Optimizing Service Operations with Price- and Density-Dependent Demand: A Copula-Based Approach

Abstract

We study a service provider’s pricing and density decisions when customers are heterogeneous both in their valuation and in their sensitivity to crowd density, the latter originating from, e.g., safety concerns in a pandemic or a desire for privacy and exclusivity. We develop a novel copula-based framework to model such multidimensional preferences and their dependence structure. We analytically characterize the provider's optimal price and density limit. For all but severely density-sensitive customer populations, (i) the provider optimally serves all segments of the market and activates the price (rather than the density) to regulate the demand and (ii) as customers' valuations and density tolerances become more positively dependent, the provider earns higher revenue and optimally increases its service density. By contrast, the optimal price may be decreasing, increasing, or decreasing followed by increasing. On the other hand, with severely density-sensitive customers, it may be optimal to partially cover the market and to activate density to regulate the demand. Finally, a calibrated numerical study based on a choice experiment on train travel preferences during the COVID-19 pandemic validates our model assumption and results. Our findings offer prescriptive guidelines for service providers in the presence of density-sensitive demand (especially in the aftermath of a pandemic). We also put forth an explanation for the stringent gatekeeping practices of exclusive private clubs, and suggest a new lever to fight the ongoing inflation in the service industry, namely by reshaping the dependence between density sensitivity and valuation. Notably, service providers’ strategy to regain customers after the pandemic could conflict with government efforts to fight inflation.

Keywords: service operations, revenue management, copulas, pandemic, exclusivity/privacy