properties of bid-price controls and performed an asymptotic analysis. Bertsimas and Popescu (2003) examined adaptive nonadditive bid-price controls and showed that they performed better than the traditional ones. For a detailed review of other earlier work (prior to 2004) on network RM see the book by Talluri and van Ryzin (2004). Some recent work has focused on dynamic bid-prices and highlighted connections between bid-prices and the approximate dynamic programming approach to network RM through linear programming (see Adelman (2007) and Farias and Van Roy (2006) ). Meissner and Strauss (2008) apply a linear programming-based approximation methodology to find dynamic inventory-sensitive bid prices for network RM in the presence of consumer choice. Simulation-based methods (stochastic gradients) have also been used to optimize bid-price controls (see Topaloglu (2008) ) as well as virtual nesting controls (see van Ryzin and Vulcano (2008) ). We employ a combination of these latest ideas for cargo RM, and discuss the refinements necessary for LP-based computation of dynamic bid prices. Kasilingam (1996) explores similarities and differences between passenger and cargo capacity management in the airline context and discusses potential modelling approaches to the problem. Thorough coverage of the details of airline cargo management practices and proposals for RM systems can be found in Blomeyer (2006) . Several recent papers explore a single-leg version of the problem. Huang and Hsu (2005) tackle the problem of uncertain cargo capacity in one dimension, and Luo et al. (2009) and Moussawi and Cakanyildirim (2005) study two-dimensional cargo overbooking models. The Luo et al. article considers a cost minimization objective and shows that the overbooking limit of the one-dimensional version of the problem is replaced by a curve encircling the acceptance region in the 2-dimensional version. Moussawi and Cakanyildirim consider a profit optimization objective and both aggregate and detailed formulations. The aggregate formulation serves as a lower bound, and the article also provides an upper bound. Popescu et al. (2006) provide a novel show-up rate estimation method for cargo overbooking and test it on real data. Amaruchkul et al. ( 2007) consider a cargo booking problem in which the state of the system is represented by the vector of numbers of accepted shipments classified by their type (an approach also taken in the current paper). The exact weights and volumes of shipments are unknown at the time of booking but their distributions are determined by the shipment type. The article develops a heuristic based on decomposition of the problem into independent weight and volume components. Several upper bounds and heuristics are compared. A continuous-time formulation more appropriate for the ocean transport setting is studied in Xiao and Yang (2006). The network version of the cargo problem has received much less attention. Sandhu and Klabjan (2006) consider an integer programming formulation for the fleeting problem that includes an origin-destination bid-price booking control component for air cargo on a network. However, the bid-price controls discussed in the paper are static and ignore the stochastic aspect of the booking problem, and the performance of the resulting booking controls is not evaluated. Pak and Dekker (2005) discuss a bid-price control policy for the case of two-dimensional capacity on each flight. Shipments are considered individually, the weight and volume requirements are known exactly, and each booking request explicitly specifies an itinerary that the shipment has to follow. The complexity of the algorithm proposed by the authors grows exponentially with the number of flights. Popescu (2006) also considers booking requests in the form of explicit itineraries rather than origin-destination (OD) pairs. A distinctive feature of this work is a decomposition of the problem Levina, Levin, McGill, and Nediak: Network Cargo Capacity Management Article forthcoming in Operations Research; manuscript no. 3 into small and large cargo capacity allocation subproblems. In the "large cargo" component, each shipment is represented by its weight and volume individually in the state description. The main contributions of the current paper follow. 1. We propose an operational stochastic dynamic booking/shipping control model for air cargo capacity management on a network. 2. The model captures several sources of uncertainty inherent to air cargo: (a) random booking arrival process, (b) uncertain capacity of flights, and (c) capacity requirements of shipments which are uncertain at the time of booking. 3. The model structure explicitly captures the periodic nature of the flight schedule. This distinguishes it from existing stochastic dynamic models in network RM. 4. We develop a solution methodology that employs dynamic capacity-dependent controls and a linear programming-based approximation procedure. 5. The proposed solution procedure for approximating this linear program is flexible and can include information about past booking/shipping policies in the solution process. In the next section, we introduce the model formulation. In §3, we discuss the challenges posed by the model, the solution approaches used to tackle them, and the structural properties of the resulting policies. We provide numerical illustrations of this solution methodology in §4, and conclude in §5. Levina, Levin, McGill, and Nediak: Network Cargo Capacity Management Article forthcoming in Operations Research; manuscript no. 23 S. Luo, M. Cakanyildirim, and R. Kasilingam. Two-dimensional cargo overbooking models. EJOR, 197(3): 862-883, 2009. P. Marbach, O. Mihatsch, and J. Tsitsiklis. Call admission control and routing in integrated service networks using neuro-dynamic programming.