Abstract
We are considering the problem of scheduling a given number of outpatients to a medical service facility with two resources servicing outpatients, inpatients, and emergency patients. Each of the three patient classes has associated classspecific arrival processes and costfigures. The objective is to maximize the total expected reward which is made of revenues for served patients, costs for letting patients wait, and costs for denial of service. For this problem we propose a generalization of the wellknown Bailey–Welch rule as well as a neighborhood search heuristic. We analyze the impact of different problem parameters on the total reward and the structure of the derived appointment schedules and address the question of the number of outpatients to be scheduled. The results show that the generalized Bailey–Welch rule performs astonishingly well over a wide range of problem parameters.
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A Appendix
A Appendix
1.1 A.1 Bounds for the number of neighborhood schedules
We assume N _{ sched } > 2 because otherwise the set of neighborhood schedules would be empty due to Eqs. 1–2. To specify an upper bound, one has to distinguish two cases:

1.
N _{ sched } ≥ N + 2: Let the schedule be such that a _{1} ≥ 3 and a _{ i } ≥ 1 for i = 2, ..., N. In this case to construct a neighborhood schedule, every integer a _{ i } with i = 1,..., N can be chosen to be reduced by one. Then, to derive a neighborhood schedule, every integer a _{ j } with i ≠ j can be chosen to be incremented by one. Applying this procedure, N ·(N − 1) different neighborhood schedules can be found.

2.
N _{ sched } < N + 2: Without loss of generality we assume that a _{1} = 3 and a _{ i } = 1 for i = 2, ..., N _{ sched } − 2. Every integer a _{ i } with i = 1, ..., N _{ sched } − 2 can be chosen to be reduced by one. Note that integers a _{ i } with i = N _{ sched } − 1,..., N can not be chosen. Here we have a _{ i } = 0. Again, to derive a neighborhood schedule, every integer a _{ j } with i ≠ j can be chosen to be incremented by one. Applying this procedure, (N _{ sched } − 2) ·(N − 1) different neighborhood schedules can be found.
A lower bound on the number of neighborhood schedules can be calculated in the following way. Since we assumed N _{ sched } > 2, there is at least one integer a _{ i } which can be chosen to be reduced by one. And again, every integer a _{ j } with i ≠ j can be chosen to be incremented by one. Hence, the lower bound on the number of neighborhood schedules is given by N − 1.
1.2 A.2 System utilization
For a given appointment schedule a, we calculate the expected system utilization as follows: First the expected number of scheduled outpatients E(s _{ i }), non–scheduled inpatients E(n _{ i }) and emergency patients E(e _{ i }) arriving in the time interval (t _{ i − 1}, t _{ i }] is calculated for i = 1, ..., N. To the sum E(z _{ i }) = E(n _{ i }) + E(s _{ i }) + E(e _{ i }) of the expected number of patients arrived in (t _{ i − 1},t _{ i }] the number of patients not served at time t _{ i − 1} (called the remainder r _{ i − 1}) is added. Then the remainder r _{ i } for time t _{ i } is updated to r _{ i } = max {0, E(z _{ i }) + r _{ i − 1} − 2}. The expected utilization E(u _{ i }) for slot i is calculated to
Finally, the expected utilization of the system is calculated to
Table 18 gives an example for the calculation of the expected utilization.
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Sickinger, S., Kolisch, R. The performance of a generalized Bailey–Welch rule for outpatient appointment scheduling under inpatient and emergency demand. Health Care Manag Sci 12, 408–419 (2009). https://doi.org/10.1007/s1072900990987
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DOI: https://doi.org/10.1007/s1072900990987