Heuristic solutions for nonlinear dynamic pricing in the presence of multiunit demand and two-dimensional customer heterogeneity

In this paper, we introduce a nonlinear dynamic pricing model in the presence of multiunit demand, enabling firms to quote separate prices for each batch size. This approach diverges from traditional models by accounting for two-dimensional customer heterogeneity in product attraction and batch size preference, each modeled by separate random variables in the calculation of customers’ willingness-to-pay. The underlying customer choice model results in a complex formulation of purchase probabilities, necessitating considerable effort for refinements to derive a manageable expression. We present optimality conditions for the state-wise optimization problem and introduce a modified formulation with reduced complexity that serves as an upper bound. We also prove that under specific conditions, the optimal solution to the modified model is optimal for the original problem. In our numerical study, these conditions were consistently met, offering a practical alternative for determining optimal prices. To address the computational challenges of solving the problem to optimality, we develop three efficient heuristics with significantly reduced runtimes. Benchmarking these heuristics against the optimal solution and other mechanisms demonstrates their near-optimal performance. We also evaluate the revenue potential of nonlinear, piecewise linear, and linear pricing schemes, providing firms with tools to weigh revenue maximization against pricing simplicity to inform strategic decisions. Notably, our analysis highlights the strong performance of piecewise linear pricing, offering a practical and easy-to-communicate alternative to full nonlinear pricing while achieving remarkably high revenues.