We analyze a model of airline overbooking in which customer cancellations and no-shows are explicitly considered. We model the reservations process as a continuous-time birth-and-death process with rewards representing the fares received and refunds paid and a terminal-value function representing the bumping penalty. The airline controls the reservation acceptance (birth) rate by declining reservation requests. Assuming that the fares and refunds are piecewise-constant functions of the time to flight, we demonstrate that a piecewise-constant booking-limit policy is optimal, i.e., at all times, the airline accepts reservation requests up to a booking limit if the current number of reservations is less than that booking limit, and declines reservation requests otherwise. When the fare is constant over time or falls toward flight time, the optimal booking limit falls toward flight-time.
Consider a multiperiod airline overbooking problem that relates to a single-leg flight and a single service class. Passengers may cancel their reservations at any time, including being no-shows at flight-time. At that time, the airline bumps passengers in excess of flight capacity and pays a penalty for so doing. We give conditions on the fares, refunds, and distributions of passenger demand for reservations and cancellations in each period, and on the bumping penalty function, that ensure that a booking-limit policy is optimal, i.e., in each period the airline accepts reservation requests up to a booking limit if the number of initial reservations is less than that booking limit, and declines reservation requests otherwise. The optimal booking limits are easily computed. We give conditions under which the optimal booking limits are monotone in the time to flight departure. The model is applied to the discount allocation problem in which lower fare classes book prior to higher fare classes. E ffective yield management can save airlines hundreds of millions of dollars each year (Smith et al. 1992). Two of the more important among the airline yield management problems are the overbooking and the seat-or discount-allocation problems. In this paper we introduce two closely related models that address the airline overbooking problem for a single-leg flight with a single fare class. The second model can also be applied to the seatallocation problem in which the airline determines when to refuse reservation requests from lower fare class customers in order to protect seats for subsequent requests from higher fare classes.The two overbooking models presented in this paper are dynamic in nature and include customer reservation requests, cancellations, and no-shows explicitly. They are dynamic because in determining the booking rules the models consider not only the reservations currently on hand and the likelihood of such reservations canceling prior to flight time or being a no-show at flight time, but also the possibility of future customer reservation requests and subsequent cancellations.While dynamic models of airline (and the related hotel) overbooking have been presented previously, in this paper we extend the earlier work by developing the theory of the structure of the optimal solution. The major advance is an investigation of the circumstances under which a bookinglimit policy is optimal, i.e., in each period the airline accepts reservation requests up to a booking limit if the number of initial reservations is less than that booking limit, and declines reservation requests otherwise. By applying the theory of total positivity we are able to give sufficient conditions for the optimality of booking limit policies. These conditions are less restrictive than those imposed by previous researchers, and the results obtained are more powerful. For example, we allow the distribution of requests for reservations to depend on the number of current bookings, and we allow for more general cancellation distributions than sim...
Online advertising is a large and rapidly growing business. The major players in the space, namely advertisers, publishers, and ad exchanges, are developing increasingly sophisticated systems, methods and tools to facilitate, manage, optimize and report on the performance of online advertising marketplaces and campaigns. Developing solutions that are both mathematically sound and practical draws on techniques from a variety of disciplines including machine learning, stochastic optimal control, information retrieval, data mining, natural language processing, and econometrics. In this paper, we provide an overview of the online advertising space, and identify, frame, and describe solution approaches to some of the major computational challenges in the space. We describe specific examples from industry applications, including ad inventory auctions, bidding and allocation strategies for ad inventory, inventory targeting, banner and landing page optimization, and performance estimation.
Consider a multi‐period multi‐fare class airline overbooking problem that relates to a single‐leg flight. Passengers may cancel their reservations at any time, including being no‐shows at flight‐time. Canceling passengers receive a refund that depends on their fare class, e.g., supersaver, coach, etc. At flight‐time, the airline bumps passengers in excess of flight capacity and pays a penalty for so doing. A continuous state‐space dynamic programming model is developed in which the state is the numbers of reservations currently on hand in each fare class. In each period, reservation requests occur in only one fare class and the fraction of reservations canceling in each class is independent of the number of reservations therein. A booking‐limit policy is optimal, i.e., in each period the airline accepts reservation requests up to a booking limit if the number of initial reservations in the fare class is less than the booking limit, and declines reservation requests otherwise. The booking limits for each class depend on the numbers of reservations in the other classes. When there are two fare classes the optimal booking limits in each class decrease with the number of reservations in the other class. © 1996 John Wiley & Sons, Inc.
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