On planning the optimal supply of hospital beds

Due to the highly stochastic nature of admission demand, hospitals must have a reservation capacity acting as a buffer. This study suggests how to plan optimal bed capacity, in order to minimize both the holding cost caused by empty beds, and the shortage cost, stemming from rejecting or queuing patients. These two objectives are conflictual, because holding cost increases whereas shortage cost decreases with respect to the available beds. A second best solution is proposed: holding cost is minimized subject to an upper bound for rejection and for average waiting periods, that is imposing a constraint on shortage cost. The utilized queuing model, due to Fandel–Schmidt, is a composed one: M/M/s/s and M/M/s for urgent and non-urgent patients, respectively, including also priority for emergency cases. The model yields the simultaneous determination of the optimal amount of beds to reserve for both kinds of patients and of the emergency beds. The application to 51 departments of an Italian hospital allows a consistent bed saving, through a fine tuning of the occupancy rate following the characteristics of the arrival process and of the length of stay.