Staffing and capacity planning for vaccination centers under uncertainty: A fuzzy queueing network optimization approach

Large-scale vaccination campaigns require operational configurations capable of maintaining service continuity under uncertain demand, variable service times, and limited staffing availability. This study proposes a fuzzy queueing network optimization framework to support staffing and capacity planning in mass vaccination centers. Unlike deterministic or purely stochastic approaches, the proposed model represents uncertain operational parameters through triangular fuzzy numbers, allowing expert-based estimates to be incorporated when historical data are scarce or incomplete. The fuzzy formulation is transformed into a nonlinear optimization problem using an α-cut methodology. Two operational objectives are analyzed: minimizing the total number of servers and maximizing the patient scheduling rate. The framework is evaluated using a three-station Jackson-type vaccination network representing verification, vaccination, and final registration stages. The results show that the proposed approach identifies bounded staffing intervals rather than single-point solutions. In the base configuration, the required number of servers ranges from 15 to 21 under maximum uncertainty and converges to 18 as uncertainty decreases. Similarly, the feasible patient scheduling rate ranges from 0.8 to 2.5 patients per minute and converges to approximately 1.6 patients per minute. The results also reveal nonlinear relationships among demand, staffing levels, waiting times, and queue lengths, indicating that congestion risk may arise not only under extreme demand but also under intermediate uncertainty conditions. Also, we adapt our proposal to analyze mass vaccination centers considering previous studies. These findings demonstrate that fuzzy queueing optimization can provide interpretable and actionable decision-support information for resilient vaccination-center planning.