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An integrated analysis of capacity allocation and patient scheduling in presence of seasonal walk-ins

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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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References

  • Ahmadi-Javid A, Jalali Z, Klassen K (2017) Outpatient appointment systems in healthcare: a review of optimization studies. Eur J Oper Res 258(1):3–34

    Article  MathSciNet  MATH  Google Scholar 

  • Balasubramanian H, Muriel A, Wang L (2012) The impact of provider flexibility and capacity allocation on the performance of primary care practices. Flex Serv Manuf J 24(4):422–447

    Article  Google Scholar 

  • Borgman NJ, Vliegen IMH, Boucherie RJ, Hans E (2017) Appointment scheduling with unscheduled arrivals and reprioritization. Flex Serv Manuf J. https://doi.org/10.1007/s10696-016-9268-0

    Article  Google Scholar 

  • Cayirli T, Gunes ED (2014) Outpatient appointment scheduling in presence of seasonal walk-ins. J Oper Res Soc 65(4):512–531

    Article  Google Scholar 

  • Cayirli T, Veral E (2003) Outpatient scheduling in health care: a review of literature. Prod Oper Manag 12(4):519–549

    Article  Google Scholar 

  • Cayirli T, Yang KK (2014) A universal appointment rule with patient classification for service times, no-shows and walk-ins. Serv Sci 6(4):274–295

    Article  Google Scholar 

  • Cayirli T, Yang KK, Quek SA (2012) A universal appointment rule in the presence of no-shows and walk-ins. Prod Oper Manag 21(4):682–697

    Article  Google Scholar 

  • Creemers S, Lambrecht M (2010) Queueing models for appointment-driven systems. Ann Oper Res 178(1):155–172

    Article  MathSciNet  MATH  Google Scholar 

  • Dobson G, Hasija S, Pinker EJ (2011) Reserving capacity for urgent patients in primary care. Prod Oper Manag 20(3):456–473

    Article  Google Scholar 

  • Forjuoh SN, Averitt WM, Cauthen DB, Couchman GR, Symm B, Mitchell M (2001) Open-access appointment scheduling in family practice: comparison of prediction grid with actual appointments. J Am Board Family Pract 14(4):259–265

    Google Scholar 

  • Gupta D, Denton B (2008) Appointment scheduling in health care: challenges and opportunities. IIE Trans 40(9):800–819

    Article  Google Scholar 

  • Gupta D, Wang L (2008) Revenue management for a primary-care clinic in the presence of patient choice. Oper Res 56(3):576–592

    Article  MATH  Google Scholar 

  • Ho CJ, Lau HS (1992) Minimizing total cost in scheduling outpatient appointments. Manag Sci 38(12):1750–1764

    Article  MATH  Google Scholar 

  • Klassen KJ, Rohleder TR (1996) Scheduling outpatient appointments in a dynamic environment. J Oper Manag 14(2):83–101

    Article  Google Scholar 

  • Klassen KJ, Rohleder TR (2004) Outpatient appointment scheduling with urgent clients in a dynamic, multi-period environment. Int J Serv Ind Manag 15(2):167–186

    Article  Google Scholar 

  • Koeleman PM, Koole GM (2012) Optimal outpatient appointment scheduling with emergency arrivals and general service times. IIE Trans Healthc Syst Eng 2(1):14–30

    Article  Google Scholar 

  • Kolisch R, Sickinger S (2008) Providing radiology health care services to stochastic demand of different customer classes. OR Spectr 30(2):375–395

    Article  MathSciNet  MATH  Google Scholar 

  • Kopach R, DeLaurentis PC, Lawley M, Muthuraman K, Ozsen L, Rardin R, Wan H, Intrevado P, Qu X, Willis D (2007) Effects of clinical characteristics on successful open access scheduling. Health Care Manag Sci 10(2):111–124

    Article  Google Scholar 

  • Kortbeek N, Zonderland ME, Braaksma A, Vliegen IMH, Boucherie RJ, Litvak N, Hans EW (2014) Designing cyclic appointment schedules for outpatient clinics with scheduled and unscheduled patient arrivals. Perform Eval 80:5–26

    Article  Google Scholar 

  • Luo J, Kulkarni VG, Ziya S (2015) A Tandem Queueing model for an appointment-based service system. Queueing Syst 79(1):53–85

    Article  MathSciNet  MATH  Google Scholar 

  • Ma X, Sauré A, Puterman ML, Taylor M, Tyldesley S (2015) Capacity planning and appointment scheduling for new patient oncology consults. Health Care Manag Sci 19(4):347–361

    Article  Google Scholar 

  • Morikawa K, Takahashi K (2016) Scheduling appointments for walk-ins. Int J Prod Econ 190:60–66

    Article  Google Scholar 

  • Murray M, Berwick DM (2003) Advanced access: reducing waiting and delays in primary care. J Am Med Assoc 289(8):1035–1040

    Article  Google Scholar 

  • Murray M, Tantau C (1999) Redefining open access to primary care. Managed Care Q 7(3):45–55

    Google Scholar 

  • Murray M, Tantau C (2000) Same-day appointments: exploding the access paradigm. Family Pract Manag 7(8):45–50

    Google Scholar 

  • Patrick J, Puterman ML (2007) Improving resource utilization for diagnostic services through flexible inpatient scheduling: a method for improving resource utilization. J Oper Res Soc 58(2):235–245

    Article  MATH  Google Scholar 

  • Patrick J, Puterman ML, Queyranne M (2008) Dynamic multipriority patient scheduling for a diagnostic resource. Oper Res 56(6):1507–1525

    Article  MathSciNet  MATH  Google Scholar 

  • Qu X, Shi J (2009) Effect of two-level provider capacities on the performance of open access clinics. Health Care Manag Sci 12(1):99–114

    Article  Google Scholar 

  • Qu X, Shi J (2011) Modeling the effect of patient choice on the performance of open access scheduling. Int J Prod Econ 129:314–327

    Article  Google Scholar 

  • Qu X, Rardin RL, Williams JAS, Willis DR (2007) Matching daily healthcare provider capacity to demand in advanced access scheduling systems. Eur J Oper Res 183(2):812–826

    Article  MATH  Google Scholar 

  • Qu X, Peng Y, Shi J, LaGanga L (2015) An MDP model for walk-in patient admission management in primary care clinics. Int J Prod Econ 168:303–320

    Article  Google Scholar 

  • Ratcliffe A, Gilland W, Marucheck A (2012) Revenue management for outpatient appointments: joint capacity control and overbooking with class-dependent no-shows A. Flex Serv Manuf J 24(4):516–548

    Article  Google Scholar 

  • Rohleder TR, Klassen K (2002) Rolling horizon appointment scheduling: a simulation study. Health Care Manag Sci 5(3):201–209

    Article  Google Scholar 

  • Sauré A, Patrick J, Tyldesley S, Puterman ML (2012) Dynamic multi-appointment patient scheduling for radiation therapy. Eur J Oper Res 223(2):573–584

    Article  MathSciNet  MATH  Google Scholar 

  • Wang WY, Gupta D (2011) Adaptive appointment systems with patient preferences. Manuf Serv Oper Manag 13(3):373–389

    Article  Google Scholar 

  • Wang S, Liu N, Wan G (2017) Managing appointment-based services in the presence of walk-in customers. Working paper, Columbia University

  • Zacharias C, Armony M (2017) Joint panel sizing and appointment scheduling in outpatient care. Manag Sci 63(11):3978–3997

    Article  Google Scholar 

  • Virji A (1990) A study of patients attending without appointments in an urban general practice. Br Med J 301:22–26

    Article  Google Scholar 

Download references

Acknowledgements

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

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Correspondence to Tugba Cayirli.

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 It 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 Pw 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 Pw, ρ and the seasonality level that determines the seasonality index It. 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

$$T = P_{w} T + \left( {1 - P_{w} } \right) T = \frac{{\lambda_{w} }}{\rho } + \frac{{\lambda_{s} }}{\rho }$$
(6)

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

$$\uplambda_{w}^{t} = \lambda_{w} I_{t} = \rho P_{w} T I_{t}$$
(7)

Similarly, given the overall average call rate for scheduled appointments (λs), the average arrival rate of calls for day t is calculated as follows:

$$\uplambda_{s}^{t} = \lambda_{s} I_{t} = \rho \left( {1 - P_{w} } \right) T I_{t}$$
(8)

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:

$$R_{t} = P_{w} TI_{t }\upgamma = \frac{{\uplambda_{w}^{t} }}{\rho }\upgamma = \frac{{\lambda_{\text{w}} I_{t} }}{\rho }\upgamma$$
(9)

Using Equations (6) and (9), the decision variable γ may be interpreted through its impact on the daily average booking limit Nt, represented as follows:

$$N_{t} = T - R_{t} = \frac{{\lambda_{s} }}{\rho } + \frac{{\lambda_{w} }}{\rho } - \frac{{\lambda_{\text{w}} I_{t} }}{\rho }\gamma = \frac{{ \lambda_{s} }}{\uprho} + \lambda_{w} \frac{{1 - I_{t} \varUpsilon }}{\rho }$$
(10)

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 Nt 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”, \(\lambda_{w}\)It 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, Pw = 80%) compared to Env_2 (DL = 9SEAS = Low, Pw = 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 cW/cL. This revises the original total cost equation TCInt (Eq. 3) as follows:

$$TC_{Int} = c_{W} (LT/\upomega + WT_{T} ) \, + c_{O} \left( {OT} \right)$$
(11)

Figure 6 presents the efficient frontiers for two values of ω = 1 and 5 for the “high” environment with DL = 99%, SEAS = H, Pw = 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.

Fig. 6
figure 6

Efficient frontiers. a ω = 1, b ω = 5

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.

Table 7 Best appointment systems for different ω = 0.5, 1, 2, 5, 10

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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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