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

## TL;DR

This paper extends dynamic programming methods for revenue management in attended home delivery by establishing the concavity and fixed point properties of the value function, enabling scalable approximation techniques.

## Contribution

It introduces a novel concave extension of the value function in dynamic programming for revenue management, facilitating scalable approximate solutions.

## Key findings

- The value function admits a continuous, concave extension under certain conditions.
- The dynamic programming operator has a unique fixed point.
- Numerical examples illustrate the practical applicability of the theoretical results.

## Abstract

We study the approximate 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, convex optimisation and discrete convex analysis to show that the underlying dynamic programming operator has a unique fixed point. Moreover, we also show that -- under certain assumptions -- for all time steps in the dynamic program, the value function admits a continuous extension, which is a finite-valued, concave function of its state variables. This result opens the road for achieving scalable implementations of the proposed formulation, as it allows making informed choices of basis functions in an approximate dynamic programming context. We illustrate our findings using a simple numerical example and conclude with suggestions on how our results can be exploited in future work to obtain closer approximations of the value function.