Single-Product Pricing via Robust Optimization

We present a robust optimization approach to the problem of pricing a capacitated product over a finite time horizon in the presence of demand uncertainty. This technique does not require the knowledge of the underlying probability distributions, which in practice are difficult to estimate accurately, and instead models random variables as uncertain parameters belonging to a polyhedral uncertainty set. A novelty of the proposed approach is that, instead of imposing an upper bound on the number of uncertain parameters that can reach their worst-case value, which is known as a budget-of-uncertainty constraint and has received much attention in the robust optimization literature, we introduce a budget of resource consumption by the uncertainty. This budget limits the amount of the resource that can be used by the random part of the cumulative demand, and allows us to derive key insights on the structure of the optimal solution for a broad class of nominal demand functions. We establish the existence of a reference price for the product and show that this new parameter plays a crucial role in understanding the impact of uncertainty on the optimal prices. In particular, it is not always optimal to decrease prices when demand is uncertain. Whether it is optimal or not will instead depend on whether the price at a given time period is above or below the reference price, and whether the maximal amount of uncertainty at that time exceeds a threshold. We compare the optimal solution in the cases of additive and multiplicative uncertainty and analyze the problem with linear nominal demand in detail. Numerical results are encouraging.