Paper: Robust Dynamic Pricing with Two Substitutable Products | Manufacturing & Service Operations Management

We consider a practical dynamic pricing problem with two substitutable products involving a number of business rules commonly seen in practice. Demand substitution exists between the two products (interproduct substitution) and may also exist across different time periods (intertemporal substitution). However, there is limited demand information such that the underlying probability distributions of the demand cannot be characterized precisely. We use an interval to represent, respectively, the demand for each individual product in each period, the aggregate demand for the two products in each period, and the total aggregate demand for the two products across multiple time periods. We propose a robust optimization model for this problem to maximize the worst-case total revenue. For the problem with interproduct demand substitution only, we develop a dynamic programming algorithm and show that the search spaces in the DP can be reduced greatly, which enables the algorithm to generate optimal solutions in a reasonable amount of time. For the problem with both interproduct and intertemporal demand substitutions, we develop a more complex dynamic programming algorithm and design a fully polynomial time approximation scheme that guarantees a proven, near optimal solution in a manageable computation time for practically sized problems. Our computational results show that, compared to a risk-neutral approach, our robust optimization approach can decrease the variance of the revenue at a small expense of the average revenue. We also generate a number of managerial insights: (i) none of the key structural properties commonly studied in the pricing literature hold for our problem; (ii) the revenue impact of ignoring intertemporal demand substitution when such substitution exists can be quite significant; and (iii) under- or overestimating the bounds of the demand intervals or imposing moderate business rules leads to relatively small revenue loss, typically less than 3%.