Impact of productivity on cross-training configurations and optimal staffing decisions in hospitals

Cross‐training of nursing staff has been used in hospitals to reduce labor cost, provide scheduling flexibility, and meet patient demand effectively. However, cross‐trained nurses may not be as productive as regular nurses in carrying out their tasks because of a new work environment and unfamiliar protocols in the new unit. This leads to the research question: What is the impact of productivity on optimal staffing decisions (both regular and cross‐trained) in a two‐unit and multi‐unit system. We investigate the effect of mean demand, cross‐training cost, contract nurse cost, and productivity, on a two‐unit, full‐flexibility configuration and a three‐unit, partial flexibility and chaining (minimal complete chain) configurations under centralized and decentralized decision making. Under centralized decision making, the optimal staffing and cross‐training levels are determined simultaneously, while under decentralized decision making, the optimal staffing levels are determined without any knowledge of future cross‐training programs. We use two‐stage stochastic programming to derive closed form equations and determine the optimal number of cross‐trained nurses for two units facing stochastic demand following general, continuous distributions. We find that there exists a productivity level (threshold) beyond which the optimal number of cross‐trained nurses declines, as fewer cross‐trained nurses are sufficient to obtain the benefit of staffing flexibility. When we account for productivity variations, chaining configuration provides on average 1.20% cost savings over partial flexibility configuration, while centralized decision making averages 1.13% cost savings over decentralized decision making.