## Abstract

This study analyzes two decision levels in appointment system design in the context of clinics that face seasonal demand for scheduled and walk-in patients. The macro-level problem addresses *access rules* dealing with capacity allocation decisions in terms of how many slots to reserve for walk-ins and scheduled patients given fixed daily capacity for the clinic session. The micro-level problem addresses *scheduling rules* determining the specific time slots for scheduled arrivals. A fully-integrated simulation model is developed where daily demand actualized at the macro level becomes an input to the micro model that simulates the in-clinic dynamics, such as the arrivals of walk-ins and scheduled patients, as well as stochastic service times. The proposed integrated approach is shown to improve decision-making by considering patient lead times (i.e., indirect wait), direct wait times, and clinic overtime as relevant measures of performance. The traditional methods for evaluating appointment system performance are extended to incorporate multiple trade-offs. This allows combining both direct wait and indirect wait that are generally addressed separately due to time scale differences (minutes vs. days). The results confirm the benefits of addressing both decision levels in appointment system design simultaneously. We investigate how environmental factors affect the performance and the choice of appointment systems. The most critical environmental factors emerge as the demand load, seasonality level, and percentage of walk-ins, listed in the decreasing order of importance.

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## Acknowledgements

This research is funded by the Scientific and Technological Research Council of Turkey with TUBITAK 3501 Grant 109K451.

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## Appendices

### Appendix 1: Formulation of the demand process with seasonality and demand load effects

In this section we formulate the demand process for walk-ins and scheduled patients, both following the same demand pattern based on a multiplicative seasonality index *I*_{t} for day *t*. This demand process is used to generate calls and walk-ins in the simulation model.

Daily number of walks-ins and scheduled patients are modelled as Poisson processes with yearly average rates \(\lambda_{w}\) and \(\lambda_{s}\) respectively. Let *T* be the available capacity in number of slots per day, and *P*_{w} be the percentage of walk-ins in the total demand, i.e., \(P_{w} = \lambda_{w} /\left( {\lambda_{w} + \lambda_{s} } \right).\). The average utilization or demand load, \(\rho = \left( {\lambda_{w} + \lambda_{s} } \right)/T\) is assumed to be less than 1.

In the simulation experiments *T* is fixed. Environmental parameters are *P*_{w}, ρ and the seasonality level that determines the seasonality index *I*_{t}. The; impact of these are investigated with a full factorial experimental design. In the following, we derive the demand rates and reservation levels in terms of the environmental factors.

Using the definitions of \(\rho\) and \(P_{w}\) given above, the daily number of slots *T* can be written as

Using (6) and multiplying the average rates with the seasonality index, we can write the daily demand rate for walk-ins as:

Similarly, given the overall average call rate for scheduled appointments (λ_{s}), the average arrival rate of calls for day *t* is calculated as follows:

Our simulation model uses \(\uplambda_{w}^{t} \,{\text{and}}\,\uplambda_{s}^{t}\). for generating seasonal demand patterns for walk-ins and scheduled patients who call for appointments. However, our access rules define the reservation level on a daily basis based on \(\uplambda_{w}^{t}\), which is adjusted by coefficient γ. As a result, the daily reservation level for walk-ins is represented as follows:

Using Equations (6) and (9), the decision variable γ may be interpreted through its impact on the daily average booking limit *N*_{t}, represented as follows:

The benchmark yearly average booking limit *N* in the case when there is no seasonal adjustment can be found by setting \(I_{t} \gamma =\) 1. The second part of (10) represents the change in the booking limit as a result of adjustment for seasonality of walk-ins. When \(I_{t} \gamma > 1\), the daily booking limit *N*_{t} is less than the yearly average booking limit *N*, since \(\lambda_{w} \frac{{1 - I_{t} \varUpsilon }}{\rho } < 0\). When *P* high (i.e., \(\lambda_{w}\). is high) and seasonality level is “High”, \(\lambda_{w}\)*I*_{t} is expected to be very high for some days, and therefore, we expect the best γ to decrease in order to alleviate the change on the booking limit. Our numerical examples in Sect. 5.3 confirm this expectation. It is obseed that γ is lower in Env_1 (*DL* = 99%, *SEAS* = High, *P*_{w} = 80%) compared to Env_2 (*DL* = 9*SEAS* = Low, *P*_{w} = 40%), and similarly, in Env_3 compared to Env_4 for *DL* = 80% (See Table 5).

### Appendix 2: Extended efficient frontier analysis for trade-offs between wait time, lead time and overtime

This section extends the analysis in Sect. 5 by plotting results on the efficient frontier when all trade-offs are included. For this purpose, patient’s lead time and wait time measures are combined into a single measure for a given ω-ratio defined as *c*_{W}/*c*_{L}. This revises the original total cost equation *TC*_{Int} (Eq. 3) as follows:

Figure 6 presents the efficient frontiers for two values of ω = 1 and 5 for the “high” environment with *DL* = 99%, *SEAS* = H, *P*_{w} = 80% (Env_1 in Sect. 5). Performance of all appointment systems are tested, inluding 18 appointment systems (i.e., combinations of End, Even, EndEven rules with Seas_Adj with γ = 0.85-1.00 and No_SeasAdj). For a given ω, the final choice depends the cost ratio, β, as indicated by different slopes on the efficient frontiers. We note that choosing two cost ratios (ω and β), automatically sets the third one (α). For example, if ω = 5, β = 30, then α = 150.

From Fig. 6, for a given choice on ω, only a limited set of appointment systems remain on the efficient frontier, while the rest—including the benchmark policies of No_SeasAdj are inferior. The β-values indicate the range at which each appointment system becomes the best choice. When ω = 1, the decision-maker will choose SeasAdj_1.00_End if β ≥ 269, SeasAdj_1.00_EndEven if 95 ≤ β < 269, SeasAdj_0.95_EndEven if 26 ≤ β < 95, SeasAdj_0.95_Even if 17 ≤ β < 26, SeasAdj_0.90_Even if 5 ≤ β < 17, and SeasAdj_0.85_Even if 1 ≤ β < 5. Similarly, when ω = 5, the best choice is SeasAdj_1.00_End if β ≥ 269, SeasAdj_1.00_EndEven if 28 ≤ β < 269, SeasAdj_1.00_Even if 28 ≤ β < 7, and SeasAdj_0.95_Even if 1 ≤ β < 7. These results are parallel with those in Sect. 5.2 (see Fig. 3), where β-values are fixed at 1.5 and 30 for the patient versus physician-centered clinics.

The complete results on β-values calculated for ω = 0.5, 1, 2, 5 and 10 are included in Table 7. Although 18 appointment systems are included in the analysis, parallel to in Sect. 5, Table 7 only lists those that appear on the efficient frontiers as the best performing ones for the illustrated Env_1. From Table 7, as ω increases, smaller sets of appointment systems remain on the efficient frontiers. This is also observed in Fig. 6, where there are four best appointment systems for ω = 5, as opposed to seven for ω = 1. As ω increases, the frontiers are increasingly dominated by those appointment systems with larger coefficient γ (e.g., SeasAdj_1.00). These rules reserve more capacity for walk-ins, and therefore less capacity for scheduled patients, resulting in longer lead times. Thus they are more preferable when ω and (indirectly) α increase for a given β, indicating a lower preference for patient lead times compared to direct wait times and overtime.

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Cayirli, T., Dursun, P. & Gunes, E.D. An integrated analysis of capacity allocation and patient scheduling in presence of seasonal walk-ins.
*Flex Serv Manuf J* **31**, 524–561 (2019). https://doi.org/10.1007/s10696-017-9304-8

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DOI: https://doi.org/10.1007/s10696-017-9304-8

### Keywords

- Operations research in healthcare
- Appointment scheduling
- Capacity allocation
- Simulation
- Demand seasonality
- Walk-ins