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. Furthermore, we derive results on the monotonicity of prices with respect to the number of orders placed in our setting. These results open 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 on a low-dimensional and an industry-sized numerical example using real-world data, for which we derive an approximately optimal pricing policy based on our theoretical results.

中文翻译：

---

优化交货时间段定价的动态编程框架

摘要 我们研究了在有人值守的送货上门的情况下收入管理的动态规划方法。我们利用马尔可夫决策问题的动态规划理论的结果来表明基础贝尔曼算子具有唯一的不动点。然后我们为得到的不动点提供了一个封闭形式的表达式，并表明它允许一个自然的解释。此外，我们还表明 - 在某些技术假设下 - 具有离散域和连续余域的价值函数在每个时间步允许连续扩展，这是其状态变量的有限值凹函数。此外，我们根据我们设置的订单数量得出价格单调性的结果。这些结果为实现所提出的公式的可扩展实现开辟了道路，因为它允许在近似动态规划上下文中对基函数进行明智的选择。我们使用真实世界的数据在一个低维和行业规模的数值示例上说明了我们的发现，我们根据我们的理论结果推导出了近似最优的定价策略。