A Markov Chain for Rental Systems with Time-Dependency and Substitution of Assets

Society currently experiences a growing interest in renting products instead of exclusively owning them. Simultaneously, the increasing availability of data allows rental companies to analyze their operations more effectively.As a result, we aim to support rental companies by providing a method for evaluating and optimizing the occupancy of their assets. Our approach accounts for rental systems of time-dependent customer behavior, where customers make substitute rentals during shortages. Specifically, we provide a continuous-time Markov chain assuming arrivals occur according to Poisson processes and rentals follow phase-type distributed rental times. We further provide a heuristic optimization algorithm for minimizing the inventory constrained by an upper shortage probability. We tested our approach in several numerical experiments and an application to a company case. The experiments indicated that our model adequately reflects the occupancy distribution of assets and minimizes the capacities to near-optimality. The company case resulted in a reduction of the inventory while simultaneously reducing the maximum shortage probability