Appointment scheduling has many applications (e.g., surgery scheduling, airport gate scheduling, container vessel dockings and radiation therapy bookings) and it has a direct and significant operational and economic impact. For example, in healthcare, surgical departments are one of the main drivers of hospital costs and revenue, and appointment scheduling is used to book surgeries. Effective scheduling not only enables patients' timely access to care but also enables more efficient operations. This becomes especially important as healthcare costs and demand are on the rise in many countries. We study appointment scheduling where there are jobs (e.g., patients, container vessels, airplanes) with random processing durations, an expensive processor (e.g., a doctor, dock crane, airport gate) and significant costs for processor idle time, processor overtime, and job waiting. The goal is to determine an appointment schedule that minimizes a measure of total costs as the objective. The appointment scheduling problem has been well studied in the literature with the expected cost objective. Almost all papers in the literature on appointment scheduling use the expected cost criterion, which may not be suitable when risk measures and/or service levels are considered. In this paper, we study this problem with a new objective: minimization of any quantile of the cost distribution, e.g., median, 90th percentile. We obtain theoretical results for some special cases and develop an algorithm for the general case. Our algorithm does not require a specific distribution assumption and can work directly with data samples. We present numerical examples with real data on surgeries. Our results show that allocated schedules based on the quantile objective with identical jobs are different than the ones generated by the expected cost objective and they do not show the well-known dome-shaped pattern but a semi-dome-shaped pattern which first increases (like the dome-shaped pattern) but then its decrease is not monotone (unlike the dome-shaped pattern). To the best of our knowledge, this is the first paper on appointment scheduling problem with the objective of the quantile function minimization.
