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.

Amid the rapid growth of online retail, last-mile delivery faces significant challenges, including the cost-effective delivery of goods to all delivery locations. Our work contributes to this stream by applying dynamic pricing techniques to effectively model the possible involvement of the crowd in fulfilling delivery tasks. The use of occasional drivers (ODs) as a viable, cost-effective alternative to traditional dedicated drivers (DDs) prompts the necessity to focus on the inherent challenge posed by the uncertainty of ODs’ arrival times and willingness to perform deliveries. We introduce a dynamic programming framework that offers individualized bundles of a delivery task and compensation to ODs as they arrive. This model, akin to a reversed form of dynamic pricing, accounts for ODs’ decision-making by treating their acceptance thresholds as a random variable. Therefore, our model addresses the dynamic and stochastic nature of OD availability and decision-making. We analytically solve the stage-wise optimization problem, outline inherent challenges such as the curses of dimensionality, and present structural properties. Tailored to meet these challenges, our approximation methods aim to accurately determine avoided costs, which are a key factor in calculating optimal compensation. Our simulation study reveals that the savings generated by involving ODs in deliveries can be significantly increased through our individualized dynamic compensation policy. This approach not only excels in generating savings for the firm but also provides a utility surplus for ODs. Additionally, we demonstrate the applicability of our approach to scenarios with time windows and illustrate the trade-off that arises from time window partitioning.

Dynamic Pricing, enabled by technological developments, is gaining more importance in fields beyond the airline industry, including retail, where neglecting multiunit demand leads to suboptimal prices and lost revenues. In these fields, nonlinear pricing is a common static pricing strategy that explicitly takes multiunit demand into account but lacks the possibility to dynamically adapt prices. In this paper, we bring the strengths of both pricing strategies together by combining them to multiunit dynamic pricing. We formulate the corresponding stage-wise optimization problem. To account for customers' preferences regarding batch size, we adapt an adequate customer choice model based on (random) willingness-to-pay. The willingness-to-pay is defined by a combination of customer’s attraction to and consumption of the product. These two aspects of customers’ preferences are private information, but the firm may have (partial) access to the information of the current customer. The firm is monopolistic and can price-discriminate between different order sizes by quoting nonlinear batch prices. This work investigates three cases of what information is observable: attraction to the product, consumption of the product, or both. We solve the resulting optimization model analytically and derive closed-form expressions of the optimal solution in two of the cases. Moreover, we proof the desirable monotonicity in time and capacity is still intact. Building on this monotonicity, we show dynamics of the optimal pricing policy. Finally, we examine the value of information in a numerical study to gain managerial insights regarding the importance of knowing customers’ preferences.