Designing variable-sized block appointment system under time-varying no-shows

Abstract

We consider the tactical problem of optimizing the capacity of a block appointment system with time-varying no-show and random service duration. The problem characteristics are motivated by a real-life case study of two outpatient specialty clinics in the US, where time-of-day variation in appointment attendance was observed. We aim to determine the block size (patients to be assigned in each time period) such that the weighted sum of block-wise waiting- and idle-times (or total cost) are minimized. First, we consider optimizing the capacity of a single block appointment system (SBAS) as a special case of the inverse newsvendor problem, and develop an analytical closed-form solution under the assumption of normally distributed service time. Also, a stochastic integer programming (SIP) model is developed to solve SBAS for any service time distribution. Subsequently, the SIP model is extended to determine the block size of the variable-sized multi-block appointment system (VSMBAS) by treating it as a sequential inverse newsvendor problem. Owing to the computational complexity of the SIP, we employ sample average approximation to estimate the expected total cost. Numerical studies considered several realistic clinic settings, and the results demonstrated that integrating time-varying no-shows for block size determination will considerably improve schedule efficiency as opposed to ignoring it. We also found the cost ratios (waiting-time to idle-time penalty), service time variation, and no-show pattern have a substantial influence on the block size of VSMBAS. Finally, we provide several practical implications based on our analysis.