Appointment scheduling and real-time sequencing strategies for patient unpunctuality

Abstract

Patient unpunctuality causes perturbations in healthcare operations, compromising productivity and service quality. In this paper, we propose an approach that mitigates the negative impacts of unpunctuality using both appointment scheduling and real-time sequencing taking into account patient unpunctuality, no-shows, random service durations, and multiple providers. The objective is to minimize the total cost incurred by patient waiting and provider overtime. An optimal real-time sequencing strategy is established to serve the waiting patient with the smallest “LAR” index, which is defined as the Larger of Appointment time and Real arrival time for a patient. The optimal appointment schedule is determined by a simulation optimization approach with unbiased gradient estimators. Sample path discontinuities are smoothed by smoothed perturbation analysis. Properties of the optimal real-time sequencing strategies are used for the efficient sample path gradient estimation. Extensive experiments demonstrate the effectiveness of the proposed algorithm. Using real data, numerical experiments illustrate that the optimal appointment schedule depends on the system parameters and differs significantly from those of the existing literature. Specifically, the pattern of the appointment schedule is determined by the number of providers and the real-time sequencing strategy. The length of the appointment intervals is sensitive to the degree of unpunctuality and no-shows. Compared with the schedules in the previous studies, our schedule can achieve a significant cost reduction. Further, the optimal real-time sequencing strategy outperforms the commonly-used strategies in practice (e.g., appointment order, arrival order). Managerial insights are also provided for hospital managers to schedule unpunctual patients.

Full text

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Appointment scheduling and real-time sequencing strategies for patient unpunctuality
Xingwei Pana, Na Gengb*, Xiaolan Xiec,d
aDepartment of Industrial Engineering and Management, Shanghai Jiao Tong University, Shanghai 200240, China
bSino-US Global Logistics Institute, Shanghai Jiao Tong University, Shanghai 200030, China
cMines Saint-Etienne, Univ Clermont Auvergne, CNRS, UMR 6158 LIMOS, Centre CIS, F-42023 Saint-Etienne, France
dAntai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200030, China
*Corresponding author: Na Geng, gengna@sjtu.edu.cn
Abstract
Patient unpunctuality causes perturbations in healthcare operations, compromising productivity and
service quality. In this paper, we propose an approach that mitigates the negative impacts of
unpunctuality using both appointment scheduling and real-time sequencing taking into account patient
unpunctuality, no-shows, random service durations, and multiple providers. The objective is to
minimize the total cost incurred by patient waiting and provider overtime. An optimal real-time
sequencing strategy is established to serve the waiting patient with the smallest “LAR” index, which is
defined as the Larger of Appointment time and Real arrival time for a patient. The optimal appointment
schedule is determined by a simulation optimization approach with unbiased gradient estimators.
Sample path discontinuities are smoothed by smoothed perturbation analysis. Properties of the optimal
real-time sequencing strategies are used for the efficient sample path gradient estimation. Extensive
experiments demonstrate the effectiveness of the proposed algorithm. Using real data, numerical
experiments illustrate that the optimal appointment schedule depends on the system parameters and
differs significantly from those of the existing literature. Specifically, the pattern of the appointment
schedule is determined by the number of providers and the real-time sequencing strategy. The length of
the appointment intervals is sensitive to the degree of unpunctuality and no-shows. Compared with the
schedules in the previous studies, our schedule can achieve a significant cost reduction. Further, the
optimal real-time sequencing strategy outperforms the commonly-used strategies in practice (e.g.,
appointment order, arrival order). Managerial insights are also provided for hospital managers to
schedule unpunctual patients.
Keywords: OR in health services; appointment scheduling; real-time sequencing; patient unpunctuality;
smoothed perturbation analysis

2
1. Introduction
The demand for medical care is growing due to many reasons, e.g., the aging society. Timely
service is critical for the patients and challenging for the hospital managers. Appointment scheduling is
one of the most effective ways to match the demand with supply, and thus to reduce outpatients’ waiting
times. In this context, the overarching assumption in the literature is that patients arrive punctually at
their appointment time, e.g., Denton and Gupta (2003), Robinson and Chen (2003, 2010), Kaandorp
and Koole (2007), Begen and Queyranne (2011), etc. However, patient unpunctuality, i.e., arriving
either earlier or later than the appointment time, is prevalent all over the world. Empirical studies have
reported that, while some patients would arrive 17 min ahead of the appointment on average, with a
standard deviation of 30 min (Cox, Birchall, and Wong, 1985; Cayirli, Veral, and Rosen, 2006; Klassen
and Yoogalingam, 2014), others would arrive 10 min late on average, with a maximum of 2 hours’
delay (Zhu, Chen, Leung, and Liu, 2018). The unpunctual arrivals result in system congestion or
provider idleness. Hence, the performance of the appointment system is undermined by unpunctuality
(White and Pike, 1964; Cayirli, Veral, and Rosen, 2006; Tai and Williams, 2012).
Patient unpunctuality affects not only the appointment system, but also real-time sequencing
strategies. When a provider is available, who is served next, a late patient with earlier appointment time
or an early patient with later appointment time? It is not an easy decision. Offering higher-priority
services to patients upon their early or late arrivals may encourage them to again arrive earlier or later
in future. Many papers assume that providers serve unpunctual patients by strictly following the
appointment order (AO) to preserve the tractability of appointment scheduling optimization. Under the
AO strategy, providers will keep idle until a late-arriving patient with a smaller appointment number
arrives, even if an early-arriving patient with a larger appointment number is waiting. Hence, AO is
likely to result in more idleness/overtime for providers and longer waiting time for early patients,
meanwhile unintentionally reward late patients (Deceuninck, De Vuyst, and Fiems, 2019). The first-in-
first-out (FIFO) strategy is a popular strategy in practice. In contrast to AO, FIFO encourages the early
arrivals by neglecting the appointment order. If AO or FIFO is adopted, patients may get the idea that
they can ‘beat’ the appointment system by arriving excessively early or late (Jiang, Tang, and Yan,
2019). This may create unnecessary congestion and compromise the service quality. Therefore, to
mitigate the negative effects of early and late arrivals, it is necessary to simultaneously optimize the
appointment schedule and real-time sequencing strategies.

3
To this end, this paper addresses the joint optimization of the appointment schedule and real-time
sequencing strategies, where patient unpunctuality, no-shows, random service durations, and multiple
providers are considered. The appointment schedule determines the optimal appointment times by
considering unpunctuality of arrivals, while the real-time sequencing strategy determines the optimal
service sequence. The objective is to minimize the total cost, i.e., the weighted sum of patient waiting
cost and provider overtime cost. To achieve this objective, an optimal real-time sequencing strategy is
proven to be the smallest LAR strategy, where the LAR index is defined as the Larger of Appointment
time and Real arrival time for a patient. This strategy implies that the waiting patient with the smallest
LAR index is served first. By implementing the smallest LAR strategy, a simulation optimization
approach with unbiased gradient estimators is proposed to determine the optimal appointment schedule.
The unbiased gradient estimation turns out to be more difficult due to the discontinuous sample path
cost function under the smallest LAR strategy. To fix this problem, smoothed perturbation analysis
(SPA) and properties of the optimal real-time sequencing strategies are applied to the unbiased gradient
estimators that can be determined on a single sample path, which is a salient feature ensuring the
effectiveness of our algorithm. Extensive experiments show that our algorithm converges to the same
objective value irrespective of the initial solution used. Our schedule under the smallest LAR strategy
is found to exhibit a peak pattern for better balancing patient waiting and provider overtime, i.e.,
appointment intervals have one or several peaks. This pattern differs from the dome pattern (the
appointment interval first increases, then remains constant, and finally decreases) exhibited by the
optimal schedule of Jiang, Tang, and Yan (2019) with the AO strategy. Further, multiple patients are
appointed in both the initial block and the final block to relieve the side effects caused by unpunctuality.
The major contributions of this paper can be summarized as follows.
(a) An integrated framework is proposed for the joint optimization of appointment scheduling and
real-time sequencing strategy with patient unpunctuality. This paper proposes the smallest LAR
strategy and proves its optimality for the real-time sequencing of unpunctual patients. Then the
appointment schedule is optimized by adopting the smallest LAR strategy for real-time sequencing;
by contrast, prior appointment scheduling studies have typically adopted AO or FIFO.
(b) An efficient simulation optimization approach is devised based on unbiased gradient estimators.
To our knowledge, this paper also represents the first study pioneering not only the adoption of
SPA in the appointment scheduling, but also the exploration of the properties of the optimal real-

4
time sequencing to establish unbiased gradient estimators that can be efficiently evaluated. Under
the smallest LAR strategy, the optimal appointment schedule is rapidly attained by our algorithm,
which can be easily extended to other strategies (e.g., AO, FIFO).
(c) Extensive numerical experiments are performed under real data to demonstrate the benefits of the
proposed appointment schedule and real-time sequencing strategy. The superiority of our schedule
and strategy grows as the unpunctuality increases. A significant cost reduction (51% on average
and max 68.7%) is achieved if the schedule of Jiang, Tang, and Yan (2019) or Pan, Geng, and Xie
(2020) is switched to our schedule. The smallest LAR strategy outperforms the commonly-used
strategies, e.g., this strategy improves AO by 31% to 86.5%.
The remainder of this paper is organized as follows. Section 2 reviews the relevant literature.
Section 3 characterizes the problem, the optimal real-time sequencing strategies, and a stochastic
gradient descent algorithm for appointment scheduling. Section 4 proposes the sample path gradient
estimation in this algorithm. Section 5 presents numerical experiments. Section 6 concludes this paper.
Further, Appendix A in the supplemental file gives the theoretical proofs, Appendix B summarizes the
notations and Appendix C shows figures and tables of the numerical experiments.
2. Literature review
This paper falls into the field of patient scheduling, including appointment scheduling that
determines patients’ appointment times and real-time sequencing that determines the service sequence
of patients. Readers can refer to Cayirli and Veral (2003), Gupta and Denton (2008), and Ahmadi-Javid,
Jalali, and Klassen (2017) for reviews.
Since the pioneering work of Bailey (1952), copious literature has been generated on appointment
scheduling on the assumption of patient punctuality. In the single-provider setting with punctuality, the
optimal schedule exhibits a dome pattern (Denton and Gupta, 2003; Hassin and Mendel, 2008;
Robinson and Chen, 2010; Erdogan and Denton, 2013). Such a schedule has been shown to outperform
others in the literature. Conversely, in the multi-provider setting, the optimal schedule exhibits a zigzag
pattern, i.e., the appointment interval increases and decreases alternately (Zhang and Xie, 2015).
However, these studies have not accounted for unpunctuality which jeopardizes the appointment system.
Patient unpunctuality has been considered in only few studies. In this context, simulation studies
are one stream of the literature focusing on variations in the system performance. White and Pike (1964)
were the first to investigate the influence of unpunctuality on the system performance. Patients have

5
been observed to wait longer when they arrive more unpunctually, given the fixed provider idleness.
Vissers and Wijngaard (1979) identified five variables to characterize an appointment system, including
two unpunctuality factors. In exploring the effects of various factors, Cayirli, Veral, and Rosen (2006,
2008) found that unpunctuality was one of the major factors compromising the system performance.
Creemers, Lambrecht, Beliën, and Van den Broeke (2020) proposed an analytical model to assess the
performance of different appointment scheduling rules by considering customer unpunctuality, no-
shows, service interruptions, and delay in session start time.
In other studies considering patient unpunctuality, analytical methods are developed to optimize
the appointment schedule. Jouini and Benjaafar (2009) proposed an exact analytical approach for a
single-provider system. On the assumption that patients were served by the AO strategy, they observed
that an equally-spaced heuristic performed favorably in both punctual and unpunctual systems. Klassen
and Yoogalingam (2014) used a simulation optimization framework to optimize the appointment
schedule for unpunctual patients in a single-provider setting. They incorporated a scatter search
algorithm with Tabu search. The results revealed that increasing intervals and increasing clustering
schedules benefited the unpunctual system. Zhu, Chen, Leung, and Liu (2018) attempted to improve
the design of appointment scheduling with unpunctuality. Upon establishing an analytical model, they
proceeded to demonstrate the optimality of a fixed-interval policy for a simplified two-patient model.
Deceuninck, Fiems, and De Vuyst (2018) proposed a modified Lindley recursion approach for the
system performance of a discrete time-single provider system and a local search algorithm to improve
the appointment schedule. Numerical results indicated that an increase in unpunctuality would exert
negative impacts on the appointment system. Jiang, Tang, and Yan (2019) addressed appointment
scheduling by formulating a stochastic programming model for a single-provider system, which was
solved by using Benders decomposition (BD). Numerical results showed that the optimal appointment
intervals exhibited the same dome pattern as systems with punctuality, but the first interval became
different when considering unpunctuality. Both Deceuninck, Fiems, and De Vuyst (2018) and Jiang,
Tang, and Yan (2019) assumed patients were served by AO to preserve tractability. Pan, Geng, and Xie
(2019, 2020) focused on the appointment scheduling with unpunctuality and multiple servers. They
respectively developed BD and a stochastic approximation method to determine the appointment time.
The AO strategy is also used in the two studies to schedule unpunctual patients. Different from the
above studies that determine the appointment time for each patient, Zacharias and Yunes (2019)

6
formulate a single-provider queueing model and use the discrete optimization to determine the number
of patients appointed in each slot. Unpunctual patients are served in FIFO order. To the best of our
knowledge, the majority of these papers have considered only a single provider and taken AO to be the
real-time sequencing strategy for tractability. Additionally, most papers have used heuristic algorithms
which cannot ensure the optimality of the solution.
Another relevant stream of literature concerns real-time sequencing. When patients arrive
punctually at their appointment times, the arrival order is identical to the appointment order, and thus
the AO strategy is adopted as the real-time sequencing strategy in serving patients. However, when
patients arrive unpunctually, complications arise from the real-time sequencing. Recent studies have
explored strategies from several aspects: (i) penalizing late patients; (ii) encouraging patient punctuality;
and (iii) sequencing patients. Cayirli, Veral, and Rosen (2006, 2008) proposed a scheduling strategy to
penalize late patients, i.e., the late patients would be delayed to an extent proportional to the magnitude
of their lateness. Klassen and Yoogalingam (2014) also suggested some scheduling strategies for late
patients and found that a back-of-queue (BQ) strategy would lead to a similar performance with the one
without penalty. Glowacka et al. (2017) proposed a hybrid strategy to encourage punctuality, which
offered the highest priority to the most punctual patients. Samorani and Ganguly (2016) explored
whether a provider should consult an early patient right away (preempt strategy) or wait for the next
appointed patient (waiting strategy). They indicated that the waiting strategy outperformed the always-
preempt strategy in high-service-level clinics, whereas the two strategies exhibited similar
performances in clinics with overbooking or short service durations.
The papers most closely related to our work are that of Klassen and Yoogalingam (2014), Jiang,
Tang, and Yan (2019), and Pan, Geng, and Xie (2020). Table 1 presents the similarities and differences.
Apart from the three relevant papers, the majority of the papers on scheduling of unpunctual patients
did not simultaneously consider appointment scheduling and real-time sequencing strategy. Jiang, Tang,
and Yan (2019) established stochastic programming models and proposed BD to determine the
appointment times. Pan, Geng, and Xie (2020) developed the stochastic gradient descent (SGD)
algorithm and the unbiased gradient estimation in SGD is derived from infinitesimal perturbation
analysis. To preserve the tractability, both papers implemented the AO strategy that results in longer
idle time for providers and longer waiting time for early patients. Klassen and Yoogalingam (2014)
used a heuristic algorithm to search the appointment times and proposed some heuristic strategies for

7
penalizing late patients. Compared with these studies, this paper simultaneously addresses appointment
scheduling and real-time sequencing by considering patient unpunctuality, no-shows, and random
service durations in a multi-provider setting. The smallest LAR strategy is first established in this paper
to be optimal whereas commonly-used AO or FIFO are not. The smallest LAR strategy leads to
discontinuity of the sample cost function and the solution approaches in the previous work (e.g., BD or
SGD with IPA gradient estimator) do not apply. Smooth perturbation analysis is used to remedy the
discontinuity, to derive unbiased gradient estimators and to devise converging SGD.
Table 1 Similarities and differences of the related studies
Papers
Decision
Setting
Sequencing
strategy
Model
Cost function
Approach
Klassen &
Yoogalingam
(2014)
Appointment
times
Single
provider
Heuristic
strategies
Simulation
optimization
Discontinuous
Scatter search
with tabu
search
Jiang, Tang &
Yan (2019)
Appointment
times
Single
provider
AO
Stochastic
programming
Continuous
BD
Pan, Geng &
Xie (2020)
Appointment
times
Multiple
providers
AO
Simulation
optimization
Continuous
SGD with
IPA gradient
estimator
This paper
Appointment
times and real-
time sequencing
Multiple
providers
Optimal
strategy
Simulation
optimization
Discontinuous
SGD with
SPA gradient
estimator
3. Problem formulation and solution
Section 3.1 presents the problem of appointment scheduling and real-time sequencing with
unpunctuality. Separately, Section 3.2 characterizes the optimal real-time sequencing strategies, and
Section 3.3 gives a stochastic gradient descent (SGD) algorithm for appointment scheduling.
3.1. Problem setting
This paper considers appointment scheduling and real-time sequencing for unpunctual patients by
incorporating stochastic service durations and no-shows in an outpatient clinic. The appointment
schedule determines the appointment times and the real-time sequencing determines the service
sequence. The objective is to minimize the total cost, i.e., the weighted sum of patient waiting cost and
provider overtime cost. Overall, the optimal appointment schedule and real-time sequencing strategy
are explored to minimize the total cost.
The assumptions concerning providers are as follows.
Assumption A1. The working session starts at time 0 and m identical providers serve a total number of
n appointment patients with independent and identically distributed (i.i.d.) random service durations.
In this context, patients are a priori all similar. What is important is the number of appointments to

8
schedule, but the exact complete set of patients is not required. When the appointment schedule is made,
the arriving patient is usually scheduled in the earliest available slot. This can be applied to most
hospitals as the hospitals tend to run at capacity.
Assumption A2. Upon arrival, patients wait in a single queue and are dynamically assigned to available
providers (Zhang and Xie, 2015; Castaing, Cohn, Denton, and Weizer, 2016).
Such an assumption is motivated by common observations in existing systems; e.g., Computed
Tomography examinations in the radiology department consist of multiple identical servers and a single
queue; a Chemotherapy room is composed of multiple identical infusion chairs and a single queue.
Assumption A3. The providers follow a work-conserving discipline (i.e., non-idle policy) to serve
patients, where a provider completing a service instantly starts the service of a new patient if the queue
is not empty, to avoid the idleness and overtime as much as possible.
Assumption A3 is often made in the literature (Deceuninck, Fiems, and De Vuyst, 2018;
Deceuninck, De Vuyst, and Fiems, 2019; Zhu, Chen, Leung, and Liu, 2018; Zacharias and Yunes, 2019).
It leads to out-of-order services for the unpunctual system. For example, if the patient of the 10-th
appointment arrives before the patient of the 9-th appointment and the queue is empty and a provider is
available, then patient 10 is seen immediately before patient 9 (Klassen and Yoogalingam, 2014). It is
reasonable as the daily outpatient demand is high in many hospitals. One of our university-affiliated
hospitals has about 15,000 outpatient visits daily. Apart from the example of Chinese hospitals, prior
studies in other countries also face the similar problems caused by patient unpunctuality; e.g., American
hospitals in Cayirli, Veral, and Rosen (2006, 2008), Canadian hospitals in Klassen and Yoogalingam
(2014). Therefore, this work-conserving discipline is commonly used, which is beneficial for hospitals
to serve as many patients as possible and provide the timely service for patients.
In addition to providers, some assumptions about patients are also needed.
Assumption A4. Patients may arrive unpunctually with respect to their appointment times.
Let Ai be the appointment time of patient i and Ri be the real arrival time. Let ui = Ri − Ai be the
random unpunctuality (e.g., ui =10 min denotes a patient is late for 10 minutes). If patient i arrives
earlier, ui is negative; otherwise, ui is positive.
Assumption A5. Patients are indexed by i and sorted according to the appointment times, i.e., patient
i+1 has a larger appointment time than patient i. Further, out-of-order arrivals are allowed for, e.g.,
patient i+1 can arrive earlier than patient i.

9
Assumption A6. Unpunctuality is assumed to have a bounded support called unpunctual time window
(UTW), where the appointment time is used as the base point and U is the allowance for
earliness/lateness (Deceuninck, Fiems, and De Vuyst, 2018). Thus, ui belongs to [-U, U]. The symmetric
UTW is assumed for simplicity and all results can extend to the general UTW.
Assumption A7. No-shows are considered in this paper. A patient with Ri = Ai + U is a no-show patient
with zero service duration and no waiting cost. This treatment of no-shows is consistent with prior
studies (Chen and Robinson, 2014; Zhu, Chen, Leung, and Liu, 2018; Jiang, Tang, and Yan, 2019).
The objective is to minimize the total cost incurred by patient waiting and provider overtime. The
waiting time of a punctual patient is equal to the service start time minus the appointment time. A
different definition is needed to measure the service quality of unpunctual patients. The waiting time
before the appointment time (promised start time) should not be included for each early patient, whereas
that before the real arrival time should not be considered for late patients. Otherwise, the waiting time
will be exaggerated. Hence, the waiting time is measured starting from the appointment time for an
early patient and from the arrival time for a late patient (Cayirli, Veral, and Rosen, 2006, 2008;
Samorani and Ganguly, 2016; Deceuninck, Fiems, and De Vuyst, 2018; Jiang, Tang, and Yan, 2019).
The waiting cost rate is scaled to 1 and providers working beyond the given clinic end time T incurs the
overtime cost at rate α. Let g, O and Wi denote the total cost, the overtime and the waiting time of patient
i. Thus we have
1
n
i
i
g W O


  

,
()
i i i i
W S A R 
  
,
 
max i
in
O C T 


, where Si and Ci are
the service start time and completion time of patient i, and
max{ , }, ( ) max( ,0)x y x y x x

  
.
3.2. Optimal real-time sequencing strategies
This subsection considers a given appointment schedule and addresses the real time scheduling
that determines the patient to serve when a provider is idle and the waiting queue is not empty. The real-
time sequencing decision at time t relies on all information available at t, i.e., the appointment schedule
and occurrence times of all past arrivals, past service starts and past service completions. According to
the linear cost structure (i.e., the weighted sum of patient waiting and provider overtime) and the work-
conserving discipline (i.e., non-idle policy to minimize provider overtime), an optimal real-time
sequencing strategy emerges to be an index rule.
Proposition 1. Under the work-conserving assumption, there exists an optimal real-time sequencing
strategy that serves the waiting patient with the smallest LAR index first when a provider is available

10
and the waiting queue is not empty, where the LAR index of patient i is defined as the Larger of
Appointment time and Real arrival time, i.e.,
 
max ,
i i i i i
LAR A R A R  
.
Proof. Under the work-conserving assumption, we consider an optimal sequencing strategy π* that
starts to serve patient j at time t and there exists another patient i such that LARi < LARj and Ri ≤ t, Rj
≤ t. Assume by contradiction that serving i at t is not optimal. Let Si > t be the starting time of patient
i. Consider a feasible strategy π that exchanges i and j. Since service durations of i and j are i.i.d. random
variables, we also switch service durations of i and j. Hence, π* and π have exactly the same event times
except the switched service sequence of i and j. Compare the total cost of the two strategies, i.e.,
weighted sum of patient waiting time and provider overtime). The same event times imply that provider
overtime is identical under the two strategies and hence we only need to focus on patient waiting time,
i.e.,
 
   
 
j i i i i j
g g t LAR S LAR t LAR S LAR



        
. Since
   
xy


is Schur-
convex (Table 2 in Marshall and Olkin, 1979), then
   
max , max ,
j i i i i j
t LAR S LAR t LAR S LAR    
and
j i i i i j
t LAR S LAR t LAR S LAR      
imply
0gg


which contradicts the assumption that serving patient i at t is not optimal. Q.E.D.
From Proposition 1, among the waiting patients, the appointment time of early arrivals and the
arrival time of late arrivals are used for real-time sequencing. Thus, patients are implicitly encouraged
to arrive punctually. Due to the identical waiting cost rate for all patients, the smallest LAR strategy is
not the only optimal real-time sequencing strategy. For example, the Appointment Order Work
Conserving (AOWC) strategy has been propounded by Deceuninck, Fiems, and De Vuyst (2018), where
the provider chooses the waiting patient with the earliest appointment time. The optimality of AOWC
is proven in Appendix A.1. LAR and AOWC have some similarities; e.g., both strategies will choose
the same patient to be served next if all the waiting patients are early arrivals. But AOWC may incur a
worse waiting distribution among patients, i.e., shorter waiting of some patients at the expense of longer
waiting of others. Therefore, the smallest LAR strategy is more beneficial to the individual patients than
AOWC as it can minimize the individual waiting time and potentially encourage patient punctuality.
Further, the optimality of the smallest LAR strategy holds for all convex waiting cost functions as the
proof of Proposition 1 trivially extends.
Besides the AOWC strategy, the smallest LAR strategy also has significant advantages over the

11
other strategies. First, LAR is expected to result in shorter provider overtime and shorter patient waiting
time than AO (appointment order). The former is because the work-conserving discipline is adopted in
LAR but not in AO, while the latter is because AO makes some early-arriving patients with a larger
appointment number wait to get service after some late-arriving patients with a smaller appointment
number. For example, the (i+1)-th appointed patient must be served after the i-th appointed patient
under the AO strategy, even if patient i+1 has arrived and patient i has not arrived. Although some prior
studies use AO for preserving tractability in modeling (Jouini and Benjaafar, 2009; Deceuninck, Fiems,
and De Vuyst, 2018; Jiang, Tang, and Yan, 2019), AO performs much worse than LAR. Second,
compared with FIFO, LAR is expected to incur the same overtime but shorter patient waiting time. The
identical overtime is because both FIFO and LAR are based on the work-conserving discipline, while
shorter patient waiting time is mainly due to the definition of waiting time in this paper. To encourage
the punctuality of patients, the waiting time of early patients before the appointment time is ignored in
the objective function. Thus, FIFO could lead to longer waiting time, especially for late patients. The
performance of FIFO is inferior to LAR and the execution of FIFO disregards the goal of appointment
scheduling, which indirectly encourages patients to arrive unpunctually. It is worth noting that these
benefits of the smallest LAR strategy are also observed in the numerical experiments (see Section 5).
3.3. Appointment schedule optimization by stochastic gradient descent
This subsection is devoted to the optimization of the appointment schedule. We devise a simulation
optimization method (i.e., SGD) that generates random sample paths or scenarios (“sample path” and
“scenario” are used interchangeably) per iteration and iteratively searches for optimal solutions by using
sample path gradients to determine the improving directions. Given an appointment schedule A = [A1,…,
An] and sample path ω with uncertain patient unpunctuality, service durations, and no-shows, the
operation of the system is as follows. At time 0, the providers start to work and at most m waiting
patients with the m smallest LAR indices are served. As soon as a provider is available in the concurrent
presence of waiting patients, this provider instantly starts to serve the waiting patient with the smallest
LAR index. Finally, the sample path cost function g(A, ω) is determined as follows:
 
, ( ) ( )
i
iI
g W O
   

  

A
(1)
 
( ) ( ) ( ) ,
i i i i
W S A R i I
  

    
(2)

12
 
( ) max ( )
i
iI
O C T




(3)
where I denotes the set of patients. The objective is to minimize the average cost within the set
 
1
|0
n
mn
A A A       A
:
   
min = ,Gg

 

AAA
(4)
Based on the formulation, the SGD algorithm is described as follows.
Stochastic Gradient Descent Algorithm
Step 1. Initializations. Randomly generate the initial appointment times A(0).
Step 2. Iterations. For q := 1 to N do:
(a) Randomly generate D independent sample paths of the system
 
 
1,
q


A
.
(b) Update the gradient
 
 
1,
q
g




A
and the improving direction
   
 
11
,
qq
g



  
dA
.
(c) Update the appointment times for the next iteration
 
( ) ( 1) ( ) ( 1)
:
q q q q



  A A d
, where
 
0/
qq


is the step size (
0

is the initial step size), and
 
arg min
y
x y x

  
is the
orthogonal projection into the set

.
Step 3. Return A(N), stop.
Since it is generally impossible to directly derive the gradient
 
 
1,
q
g




A
(Zhang and Xie,
2015), we estimate it from the unbiased gradient estimator
 
 
1,
q
g





A
. The derivation of the
gradient estimator is presented in Section 4 via perturbation analysis.
4. Sample path gradient estimation
This section presents the derivation of the sample path gradient estimator. Sections 4.1 to 4.4
consider the case without no-shows and Section 4.5 extends to the case with no-shows. Section 4.1
describes the sample path representation. In Section 4.2, the sample path gradient from infinitesimal
perturbation analysis (IPA) is presented while, in Section 4.3, SPA is employed to correct the bias of the
IPA gradient estimator by smoothing the discontinuity of the sample path cost function. Section 4.4
addresses the efficient computation of the discontinuity. The correctness of the SPA gradient estimator
is proven in Appendixes A.3 to A.5. Note that the major notations of this paper are also summarized in
Appendix B for a quick review.

13
4.1. Sample path representation
This subsection gives the sample path in details. Each sample path is generated from appointment
times A = [A1,…, An], and random variables: u = [u1,…, un] and p = [p1,…, pn], where ui is unpunctuality
of the i-th appointed patient and pi is service duration of the i-th started patient. Service durations are
assigned to the patients according to the service sequence, which is equivalent to assigning i.i.d. service
durations directly to patients. Note that assigning service durations based on the service sequence
ensures the sample path continuity that is impossible if service durations are assigned to patients.
When performing the discrete-event simulation, three types of events are considered: appointments,
patient arrivals, and service completions. The “appointment” events do not exert any direct effect on
the service sequence, but they are needed to ensure the continuity of the cost function. The notations
and system dynamics are as follows:
ek
event k
tk
time epoch of event k, and
-
k
t
(
+
k
t
) denotes the time immediately before (after) tk
τk
time duration between tk-1 and tk, i.e., τk = tk - tk-1
Qk
queue of patients at time
-
k
t
with the initial queue Q0
Ek
set of active events at time
+
k
t
with the initial set E0
rk(e)
occurrence time epoch of event e  Ek
At time t0 = 0, patients who arrive before time 0 form the initial queue Q0 = {i: Ri ≤ 0}. A total of
|Q0|m patients in Q0 with the smallest LAR index are removed and completion events {t0+pi} become
active, thus
   
 
0 1 0
, , : 0 :completion events of services started at
m n i i j
E A A R R C t

   
.
Based on Ek, event k+1 is determined by
 
1arg min
k
kk
eE
e r e


and occurs at time
 
11k k k
t r e


. The
next state depends on the event and a new service starts if ek+1 is a completion event and the queue is
not empty or it is an arrival event and the queue is empty. Specifically,
(a) Case ek+1 = Ai:
 
11
\
k k k
E E e


and
21kk
QQ


;
(b) Case ek+1 = Ri: If at least one provider is available, then
 
 
1 1 1 ( )
\
k k k i k i
E E e C t p

  
   
and
21kk
QQ


, where

(i) denotes the order in which i is served; otherwise,
 
11
\
k k k
E E e


,
 
21kk
Q Q i


;
(c) Case ek+1 = Cj: If Qk+1 is not empty, then serve patient i in Qk+1 with the smallest LAR index,
 
 
1 1 1 ( )
\
k k k i k i
E E e C t p

  
   
,
 
21
\
kk
Q Q i


; otherwise,
 
11
\
k k k
E E e


,

14
21kk
QQ


.
As a result, the complete sample path can be denoted as (Q0, e1, t1, …, eK, tK) with a total number of
events of K = 3n – m – |Q0|. The sample path cost function can be determined below:
     
i i i K
iI
g S A R t T



    

A
(5)
Fig. 1 gives two examples of four patients with the same appointment and arrival times but different
service times. First consider Fig. 1a. At time 0, the queue is
0{2}Q
and patient 2 is served. Hence
events of
0 3 4 1 3 4 2
{ , , , , , }E A A R R R C
are active and the queue becomes empty, i.e.,
1
Q


. The
next event is the arrival of patient 1
11
eR
at time
11
tR
. Patient 1 is served and the events and the
queue are
1 3 4 3 4 2 1
{ , , , , , }E A A R R C C
and
2
Q


. Repeating the above leads to the sample path
 
 
0 1 1 2 2 3 2 4 4 5 3 6 4 7 4 8 1 9 3
2 , , , , , , , , ,Q e R e C e A e R e R e A e C e C e C         
. Patient 4 is
served upon arrival at R4 as provider 1 is available. The case of Fig. 1b is different as patients 3 and 4
are waiting when provider 1 becomes idle at C2 and patient 3 having the smallest LAR index starts next.
(a) Completion before new arrivals (b) Completion after new arrivals
Fig. 1. Examples of the sample path
4.2. Infinitesimal perturbation analysis
According to the above sample path representation, this subsection performs the infinitesimal
perturbation analysis (IPA), which is a simple and usually unbiased gradient estimation of discrete-
event dynamic systems as shown in Fu, Gong, Hu, and Li (2013). In our system, the appointment vector
A is considered as a function of some real number

with
*i
A


for some patient i* and

is the
infinitesimal perturbation of

.
Assumption A8. The distribution functions
 
i
u
Fx
and
 
i
p
Fx
of unpunctuality ui and service
duration pi are continuous in x and without probability mass.

15
Based on A8, the derivatives of event epochs tk, τk, and rk(e) are determined in Theorem 1.
According to Theorem 1, Corollary 1 gives the sample path gradients of service start time Si, waiting
time Wi, and overtime O. Finally, the IPA gradient estimator of the cost function is presented.
Theorem 1. Under A8, tk, τk, and rk(e) are almost surely (a.s.) linear in a neighborhood of

and
 
* * *
, ,
,
k i i k i i
k1 if e R A or e C i BP
dt
d0 otherwise

  




,
   
* * *
, ,
,
i i i i
k1 if e R A or e C i BP
dr e
d0 otherwise

  




,
1k k k
d dt dt
d d d

  


, where BPi corresponds to a set of patients served without interruption by the same
provider who first serves patient i at Ri and then others, i.e., the busy period initiated by the arrival of
patient i is denoted by
 
1
0
, 0 : , , ll
i l i i i i
BP i I l i i S R S C

     
.
Proof. Direct consequence of the definitions of events and Lemma 1 (in Appendix A.2) on the constant
sample path (Q0, e1, e2, …, ek) and differentiability of tk, τk, and rk(e) in the neighborhood of

. Q.E.D.
Corollary 1. Under A8,
 
si
idt
dS
dd


,
   
 
,
,
i
ii
i
ii
dS
S LAR if i i
dW d
dS LAR otherwise








1
1
,
 
 
*
&
K K i i
dO t T e C i BP
d


   1
, and
       
i
iI
dg dW dO
IPA d d d
  

  



, where
()x1
denotes the indicator function equal to 1 if x is true and 0 otherwise, and
()si
is the event index at
which the service of patient i starts, i.e.,
 
isi
St
.
The usefulness of the above IPA estimator
 
dg d

for any sample performance measure g
depends on whether it is unbiased, i.e. whether
   
d g d dg d
   

   
   
. For this purpose, let
     
y y y
  
    
denote the finite difference for any function y. Proposition 2 establishes
the Lipschitz continuity of the ordered service start/completion times and overtime, leading to the
unbiasedness of the sample path gradient of the overtime in Corollary 2. Proposition 3 extends the
Lipschitz continuity to a more general setting. However, Proposition 4 indicates that event order
changes lead to the discontinuity of the total waiting time and hence that of the sample path cost function.
Therefore, the IPA estimator in Corollary 1 is biased (see Appendix A.5 for a counter-example) and the
method of SPA in Section 4.3 is used to overcome the biasedness.
Proposition 2.
 
 
 
   
0 , 0 , 0 , 0
ii
S C O
  
             
, where S[i] and C[i] are

16
respectively the i-th start time and completion time.
Proof. The non-negativity of the three differences is trivial as from Theorem 1, and
[]i
S
,
[]i
C
and O
are non-decreasing in

. We complete the proof by showing the Lipschitz continuity, i.e.,
 
y

  
for y being any of these performance functions. First, Ri, Ai and
ii
AR
are Lipschitz-continuous as
 
 
 
 
 
 
 
, ,
i i i i
R i i A i i A R i i
  
  
            1 1 1
. The Lipschitz continuity of
 
[]i
C

follows directly from that of
 
[]i
S

as
 
[]i
C

is the i-th minimum of
 
[] ,
jj
S p j i

  
and that of O follows from that of
[]i
C
and Lemma 2 (in Appendix A.2) as
 
[]n
O C T 

. In the
remaining, we focus on the proof of
 
 
i
S

  
. Let
th min
ij
iR


be the i-th arrival time. Thus,
 
 
[ ] [ ]
max ,
i i i m
SC




and
[]i
C
is the order statistics of
[]ii
Sp
.
(a) Consider the first service,
   
11
   
    
by the Lipschitz continuity of Ri and Lemma
2. Combining it with
[1 ] 0
m
C
, from Lemma 2 and
 
 
[1] 1 [1 ]
max , m
SC




,
 
[1]
S

  
.
Assume
 
 
,
i
S i j

    
.
(b) Consider the (j+1)-st service,
 
 
[ 1] 1 [ 1 ]
max ,
j j j m
SC


   

.
   
11jj
   

    
by the
Lipschitz continuity of Ri and Lemma 2. Further,
[ 1 ]jm
C
is the (j+1-m)-th minimum among all
[] ,
ii
S p i j  
. By the Lipschitz continuity of
[]
,
i
S i j
and Lemma 2,
 
[ 1 ]jm
C


  
.
Applying Lemma 2 again,
 
 
[ 1] 1 [ 1 ]
max ,
j j j m
SC


   

is also Lipschitz continuous, i.e.,
 
 
1j
S


  
. By induction,
 
 
,
i
S i I

    
. Q.E.D.
Corollary 2.
     
0
lim OO
dO
d




       
    

, i.e.,
 
dO
d


is unbiased.
Proposition 3 (Lipschitz continuity). Let x = (R1, A1, p1, …, Rn, An, pn). If
 
0,
j
xj

    
with
0
, then
     
[ ] [ ]
0 1 , 0
ii
S i C i

        
.
Proof. Similar to the proof of Proposition 2 by considering the fact
 
   
01
i
Si

    
implying
 
   
0i
i
S p i

     
. Q.E.D.

17
Proposition 4. The total waiting time and the sample path cost function are not continuous in

.
Proof. As the overtime is continuous by Proposition 2, the sample path cost function is not continuous
if the total waiting time is so. Consider a system with m = 2, n = 4 and a sample path such that:
       
1 2 3 4
0A A A A
   
   
,
       
1 4 3 2
0R R R R
   
   
,
 
 
 
 
 
   
 
 
2
1 2 3 4
S S S R S
    
   
.
The patient service sequence at

is 1→3→4→2 and
 
 
   
 
   
4
42
34
1i
iW S A S R
    
   

.
In the neighborhood of

such that
 
30R

  
, the patient service sequence is 1→4→3→2 and
 
 
   
 
         
44
3 2 4 3
34
11
ii
ii
W S A S R W A A
       

                

which implies the discontinuity of
 
4
1i
iW



. Q.E.D.
4.3. Smoothed perturbation analysis
When the IPA gradient estimator is biased, various perturbation analysis methods have been
introduced to obtain unbiased gradient estimators; e.g., SPA. As shown in Fu, Gong, Hu, and Li (2013),
SPA is a sample path approach for gradient estimation based on conditional Monte Carlo, which is quite
general and in principle it can be applied to any stochastic system. The fundamental notion underlying
SPA is the use of conditional expectation to calculate the conditional probability of an event order
change in the sample path and the resultant expected effect of such a change (Fu and Hu, 1992). To
overcome the biasedness of the IPA estimator due to the discontinuity of the waiting time in our setting,
we employ SPA based on the conditional distribution of the gradient estimation on some partial sample
paths z. The SPA estimator represents the aggregate of the contributions from IPA and SPA.
The SPA relies on the analysis of the impact of event order changes on the discontinuity of the cost
function when the system parameter changes from

to

+ . Two types of event order changes are
considered in our system:
(a) type-I: the arrival event before time 0 (Rj < 0), i.e.,
   
0, 0
jj
RR

   
. The session starts
at time 0, Rj < 0 indicates that patients arrive before 0.
(b) type-II: adjacent events (ek, ek+1), i.e.,
   
kk
ee

  
. Only adjacent event order changes are
included since the non-adjacent ones exert only negligible effects in Theorem 3 in Appendix A.3.
We assume the infinitesimal perturbation
0
and consider only the right-hand derivative of

18
the SPA estimator. All results can extend to the left-hand derivative. The event order change is critical
if it incurs non-zero SPA contribution. For a type-I event order change (Rj < 0), by conditioning on the
sample path
 
 
, , \
i i j
z R p i I R  
, we have the following SPA contribution from Fu and Hu (1992):
 
 
 
,,
0
0
0
j
jj
jj
Rj
PP R DNP R
RR
fR
ggF









(6)
where
,,
jj
PP R DNP R
gg
is the effect on cost function g of event order changes with respect to Rj, DNP
with arrival immediately before 0 (Rj = 0–) denotes the degenerated nominal path, and PP with arrival
immediately after 0 (Rj = 0+) denotes the perturbed path. Further,
 
j
R
fx
and
 
j
R
Fx
are the density
function and distribution function of Rj while
   
 
1
0
00
jj j
R R j R
f F R




  
is the conditional
probability of type-I change. A type-I change is critical if the event order change leads to discontinuity
,,
0
jj
PP D RNPR
gg
and patient i is served at time 0 in DNP but not in PP, i.e. |Q0| > m,
 
0:ij
i Q A A m  
.
For a type-II event order change (ek, ek+1), event e is associated with either an arrival Rj denoted as
e~Rj, an appointment Aj denoted as e~Aj, or the v-th service completion C[v] denoted as e~C[v]:
(a) Case (ek, ek+1) with ek ~ Ai or ek+1 ~ Aj: The event order change is either impossible or not critical
and does not generate the discontinuity.
(b) Case (ek, ek+1) ~ (C[v-1], C[v]): The event order change is not critical and incurs no discontinuity.
(c) Case (ek, ek+1) ~ (Rj, C[v]) associated to the same patient: The event order change is impossible.
(d) Case (ek, ek+1) ~ (C[v], Rj): We have the following SPA contribution by conditioning on sample path
   
 
0 0 1
, , , ,..., , , \
k i i k k k j
z R p i I e t e t e R

   
:
 
 
1
,,
0
k
jk
j
k
PP k DNP k
Rt
R
t
g g f












(7)
where ek+1 occurs immediately after (before) ek at
+
k
t
in DNP (at
-
k
t
in PP),
   
   
1
j
k
jj
Rk
R k R k
f t x
fx
F y F t




with yk being the second smallest event time active at
+
k
t
. If there

19
is exactly one active event at
+
k
t
, then
k
y
. The event order change is critical if |Qk| > 0,
k
jk h
A t LAR
, and
,,
0
PP k DNP k
gg
, where
k
h
is the head of Qk, i.e.,
arg min
k
ki
iQ
h LAR


.
(e) Case (ek, ek+1) ~ (Rj’, Rj): The SPA contribution is the same as the previous case with
   
 
0 0 1
, , , ,..., , , \
k i i k k k j
z R p i I e t e t e R

   
. The event order change is critical if mk = 1 and
,,
0
PP k DNP k
gg
, where mk is the number of available providers at
-
k
t
.
(f) Case (ek, ek+1) ~ (Rj, C[v]) with different patients: The SPA contribution with sample path
   
 
0 0 1 1
, , , ,..., , , \
k i i k k k j
z R p i I e t e e t R

   
is:
 
 
1
1
1
,,
0
k
jk
jk
PP k DNP k
Rt
Rt
g g f













(8)
where ek occurs immediately before (after) ek+1 at
1-
k
t
in DNP (
+
1k
t
in PP),
   
   
1
1
11
j
k
jj
Rk
R k R k
f t x
fx
F t F t






. The event order change is critical if |Qk| > 0,
k
jk h
A t LAR
, and
,,
0
PP k DNP k
gg
.
Since
*i
A


, from Lemma 3 in Appendix A.3, we can restrict to event order changes such that
k
eE


and
1k
eE


with
 
 
* * [ ] *
,:
i i i i
E R A C i BP
  
. We can further restrict to
k
eE
and
1k
eE

with
 
*
\i
E E A


. From Theorem 1, the SPA estimator can be simplified as follows:
   
   
 
 
   
 
 
 
   
 
 
* * * *
**
0
* * 0 0 * , ,
[ ] 1 , ,
* 1 , ,
1
lim
0, , : 0 0
, , ,
, , 1
i i i i
j
k
jj
ii
i i i i PP R DNP R R R
Rk
k v k j k j k PP k DNP k
h
R k R k
R
k i k j k PP k DNP k
g A g A
IPA A R Q m i Q A A m g g f F
ft
e C E e R E Q A t LAR g g F y F t
f
e R e R E m g g




  




       



        
      
1
1
1
 
   
 
 
 
   
*
**
1
1
1
* 1 [ ] * 1 , ,
11
, , ,
j
jj
i
k
ii
Kk
kR k R k
Rk
k i k v k i k PP k DNP k
h
R k R k
t
F y F t
ft
e R e C E Q A t LAR g g F t F t

























       






1
(9)
The unbiasedness of this SPA estimator is validated in Appendix A.3, which follows the proof of
Fu and Hu (1992) and performs modifications according to our problem. Instead of directly proving the

20
correctness for our original system

, we first define an approximation system
()


that
asymptotically converges to

, then prove the correctness of the SPA estimator of
()


, and finally
prove that the SPA estimator of
()


asymptotically converges to that of

. Numerical evidence is
given in Appendix A.5 to complement the proof.
4.4. Evaluation of the discontinuity due to event order changes
The efficiency of the SGD algorithm is significantly affected by the computation of the
discontinuity
,,PP k DNP k
gg
in the derivation of the SPA estimator. This subsection proposes the fast
evaluation of the discontinuity
,,PP k DNP k
gg
, which is our original work. It relies on the properties of
the optimal real-time sequencing. First, we propose a modified LAR strategy in Proposition 5 that
extends its original counterpart.
Proposition 5. The modified LAR strategy, which modifies the LAR index of a given patient i at and
after time t by
ii
AR


such that
ii
R t R

, is optimal among all work-conserving disciplines. In
other words, the LAR strategy and modified LAR strategy are equivalent.
Proof. The modified index
ii
AR


has an effect on the schedule only at or after
i
t t R

. We
compare the two systems

and


having the same schedule up to
-t
but with Ri replaced by
i
R
at and after
t
in


. As a result, the sequence of times corresponding to the service start/completion
is the same for both systems. Further any feasible schedule of

is feasible for


and vice versa.
We compare the total cost of the two systems for a given feasible strategy π, i.e.,
   
i i i i i i
g g S A R S A R



      
. By definition,
i
St


.
(a) If
i
At


, then
i i i
S A R

,
i i i
S A R
,
i i i i
g g A R A R


    
;
(b) If
i
At


, then
i i i
A R A


,
i i i
A R A
,
0gg


.
In all cases,
gg


is equal to a constant independent of
π
. As a result, any optimal strategy of


is optimal for

and vice versa. From Proposition 1, both the LAR strategy and modified LAR
strategy are optimal for the two systems. Q.E.D.
The following result simplifies the computation of the SPA contribution. To aid our discussion, the
original sample path is called the nominal path and denoted by NP.

21
Proposition 6. Consider a critical event order change at time t°+ and let Rj be the arrival time used to
smooth the effect of event order changes. Apply the modified LAR strategy to both DNP and PP with
PP DNP
jj
R R t  
replaced by Rj beyond time t°+. Then, for DNP and NP, the following three quantities
are the same: service start times S[i]; service completion times C[i]; and service order. For DNP and PP,
S[i] and C[i] are the same, but patients are served in different orders.
Proof. By construction, patients start at the same time and in the same order in DNP and NP and hence
they have the same C[i]. From Proposition 3, S[i] and C[i] of DNP are the same as those of PP. Q.E.D.
Propositions 5 and 6 are explored in the following subsection for the efficient computation of the
discontinuity
,,k PP k DNP k
d g g
due to the order change of an arrival Rj and another event by
comparing the queues of DNP and PP at the service start time S[i] of NP.
A. Case Rj < 0
Step 1. By Proposition 5, replacing
0
PP DNP
jj
RR
by Rj < 0 does not change the optimality. As a
result, QDNP with the modified LAR strategy is equivalent to QNP.
Step 2. Let S[i] be the i-th patient starting time in NP with
[0] 0S
and
[1] 0S
. Let li be the (unique)
patient started at S[i] for i > 0.
Step 3. At time 0+, QDNP and QPP differ by two patients,
 
\
DNP PP
Q Q j

,
 
\
PP DNP
Q Q j
.
Through the LAR strategy,
jj
LAR LAR 

. Initialize dk = 0, i = 0.
Step 4. Find the first patient li’ in NP such that
',
i
lj
LAR LAR i i


, where li’ denotes the patient started
at
 
'i
S
and set
 
 
 
 
'
''
i
PP DNP
k k j l
ii
d d S LAR S LAR

    
.
Step 5. If li’ = j’, then stop. Otherwise, set j = li’ and i = i’, and then repeat Step 4.
B. Case (ek, ek+1) ~ (C[v], Rj)
Step 0. If mk > 0 or no patient starts at tk or tk = S[i] and
i
j k l
A t LAR
, then dk = 0 and stop.
Step 1. By Proposition 5, replacing
PP DNP
jjk
R R t
by Rj at or after tk+2 (i.e., the next decision epoch)
does not change the optimality. Thus, QDNP with the modified LAR strategy is equivalent to QNP.
Step 2. At time
+
k
t
, QDNP and QPP differ by two patients,
 
\
DNP PP
Q Q w

,
 
\
PP DNP
Q Q w
with
w’ = j and w = li . It should be noted that patient w’ exists in QDNP but has already been served in PP,

22
whereas patient w exists in QPP but has already been served in DNP. Initialize,
   
PP DNP
k k w k w
d t LAR t LAR


   
.
Step 3. Compare LARw and LARw’:
(i) If
ww
LAR LAR 

, then
Step 3a. Find the first li’ in NP such that
',
i
lw
LAR LAR i i


. As DNP serves li’ and PP serves w,
 
 
 
 
'
''
i
PP DNP
k k w l
ii
d d S LAR S LAR

    
.
Step 3b. If li’ = w’, then stop. Otherwise, set w = li’ and i = i’, and then repeat Step 3a.
(ii) If
ww
LAR LAR 

, then
Step 3c. Find li’ = w’ in NP and the head of queue hi’ at time
+
[ ']i
S
.
Step 3d. Let hi be the head of the queue at time
+
[]i
S
. If
'i
h


or
'i
hw
LAR LAR
,
 
 
 
 
'
''
i
PP DNP
k k w l
ii
d d S LAR S LAR

    
as DNP serves li’ and PP serves w, and stop.
Step 3e. Otherwise,
 
 
 
 
''
''
ii
PP DNP
k k h l
ii
d d S LAR S LAR

    
as DNP serves li’ and PP serves
the head of the queue hi’, set w’ = hi’ and i = i’, and repeat Step 3c.
C. Case (ek, ek+1) ~ (Rj’, Rj)
If
1
k
m
, then dk = 0 and stop. Otherwise, dk can be determined similarly as Steps 1-3 of Case B
by starting with
 
\
DNP PP
Q Q j
and
 
\
PP DNP
Q Q j

.
D. Case (ek, ek+1) ~ (Rj, C[v])
If |Qk| = 0, then dk = 0 and stop. Otherwise, by Proposition 5, replacing
1
PP DNP
jjk
R R t 

by Rj at
and after tk+1 does not change the optimality. Let li be the patient started at tk+1 in NP, i.e., S[i] = tk+1. If
i
lj
, then stop. Otherwise, the rest remains the same as Case B by starting with
 
\
DNP PP k
Q Q h
and
 
\
PP DNP
Q Q j
, where
k
h
is the head of QNP at
-
k
t
.
4.5. Extension to no-shows
This subsection extends the discussion to the case with no-shows. The sample path cost function
becomes
     
Y
i
iI
g W O
   

  

, where
 
 
max max , max
YN
ii
i I i I
O C A U



, IY denotes the set of
show-up patients, and IN is the set of no-shows. The simulation of the system with no-shows and the

23
estimation of its sample path cost function, its sample path gradient, and its SPA contribution are exactly
identical to the case without no-shows by simply replacing set I by IY. The only exception is the overtime
O and its sample path gradient that becomes:
 
 
 
max ,max
N
Ki
iI
O t A U T



  
(10)
   
*
**
1, if max and
1, if , max ,
0, otherwise
N
N
K i K i i
iI
N
i i K i
iI
t T A U e C i BP
dO i I A A t T A U
d





    


     




(11)
All results of Section 4.2 hold for the no-show case. In particular, the sample path gradient of O is
unbiased. Thus, the correctness of the SPA estimator for the total waiting time relies on
 
 
 
Pr
Y
YY
I
g I g I




 

and
 
 
 
Pr
Y
Y
Y
I
d g I
dg I
dd





   

. For sample paths where
patient i* is a no-show,
 
0, :
Y
YY
d g I I i I
d





  
. For sample paths where i* shows up, results of
Sections 4.2 to 4.4 and Appendix A.3 apply to the unbiased estimation of
 
Y
d g I
d




. To sum up,
   
 
     
 
 
 
 
 
 
 
 
 
*
1
**
0
* * * * 0 * *
0
1
1 * 1 * 1
0
1
1
lim
0 lim , 0 0
, lim , ,
i
k
ii
i i i i i i R
Y
K
k k i k k k i k k t k
k
g A g A
IPA A R g A R B R g A f
iI
e E e E g A t t B g A t t f t




  



  




     








       





1
1
1
(12)
5. Numerical experiments
A series of experiments are performed to investigate three key issues: (i) the effectiveness of the
SGD algorithm; (ii) the impact of various factors on the optimal appointment schedule; and (iii) the
benefits of the optimal strategy and schedule. All programs are coded in C++ and run on the Intel Xeon
E5-2670 2.60 GHz CPU and 64 GB RAM under Linux.
5.1. Experimental setting
Real data collected from the partner hospital is used to support the experiments. We have 147,000
valid records of patient visits from January 2013 to May 2016, which are also used in our prior research
(i.e., Pan, Geng, and Xie, 2020). The basic experiment is set for twenty patients and two providers in a

24
session with eight time periods, i.e., n = 20, m = 2, and T = 8 periods. The service duration is lognormally
distributed with an expected value of μs = 1 period and a standard deviation of σs = 0.5. Unpunctuality
in our partner clinic follows a normal distribution N(-0.5, 42) truncated by [-U, U] with U = 3. The
normal distribution and the empirical distribution of the actual unpunctuality are given in Fig. C-1 (see
Appendix C). The normal distribution is widely used in the literature to describe unpunctuality, e.g.,
Cayirli, Veral, and Rosen (2006, 2008), Klassen and Yoogalingam (2014), Zhu, Chen, Leung, and Liu
(2018), Deceuninck, Fiems, and De Vuyst (2018), and Jiang, Tang, and Yan (2019). In the base case,
the no-show probability is set as Pns = 0.2 and the overtime cost rate is α = 15, which is similar to
previous studies, e.g., Chen and Robinson (2014), Jiang, Tang, and Yan (2019), Pan, Geng, and Xie
(2020). The following sensitivity analyses are performed to explore the behavior of the optimal
appointment schedule (note that the values of the base case are bolded):
(a) Impact of unpunctuality factors: (i) expectation of unpunctuality with μu = {-3, -1.5, -0.5, 1, 2.5};
(ii) standard deviation of unpunctuality with σu = {1, 2, 3, 4, 5}; and (iii) allowance in UTW with
U = {1, 2, 3, 4, 5}.
(b) Impact of other uncertainty factors: (i) no-show probability with Pns = {0, 0.1, 0.2, 0.3, 0.4}; and
(ii) standard deviation of service duration with σs = {0.05, 0.25, 0.5, 0.75, 1}.
(c) Impact of system factors: (i) overtime cost rate with α = {5, 10, 15, 20, 25}; (ii) number of providers
with m = {1, 2, 3, 4, 5}, correspondingly T = {16, 8, 5.33, 4, 3.2}; and (iii) number of patients with
n = {10, 20, 30, 40, 50}, correspondingly T = {4, 8, 12, 16, 20}.
In addition to these environmental parameters, key algorithmic parameters are also carefully
selected to validate the effectiveness of the SGD algorithm. We use D realizations to estimate the
improving direction in Step 2(a) of SGD. We test with D = {1, 10, 100, 1000} under a fixed number
ND =106 and find D = 10 to achieve the best solution. The conditions
2
( ) ( ) ( ) ( )
11
0, lim 0, ,
q q q q
qqq
   

 

     


should be satisfied for the step size in Step 2(b),
thereby ensuring the convergence of SGD for a continuously-differentiable function (Theorem 6.3.1 in
Kushner and Clark, 1978). We test with
0

= {0.1, 1, 5, 10} and obtain the best results with
0

= 5.
The effectiveness of SGD is shown in extensive experiments by verifying that it converges to the
same objective value, regardless of the initial solution, which is similar to Zhang and Xie (2015). As

25
SGD is known to converge to the local optimum, the effectiveness is established if SGD converges to
the same objective value irrespective of the initial solution used. For each of the 33 instances in the
sensitivity analysis, SGD starts from 50 random initial solutions. Finally, 50 final solutions are obtained
under each instance and the following indicators are used to support the effectiveness of SGD:
(a) Dispersion of the stopping solutions A(N) is the sum of the standard deviations of 50 stopping
solutions; i.e.,
 
 
Disp N
i
iI A



. (The dispersion of the initial solutions is 15.)
(b) Average of 50 objective values (Obj) is computed; i.e.,
 
 
 
Avg N
Obj

A
.
(c) Standard deviation of 50 objective values is calculated; i.e.,
 
 
 
Std N
Obj

A
.
(d) Coefficient of variation of the objective value is determined by the standard deviation and the
average; i.e.,
 
 
 
 
 
 
CV Std Avg NN
Obj Obj

AA
.
These indicators are shown in Table C-1 (see Appendix C). For the majority of the instances, “Disp”
of the stopping solutions is in close proximity to zero (≤0.5), which reveals that 50 stopping solutions
are quite similar. Moreover, “Std” of the objective value is quite close to zero (≤0.01) for all the
instances and “CV” is almost equal to zero (≤0.05%). Thus, the variability of the objective value is quite
low and negligible, which implies that SGD converges to the same objective value under different
initializations. Therefore, SGD has been demonstrated to effectively achieve optimal solutions.
The CPU time of one single run is about 2.5 min for a 20-patient instance and 5.8 min for a 50-
patient instance. Notwithstanding its dependence on the number of patients, the CPU time is desirably
short, thus reflecting the efficiency of SGD. The CPU time is also affected by the total number of
replications and can be shortened by reducing ND, e.g., ND =105.
5.2. Sensitivity analyses
This subsection performs the sensitivity analyses to draw the insights for managing unpunctual
appointments: (i) analyzing the properties of the appointment schedule, and (ii) exploring the impacts
of various factors on the appointment times.
5.2.1. Properties of optimal appointment schedule
We consider the appointment scheduling with unpunctuality (AS-U) under the AO strategy and
LAR strategy, denoted respectively by AS-U(AO) and AS-U(LAR). The AO strategy follows the
definition in prior studies (e.g., Jiang, Tang, and Yan, 2019) and is not based on the work-conserving

26
discipline, while the LAR strategy is achieved under the work-conserving discipline. We also
benchmark AS-U with appointment scheduling with punctuality (AS-P). AS-P is paired with AO since
both the AO and LAR strategies are exactly equivalent for the punctual system, where the parameters
of unpunctuality are set as zero, i.e., μu = 0, σu = 0, U = 0. AS-U(LAR) is determined by our SGD
algorithm with the SPA estimator, whereas both AS-P and AS-U(AO) are derived from Pan, Geng, and
Xie (2020) that use SGD with the IPA estimator. Note that the effectiveness of SGD with the IPA
estimator has been validated in Pan, Geng, and Xie (2020).
The three schedules, AS-P, AS-U(AO) and AS-U(LAR), are compared under different four cases
with single/multiple providers and with/out no-shows. Each schedule for each case is determined by the
corresponding algorithm and their performance is evaluated by a long discrete-event simulation of 106
replications. Table 2 presents the average performance metrics of the three schedules under the four
typical cases; i.e., total cost (TC), total patient waiting time (WT), provider overtime (OT). On the one
hand, the result reveals that patient unpunctuality does jeopardize the system performance since both
AS-U(AO) and AS-U(LAR) lead to more TC than AS-P. It is necessary to consider unpunctuality in the
design of the appointment schedule. On the other hand, compared with AS-U(AO), AS-U(LAR) can
achieve a greater cost reduction, whose WT and OT are quite close to those of AS-P. Therefore, the
optimal strategy (i.e., LAR) can help to alleviate the negative influence of unpunctuality.
The features of the optimal appointment intervals are also investigated, which is the time interval
between two consecutive appointments; i.e., ai = Ai+1 − Ai. Fig. 2 illustrates the optimal appointment
intervals under the four typical cases. We define n1 as the number of patients appointed in the initial
block; i.e., initial block size. The specific observations are as follows:
(a) 1-provider without no-shows: The intervals of both AS-P and AS-U(AO) exhibit the same
dome pattern, whereas AS-U(AO) has the overbooking (see ai≈0) at the beginning of the session (i.e.,
n1 = 2) due to patient unpunctuality. However, AS-U(LAR) is quite different from AS-P and AS-U(AO).
On the one hand, the intervals of AS-U(LAR) appear to be a multi-peak pattern. On the other hand, the
overbooking exists in both the initial block and the final block.
(b) 1-provider with no-shows: Similar to Case (a), AS-P and AS-U(AO) have the dome pattern,
while AS-U(LAR) follows the multi-peak pattern. No-shows lead to shorter appointment intervals for
the three schedules to reduce provider idling. Especially, AS-U(AO) appoint more patients in the initial
block than AS-P. AS-U(LAR) still appoints multiple patients in both the initial block and the final block.

27
(c) 2-provider without no-shows: Due to the multiple providers, the pattern and the block size
become different from those in Case (a). On the one hand, AS-P and AS-U(AO) tend to be a slightly
zigzag pattern with the more obvious overbooking in the initial block. On the other hand, AS-U(LAR)
exhibits the single-peak pattern with more patients appointed in the initial block and the final block.
(d) 2-provider with no-shows: The three schedules have a similar pattern to those of Case (c).
AS-P and AS-U(AO) have the slightly zigzag pattern, whereas AS-U(LAR) follows the single-peak
pattern. No-shows are taken into account in this case, which leads to more obvious overbooking in the
initial block. Further, AS-U(LAR) appoints more patients in the final block.
Table 2 Average performance metrics of different schedules under different settings
m
Pns
TC
WT
OT
AS-P
AS-U
(AO)
AS-U
(LAR)
AS-P
AS-U
(AO)
AS-U
(LAR)
AS-P
AS-U
(AO)
AS-U
(LAR)
1
0.0
94.6
133.7
100.3
23.6
44.7
28.5
4.7
5.9
4.8
1
0.2
41.0
83.9
42.9
19.5
46.4
21.0
1.4
2.5
1.5
2
0.0
50.8
90.4
53.6
10.8
31.9
13.5
2.7
3.9
2.7
2
0.2
23.3
70.7
24.1
9.3
34.5
10.1
0.9
2.4
0.9
(a) 1-provider without no-shows (b) 1-provider with no-shows
(c) 2-provider without no-shows (d) 2-provider with no-shows
Fig. 2. Optimal appointment intervals under different settings
The above findings show that the major distinction between these schedules is the dome pattern
under the AO strategy and the peak pattern under the smallest LAR strategy. The dome pattern is
recognized in the prior studies (e.g., Jiang, Tang, and Yan, 2019), but the peak pattern is observed in
this paper. The possible explanations are given as follows. Firstly, compared with AS-U(AO), AS-
0.0
1.0
2.0
3.0
1 3 5 7 9 11 13 15 17 19
ai
i
m=1, Pns=0 AS-P
AS-U(AO)
AS-U(LAR)
0.0
1.0
2.0
3.0
1 3 5 7 9 11 13 15 17 19
ai
i
m=1, Pns=0.2 AS-P
AS-U(AO)
AS-U(LAR)
0.0
1.0
2.0
3.0
1 3 5 7 9 11 13 15 17 19
ai
i
m=2, Pns=0 AS-P
AS-U(AO)
AS-U(LAR)
0.0
1.0
2.0
3.0
1 3 5 7 9 11 13 15 17 19
ai
i
m=2, Pns=0.2 AS-P
AS-U(AO)
AS-U(LAR)

28
U(LAR) appoints more patients at the beginning of the session to fill the system for reducing provider
idleness caused by patient unpunctual arrivals. If AS-U(AO) appoints the same number of patients as
AS-U(LAR), the AO strategy will result in longer total patient waiting time than the smallest LAR
strategy because some early patients have to get service after the prior-appointed patients. Secondly,
some “peaks” are necessary for the AS-U(LAR) schedule to balance patient waiting time. For the short
intervals, it is more likely to incur longer patient waiting time since patients arrive more intensively.
Hence, following some short intervals, peaks will reserve a long interval between patients, which can
reduce patient waiting. Especially for the single-provider case, each peak allows the system to absorb
any patient queue that might have built up because of late arrivals and/or long service times, and not
pass on the delays to patients scheduled later. Thirdly, at the end of the session, patients are usually
assigned to the similar appointment time (see ai≈0) under AS-U(LAR) for reducing provider overtime
as much as possible. But if AS-U(AO) does so, patient waiting time will be much longer due to patient
unpunctuality. Therefore, the AS-U(LAR) schedule can better balance patient waiting time and provider
overtime when patients arrive unpunctually.
To conclude, managerial insights are provided for hospital managers to manage unpunctual patients
in a multi-provider setting with no-shows. Firstly, to reduce the negative influence of patient
unpunctuality as much as possible, it is significant for managers to explore the appointment schedule
with the optimal real-time sequencing strategy (i.e., the smallest LAR strategy). Secondly, the number
of providers and the sequencing strategy simultaneously determine the pattern of the appointment
intervals: (i) Based on the same sequencing strategy, multiple providers incur a zigzag pattern whereas
single provider generates a dome pattern; (ii) Under the fixed number of providers, the smallest LAR
strategy leads to a multi-peak or single-peak pattern whereas the AO strategy results in a dome or zigzag
pattern. The last but not the least, the length of the appointment intervals is quite sensitive to the degree
of both unpunctuality and no-shows: (i) a large initial block size is set to ease the negative impact of
unpunctuality; (ii) shorter intervals are created to reduce provider idling caused by no-shows.
5.2.2. Impact of various factors
In the following experiments, the characteristics of schedule AS-U(LAR) are explored in the multi-
provider setting with no-shows. Fig. 3 illustrates the optimal appointment times under various factors,
while Table 3 presents the corresponding performance metrics. The findings are described as follows.
Impact of unpunctuality factors (Fig. 3a to Fig. 3c): The increase of the expectation of

29
unpunctuality, the standard deviation of unpunctuality or the allowance in UTW implies that patients
tend to arrive more unpunctually, which inevitably leads to an increase in TC. To relieve the side effects
Table 3 Performance metrics of schedule AS-U(LAR) under various factors
Factors
TC
WT
OT
Factors
TC
WT
OT
Factors
TC
WT
OT
Factors
TC
WT
OT
μu
-3.0
22.8
9.1
0.9
U
1
23.8
9.4
1.0
σs
0.05
19.2
8.7
0.7
m
1
42.9
21.0
1.5
-1.5
23.6
9.7
0.9
2
23.9
9.8
0.9
0.25
20.5
9.0
0.8
2
24.1
10.1
0.9
-0.5
24.1
10.1
0.9
3
24.1
10.1
0.9
0.50
24.1
10.1
0.9
3
20.8
8.7
0.8
1.0
24.9
10.6
1.0
4
24.6
10.9
0.9
0.75
28.8
11.4
1.2
4
20.4
8.4
0.8
2.5
25.7
11.0
1.0
5
27.3
14.2
0.9
1.00
33.7
12.4
1.4
5
20.7
6.7
0.9
σu
1
23.2
10.6
0.8
Pns
0.0
53.6
13.5
2.7
α
5
13.5
7.3
1.2
n
10
15.3
5.5
0.6
2
23.7
9.8
0.9
0.1
38.3
12.9
1.7
10
19.2
8.9
1.0
20
24.1
10.1
0.9
3
24.0
10.0
0.9
0.2
24.1
10.1
0.9
15
24.1
10.1
0.9
30
34.4
16.6
1.2
4
24.1
10.1
0.9
0.3
13.6
6.9
0.4
20
28.8
11.0
0.9
40
45.5
23.7
1.4
5
24.2
10.2
0.9
0.4
7.0
4.3
0.2
25
33.1
11.7
0.9
50
57.2
31.7
1.7
(a) Expectation of unpunctuality (b) Standard deviation of unpunctuality
(c) Allowance of UTW (d) No-show probability
(e) Standard deviation of service durations (f) overtime cost rate
0
2
4
6
8
1 3 5 7 9 11 13 15 17 19
Ai
i
μu=-3.0
μu=-1.5
μu=-0.5
μu=1.0
μu=2.5
0
2
4
6
8
1 3 5 7 9 11 13 15 17 19
Ai
i
σu=1
σu=2
σu=3
σu=4
σu=5
0
2
4
6
8
1 3 5 7 9 11 13 15 17 19
Ai
i
U=1
U=2
U=3
U=4
U=5
0
2
4
6
8
1357911 13 15 17 19
Ai
i
Pns=0.0
Pns=0.1
Pns=0.2
Pns=0.3
Pns=0.4
0
2
4
6
8
1 3 5 7 9 11 13 15 17 19
Ai
i
σs=0.05
σs=0.25
σs=0.50
σs=0.75
σs=1.00
0
2
4
6
8
1 3 5 7 9 11 13 15 17 19
Ai
i
α=5
α=10
α=15
α=20
α=25

30
(g) Number of providers (h) Number of patients
Fig. 3. Optimal appointment times of schedule AS-U(LAR) under various factors
resulted from patient unpunctuality (especially late arrivals) on the system performance, the optimal
appointment times Ai are set earlier to accommodate more unpunctual arrivals.
Impact of other uncertainty factors (Fig. 3d to Fig. 3e): No-show behaviors and random service
durations represent the other uncertainty factors besides unpunctuality. On the one hand, under a fixed
number of patients, a larger no-show probability indicates that fewer patients will come to get service.
Hence, the appointment times are generated earlier to make patients arrive more compactly to reduce
TC, WT and OT. On the other hand, an increase in the standard deviation of service durations leads to
a more stochastic service process, which negatively affects all the performance metrics. To reduce its
negative impact, the appointment times are created slightly earlier.
Impact of system factors (Fig. 3f to Fig. 3h): We explore the effects of the overtime cost rate,
number of providers, and number of patients. First, as the overtime cost rate increases, the appointment
times become more compact to minimize OT at the sacrifice of TC and WT. Second, when more
providers become available, the earlier appointment times can make the best use of providers to reduce
all the performance metrics. Third, the change of the number of patients appears to exert negligible
effects on the tendency of appointment times. Thus, our schedule under an appropriate number of n
(e.g., n = 50) can be used in the case of random patient requests for appointments.
5.3. Performance comparison
This subsection performs the comparison of real-time sequencing strategies and appointment
schedules: (i) assessing the benefits of the smallest LAR strategy by comparing various strategies, and
(ii) comparing our schedule and those of Jiang, Tang, and Yan (2019) and Pan, Geng, and Xie (2020).
These strategies and schedules are evaluated by a long discrete-event simulation of 106 replications.
5.3.1. Benefits of optimal real-time sequencing strategy
Since the smallest LAR strategy is our novel contribution, the need remains to assess its benefits.
0
5
10
15
20
1357911 13 15 17 19
Ai
i
m=1
m=2
m=3
m=4
m=5
0
5
10
15
20
1 6 11 16 21 26 31 36 41 46
Ai
i
n=10
n=20
n=30
n=40
n=50

31
A total of 33 instances are used to evaluate the performance of various strategies under the same
schedule AS-U(LAR). Strategies such as LAR, FIFO, AO, and BQ(δ) are compared, where BQ(δ) is
similar to AO, but patients who are late more than δ periods are rescheduled to the back of the queue
(Klassen and Yoogalingam, 2014). LAR and FIFO are based on the work-conserving discipline, but AO
and BQ(δ) are not. Table 4 gives the performance metrics, including the total cost (TC) assessed by
LAR, gap (%) =
 
1 100
LAR Other
TC TC
, total patient waiting (WT), and provider overtime (OT).
Table 4 Performance metrics of different real-time sequencing strategies under schedule AS-U(LAR)
Factors
TCLAR
GAP (%)
WT
OT
FIFO
BQ
AO
LAR
FIFO
BQ
AO
LAR
FIFO
BQ
AO
μu
-3.0
22.8
5.3
27.6
72.3
9.1
10.4
13.0
35.6
0.9
0.9
1.2
3.1
-1.5
23.6
4.4
28.5
70.9
9.7
10.8
14.1
36.0
0.9
0.9
1.3
3.0
-0.5
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
1.0
24.9
3.3
30.7
68.6
10.6
11.5
16.1
36.6
1.0
1.0
1.3
2.9
2.5
25.7
2.7
32.3
67.1
11.0
11.7
17.3
36.8
1.0
1.0
1.4
2.7
σu
1
23.2
2.4
27.4
71.7
10.6
11.2
14.1
36.0
0.8
0.8
1.2
3.1
2
23.7
3.7
28.9
70.5
9.8
10.7
14.0
35.1
0.9
0.9
1.3
3.0
3
24.0
3.8
29.1
70.3
10.0
11.0
14.6
36.0
0.9
0.9
1.3
3.0
4
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
5
24.2
3.8
29.7
70.0
10.2
11.2
15.3
36.6
0.9
0.9
1.3
2.9
U
1
23.8
0.4
6.5
31.0
9.4
9.6
10.3
14.7
1.0
1.0
1.0
1.3
2
23.9
2.1
16.8
56.0
9.8
10.4
12.2
24.1
0.9
0.9
1.1
2.0
3
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
4
24.6
5.5
41.5
76.6
10.9
12.4
18.1
48.9
0.9
0.9
1.6
3.8
5
27.3
6.4
44.4
78.6
14.2
16.1
22.5
63.0
0.9
0.9
1.8
4.3
Pns
0.0
53.6
2.5
13.7
45.7
13.5
14.9
17.6
35.1
2.7
2.7
3.0
4.2
0.1
38.3
2.9
22.3
59.0
12.9
14.0
18.1
38.6
1.7
1.7
2.1
3.7
0.2
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
0.3
13.6
5.5
37.8
79.5
6.9
7.7
11.2
31.9
0.4
0.4
0.7
2.3
0.4
7.0
7.1
47.4
86.5
4.3
4.9
7.9
27.0
0.2
0.2
0.4
1.7
σs
0.05
19.2
5.4
36.3
76.2
8.7
9.8
13.7
36.2
0.7
0.7
1.1
3.0
0.25
20.5
5.0
34.2
74.3
9.0
10.1
13.9
35.8
0.8
0.8
1.2
2.9
0.50
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
0.75
28.8
2.9
24.5
64.9
11.4
12.2
16.1
36.9
1.2
1.2
1.5
3.0
1.00
33.7
2.4
20.6
60.1
12.4
13.2
17.0
37.5
1.4
1.4
1.7
3.1
α
5
13.5
5.8
33.3
73.3
7.3
8.1
11.7
33.3
1.2
1.2
1.7
3.4
10
19.2
4.7
30.6
70.8
8.9
9.8
13.5
34.7
1.0
1.0
1.4
3.1
15
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
20
28.8
3.3
28.2
69.6
11.0
12.0
15.9
37.3
0.9
0.9
1.2
2.9
25
33.1
3.0
27.2
69.2
11.7
12.7
16.6
37.9
0.9
0.9
1.2
2.8
m
1
42.9
1.6
12.4
51.8
21.0
21.7
24.0
44.0
1.5
1.5
1.7
3.0
2
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
3
20.8
5.2
33.5
70.8
8.7
9.9
12.8
33.2
0.8
0.8
1.2
2.5

32
4
20.4
3.5
35.3
70.2
8.4
9.2
12.7
32.0
0.8
0.8
1.3
2.4
5
20.7
0.5
43.0
69.5
6.7
6.8
13.6
29.6
0.9
0.9
1.5
2.5
n
10
15.3
1.5
28.3
67.2
5.5
5.7
7.5
16.2
0.6
0.6
0.9
2.0
20
24.1
3.9
29.5
70.1
10.1
11.1
15.0
36.3
0.9
0.9
1.3
3.0
30
34.4
4.6
26.1
67.5
16.6
18.2
23.3
56.5
1.2
1.2
1.6
3.3
40
45.5
4.8
24.1
65.7
23.7
26.0
32.5
78.0
1.4
1.4
1.8
3.6
50
57.2
4.8
22.7
64.1
31.7
34.5
42.6
100.5
1.7
1.7
2.1
3.9
Min.
7.0
0.4
6.5
31.0
4.3
4.9
7.5
14.7
0.2
0.2
0.4
1.3
Max.
57.2
7.1
47.4
86.5
31.7
34.5
42.6
100.5
2.7
2.7
3.0
4.3
Avg.
26.3
3.8
29.0
68.0
11.2
12.2
16.1
38.6
1.0
1.0
1.4
3.0
Overall, the ranking of the strategies is as follows: LAR (most favorable), FIFO, BQ(1), and AO
(least favorable). Table 4 shows that LAR improves AO by 31% to 86.5%, BQ(1) by 6.5% to 47.4%,
and FIFO by 0.4% to 7.1%. The results imply that LAR always performs the best since it generates
shorter WT and OT than the other strategies. The reasons are as follows: (i) It is based on the work-
conserving discipline which aims at reducing provider idleness/overtime as much as possible; (ii) Given
the definition of waiting time in the setting of patients’ unpunctuality, choosing the smallest LAR-
indexed patient can effectively minimize the waiting time of each individual patient, which leads to the
minimization of the average waiting time. Specifically, compared with FIFO, the two strategies are
based on the work-conserving discipline and hence have the identical OT. But LAR could incur shorter
WT than FIFO since the major purpose of LAR is to minimize patient waiting. The performance of
FIFO is inferior to LAR and the execution of FIFO disregards the goal of appointment scheduling,
which indirectly encourages patients to arrive unpunctually. Compared with AO and BQ, the benefit of
LAR is more obvious. AO and BQ are not based on the work-conserving discipline, which leads to
longer OT than LAR. AO also incurs the longest WT since early patients have to get service after late
patients. Some prior studies use AO for preserving tractability (Jouini and Benjaafar, 2009; Deceuninck,
Fiems, and De Vuyst, 2018; Jiang, Tang, and Yan, 2019), but AO is the worst among these strategies.
BQ performs better than AO because it imposes the penalty on late patients. The cost of BQ relies on
parameter δ: δ = {0.5, 1.0, 1.5, 2.0} is tested and δ = 1 helps BQ achieve a lower TC. Finally, the
smallest LAR strategy is considered as the most favorable one for scheduling unpunctual patients.
5.3.2. Strengths of optimal appointment schedule
We benchmark our optimal schedule AS-U(LAR) (denoted as OPT) with the single-provider
schedule AS-U(AO) (denoted as JIANG) of Jiang, Tang, and Yan (2019) and the multi-provider
schedule AS-U(AO) (denoted as PAN) of Pan, Geng, and Xie (2020). OPT is derived from our SGD

33
algorithm with the SPA estimator under the smallest LAR strategy, whereas JIANG is determined by
BD under AO, and PAN is solved by SGD with the IPA estimator under AO. In Jiang, Tang, and Yan
(2019), BD is used for solving only the single-provider instance; hence, the performance of OPT and
JIANG is compared in a single-provider setting. SGD is more flexible than BD and is suitable for both
the single and multiple providers, thus OPT and PAN are compared under all the 33 instances in the
sensitivity analyses. OPT is evaluated by LAR, whereas JIANG and PAN are evaluated by AO since
the authors adopt AO. The performance metrics of the three schedules are given in Tables 5 and C-2
(Table C-2 is in Appendix C), including WT, OT, TC, and gap (%) =
 
1 100
OPT Other
TC TC
. Overall,
OPT not only improves PAN by 34.7% to 68.7% with an average of 51%, but also improves JIANG by
12.4% to 68.0% with an average of 47.9%. OPT desirably incurs shorter WT and OT for all the instances
than JIANG and PAN. Further, the benefit of OPT increases with more unpunctual patient arrivals.
Table 5 Performance metrics of schedule OPT and PAN
Factors
WT
OT
TC
GAP(%)
PAN
OPT
PAN
OPT
PAN
OPT
μu
-3.0
36.5
9.1
2.3
0.9
70.9
36.5
48.5
-1.5
35.3
9.7
2.4
0.9
70.8
35.3
50.1
-0.5
34.5
10.1
2.4
0.9
70.7
34.5
51.2
1.0
33.2
10.6
2.5
1.0
70.5
33.2
53.0
2.5
31.7
11.0
2.6
1.0
70.2
31.7
54.8
σu
1
39.0
10.6
2.1
0.8
70.2
39.0
44.5
2
36.1
9.8
2.3
0.9
70.4
36.1
48.7
3
35.0
10.0
2.4
0.9
70.6
35.0
50.5
4
34.5
10.1
2.4
0.9
70.7
34.5
51.2
5
34.3
10.2
2.4
0.9
70.8
34.3
51.5
U
1
14.3
9.4
1.2
1.0
32.9
14.3
56.7
2
24.6
9.8
1.7
0.9
50.3
24.6
51.2
3
34.5
10.1
2.4
0.9
70.7
34.5
51.2
4
44.8
10.9
3.2
0.9
92.7
44.8
51.6
5
55.3
14.2
4.0
0.9
115.7
55.3
52.2
Pns
0.0
31.9
13.5
3.9
2.7
90.4
31.9
64.7
0.1
34.7
12.9
3.2
1.7
82.6
34.7
58.0
0.2
34.5
10.1
2.4
0.9
70.7
34.5
51.2
0.3
32.6
6.9
1.6
0.4
56.5
32.6
42.4
0.4
27.4
4.3
1.0
0.2
42.0
27.4
34.7
σs
0.05
32.7
8.7
2.3
0.7
66.7
32.7
50.9
0.25
33.2
9.0
2.3
0.8
67.7
33.2
50.9
0.50
34.5
10.1
2.4
0.9
70.7
34.5
51.2
0.75
36.0
11.4
2.6
1.2
74.6
36.0
51.7
1.00
37.4
12.4
2.8
1.4
78.7
37.4
52.5
α
5
26.2
7.3
3.4
1.2
43.4
26.2
39.7
10
31.6
8.9
2.7
1.0
58.2
31.6
45.7
15
34.5
10.1
2.4
0.9
70.7
34.5
51.2
20
36.6
11.0
2.3
0.9
82.5
36.6
55.7
25
38.1
11.7
2.2
0.9
93.8
38.1
59.3
1
46.4
21.0
2.5
1.5
83.9
46.4
44.7

34
2
34.5
10.1
2.4
0.9
70.7
34.5
51.2
m
3
29.7
8.7
2.5
0.8
67.1
29.7
55.7
4
26.5
8.4
2.6
0.8
65.2
26.5
59.3
5
24.0
6.7
2.7
0.9
63.9
24.0
62.5
10
14.3
5.5
2.1
0.6
45.6
14.3
68.7
20
34.5
10.1
2.4
0.9
70.7
34.5
51.2
n
30
54.5
16.6
2.8
1.2
96.6
54.5
43.6
40
74.0
23.7
3.3
1.4
122.8
74.0
39.8
50
92.3
31.7
3.8
1.7
149.3
92.3
38.2
Min.
14.3
4.3
1.0
0.2
32.9
14.3
34.7
Max.
92.3
31.7
4.0
2.7
149.3
92.3
68.7
Avg.
36.4
11.2
2.5
1.0
73.8
36.4
51.0
6. Conclusions and perspectives
This paper proposes a new analytical framework for the multi-provider outpatient scheduling with
patient unpunctuality and no-shows. The goal is to balance patient waiting time and provider utilization
by jointly optimizing the appointment schedule and real-time sequencing strategies. An optimal real-
time sequencing strategy is first established to be the smallest LAR strategy, under which the
appointment schedule is derived from a simulation optimization approach with unbiased gradient
estimators. The technique of SPA is used to overcome the discontinuity of the sample path cost function.
Extensive experiments reveal that this algorithm can converge to the same objective value whatever the
initialization is. The optimal appointment schedule under the smallest LAR strategy is shown to be a
peak pattern to better balance patient waiting and provider overtime, which differs from the dome-
shaped schedule without the smallest LAR strategy. The benefits of our schedule are fully validated by
the comparison with schedules in prior studies, e.g., Jiang, Tang, and Yan (2019) and Pan, Geng, and
Xie (2020). A significant cost reduction (51% on average and max 68.7%) is achieved if the schedule
of Pan, Geng, and Xie (2020) is switched to our schedule. Further, the benefits of the smallest LAR
strategy are assessed by comparing with other strategies, i.e., FIFO, BQ, and AO. Numerical results
show LAR outperforms these strategies, e.g., LAR improves AO by 31% to 86.5%. Our schedule and
strategy performs better since it incurs shorter patient waiting and provider overtime.
This paper provides managers with insightful guidance on how to determine the appointment
schedule and real-time sequencing strategies for unpunctual patients. On the one hand, managers should
adopt the smallest LAR strategy instead of AO or FIFO. AO performs the worst and FIFO potentially
encourages unpunctuality. The smallest LAR strategy behaves well from two aspects: (i) It is based on
the work-conserving discipline which aims at reducing provider idleness/overtime as much as possible;
(ii) Given the definition of waiting time in the setting of patients’ unpunctuality, choosing the smallest

35
LAR-indexed patient can effectively minimize the waiting time of each individual patient, which leads
to the minimization of the average waiting. On the other hand, when designing the appointment schedule,
managers should pay more attention to the system setting. For example, the pattern of the appointment
schedule is determined by the number of providers and the real-time sequencing strategy. And the length
of the appointment intervals is quite sensitive to the degree of unpunctuality and no-shows.
Future research can be extended in several directions. One direction is to relax the work-conserving
assumption. As shown by Samorani and Ganguly (2016), providers could stay idle to wait for an
appointment when early patients are waiting. Another direction concerns the incorporation of more
factors such as walk-in patients and service cancellation. These factors will make the systems better
resemble the realistic contexts and desirably benefit the design of the appointment schedule.
Funding
This research is sponsored by Research Grants from National Natural Science Foundation of China
(72031007, 71972129, 71671111), and “Shuguang Program” (20SG13) supported by Shanghai
Education Development Foundation and Shanghai Municipal Education Commission.
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