We analyze joint overbooking and capacity control decisions in the presence of class-dependent no-shows. Assuming immediate appointment requests attend with certainty, we derive a simple
expression for the optimal booking limit for immediate requests as a function of previously booked appointments which may no-show. Due to no-show heterogeneity, the prot function may not be unimodal in the number of advance appointments accepted, making it necessary to use computationally intensive search methods to derive the optimal advance booking limit. We develop upper and lower bounds which greatly reduce the size of the search space. We perform sensitivity analysis to investigate how model parameters aect the optimal expected prot and policy. We compare ten policies developed from this paper, previous models from the literature, and popular outpatient appointment scheduling methods. The policies we develop perform extremely well compared to the optimal.
Our results show that expected prots are improved by allocating total capacity and over- booking to distinct demand classes with dierent no-show rates, even with constant revenue per patient. Compared to a rst-come-rst-serve (FCFS) allocation, expected prots increase 12.3% on average, across a numerical study of 198 scenarios, when incorporating capacity con- trol with no overbooking. Expected prots increase 12.6% using a single overbooking limit and maintaining a FCFS allocation. When jointly incorporating capacity control and overbooking, the optimal policy improves expected prots 17.8% over FCFS. On average, expected prots for the lower and upper bounds perform within 2.24% and 1.58% of optimal, respectively. We also develop a Marginal Revenue (MR) policy which quickly and accurately approximates the optimal solution with mean and median percent errors less than 0.3%.
The implications of our results demonstrate how appointment allocation and overbooking decisions should consider dierences in patient attendance rates, forecasts of immediate appoint- ment requests, and revenues relative to overbooking costs. On average, across our numerical study the optimal policy protects 58.3% of its appointment capacity for immediate requests with
a range of 12.5% to 100%. In 147 of 198 scenarios, the optimal policy allows future appoint- ments to overbook. On average, the optimal policy sets its advance booking limit at 56.19% of capacity with an average overbooking pad of 14.6% of capacity. On average, 25.8 total ap- pointments (107.5% of capacity) are booked (11.1 advance and 14.7 immediate). When the advanced booking limit is not reached, the clinic increases the number of immediate requests it is willing to accept, but at less than a one-to-one ratio because attendance rates are higher for immediate requests. The optimal policy generally books more advance appointments as revenues increase relative to costs and fewer advance appointments as advance attendance rate or demand for immediate appointments increases. Given the variability of optimal policy values between scenarios, we show the value of using bounds and approximations provided in this work to determine the best allocation for the setting, instead of naive FCFS allocations or crude guidelines from practice - e.g. 25-35% future appointments and 75% immediate appointments (Qu et al. (2007), Green et al. (2007)).
A pure open access policy (POA), which reserves 100% of appointment capacity for immedi- ate requests, does not perform well on average with expected prots 23.0% below optimal. POA can be optimal if the advance appointment attendance rate is small relative to the immediate demand distribution and the marginal overtime cost is not too far below the marginal revenue. We develop policies, such as MR policy, which generalize pure open access and outperform it under more general assumptions.
Our sensitivity analysis shows that clinics can achieve higher expected prots by making eorts to impact important model parameters. Optimal expected prots increase with the appointment attendance rate, the mean demand for immediate requests, and the p/h ratio.
ment reminders, or by increasing demand for immediate appointments by encouraging patients to schedule closer to the desired appointment date.
Extensions to this work could incorporate features such as cancellations, rejection costs, multiple booking attempts, and dynamic models. Cancellations have been incorporated in pre- vious revenue management models assuming static no-show rates and memory-less distribution of time until cancellation. If heterogeneity in attendance rates is partly due to cancellations, in- tuition from our model suggests clinics would increase booking limits, since recourse is available for some appointment waste. Often no explicit rejection costs are incurred in reality, but these parameters could model the eects of competition or increased appointment delay for patients. Future work could also assume rejected requests make additional booking attempts at a later stage or appointment date. Our model can also be extended into a dynamic model to allow for simultaneous booking of dierent patient classes, such as new or return patients. Empirical studies have investigated patient attributes that determine probability of a no-show, but clinics may nd it ethically inappropriate to use personal attributes for appointment allocation. Addi- tionally, future research could extend our model to more than two booking stages, but no-show heterogeneity requires tracking reservations from each class; leading to exponential growth in the state space.
As clinic management technology develops, more clinics gain the ability to implement com- plex appointment allocation policies. While recent work in operations sciences has developed eective policies, there is still opportunity for improvement translating research into practice. Managers and researchers should continue to collaborate on developing eective policies and methods for using real-time data to drive decisions.
Appendix A
Competition
A.1 Proofs of Key Results