Optimal sequencing using a scheduling heuristic

The problem of finding the optimal appointment schedule has received significant consideration in the literature on healthcare optimization. Due to randomness, determining when clients should be scheduled in order to strike a balance in terms of idle times, waiting times, and overtime is a non-trivial task. Furthermore, it is often solely studied in the dimension of scheduling, whereas the sequencing decision in which heterogeneous clients (e.g., new and return clients) are to arrive is at least of equal importance. The approach presented integrates an approximation method with a scheduling heuristic that tackles this joint objective of sequencing and scheduling.

A moment-iteration method is employed to construct appointment schedules. This method permits analytical solutions and thereby is incredibly fast to solve and optimize. Next, a sequential optimization approach is applied which aligns well with the moment-iteration method and permits per-client control of costs. Moreover, within this approach, the classical surgery scheduling problem of minimizing earliness and tardiness by determining due dates is equivalent to the appointment scheduling problem, i.e., in a sequentially optimal schedule the inter-arrival time equals the previous client’s due date.

Finally, for the case of clients having heterogeneous service-time distributions, rigorous sequencing rules are derived in terms of the first two moments. The focus lies on non-identical exponential service times and log-normal service times, as commonly found in healthcare modelling. The smallest-mean/variance-first rule is sequentially optimal. For cases in which the mean and variance move in opposite directions contour plots are provided to aid practitioners in sequencing clients.