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We study the approximate dynamic programming approach to revenue managementin the context of attended home delivery. We draw on results from dynamicprogramming theory for Markov decision problems, convex optimisation anddiscrete convex analysis to show that the underlying dynamic programmingoperator has a unique fixed point. Moreover, we also show that -- under certainassumptions -- for all time steps in the dynamic program, the value functionadmits a continuous extension, which is a finite-valued, concave function ofits state variables. This result opens the road for achieving scalableimplementations of the proposed formulation, as it allows making informedchoices of basis functions in an approximate dynamic programming context. Weillustrate our findings using a simple numerical example and conclude withsuggestions on how our results can be exploited in future work to obtain closerapproximations of the value function.

A Concave Value Function Extension for the Dynamic Programming Approach to Revenue Management in Attended Home Delivery