A stochastic approximation approach for managing appointments in the presence of unpunctual patients, multiple servers and no-shows

Patient unpunctuality significantly disrupts the operations of healthcare facilities, reduces provider productivity, and increases healthcare costs. To alleviate the negative impact of unpunctual patients, this study addresses the appointment scheduling (AS) in the simultaneous presence of unpunctual patients, multiple servers, and no-shows. To determine the appointment schedule, we propose a two-stage stochastic mixed-integer programming model to minimise the total cost incurred by patient waiting and clinic overtime. It becomes challenging for a standard solver to solve this model due to the dynamic patient-to-server assignment decisions that are proactively anticipated in the determination of appointment times. To deal with this problem, a stochastic approximation algorithm is proposed under unbiased gradient estimators. The effectiveness and efficiency of this algorithm are validated in extensive numerical experiments that compare it with Benders decomposition and a heuristic algorithm. Further, the features of the optimal appointment schedule are analysed: (i) the shape of the appointment intervals relies on the number of servers; (ii) the length of intervals is sensitive to no-shows; (iii) the initial block size is greatly affected by patient unpunctuality. Managerial insights are also provided for hospital managers to schedule unpunctual patients in practice.