Dynamic Programming for Optimal Delivery Time Slot Pricing

TL;DR

This paper applies dynamic programming theory to optimize delivery time slot pricing, providing a fixed point solution and continuous extension of the value function, facilitating scalable approximations in revenue management.

Contribution

It introduces a closed-form fixed point solution for the Bellman operator and demonstrates the continuous, concave extension of the value function under certain conditions.

Findings

Unique fixed point of the Bellman operator established

Closed-form expression for the fixed point derived

Value function admits a continuous, concave extension

Abstract

We study the dynamic programming approach to revenue management in the context of attended home delivery. We draw on results from dynamic programming theory for Markov decision problems to show that the underlying Bellman operator has a unique fixed point. We then provide a closed-form expression for the resulting fixed point and show that it admits a natural interpretation. Moreover, we also show that -- under certain technical assumptions -- the value function, which has a discrete domain and a continuous codomain, admits a continuous extension, which is a finite-valued, concave function of its state variables, at every time step. This result opens the road for achieving scalable implementations of the proposed formulation in future work, as it allows making informed choices of basis functions in an approximate dynamic programming context. We illustrate our findings on a simple numerical example and provide suggestions on how our results can be exploited to obtain closer approximations of the exact value function.