A Review of Optimization Studies for System Appointment Scheduling

Citation: Niu, T.; Lei, B.; Guo, L.;
Fang, S.; Li, Q.; Gao, B.; Yang, L.; Gao,
K. A Review of Optimization Studies
for System Appointment Scheduling.
Axioms 2024, 13, 16. https://doi.org/
10.3390/axioms13010016
Academic Editors: Omar Rojas,
Guillermo Sosa and Manuel
Arana-Jimenez
Received: 1 November 2023
Revised: 13 December 2023
Accepted: 22 December 2023
Published: 25 December 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
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axioms
Review
A Review of Optimization Studies for System
Appointment Scheduling
Tiantian Niu 1 , Bingyin Lei 1, Li Guo 2, Shu Fang 3, Qihang Li 4, Bingrui Gao 5, Li Yang 6 and Kaiye Gao 7,8,*
1 School of Economics and Management, Beijing Information Science & Technology University,
Beijing 100192, China
2 School of Economics and Management, University of Science & Technology Beijing, Beijing 100083, China
3 School of Economics and Management, Beijing University of Technology, Beijing 100124, China
4 School of Mathematics and Information Science, Zhongyuan University of Technology,
Zhengzhou 450007, China
5 School of Education, Johns Hopkins University, Baltimore, MD 21218, USA
6 School of Reliability and Systems Engineering, Beihang University, Beijing 100083, China
7 School of Economics and Management, Beijing Forestry University, Beijing 100091, China
8 Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University,
Hong Kong 999077, China
* Correspondence: kygao@foxmail.com or kygao@bistu.edu.cn
Abstract: In the face of an increasingly high-demand environment for outpatients, achieving a
balance between allocation of limited medical resources and patient satisfaction has considerable
social and economic benefits. Therefore, appointment scheduling (AS) system operation is used
in clinics and hospitals, and its operation optimization research is of great significance. This study
reviews the research progress on appointment scheduling system optimization. Firstly, we classify
and conclude the existing appointment scheduling system structures and decision-making frame-
works. Subsequently, we summarize the system reliability optimization framework from three
aspects: appointment scheduling system optimization objectives, decision variables and constraints.
Following that, we methodically review the most applied system optimization algorithms in different
appointment scheduling systems. Lastly, a literature bibliometric analysis is provided. During our
review of the literature, we observe that (1) optimization methods in ASs predominantly involve the
application of genetic algorithms and simulation optimization algorithms; (2) neural networks and
deep learning methods are core technologies in health management optimization; (3) a bibliometric
analysis reveals a heightened interest in the optimization technology of ASs within China compared
to other nations; and (4) further advancements are essential in the comprehensive optimization of
the system, exploration of practical usage scenarios, and the application of advanced simulation and
modeling techniques in this research.
Keywords: appointment scheduling; optimization algorithm; healthcare; system structure; queuing
theory; literature bibliometric
MSC: 90B22
1. Introduction
Healthcare extends beyond medical care and encompasses the delivery of services
aimed at promoting, preserving, monitoring, or restoring health for individuals or commu-
nities through the expertise of healthcare professionals or services [1]. The development of
the healthcare system is more important over the past few decades, and it is facing increased
costs, increasing number of users, a more conscious population about healthcare issues,
and increased demand for quality healthcare services [2]. Most countries are experiencing
ageing and population growth, which led to an oversupply of doctors, unequal distribution
Axioms 2024, 13, 16. https://doi.org/10.3390/axioms13010016 https://www.mdpi.com/journal/axioms

Axioms 2024, 13, 16 2 of 30
of resources and inefficiency. This shows that the experience of the healthcare system affects
the economy and prosperity of cities around the world [3–5].
The scheduling of outpatient departments has become more important in healthcare
systems aiming at the efficient utilization of available hospital resources to provide satisfac-
tory services to patients. Patient access to healthcare services and extended wait durations
have been identified as major problems in ambulatory care systems [6,7]. It affects the
productivity, pressure, and effectiveness of medical staff, as well as the cost and availability,
quality, and efficiency of care. Various factors can impact the performance of outpatient de-
livery systems, including the clinic’s environment and conditions, the adopted appointment
policies, the timeliness of patients, and the unpredictability of demand [8]. Appointment
scheduling (AS) is an indispensable part of the efficient operation of an outpatient clinic,
which improves the utilization of healthcare resources such as personnel and equipment.
The core problem of AS is to reduce patient waiting and doctor downtime by creating an
appointment system that assigns an available time slot to the patient [9]. Appointment
scheduling constitutes a challenge in allocating resources amid uncertainty within the field
of IE/OR. A decision-making process is needed to reduce operational costs and improve
health management services [10–13].
With the advancement of modernization of medical technology and the continuous
growth of patient consultation in recent years, AS will have a wider application in the future.
Therefore, theoretical value and significance are associated with the research on optimizing
AS operations. By conducting optimization calculations, an AS system is formulated to
improve the efficiency of medical operation [14–16]. There are many related studies on AS
system optimization.
As a result, various reviews concerning appointment scheduling systems have sur-
faced in recent years. For instance, Cayirli et al. [17] studies problem formulation and
modeling considerations used in the previous literature about appointment scheduling in
outpatient services. Amir et al. [18] reviewed studies focusing on decision-support tools
for the design and planning of OASs. It offered a structured approach to organizing the
latest literature based on various criteria, including a framework for classifying decisions
across strategic, tactical, and operational levels. Elizabeth et al. [19] documented patient
self-scheduling, along with its advantages and intervention characteristics. Moreover,
Marynissen et al. [20] introduced multi-appointment scheduling problems in hospitals and
used them to classify the scientific work on MASPHs published before the end of 2017. Ali
et al. [21] published a review for the complexity of the appointment scheduling problem.
Concomitantly, they presented a critical review of the simulation appointment scheduling
system (SASS) research topic, and highlighted critical challenges and outlined potential
research topics [22].
A well-designed appointment scheduling system requires many rule constraints; in
fact, missing or delaying appointments can reduce the efficiency of outpatient clinic op-
erations and increase healthcare costs. Research on the behavior of patients who missed
appointments is also an important direction. Gupta et al. [3] offered a comprehensive
synthesis of previous research related to the AS environments and decisions within primary
care clinics, specialty clinics, and hospitals. Leila et al. [23] provided a systematic literature
review (SLR) on appointment scheduling no-shows; their study analyzes the characteristics
of existing research, synthesizes results on factors impacting missed appointment rates, and
compares these findings with previous studies. Wilson et al. [24] explored absences within
the National Health Service in terms of causes (inefficiencies in appointment systems, tradi-
tional reminder failures and inconvenient scheduling), the effects of missed appointments
and explores potential strategies.
However, a comprehensive review summarizing methods and techniques for optimiz-
ing AS systems is still lacking. Therefore, to help designers and researchers in addressing
challenges and giving recommendations for the practical implementation of AS system
operations, this paper examines the previous studies on AS system optimization across
three dimensions: (1) system structure classification; (2) optimization framework; and

Axioms 2024, 13, 16 3 of 30
(3) optimization algorithms. Additionally, this paper includes a literature bibliometric
analysis of prior research on AS system optimization. This paper contributes a comprehen-
sive review dedicated to AS system optimization, unlike previous partial reviews. First,
it is a review focusing on AS system optimization, whereas previous reviews have only
partially addressed this issue. Second, optimization methods are discussed in terms of
system structure, optimization framework and optimization algorithms. Finally, this paper
includes the bibliometric analysis of the literature, serving as an additional reference to
illustrate the development of our matter.
For the literature bibliometric analysis, we adopt a bibliometric methodology. This
methodology is a quantitative approach to analyzing the scientific literature, providing
insights into the characteristics, trends, and impact of research within a particular content or
across various disciplines. This approach obtains a literature overview, identifies thematic
gaps by systematically collecting and analyzing bibliographic data, including citation
patterns, author trends, publication outputs and other relevant bibliographic elements to
understand the current state of the research and identify new research ideas [25].
This paper provided a SLR of AS system optimization. A systematic literature review
is a structured methodology widely used in academic research, healthcare, social sciences,
etc., to provide a comprehensive overview of existing knowledge, to help identify gaps
in research, and at the same time to guide future research directions. It is recognized as a
valuable tool for evidence-based decision-making.
The remainder of this study is organized as follows. Section 2 provides an overview of
the structure of AS system. Section 3 studies the optimization of AS systems in optimization
objectives, decision variables and constraints. Section 4 explores various primary algorithms
utilized. Section 5 includes the literature bibliometric analysis on the optimization of AS
system. Section 6 discusses the status and trends in the development of AS systems in
different countries. Section 7 gives a summary of the article.
2. System Structure of AS
2.1. AS Characteristics
The scheduling of appointments within outpatient departments stands as an exten-
sively investigated subject, playing a pivotal role in facilitating the efficient utilization of
clinical services. It is a topic that demands in-depth research, given its significance in ensur-
ing timely access to high-quality care for patients. In outpatient services, a central issue
for scheduling staff is how to allocate available time slots for patients, thereby reducing
patient delays and physician availability or additional time [26,27]. The role of appointment
scheduling systems in optimizing quality healthcare has been recognized as a challenging
problem [28]. Appointment scheduling systems are a well-positioned and practical research
area. Unlike the traditional way of registering for appointments, it avoids crowd gathering
as well as the unknown nature of patient scheduling, and greatly improves the services’
efficiency, which is an emerging field of research within healthcare [29–31]. The attributes
of ASs are primarily evident in two aspects:
1. Uncertainty AS performance;
Patient unpunctuality is common in many healthcare organizations, and even failure
to show up for appointments exists. Patients arriving earlier or later can complicate the
AS system design. Late or no-show patients can lead to doctor availability affecting the
waiting time of subsequent patients thus reducing efficiency [32,33]. At the same time, the
AS system considers that doctors are not always on time, and the nature of their work leads
to the possibility that they may not always arrive on time when they are on the operating
table or in emergency resuscitation ward consultations [34,35]. The AS system considers
the changing circumstances of both parties and thus conducts outpatient scheduling.
2. ASs can be extended to represent other industries.
AS system design and analysis are very complex. AS systems need to consider
nurses and other medical tests or other staff operating diagnostic machines and assisting

Axioms 2024, 13, 16 4 of 30
patients, so the appointment scheduling plan is mainly considered as a resource focused on
physician professors [36,37]. It is also comprehensive to consider that nurses and other staff
members are assigned to diagnostic resources before patients can be scheduled for further
consultation. The AS problem is a resource allocation problem, and the establishment
of AS systems can be extended to multiple other resource-limited industries. Outside of
healthcare, transportation, hospitality, restaurant reservations, etc., have been referenced
and improved in terms of operations, profit models, and customer wait times [38–40].
2.2. System Structure of AS
The prerequisite for building an AS system is to be familiar with the patient consulta-
tion process, as we all know, anyone who wants to consult needs to arrive at the hospital
outpatient clinic as early as possible, and then go to the registration desk to obtain a number
(even if you use the online advance booking needs to be in accordance with the booking
time to report to the queue and wait). After receiving your number, you will wait in the
designated viewing area and meet with the doctor when you are called. After the consul-
tation, the patient may need to undergo further tests and return to the queue to consult
with the doctor again after receiving the results [41]. Figure 1 illustrates the structure of
the process for the patient’s outpatient consultation in a clinical setting. The AS system
design is viewed as a range of decision-making hierarchies, including appointment rules,
classification of patients, and adjustments for reducing the disruptive effect of no-shows,
walk-ins, and emergency patients [42].
erating table or in emergency resuscitation ward consultations [34,35]. The AS system con-
siders the changing circumstances of both parties and thus conducts outpatient schedul-
ing.
2. ASs can be extended to represent other industries.
AS system design and analysis are very complex. AS systems need to consider nurses
and other medical tests or other staff operating diagnostic machines and assisting patients,
so the appointment scheduling plan is mainly considered as a resource focused on physi-
cian professors [36,37]. It is also comprehensive to consider that nurses and other staff
members are assigned to diagnostic resources before patients can be scheduled for further
consultation. The AS problem is a resource allocation problem, and the establishment of
AS systems can be extended to multiple other resource-limited industries. Outside of
healthcare, transportation, hospitality, restaurant reservations, etc., have been referenced
and improved in terms of operations, profit models, and customer wait times [38–40].
2.2. System Structure of AS
The prerequisite for building an AS system is to be familiar with the patient consul-
tation process, as we all know, anyone who wants to consult needs to arrive at the hospital
outpatient clinic as early as possible, and then go to the registration desk to obtain a num-
ber (even if you use the online advance booking needs to be in accordance with the book-
ing time to report to the queue and wait). After receiving your number, you will wait in
the designated viewing area and meet with the doctor when you are called. After the con-
sultation, the patient may need to undergo further tests and return to the queue to consult
with the doctor again after receiving the results [41]. Figure 1 illustrates the structure of
the process for the patient’s outpatient consultation in a clinical setting. The AS system
design is viewed as a range of decision-making hierarchies, including appointment rules,
classification of patients, and adjustments for reducing the disruptive effect of no-shows,
walk-ins, and emergency patients [42].
Figure 1. The flow structure of the patient in the hospitals.
2.2.1. Appointment Rules
Figure 1. The flow structure of the patient in the hospitals.
2.2.1. Appointment Rules
AS system operation optimization typically improves healthcare system performance
by designing AS rules. These regulations describe the duration of each appointment
interval and the number of patients (block size) to be accommodated during that interval.
By utilizing various permutations of fixed and variable appointment interval lengths,
several innovative rules have been developed to make the operation of the system more
responsive to actual consultation needs [17]. The appointment rules of the AS system
are expressed in terms of block-size (ni: the number of patients assigned per block i),
begin-block (n1:patients assigned the same appointments at the session’s outset), and
appointment interval (ai: the interval between two consecutive appointments). The block

Axioms 2024, 13, 16 5 of 30
size and the scheduling interval can be fixed or variable, so the reservation rules composed
by the combination of any of its three variables (ni, n1, ai) [43].
1. Fixed-interval rules.
The fixed-interval rule refers to equal intervals throughout the consultation. The
intervals between the first patient and the subsequent patient and between every two
subsequent patients are fixed, i.e., ai is constant. There are some differences in the other two
variables under the fixed-interval rule. Simply, the IBFI (individual-block/fixed-interval
rule) assigns individual appointments to every patient with equal intervals throughout
the consultation, meaning that ni = 1, n1 = 1 [44]. Considering the patient’s late arrival
or no-show, it is possible to increase the number of appointments at the beginning of the
meeting for the same time patient refers to n1 > 1. Avoiding the phenomenon of doctor’s
idleness improves the efficiency of healthcare workers. Bailey et al. [45,46] proposed an
individual-block system assigning two patients at the session’s start and scheduling the rest
at intervals matching the mean consultation time. This highlights the need for thorough
research into outpatient appointment scheduling (n1 = 2, ni = 1).
The multiple-block/fixed-interval rule can be interpreted as assigning m patients
to the appointment slot with a constant appointment interval. The multiple-block rule
performs better when the consultation time is shorter, and the rise in patient numbers in
the same appointment time slot allows for more flexibility in the physician’s and patient’s
time. Soriano et al. [47] studied a system in which patients are scheduled in pairs, with
intervals set at twice the mean consultation time. In fact, block-size can also be variable;
the rules allow variations in block sizes throughout the clinic session.
2. Variable-interval rules.
Rules with constant intervals are in fact unlikely to perform well in all environments,
so more research has focused on optimizing AS systems using variable interval rules. This
rule is to schedule patients individually when the interval is variable. Considerations are
made for walk-ins, no-shows, and urgent cases whenever they occur during the planning
of an AS [48]. Therefore, there is a need to adjust AS by making advanced predictions to
reduce their damaging effects. The literature suggests two solutions. The first is to book
other patients into a single slot in the presence of no-shows and maintain some vacant
slots for walk-ins [49]. The other method is to adjust the appointment interval so that it is
proportional to the probability [50]. Universal rules are proposed, for example, the” offset”
rules, the” dome” rules and the “plateau-dome” rules.
• The “offset” rule.
The “offset” rule refers to a set of rules which modifies the appointment times based
on the allocated appointment slots. A framework of different reservation rules for different
operational environments is established, which emphasizes that there is no “universally
good” rule, and that the appropriate rule needs to be selected by determining the value of
some environmental factors. Rohleder et al. [51] applied the offset as shown in Equation (1),
where the early time slots are below the average service duration and the late time slots are
even longer, which in various situations leads to a very late end of the session.
Ti =

As − k1σ(K − s) f or i and s = 1, 2, . . . , K
As + k2σ(s − K) f or i and s = K + 1, K + 2, . . . 21
(1)
where Ti is the planned time for client i appointment, as is the standard time for s, and σ is
variability in service time. The parameters K, k1, k2 are the early breakpoints, delay break-
points and the multipliers. (k1/k2) controls the earliness rate imposed on the first K patients
and the rate of lateness imposed on the remaining patients.
• The “dome” appointment rule.
The most prominent finding in the study of variable interval rules is the “dome” rule,
including absences and appointments, and patient categorization. The universal appoint-
ment rule demonstrate effective performance, independent of the clinical environment [52].

Axioms 2024, 13, 16 6 of 30
Optimal scheduling exhibits a dome pattern, whereby appointment intervals begin to grow
later in the session and then decrease. Dome mode reduces wait times compared to other
scheduling rules [53]. Considering no-shows and walk-ins, the patient count scheduled to
achieve the target T patients served at a session is calculated as follows [54].
N = T
(1−Pn+Pw) (2)
In this formula, Pn is that a patient fails to attend and Pw is the probability of coming
without an appointment.
The appointment times are determined through a two-step process employing the
subsequent formula:
Ai = max
n
0, k(i − 1)µ − σ
√
i · π
o
f or i = 1, . . . , N where π = (N + i)/(N − 1) (3)
In this formula, Ai is the patient i’s scheduled time, with the mean duration of service
µ, and variability in service time σ. The k controls the appointment intervals for different
appointment rules by setting different values. π is a parameter utilized to establish a
pattern resembling a “dome” in the appointment intervals.
K = f (N, Cv, Pn, Pw, CR) (4)
In Equation (4), Cv is the coefficient of variation; and CR is the ratio of the cost of the
doctor’s time to that of the patients’ time.
Considering the combined impact of no-shows and walk-ins on the consultation
durations, the updated formulas for the mean and variance are shown:
µ′ = (1 − Pn + Pw)µ (5)
σ′2
= (1 − Pn − Pw)(σ2
+ (Pn − Pw)2
µ2
+ Pn(1 − Pn + Pw)2
µ2
+ Pw(2σ2
+ (1 + Pn − Pw)2
µ2
(6)
• The “plateau-dome” appointment rule.
Doctors could find advantages in implementing a “plateau-dome” rule with fixed
individual blocks evenly distributed in the middle of the appointment session [55], explicitly
modelling the impact that the doctor’s interruptions may have on optimal appointment
times, and thereby affecting waiting times for patients. A traditional pattern for the high
number of interrupts and flat dome mode for a low number of interrupts performs better [56].
This was achieved by conducting an additional run with the supplementary constraint:
xi − 2xi+1 + xi+2 = 0 f or iϵ[α, β] (7)
In this formula, xi is the appointment time, and α and β are the appointment slots
corresponding to the beginning and end of the plateau portion, respectively.
2.2.2. Patient Classification
Several investigations have explored the implementation of patient classification (PC)
based on the assumption that populations can be categorized into distinct groups. For
example, in practice, a prevalent method of classifying patients is according to whether
they are new or old who have been examined for a different length of time. This allows
patients to be sorted at the time of the appointment, and diversity in consultation duration
is combined by patient categorization to adjust the length of the appointment interval
according to the characteristics of each group. Most studies are only concerned with the
scheduling of patients attending elective appointments [57]. Meersman et al. proposed the
scheduling horizon involves the time slots for elective and urgent patient categories [58],
considering the slot allocation of non-elective patients.
Cayirli et al. [59] proposed to consider generic extended “dome” rules for patient
classification to schedule and order appointments by distinguishing patients based on the

Axioms 2024, 13, 16 7 of 30
average and variability in service time. The alternative appointment system was tested by
simulation using the generic dome rules. Equation (8) represents the appointment times (Ai):
Ai = max
n
0, kA(i − 1)µ′
A −
p
iσ′
A
2 · π
o
f or (class A)patients i = 1, . . . , NA′ (8)
Ai = kA(NA)µ′
A + kB(i − NA − 1)µ′
B −
p
(NA + 1)σ′
A
2 + (i − NA − 1)σ′
B
2 · π
f or (class B) patients i = NA + 1, . . . , N
(9)
In Equations (8) and (9), class A and B are two patient groups, where class A is charac-
terized by distinct mean and standard deviation in service times µA and σA probabilities.
The parameter π contributes to forming a dome pattern within the appointment interval,
where π = (N + i)/(N − 1), N = NA + NB.
Salzarulo et al. [60] introduced a methodology for heterogeneous scheduling (HS)
wherein appointment times are established by forecasting examination durations using the
individual characteristics of the patient rather than traditional patient classifications. Sev-
eral models were calculated to evaluate how patient characteristics influence appointment
schedules. And three levels (HS1, HS2, HS3) of information usage were considered when
making appointment schedules.
2.3. AS System Decision Framework
In AS system design and planning, a range of decisions and planning determine the
main structure of the optimization research system. They can be categorized into three
types which are strategic, tactical and operational [18]. Strategic decisions are the keys in
shaping the modeling process and determining the practical applicability of the proposed
solutions. Such decisions are usually considered as inputs to the AS system. Tactical
decisions aim at determining the system structure. While operational decisions rely on the
first two to develop optimization models and rules. In Table 1, their features and related
studies are summarized.
Table 1. The features and related studies of AS system decision framework.
Decision Types Features Related Studies
Strategic decisions
(a) Long-term decisions;
Robinson et al. [61]; Qu et al. [62]; Turkcan
et al. [63]; Gocgun et al. [64]; Saghafian
et al. [65]; Sauréet al. [66]; Pérez et al. [67]
(b) Determine the main system structure;
(c) Include access policy; number of resources; policy
on acceptance of walk-ins; types of scheduling.
Tactical decisions
(a) Medium-term decisions;
Wang et al. [68]; Kong et al. [69]; Erdogan
et al. [70]; Turkcan et al. [71]; Ozen
et al. [72]
(b) Explaining how patients are placed as a whole;
(c) Maximizing resource utilization and the
accessibility of nursing services.
Operational decisions
(a) Short-term decisions;
Balasubramanian et al. [73]; Conforti
et al. [74]; Truong et al. [75]; Feldman
et al. [14]; Cayirli et al. [17]
(b) Focused on the efficient scheduling of individual
patients;
(c) Incorporating both the rule-based approach (RBA)
and the optimization-based approach (OBA).
2.3.1. Strategic Decisions
Strategic decisions are considered as inputs and are long-term decisions that determine
the main structure of the AS system. Robinson et al. [61] proposed three main types of
access strategies for scheduled patients: traditional, open access, and hybrid. Traditional
policies mean all capacity is allocated to pre-scheduled patients, which results in higher
no-show rates and longer wait times. Qu et al. [62] proposed a policy that allocates
all capacity to one-day appointments to prevent the worse impact of high absenteeism

Axioms 2024, 13, 16 8 of 30
rates under traditional policies and meet same-day demand, which bring low resource
utilization. Resource allocation is an important step in designing AS systems. The main
resources considered in many studies include clinical staff, medical equipment, rooms
and beds/chairs [63]. The number of servers and resource levels are key factors affecting
clinic profits. High resource levels can be costly and wasteful, while low levels can increase
latency services, thereby reducing care quality and clinic revenue [64].
Walk-in patients refer to patients who go to the clinic without an appointment during
the consultation period. Accepting appointments is a means of mitigating the negative
impact of no-shows while increasing modeling complexity due to the dynamic random
arrival of non-appointment patients [65]. Appointment scheduling methods are categorized
into online and offline [66]. In the offline method, arranged once all requests have been
received, whereas in the online method, patients are promptly scheduled after requests
arrive. Pérez et al. [67] proposed to categorize the patient and resource scheduling problem
into three aspects: offline, online and stochastic online. Stochastic online scheduling is an
extension of the concept of online scheduling. With the development of the Internet, online
systems are gradually becoming the subject of research.
2.3.2. Tactical Decisions
The primary purpose of approaching the decision problem at the tactical level is to
characterize the system to optimize resource utilization and the integrity of the consultation
service. For example, in allocating capacity among various groups, the decision should
consider factors such as the needs of each patient group, prioritization, probability of
absence, revenue per patient group, and patient and physician preferences [68], which
affect the absence rate and hence the system efficiency. A decrease in the scheduling
window results in a reduction of indirect wait time, consequently leading to decrease the
absence rate. This enables a more effective use of the clinic’s capacity. Excessive constraints
on scheduling windows may result in a reduced patient count, leading to a decline in clinic
revenue [69]. Issues such as appointment intervals (slots), block size, panel size, etc., are
essentially capacity allocation issues as well, affecting tactical-level decisions [70–72].
2.3.3. Operational Decisions
Operational decisions are associated with plans at the individual-patient level. Rule-
based approaches (RBA) and optimization-based approaches (OBA) are used to determine
these decisions [68]. RBA refers to a set of instructions having associated rules and pa-
rameters, but it does not guarantee the realization of optimal performance compared to
OBA. Whereas OBA specifies the level of operational decisions, its goal is to attain globally
optimal answers for operational decisions [73].
Early research on appointment scheduling usually focused on RBA [17]. In scenar-
ios with homogeneous patients, the primary scheduling rule combines two factors: the
appointment interval and block size. Appointment time handling is aimed at the issue
of scheduling a specific time for a patient to begin treatment to optimize performance
criteria. Most models proposed for the appointment time decision problem use simplifying
assumptions (e.g., offline scheduling, on-time patients, single-phase service) [74]. That time
is usually combined with the appointment date problem, which refers to determining the
appointment date for each patient [75] to satisfy restrictions on patient prioritization and
the waiting time objective. Most of the problems such as appointment date and patient
selection from list of waiting are usually solved using OBA [14].
3. Optimization Framework
There are several optimization objectives: societal benefit, economic performance, and
capacity utilization. A summary of their contributions and relevant studies can be found in
the following Table 2.

Axioms 2024, 13, 16 9 of 30
Table 2. Contribution of different optimization objectives and related studies.
Objectives Contributions Related Studies
Societal benefit
(a) Minimize the time to complete all tasks; Anderson et al. [76]; Han et al. [77];
Guido et al. [78]; Dharmadhikari et al. [79];
Cordier et al. [80]; Nazanin et al. [81];
Ma et al. [82]
(b) Maximizing patient satisfaction;
(c) Optimization of the number of patients per unit
cost of service.
Economic performance
(a) Revenue optimization model of AS; El-Sharo et al. [83]; Parizi et al. [84];
Chakraborty et al. [85,86];
Muthuraman et al. [87];
Geng et al. [88]; Patrick et al. [89];
Wang et al. [90]
(b) Stochastic overbooking model;
(c) MDP strategy for the consistent throughput;
(d) Multimodal function of no-shows and walk-ins.
Resource utilization
(a) Maximum required capacity for the demand of
outpatient physicians; Nguyen et al. [91];Zeng et al. [92];
Yan et al. [93]; Tsai et al. [94];
Jiang et al. [95,96]; Sevinc et al. [97]
(b) Determining optimal scheduling characteristics for
heterogeneous patients;
(c) Estimate of clinic utilization under the F-CAP.
Other objectives
(a) Optimize the total probability of a patient being in
an uncontrolled health state;
Savelsbergh et al. [98]; Ozen et al. [72]
Zhu et al. [99]; Erdelyi et al. [100]
(b) Improving the maximum overflow for
multi-physician practices;
(c) Minimize the costs of holding and rejecting
patients;
(d) Matching patient preferences and scheduling.
3.1. Optimization Objective
3.1.1. Societal Benefit
Many studies in the literature on AS have been devoted to minimizing pathway
completion time or maximizing patient satisfaction, which leads to better societal benefits
and is the primary goal that hospital management needs to follow. The key performance
indicators for minimizing the time to complete all tasks include the average waiting time
of patients (WAIT), the average idle time of physicians (IDLE) and the average overtime
of physicians (OVER). Anderson et al. [76] investigated an Overlapping Appointment
Scheduling (OLAS) model to minimize waiting time and physician idle time in medical
clinics, considering randomized service times. Han et al. [77] used the sum of patient
wait and physician idle costs as the system cost and introduced an analytical model and
demonstrated the optimality of a fixed-interval policy. Guido et al. [78] derived a local
search algorithm with an objective weighted average of the patient’s expected waiting time,
the doctor’s idle time, and the lateness time, and proves that the algorithm converges to
optimal scheduling, with the following objective function:
C(x) = awW(x) + αI I(x) + αLL(x) (10)
min

αwW(x) + αI I(x) + αLL(x) ∑t
xt = N
xt ∈ N0

(11)
In Equation (10), Wx, Ix, Lx are the mean waiting time, idle time, and lateness, respec-
tively. α is the weight of function. N represents the total number of patients. xt is the
number of patients scheduled at the start of interval t.
Dharmadhikari et al. [79] used performance metrics of average rewards per patient
to assess the effectiveness of the proposed block scheduling strategy with prioritization.
Cordier et al. [80] developed algorithmic tools to construct one-day schedules and optimize
these schedules to optimize the length of patient stay. Nazanin et al. [81] conducted case
studies for different healthcare settings and selected the most effective scheduling model

Axioms 2024, 13, 16 10 of 30
that fits the case in terms of the balance between three types of metrics related to patient
appointment scheduling systems: patient satisfaction, scheduler utilization, and scheduling
system cost. Accessibility was a factor of patient satisfaction, measured by the average
waiting time before connecting to the dispatch program and the average duration of call.
Ma et al. [82] examined the scheduling of outpatient appointments in a multi-server model
as a two-stage stochastic optimization problem. The initial step focus is on optimizing
doctor planning based on capacity and patient requests, and the second step optimizes
patient planning with considering uncertain service times, which first aim to maximize
patient service time per unit cost:
MaxF1 = R
Cf
(12)
R = ∑T
t=1 Qt
P
(13)
Cf = ∑
J
j=1 ∑T
t=1 xjtαjβ (14)
P = ∑
J
j=1 ∑T
t=1 xjtβ (15)
where Cf is total amount of allowances for all doctors on assignment based on time worked.
R is the hours of service each patient, Qt is total number of patient weeks, and P is the total
working hours per week. β is the fixed working hours and α indicates the unit allowance.
3.1.2. Economic Performance
The appointment scheduling system is a potentially useful tool for reducing health
management costs and maximizing medical benefits. Improving system reliability and
cost savings are the main ideas of optimization [101–104]. Profit maximization is becom-
ing a major research objective due to the need for cost efficiency and facing budget cuts.
Some hospitals maximize profits by rising the number of patients scheduled and max-
imizing contribution margins. El-Sharo et al. [83] modeled an overbooking scheduling
model for multi-provider practices to optimize patient overbooking and maximize expected
profits. Parizi et al. [84] studied a class of scheduling problems where his system is re-
warded for service provision; and costs incurred for rejected requests, delays, and overtime.
Chakraborty [85,86] proposed a sessional sequential scheduling algorithm without time
slots proving that the expected profit of scheduling is single-peaked, providing the stopping
criterion for the scheduling algorithms, with the following total expected profit:
P(Sn) = R(Sn) − C(Sn) (16)
R(Sn) = r
λ E[Zn] (17)
C(Sn) = E

n
∑
j=1
cw pjWn Sj

+ coVn
#
(18)
where set Sn = {S1, S2, . . . , Sn} indicates partial schedule after n call-ins, and Si, i = 1, 2, . . . , n
is the time of the ith “scheduled” customer. The expected cost is given by C(Sn) due to customer
waiting and overtime. R(Sn) is the expected revenue, where r is the revenue per unit time, and
the random variable Zn denotes the number of scheduled clients arriving for service after n clients
are scheduled.
Muthuraman et al. [87] built a stochastic overbooking model and developed an outpatient AS
strategy that considers patient arrival and removal probabilities from slots to maximize operational
profitability. Geng et al. [88] used the Markov decision process model to consider outpatients with
different waiting time goals and different rewards to maximize the expected benefit. Patrick et al. [89]
developed a Markov Decision Process (MDP) model that explores tradeoffs between patient-related
measures and physician-related measures. Maximizing system profit as the objective function
(revenue minus overtime for visiting patients, idle time, indirect waiting time, and referral schedule),
the MDP strategy is shown through simulation to be even better than open access in maximizing
profit and providing more consistent throughput. Wang et al. [90] developed a model for scheduling
appointments and managing capacity concurrently, taking into account no-show patients and walk-in

Axioms 2024, 13, 16 11 of 30
patients. The objective is to maximize the overall profit and the objective function is a multimodal
function, with the following total expected profit:
Max rθXsum + rNρ − W − O − I (19)
In Equation (19), r is the revenue of per patient, and Xsum is calculated as the total number
of patients in the schedule. θ is the probability of the patient arriving on time each time interval,
while ρ is the probability of walk-in patient. In this equation, walk-in patient income is considered to
maximize the economic benefits.
3.1.3. Resource Utilization
With limited healthcare resources, clinics are under tremendous pressure to the increasing
demand. Therefore, rational capacity allocation is one of the main goals of AS. Previous papers on
outpatient capacity assignment can be divided into two broad categories. One category is where
the overall capacity is a constant, the hospital must assign limited medical capacity to various types
of patients. For instance, Nguyen et al. [91] presented a mixed-integer planning model with the
aim of minimizing the maximum demand capacity to plan the demand of outpatient physicians to
achieve the service goal of patient appointment lead time. The other category, where the capacity is a
variable that needs to be solved in conjunction with the allocation scheduling problem, is categorized
into static and dynamic cases. The former means that the decision is made before the session, e.g.,
Zeng et al. [92] investigated the appointment booking problem for heterogeneous patients. The
features of optimal scheduling with heterogeneous patients are determined, and a method of local
search algorithm is proposed to for finding the locally optimized scheduling. The latter means that
appointments are made sequentially, and when a patient calls in, capacity allocation is conducted
online, and the hospital needs to decide how much capacity is available for each time slot. Yan
et al. [93] studied a sequential AS method that considers choice for patients and equity of services to
balance clinic effectiveness and patient satisfaction. Tsai et al. [94] offered a overbooking model to
consider the patients’ call-in order, who proposes two methods, convolutional estimation with upper
and lower bounds and joint cumulative estimation, to make the workload of each appointment slot
more balanced.
Jiang et al. [95,96] investigated a capacity addition strategy to mitigate the supply demand
mismatch and proposed a two-tier enumeration search algorithm based on how to allocate normal ca-
pacities between regular and same-day patients. A demand-driven approach to allocating additional
capacity at outpatient clinics is also discussed, and the capacity assignment optimization problem
under CAP is investigated to optimize the whole process by jointly setting the optimal values for
both additional and conventional capacity. Sevinc et al. [97] used a two-stage approach to optimize
the chemotherapy regimen and proposed an adaptive negative feedback scheduling algorithm for
managing system loads. Two heuristics were evaluated to optimize the utilization of infusion chairs
by assigning patients to specific infusion seats.
3.1.4. Other Objectives
In general, the literature on optimization studies of AS systems covers many aspects, as well as
the optimization of objectives through other aspects to obtain better social and economic effects.
Savelsbergh et al. [98] investigated methods based on optimization and cohorts to SA for patients
in a chronic disease management program. The objective was to minimize the overall probability of a
patient entering an uncontrolled health state. The approach considers transitions in disease control
since the last appointment time and the probability of the patient not attending the appointment.
Ozen et al. [72] used the probability of demand exceeding capacity as a measure of access capacity.
Formulating the problem of minimizing the maximum overflow of a multi-medicine practitioner
clinic as a non-linear integration programming problem describes how the frequency of overflow
varies from physician to physician and demonstrates how these supply and demand imbalances
can be minimized in the long run using real-world data from primary care practices. Zhu et al. [99]
provides a method for efficient outpatient scheduling that maximizes the matching between patient
preferences and scheduling by collecting patient choices from available time slots through special
processing to find an allocation scheme for all patients that maximizes the preferences of all patients.
Erdelyi et al. [100] exploited a stochastic approximation method to find a favorable class of protection.
The levels calculated through the stochastic approximation method which consistently outperforms
those computed using the deterministic approximation method.

Axioms 2024, 13, 16 12 of 30
3.2. Decision Variable
About the optimizing appointment scheduling’s study, the decision variables for optimization
show the strategy and actions taken to enhance the objectives [105]. Appointment time is one of the
common decision variables in the AS problem. Tito et al. [106] considered whether patient absence
was affected by appointment time using arrival time as the decision variable and investigates a
stochastic optimization problem with a random distribution of service times and patient decision-
dependent no-show behavior. Solutions are given for different patterns of absenteeism behavior,
and it is shown that disrupting the hypothesis of a fixed probability of attendance greatly alters the
scheduling scheme. Liu et al. [107] used a programmatic single-server queuing model to pattern
appointments scheduled for providers, examine demand and capacity control decisions and consider
two distinct problems, which considered service capacity as a fixed and decision variable, respectively.
Maximizing the net reward function is used as the objective and the optimal strategy is given.
Zhang et al. [108] described the server’s response to congestion as influenced by client waiting
time, the model employs the scheduled arrival time of a sequence of clients as the decision variable.
And provided replacements for the simulation optimization model and the SIP model. SIP models
have been shown to enable servers to react to congestion under certain conditions. This resolved the
challenge of scheduling appointments while considering the server’s response to congestion in the
service system.
Luo et al.’s work [109] is applicable for identifying optimal appointment strategies under varying
assumptions regarding rewards, costs, and decision variables. The aim was to balance between patient
waiting time and server utilization. It was also investigated whether a strategy performs very poorly
when it ignores interferences when the decision variable is the appointment’s number.
3.3. Constraints
Various optimization decisions about appointment scheduling systems will have different
constraints. Optimization studies of ASs are often inseparable from the optimization of operational
costs as well as time from the perspective of economic or social benefits [110]. Considering the
minimum cost of AS operation and the minimum waiting time or the minimum physician overtime
time, the constraints consist of appointment sequence, resource capacity, time constraints, etc.
3.3.1. Queue Balance Constraints
Consider the need to have maintained a normal flow of patients through the system. It is
essential to maintain a balance between the number of patients in the line at each stage and the sum
of the initial number of patients, the additional patients entering the stage, and the reduction from
patients who have exited thus far [111]. The queue balance constraints can be expressed as follows:
qj,t,p = Ij,p − Xj,t,p + Xj−1,t−dj−1,p,p ∀j, t, p (20)
where qj,t,p is the number of patients, Ij,p is the starting count of patients of type p, and Xj,t,p is the
cumulative count of patients of type p at stage j.
3.3.2. Capacity Constraints
Capacity allocation, including equipment resources, medical staff, and patient numbers, requires
constraints. The equations are as follows:
• Surgeon availability constraint:
∑
p ϵ Bs

X2,t,p − X2,t−d2,p,p

≤ ls,t ∀t, s (21)
Considering the availability and scheduling of surgeons, it is required that a sufficient number
of surgeons needs to be available to operate at each time period, where ls,t is surgeons of type s who
are currently available at time t; the values of stage index j of 1, 2 and 3 refer to the pre-, mid- and
post-run stages, respectively, where d2,p is the service duration during the operation.
• Operating room availability constraint:
∑
p

X2,t,p − X2,t−d2,p,p

≤ ot ∀t (22)
where it is ensured that the sum of surgeries per period does not exceed the available number of
operating rooms. ot represents the amount capacity of operating rooms.

Axioms 2024, 13, 16 13 of 30
3.3.3. The Completion Time Constraint
Same-day appointment patients cannot have a wasted visit, so it is necessary to stipulate that
services are available to all patients before the end of treatment. The number of patients that must be
served is determined as follows:
X3,T−d3,p,p = ∑
j
Ij,p∀p (23)
The constraint indicated by completion time determines the time block for the discharge of the
last patient.
Myt ≥ ∑
p
x3,t−d3,p,p ∀t (24)
m ≥ t × yt ∀t (25)
In Equation (24), yt indicates whether the patient is discharged where it is 1 if discharged and 0
otherwise. M is an arbitrarily large number and m is the last time block, which means completion time.
4. Optimization Algorithms
Numerous studies have been undertaken to determine the optimal allocation of AS systems to
reduce operational and time costs, improve patient satisfaction and system reliability dependability.
Various algorithms are applied to the optimization of AS systems, with genetic algorithms (GAs)
being the most utilized. For instance, Braune et al. [112] proposed a combination of GAs and Monte
Carlo simulation to heuristically solve a stochastic optimization model developed for planning the
appointment times of healthcare units under uncertain activity durations, allowing to minimize
the waiting time of patients while maximizing the use of resources. Fan et al. [113] considered
patient preferences for highly qualified general practitioners and specialist doctors. By analyzing
real data from hospital outpatient clinics, a behavioral pattern was derived in which the patient’s
tolerance limit adapted to the expected waiting time. A simulation optimization framework for
maximizing clinic benefits and minimizing patient dissatisfaction is proposed. Utilizing multi-
objective optimization and a genetic algorithm, a simulation budget allocation approach is integrated
to derive an approximate Pareto scheme for joint capacity planning and patient scheduling across
multiple servers. Alizadeh et al. [114] presented an optimization problem for non-urgent outpatient
appointments, where only one machine (equipment) and a finite number of healthcare professionals
are required to satisfy the medical needs of many waiting patients.
In addition, the key to the appointment dilemma is the challenge of solving the multi-objective
optimization problem; therefore, many decisions need to be made using a multi-objective evolution-
ary algorithm. Mohammad et al. [115] improved the quality of operational efficiency and healthcare
quality by writing a MOPSO algorithm and introducing a MO-PASS architecture. Ali et al. [116]
investigated a multi-criteria approach in appointment scheduling by WOA. optimization for hospital
management quality and patient satisfaction. Qiu et al. [117] studied the AS challenge for a single
diagnostic device, namely, a magnetic resonance imaging (MRI) device. The issue is expressed as a
two-stage stochastic modelling of programming. An enhanced evolutionary algorithm for multiple
objectives is subsequently introduced within the framework of multi-objective evolutionary algorithm
based on decomposition (MOEA/D) to simulate uncertainty by constructing several different scenar-
ios, and the validity of the presented model is evaluated by comparing it with the typical NSGA-II
and MOEA/D algorithms using real healthcare data. Meersman et al. [58] utilized a multi-objective
archived simulated annealing heuristic algorithm to design a Pareto-cyclic appointment schedule
according to the wait times with various capacity allocations for emergency patients.
There are some other methods to solve this problem. Garaix et al. [118] proposed a heuristic to
calculate the order in which patients receive treatment at outpatient chemotherapy centers called the
GRASP algorithm. It optimizes the facility’s closing time and overtime working time. It can reach
near-optimal solutions quickly and the performance of the patient-listing strategy is comparable
to more complex scheduling strategies. Corsini et al. [119] developed a hybrid harmonic search
element heuristic algorithm to simulate a multi-stage chemotherapy ward to explore the problem
of scheduling same-day chemotherapy outpatients to optimize patient waiting time. Ali et al. [120]
studied an integer planning model with a contraindicated search approach along with the addition
of a simulation model solved in the ASs. This was conducted to assess diverse patient flow control
in emergency centers, aiming to minimize patient waiting time and enhance patient satisfaction
during hospital experiences. Rajakumari et al. [121] used Convolutional Neural Networks (CNNs) for
clinician schedule analysis to develop a web-based appointment system to improve the efficiency and
quality of deliveries to minimize waiting times. Ali et al. [122] proposed a simulation-optimization
methodology for sequentially scheduled appointments to provide ideal schedules from both patient

Axioms 2024, 13, 16 14 of 30
and medical practice perspectives. The goal is to minimize the weighted expected cost associated
with direct and indirect waiting time, idle time, and overtime.
Typically, there exist specific methods for addressing optimization problems. Table 3 provides a
summary of their characteristics and related studies. The subsequent section presents the utilization
of these algorithms in optimizing appointment scheduling operations.
Table 3. The features of the most used algorithms and related research.
Algorithms Features Related Studies
GA
(a) Strong global search ability;
Braune et al. [112]; Fan et al. [113];
Alizadeh et al. [114]; Squires et al. [123]
(b) Far exceeds conventional methods in terms of solving speed
and quality;
(c) Slow convergence, weak local search ability, and long running
time.
WOA
(a) Fast convergence speed;
Mohammad et al. [115]; Ali et al. [116];
Qiu et al. [117];
(b) Parameter settings are more sensitive;
(c) Strong dependence on initial solution.
TSA
(a) Avoiding duplicate solutions during the search process and
falling into local optimal solutions;
Meersman et al. [58]; Garaix et al. [118];
Corsini et al. [119]; Ali et al. [120];
(b) Dynamically adjusting the length of the taboo list during the
search process;
(c) Maintain list and increase algorithm calculation time.
Other algorithms (GRASP, contraindicated search approach, etc.).
Rajakumari et al. [121]; Ali et al. [122];
Akbarzadeh et al. [124]
4.1. Genetic Algorithm
Genetic algorithms (GAs) offer an efficient approach to optimizing complex systems. Previous
research indicates that many scholars have employed GAs to address optimization problems in
the scheduling fields. For example, Squires et al. [124] proposed a new genetic algorithm designed
for scheduling repetitive transcranial magnetic stimulation (rTMS) appointments. The mentioned
algorithm (LSWT-GA) combines a novel survivor selection strategy with heuristic population initial-
ization. The objective of the algorithm is to enhance the operational efficiency of medical centers by
optimizing the scheduling of repetitive transcranial magnetic stimulation (rTMS) appointments.
4.1.1. Objective Function and Constraint
The primary goal of the initial aim in the experiment is to minimize the job processing time
expressed as Cmax and aim to minimize weighted flowtime Fw secondly. The problem can be under
the category of minimizing make span in a parallel machine scenario, where job processing times are
deterministic, and preemption is not allowed.
P ∥ Cmax P ∥ Fw (26)
min Cmax (27)
Fw =
x
∑
j=1
wjFj (28)
min Fw (29)
where
Fj = Cj − rj (30)
In this formula, P is a group of patients waiting for a treatment, Fj denotes the flowtime of
patient Pj; Cj is the time to finish the treatment for Pj and rj is the release time of the job.

Axioms 2024, 13, 16 15 of 30
4.1.2. Optimization Process
This study presents the new genetic algorithm used. Drawing inspiration from evolution, the
genetic algorithm relies on natural selection to choose the most suitable chromosomes, as determined
by a fitness function, for the reproduction phase of the genetic algorithm. Figure 2 illustrates the
flowchart of the optimization algorithm employed in this paper.
In this formula, P is a group of patients waiting for a treatment, j
F denotes the
flowtime of patient j
P ; j
C is the time to finish the treatment for j
P and j
r is the re-
lease time of the job.
4.1.2. Optimization Process
This study presents the new genetic algorithm used. Drawing inspiration from evo-
lution, the genetic algorithm relies on natural selection to choose the most suitable chro-
mosomes, as determined by a fitness function, for the reproduction phase of the genetic
algorithm. Figure 2 illustrates the flowchart of the optimization algorithm employed in
this paper.
Figure 2. Flowchart of genetic algorithm modules.
4.2. Whale Optimization Algorithm
The WOA algorithm is a meta-heuristic algorithm designed with both subtlety and character,
which is derived from simulating the hunting behavior of humpback whale groups in nature and
realizes the purpose of optimizing search through the process of searching, pursuing, and attacking
prey by the whale group. The WOA literature has been widely used in solving various of optimization
problems, and Ali et al. [116] explores the multi-criteria in AS were analyzed using both algorithms,
WOA and NSGA techniques. They are computed using various assumptions to meet the requirements
and aspects associated with WOA and NSGA.
4.2.1. Optimization Model
Scheduling and counseling patients for scheduling to the hospital is the issue discussed. Each
hospital section has a varying number of operators to handle distinct operations. Moreover, each
capable of providing specific services. In this system, there are n patients, categorized as either
emergency or general patients. There are several surgeries for each patient that must be performed
in different departments. In addition to this, it may diagnose a patient as an emergency at the time
of planning. Emergency patients are usually visited earlier than general patients and are given
precedence (fairness). After doing something such as seeing a doctor, the diagnosis depending on the
patient’s condition, in which case the patient is referred to an inpatient unit with limited beds.
4.2.2. Objective Function
The objective function is to minimize the average total weighted patient time and to reduce
dissatisfaction because of increased waiting time for patients.
min cw = 1
∑
n
i=1 wi
n
∑
i=1
(wi.CNi) (31)
min Z = ρ
n
∑
i=1
(CNi − ri) (32)

Axioms 2024, 13, 16 16 of 30
4.2.3. Constraint
Examination or processing of each patient’s procedure is exclusive in one location on one
ward, as shown in the formula below. And for the restriction in a sequencing assured to prevent
patients from being assigned to a segment of the surgery when another area is empty, as shown in
the Formulas (33) and (34).
M
∑
m=1
L
∑
l=2
xijlm = 1, j = 1, . . . , hi, i = 1, . . . , n, (33)
n
∑
i=1
hi
∑
j=1
xijlm ≤
n
∑
i=1
hi
∑
j=1
xij(l−1)m,m = 1, . . . , M, l = 3, . . . , L, (34)
where x is the Appointment System Decision Variables and is equivalent to 1 if the action Oij is
performed in section mth at position ith. It is also important to consider the indexes used for appoint-
ments, appointment durations, association variables, etc.
4.2.4. Model Solving
This study solves the model by introducing a local search operator to optimize the solution,
using the stochastic WOA nature of the meta-heuristic algorithm. The WOA begins with stochastic
solutions. Depending on each search factor, the search agent arbitrarily changes according to the
optimal solution. The position of the task agent can be updated through two methods. If |A| 1, a
random search agent is chosen; otherwise, the optimal solution is selected. The whale can transition
between two types of motions, either spiral or rotational, depending on the p-value. Eventually, the
algorithm concludes when it attains a predefined satisfaction criterion, where x* represents the best
solution, as shown in Figure 3.
Axioms 2024, 13, x FOR PEER REVIEW 18 of 33
4.2.4. Model Solving
This study solves the model by introducing a local search operator to optimize the
solution, using the stochastic WOA nature of the meta-heuristic algorithm. The WOA be-
gins with stochastic solutions. Depending on each search factor, the search agent arbitrar-
ily changes according to the optimal solution. The position of the task agent can be up-
dated through two methods. If |A| 1, a random search agent is chosen; otherwise, the
optimal solution is selected. The whale can transition between two types of motions, either
spiral or rotational, depending on the p-value. Eventually, the algorithm concludes when
it attains a predefined satisfaction criterion, where x* represents the best solution, as
shown in Figure 3
WOA mining and discovery is divided into two phases. In the operational phase, a
bubble net attack strategy is demonstrated the procedures of searching for agents based
on the current optimum. There are two approaches to this attack, including pride and
spiral swims. Based on a randomly selected search agent, each whale independently
changes its position during the discovery process.
Figure 3. Proposed flowchart for the whale optimization algorithm.
4.3. Tabu Search Algorithm
Tabu search selects a candidate solution from the neighborhood using a set of tech-
niques. This chosen solution is then subjected to the measurement phase within the sim-
ulation. The scheduling process implements the candidate’s solution and computes its
output through numerous replicates [120].
Figure 3. Proposed flowchart for the whale optimization algorithm.
WOA mining and discovery is divided into two phases. In the operational phase, a bubble net
attack strategy is demonstrated the procedures of searching for agents based on the current optimum.

Axioms 2024, 13, 16 17 of 30
There are two approaches to this attack, including pride and spiral swims. Based on a randomly
selected search agent, each whale independently changes its position during the discovery process.
4.3. Tabu Search Algorithm
Tabu search selects a candidate solution from the neighborhood using a set of techniques. This
chosen solution is then subjected to the measurement phase within the simulation. The scheduling
process implements the candidate’s solution and computes its output through numerous repli-
cates [120].
4.3.1. The Integer Programming Model in a Definitive Model
The optimization objective is reducing waiting periods and the cost of completion times to a
minimum. Every patient went through three phases of initial admission and surgery and recovery
phases of surgery.
min ∑
j
∑
t
∑
p
γpqj,t,p + βm (35)
where q denotes the queue limit, γ is the resource capacity content, and β denotes the patient visit or
procedure completion time. Meanwhile, patient queue balance as well as resource capacity versus
the number of patients to be served are constraints of the appointment system.
4.3.2. Algorithm Optimal Progress
The random solution called at the beginning of the algorithm is x0. And the best-found neighbor
solution, x′, is named as a pivot. Tabu tenure comes into play to navigate away from local optima
throughout the search process. An iterative update of the list is implemented to enable the algorithm
to adapt to the status of the search. This methodology ensures the comprehensive exploration
of the entire solution space while simultaneously addressing the objectives of intensification and
diversification. Additionally, in specific instances, the aspiration criterion is applied to supersede the
Tabu list. Figure 4 illustrates the proposed TS algorithm, where X* represents the best solution.
4.3.1. The Integer Programming Model in a Definitive Model
The optimization objective is reducing waiting periods and the cost of completion
times to a minimum. Every patient went through three phases of initial admission and
surgery and recovery phases of surgery.
, ,
min p j t p m
j t p
q
γ β
+
 (35)
where q denotes the queue limit, γ is the resource capacity content, and β denotes
the patient visit or procedure completion time. Meanwhile, patient queue balance as well
as resource capacity versus the number of patients to be served are constraints of the ap-
pointment system.
4.3.2. Algorithm Optimal Progress
The random solution called at the beginning of the algorithm is 0
x . And the best-
found neighbor solution, '
x , is named as a pivot. Tabu tenure comes into play to navigate
away from local optima throughout the search process. An iterative update of the list is
implemented to enable the algorithm to adapt to the status of the search. This methodol-
ogy ensures the comprehensive exploration of the entire solution space while simultane-
ously addressing the objectives of intensification and diversification. Additionally, in spe-
cific instances, the aspiration criterion is applied to supersede the Tabu list. Figure 4 illus-
trates the proposed TS algorithm, where X* represents the best solution.
The term of Tabu was determined from some preliminary experiments. The taboo
retention period specified the upper limit for the number of iterations. The maximum
computation time is set as the criterion for terminating the algorithm when no improve-
ment is observed within the specified number of iterations. The second method uses inte-
ger programming with stochastic models, which are executed separately and then com-
pared.
Figure 4. The flowchart of the Tabu search algorithm.
The term of Tabu was determined from some preliminary experiments. The taboo retention
period specified the upper limit for the number of iterations. The maximum computation time is set

Axioms 2024, 13, 16 18 of 30
as the criterion for terminating the algorithm when no improvement is observed within the specified
number of iterations. The second method uses integer programming with stochastic models, which
are executed separately and then compared.
4.4. Other Heuristic Algorithm
4.4.1. Problem Description
A heuristic optimization algorithm is an intuitively or empirically constructed algorithm that
provides a viable solution for the combinatorial optimization problem at an acceptable cost, consid-
ering computational time and space constraints. This means that the heuristic algorithm solves the
problem empirically or according to some rules, and the solution to the problem is not necessarily
optimal but is likely to be approximate (or near optimal). Akbarzadeh et al. [124] proposed a three-
phase heuristic algorithm for constructing high-quality feasible solutions using column generation,
which is further improved by local branching. The surgical case replanning and scheduling problem
with resource rescheduling is investigated in the presence of a chunk release time when the OR
planners aim to achieve equilibrium between the capacity and demand for operating rooms. The
following equation involves four components that reflect the benefits, resulting in weighted sums of
various metrics.
Min ω = −α
∑idj wid×APidj×xj
αMax + β ∑i WTi
βmax + γ NurRe
γmax + δ SurRe
δmax (36)
where the initial component compels the model to allocate surgical cases to an appropriate day.
Following this, the second component seeks to minimize waiting time. Subsequently, the next
component aims to minimize the overall nurse rescheduling cost. Finally, the last component focuses
on minimizing the deviation of the surgeon’s schedule from the Master Surgery Schedule (MSS).
The weights assigned to these different objective functions are relative and collectively sum to 1, i.e.,
α + β + γ + δ = 1.
4.4.2. Constraint
There needs to be a requirement for nurses to perform only one procedure at a time, and to
assign procedures to nurses only if they have been assigned a shift. In fact, a surgeon is restricted to
one operating room at any given time, and each surgery is limited to being conducted only once.
∑
j
NTntdj × xj ≤ ∑
v
nxnvd (37)
∑
rj
SXsbrdj × xj ≤ 1 (38)
∑
dj
APidj × xj = PXi (39)
In Equation (37), NTntdj denotes the case where nurse n received the task for schedule j at time
t on day d, xj indicates whether the scheduling is accepted or not, nxnvd indicates whether nurse n
worked on shift v on day j; SXsbrdj denotes whether surgeon s is assigned to the r-room b area on
day d of program j; APidj denotes if surgical case i is included in schedule j, day d; and PXi is the
decision variable implies that the surgery is performed within the time frame under consideration.
In addition, some other constraints should be considered, such as the order of nurse scheduling
and the number of physician surgical blocks.
4.4.3. Solution Methodology
The solution procedure named Column Generation and Local Branching Heuristic (CLH) is
proposed to address the integrated problem. Figure 5 provides an overview for the solution procedure,
consisting of three distinct steps. Firstly, the CLH algorithm employs a computationally feasible
column generation procedure to address the impracticality of generating all possible OR schedules
directly. If the solution is an integer, it is optimal; otherwise, transforming fractional solutions into
feasible integer solutions offers an upper bound on the optimal solution. If the upper bound matches
the lower, the optimal solution is identified. If not, the optimal upper bound is used as input for an
improvement step, using local branches to narrow the optimality gap.

Axioms 2024, 13, 16 19 of 30
ploys a computationally feasible column generation procedure to address the impractical-
ity of generating all possible OR schedules directly. If the solution is an integer, it is opti-
mal; otherwise, transforming fractional solutions into feasible integer solutions oﬀers an
upper bound on the optimal solution. If the upper bound matches the lower, the optimal
solution is identified. If not, the optimal upper bound is used as input for an improvement
step, using local branches to narrow the optimality gap.
Figure 5. Summary of the Column Generation and Local Branching Heuristic.
5. Bibliometric Analysis of Literature
Research on AS systems has attracted significant attention. A literature bibliometric
analysis is therefore needed to systematically collect and analyze bibliographic data to
gain a more detailed understanding of the research field and the future direction of re-
search ideas. This article provides a summary of the literature on AS optimism published
in SCI-indexed journals from 1 January 2016 to 25 August 2023, utilizing Cite Space. The
search terms employed were “appointment scheduling” and “system optimism.” The pa-
per’s representative institution and country were determined based on the first author’s
aﬃliation and location.
5.1. Methods and Data
5.1.1. Methods
To conduct the analysis, we utilized Cite Space, a Java program established in 2006
[125]. This powerful tool specializes in document analysis and bibliometric visualization,
providing valuable insights into scientific literature databases. Its notable features encom-
pass co-citation analysis for discerning paper relationships, burst detection to highlight
emerging trends and influential works, visualization tools. In contrast to tools like VOS
viewer, Cite Space uniquely provides spatiotemporal visualization, illustrating the evolu-
tion of literature citations over time. It identifies core nodes, tracks keyword trends, and
constructs citation coupling networks for intrinsic connections in the research field. With
a user-friendly interface, Cite Space stands out for its accessibility and analytical depth.
Figure 5. Summary of the Column Generation and Local Branching Heuristic.
5. Bibliometric Analysis of Literature
Research on AS systems has attracted significant attention. A literature bibliometric analysis
is therefore needed to systematically collect and analyze bibliographic data to gain a more detailed
understanding of the research field and the future direction of research ideas. This article provides a
summary of the literature on AS optimism published in SCI-indexed journals from 1 January 2016 to
25 August 2023, utilizing Cite Space. The search terms employed were “appointment scheduling”
and “system optimism.” The paper’s representative institution and country were determined based
on the first author’s affiliation and location.
5.1. Methods and Data
5.1.1. Methods
To conduct the analysis, we utilized Cite Space, a Java program established in 2006 [125]. This
powerful tool specializes in document analysis and bibliometric visualization, providing valuable
insights into scientific literature databases. Its notable features encompass co-citation analysis for
discerning paper relationships, burst detection to highlight emerging trends and influential works,
visualization tools. In contrast to tools like VOS viewer, Cite Space uniquely provides spatiotemporal
visualization, illustrating the evolution of literature citations over time. It identifies core nodes, tracks
keyword trends, and constructs citation coupling networks for intrinsic connections in the research
field. With a user-friendly interface, Cite Space stands out for its accessibility and analytical depth.
5.1.2. Data
Bibliographic information on AS was collected using the Web of Science (WOS) over a span of
seven years (2016–2023). A total of 1144 records are present in WOS, excluding reviews. WOS is a
global academic literature search tool covering a wide range of disciplines in the natural sciences,
social sciences, humanities, engineering, and technology, and includes many well-known journals
and academic publications. It provides the ability to search by keywords in multiple dimensions and
can be used to gain insights into the linkages between the literature by visual citation analysis tools,
which can help promote an in-depth understanding of the literature and scientific research. This paper
investigates a review of methods for optimizing the operation of appointment scheduling systems in
the healthcare field, so the key subject terms for the literature search were “appointment scheduling”
as well as “system optimism”. Since 2010, the digital transformation of AS systems and the spread of
healthcare information technology have driven the development of the optimization of the operation
of AS systems. And, in 2016, digital tools such as smart queuing systems, online pre-screening
and other digital tools began to be invested in healthcare system optimization research. Numerous
healthcare institutions began to realize the importance of digital transformation, and researchers
began to explore how to use big data and analytical tools to optimize AS system operations, including

Axioms 2024, 13, 16 20 of 30
improving resource utilization as well as predicting patient demand. During this period, research on
the operation optimization of AS systems may start to receive more attention and gradually develop
and improve in the following years. Therefore, literature reviews from 2016 onwards were selected
for bibliometric analysis.
5.2. Operational Results
Cite Space is employed to analyze data from various countries, institutions, journals, and
scholars, facilitating a comparative assessment of their contributions. Utilizing Cite Space, we
summarized the articles and the cumulative count of articles related to the optimization of AS
systems, as illustrated in Figure 6.
the healthcare field, so the key subject terms for the literature search were “appointment
scheduling” as well as “system optimism”. Since 2010, the digital transformation of AS
systems and the spread of healthcare information technology have driven the develop-
ment of the optimization of the operation of AS systems. And, in 2016, digital tools such
as smart queuing systems, online pre-screening and other digital tools began to be in-
vested in healthcare system optimization research. Numerous healthcare institutions be-
gan to realize the importance of digital transformation, and researchers began to explore
how to use big data and analytical tools to optimize AS system operations, including im-
proving resource utilization as well as predicting patient demand. During this period, re-
search on the operation optimization of AS systems may start to receive more attention
and gradually develop and improve in the following years. Therefore, literature reviews
from 2016 onwards were selected for bibliometric analysis.
5.2. Operational Results
Cite Space is employed to analyze data from various countries, institutions, journals,
and scholars, facilitating a comparative assessment of their contributions. Utilizing Cite
Space, we summarized the articles and the cumulative count of articles related to the op-
timization of AS systems, as illustrated in Figure 6.
Figure 6. Yearly distribution of papers and cumulative publications over the period 2016–2023.
Since 2016, there have been 1144 publications related to the optimization of AS sys-
tems. The annual publication count rose from 95 articles in 2016 to 224 articles in 2022,
indicating a substantial increase each year. The cumulative number of yearly publications
demonstrates a noteworthy upward trend, reflecting a growing scholarly interest in
healthcare optimization studies.
95
211
336
471
653
846
1070
1144
95
116
125
135
182
193
224
74
0
200
400
600
800
1000
1200
1400
0
50
100
150
200
250
2016 2017 2018 2019 2020 2021 2022 2023
Cumulative annual publications
Figure 6. Yearly distribution of papers and cumulative publications over the period 2016–2023.
Since 2016, there have been 1144 publications related to the optimization of AS systems. The
annual publication count rose from 95 articles in 2016 to 224 articles in 2022, indicating a substantial
increase each year. The cumulative number of yearly publications demonstrates a noteworthy upward
trend, reflecting a growing scholarly interest in healthcare optimization studies.
5.2.1. Country and Publisher
As shown in Figure 7, the leading nations in publications concerning the optimization technol-
ogy of ASs include: China with 449 publications, the United States with 433 publications, Canada
with 62 publications, England with 47 publications, and Germany with 46 publications. Notably,
China exhibits the highest volume of papers, indicating its prominent focus on AS optimization
technology.
5.2.1. Country and Publisher
As shown in Figure 7, the leading nations in publications concerning the optimiza-
tion technology of ASs include: China with 449 publications, the United States with 433
publications, Canada with 62 publications, England with 47 publications, and Germany
with 46 publications. Notably, China exhibits the highest volume of papers, indicating its
prominent focus on AS optimization technology.
Figure 7. Annual bar chart depicting the top ten countries based on the number of publications.
The institutions leading in the publication of papers related to optimization of AS
systems in Figure 8 are Shanghai University (66 papers), Shanghai Jiao Tong University
(59 papers), University of Shanghai for Science Technology (56 papers), Fudan University
(42 papers), US FDA (34 papers), Shanghai Maritime University (29 papers), East China
University of Science Technology (27 papers), Shanghai University of Engineering Science
449
433
62
47
46
34
32
31
27
26
25
0 100 200 300 400 500
PEOPLES R CHINA
USA
CANADA
ENGLAND
GERMANY
NETHERLANDS
FRANCE
AUSTRALIA
JAPAN
SPAIN
SINGAPORE

Axioms 2024, 13, 16 21 of 30
The institutions leading in the publication of papers related to optimization of AS systems in
Figure 8 are Shanghai University (66 papers), Shanghai Jiao Tong University (59 papers), University
of Shanghai for Science Technology (56 papers), Fudan University (42 papers), US FDA (34 papers),
Shanghai Maritime University (29 papers), East China University of Science Technology (27 papers),
Shanghai University of Engineering Science (23 papers), Tongji University (22 papers), and the
Chinese Academy of Sciences (17 papers). Among these institutions, there are nine in China, and
one in the US. This indicates China’s interest in the research of ASs optimization technology, making
extensive efforts to the development of intelligent healthcare management compared to other nations.
The institutions leading in the publication of papers related to optimization of AS
systems in Figure 8 are Shanghai University (66 papers), Shanghai Jiao Tong University
(59 papers), University of Shanghai for Science Technology (56 papers), Fudan University
(42 papers), US FDA (34 papers), Shanghai Maritime University (29 papers), East China
University of Science Technology (27 papers), Shanghai University of Engineering Science
(23 papers), Tongji University (22 papers), and the Chinese Academy of Sciences (17 pa-
pers). Among these institutions, there are nine in China, and one in the US. This indicates
China’s interest in the research of ASs optimization technology, making extensive eﬀorts
to the development of intelligent healthcare management compared to other nations.
Figure 8. Dendrogram of the top ten institutions ranked by the number of publications.
449
0 100 200 300 400 500
PEOPLES R CHINA
Figure 8. Dendrogram of the top ten institutions ranked by the number of publications.
5.2.2. Author
In this paper, Cite Space is employed to conduct a statistical analysis of authors contributing to
published research on optimization technology for ASs. The results highlight several authors who
have made a significant impact in this field, as depicted in Figure 9.
Axioms 2024, 13, x FOR PEER REVIEW
5.2.2. Author
In this paper, Cite Space is employed to conduct a statistical analysis of a
tributing to published research on optimization technology for ASs. The resul
several authors who have made a significant impact in this field, as depicted i
The number of co-occurring nodes is 290, the number of connections is 4
network density is 0.0115. Cayirli, T. has been cited 99 times, followed by La
with 96 citations; in addition, Klassen, KJ. and Robinson, LW.’s cited numbers a
and 62 times, respectively. Among them, Cayirli, T. is the most influential in
optimization research of ASs, focusing on appointment arrangement operatio
agement optimization; Laganga, LR focus on the reservation over-capacity
problem and Klassen, KJ. focuses more on the research on simulation optimiz
ods. The work conducted by these authors reflects the predominant direction
mization research of ASs.
Figure 9. Co-citation network of authors (timespan: 2016–2023; slice length = 1; g-inde
3; LBY = 5; e = 1; n = 290; E = 481).
5.2.3. Thematic Trends
This study employs Cite Space to statistically assess topic trends spannin
period. The evaluation of topic trends involves analyzing keyword burstiness,
Figure 9. Co-citation network of authors (timespan: 2016–2023; slice length = 1; g-index = 25; LRF = 3;
LBY = 5; e = 1; n = 290; E = 481).

Axioms 2024, 13, 16 22 of 30
The number of co-occurring nodes is 290, the number of connections is 481, and the network
density is 0.0115. Cayirli, T. has been cited 99 times, followed by Laganga, LR., with 96 citations; in
addition, Klassen, KJ. and Robinson, LW.’s cited numbers are 88 times, and 62 times, respectively.
Among them, Cayirli, T. is the most influential in the system optimization research of ASs, focusing
on appointment arrangement operation and management optimization; Laganga, LR focus on the
reservation over-capacity expansion problem and Klassen, KJ. focuses more on the research on
simulation optimization methods. The work conducted by these authors reflects the predominant
direction in the optimization research of ASs.
This study employs Cite Space to statistically assess topic trends spanning an 8-year period.
The evaluation of topic trends involves analyzing keyword burstiness, as depicted in Table 4. The
burstiness of keywords is measured using the “strength” based on a statistical formula to gauge
changes in keyword frequency.
Table 4. Thematic trend analysis over eight years.
Keywords Year Strength Begin End 2016–2023
patient flow 2016 4.64 2016 2017
Figure 9. Co-citation network of authors (timespan: 2016–2023; slice length = 1; g-index = 25; LRF =
3; LBY = 5; e = 1; n = 290; E = 481).
This study employs Cite Space to statistically assess topic trends spanning an 8-year
period. The evaluation of topic trends involves analyzing keyword burstiness, as depicted
in Table 4. The burstiness of keywords is measured using the “strength” based on a statis-
tical formula to gauge changes in keyword frequency.
patient flow 2016 4.64 2016 2017
discrete event simulation 2016 2.95 2016 2018
overbooking model 2017 2.8 2017 2018
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
health information technology 2019 2.48 2019 2020
electronic health record 2020 2.93 2020 2021
robust optimization 2017 2.39 2021 2023
operations research 2021 2.27 2021 2023
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
time windows 2017 2.58 2017 2018
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
metanalysis 2018 2.59 2018 2019
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
simulation 2017 2.56 2018 2019
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
digital health 2019 3.31 2019 2020
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
3; LBY = 5; e = 1; n = 290; E = 481).
patient flow 2016 4.64 2016 2017
time windows 2017 2.58 2017 2018
metanalysis 2018 2.59 2018 2019
simulation 2017 2.56 2018 2019
digital health 2019 3.31 2019 2020
COVID-19 pandemic 2021 2.27 2021 2023
COVID-19 pandemic 2021 2.27 2021 2023
As illustrated in Table 4, the lines depict the evolution of topic trends over the eight-
year period. Specifically, covering the years 2016 to 2018, patient flow, discrete event sim-
ulation, overbooking model, time windows became hot keywords. This reflects the early
trend of appointment scheduling research. The research is related to patient flow no-show
and walk-in scenario simulation analysis. From 2019 to 2020, digital health, electronic
health record and health information technology were the most popular keywords. Online
registration and consultation have become more common, so research on health manage-
ment informatization has become a trend. From 2020 to 2023, robust optimization, opera-
tions research and COVID-19 pandemic became popular keywords. Cost control require-
ments and resource allocation for outpatient appointments have become particularly im-
portant, and system optimization has become a key factor in health management consul-
tation appointments. This result reflects the increasing importance of health management
informatization and ASs system optimization research.
5.2.4. Cited Journals
In this paper, we use Cite Space to analyze the cited journal of published research on
optimization technology for ASs as shown in Figure 10. The top 3 cited journals are Euro-
pean Journal of Operational Research (262 citations), Production and Operations Manage-
ment (225 citations), BMC Health Services Research (214 citations). They belong to the
fields of operations research, production and operations management, and health services
research, respectively. These journals are important indicators in the research field and
have a profound impact on the study of operational optimization of AS systems.
As illustrated in Table 4, the lines depict the evolution of topic trends over the eight-year period.
Specifically, covering the years 2016 to 2018, patient flow, discrete event simulation, overbooking
model, time windows became hot keywords. This reflects the early trend of appointment scheduling
research. The research is related to patient flow no-show and walk-in scenario simulation analysis.
From 2019 to 2020, digital health, electronic health record and health information technology were the
most popular keywords. Online registration and consultation have become more common, so research
on health management informatization has become a trend. From 2020 to 2023, robust optimization,
operations research and COVID-19 pandemic became popular keywords. Cost control requirements
and resource allocation for outpatient appointments have become particularly important, and system
optimization has become a key factor in health management consultation appointments. This
result reflects the increasing importance of health management informatization and ASs system
optimization research.
5.2.4. Cited Journals
In this paper, we use Cite Space to analyze the cited journal of published research on optimiza-
tion technology for ASs as shown in Figure 10. The top 3 cited journals are European Journal of
Operational Research (262 citations), Production and Operations Management (225 citations), BMC
Health Services Research (214 citations). They belong to the fields of operations research, production
and operations management, and health services research, respectively. These journals are important
indicators in the research field and have a profound impact on the study of operational optimization
of AS systems.

Axioms 2024, 13, 16 23 of 30
optimization technology for ASs as shown in Figure 10. The top 3 cited journals are Euro-
pean Journal of Operational Research (262 citations), Production and Operations Manage-
ment (225 citations), BMC Health Services Research (214 citations). They belong to the
fields of operations research, production and operations management, and health services
research, respectively. These journals are important indicators in the research field and
have a profound impact on the study of operational optimization of AS systems.
Figure 10. Co-citation network of cited journals.
6. Discussion
Healthcare and appointment scheduling methods vary globally due to diverse cul-
tural, economic, and infrastructural factors. Therefore, more comments about the specifics
of the healthcare and the related methods for system appointment scheduling are neces-
sary. For instance, some countries prioritize centralized electronic systems for appoint-
ments, ensuring eﬃciency and accessibility, while others might rely more on decentral-
Figure 10. Co-citation network of cited journals.
6. Discussion
Healthcare and appointment scheduling methods vary globally due to diverse cultural, eco-
nomic, and infrastructural factors. Therefore, more comments about the specifics of the healthcare and
the related methods for system appointment scheduling are necessary. For instance, some countries
prioritize centralized electronic systems for appointments, ensuring efficiency and accessibility, while
others might rely more on decentralized or traditional methods. Additionally, cultural attitudes to-
wards healthcare, technology adoption, and government policies greatly influence these approaches.
It is crucial to consider these nuances when analyzing and comparing healthcare systems worldwide.
6.1. Specifics of Healthcare
The healthcare specifics are explained in four areas which are online platforms, e-information,
waiting lists and customization. Firstly, a range of processes and technologies are aimed at facilitating
patient access to medical services involved in AS. Many healthcare providers use online appoint-
ment platforms that allow patients to make appointments through a website or mobile application.
Secondly, it is also integrated with the electronic health record system to ensure that appointment
arrangements are synchronized with patient records, reducing the risk of medical errors, saving
costs through diagnostic examinations, reducing medication, and promoting the process of health
management informatization [126]. Lastly, when patients cannot make an appointment at the ideal
time, they can add the appointment entry to the waiting list through waiting list management. And
when a patient is unable to make a desired appointment, the appointment entry can be placed on a
waitlist through waitlist management, and patients on the waitlist receive automatic notifications
once the time becomes available. In conclusion, the result of the current rapid development of AS
is its customization and flexibility, with patients often using customizable templates for different
types of appointments and services, and the overall system could handle emergencies outside of the
regular scheduling process appointment services [127].
With the rise of telemedicine, AS systems include virtual consultation options that allow
patients to schedule and attend appointments remotely, which also leads to the efficient utilization of
healthcare resources, ensuring that future appointments are reasonable and matched with the right
care or equipment. Indeed, the system cannot operate without security and compliance, and must
adhere to healthcare privacy regulations, such as the Health Insurance Portability and Accountability
Act (HIPAA), to ensure the security of patient information. And the use of online platforms should be
secured with patient data, especially when transmitted online, which should be encrypted to prevent
unauthorized access [128].
6.2. Differences of Related Methods
Differences in the structure of healthcare systems, uneven digitalization and technological
development, resource allocation and cultural differences lead to variations in how appointments are

Axioms 2024, 13, 16 24 of 30
booked in different countries. The United States, for example, emphasizes online platforms and elec-
tronic health records, relies on insurance processes, and offers convenience and telemedicine [129,130].
Japan achieves universal health coverage, with a greater focus on prevention and early intervention,
but faces the challenges of aging. China is rapidly developing digital health technologies, including
online platforms and the integration of Chinese medicine, but the operation and maintenance require-
ments are relatively high for system calculation examples [131]. The specifics of the healthcare and
the related methods for system appointment scheduling in the different countries as shown in Table 5,
and the unique characteristics of each country’s healthcare system have shaped different appointment
booking practices and patient experiences. These specifics highlight the diverse approaches to health-
care delivery, financing, and coordination across different countries. Understanding these unique
features is essential for developing effective healthcare policies and systems tailored to the needs
of each population. It is important to note that the healthcare landscape is dynamic, and ongoing
developments may lead to changes in these methods and differences over time.
Table 5. Difference in AS system methods in global.
Country AS Methods Advantages Disadvantages
China
(a) Online platforms and mobile health
applications;
Rapid adoption of digital
technologies
Integration of traditional
Chinese medicine
Higher requirements for
system arithmetic operations
and maintenance
(b) QR code systems for check-ins;
(c) Growing adoption of telemedicine.
United States
(a) Online platforms for appointment
booking; Convenient for patients
Improved accessibility with
telemedicine
Insurance dependence
Complex verification
(b) Integration with Electronic Health
Records;
(c) Telemedicine adoption.
Canada
(a) Online appointment booking;
Universal healthcare coverage
Variation in healthcare
services across provinces
(b) Coordination of care among healthcare
providers.
Germany
(a) Integration with statutory health
insurance; Well-established healthcare
infrastructure
Gatekeeping may lead to
delays in specialty care access
(b) Gatekeeping by general practitioners.
Japan
(a) Universal health coverage;
Prevention and early
intervention
Aging population strain
healthcare resources
(b) Coordination between primary and
specialty care.
Healthcare has a high degree of specialization in developed countries, where AS systems are
used to locate appropriate specialists and information is shared through electronic records. In contrast,
some developing countries rely more on traditional queuing and on-site appointments due to lack of
infrastructure, and healthcare IT is less efficient, especially in rural areas. In addition, some countries
or regions may be wary of new technologies, perhaps due to cultural or religious differences and
data security issues. Differences in healthcare systems worldwide reflect a combination of cultural
influences, levels of technological development and resource allocation considerations.
6.3. Tendencies and Scientific Methods
Understanding the trends in healthcare systems allows for better optimization of systems to fa-
cilitate health management, with the trend across healthcare being towards streamlined appointment
scheduling and optimized technology driven. Electronic medical records are replacing traditional
paper-based medical records and shifting many clinical tasks that were once paper-based to electronic
media. For instance, many countries such as the US, Germany, France, Finland, and New Zealand
have demonstrated a tendency to integrate EHR systems by establishing EHR systems at the national
level or simultaneously establishing national health information infrastructures [132]. Thus, patient
data can be accessed in real time, facilitating efficient and centralized appointment management.
However, the fact is that the process of integrating electronic health records is long and complex,

Axioms 2024, 13, 16 25 of 30
healthcare systems with EHRs are expensive to develop as well as to maintain, and the digitization
of technology-driven health management requires a greater focus on security [133], which are the
trends and challenges for future research in healthcare AS systems.
Research on AS system optimization entails investigating and analyzing data collected from
various healthcare systems, which employed methodologies such as comparative studies, statistical
analyses, and surveys to assess the impact of different scheduling approaches on patient outcomes,
resource allocation, and overall system efficiency. Furthermore, emerging technologies like ma-
chine learning and data analytics play a pivotal role in understanding and optimizing appointment
scheduling processes. These scientific methods delve into patterns, patient preferences, and system
performances, offering insights that contribute to the continual refinement of healthcare appointment
systems on a global scale. E.g., data mining is calculated by analyzing and processing large amounts
of medical data for modelling, identifying the connections and patterns between the information thus
assisting doctors in decision-making [134]. Deep learning is applied to solve difficult problems in
health management information technology through prediction and evaluation.
7. Conclusions
In this paper, we summarized the structural components of the system based on the existing
literature review at first, and introduced the characteristics, system establishment process, and
system decision-making framework. Then, we summarized the AS system optimization framework
and reviewed the algorithms used for AS system optimization, among which we find that the
GA algorithm is the most used optimization algorithm. Additionally, a bibliometric analysis and
discussion were performed, which showed that ASs optimization has been growingly focused on
developing countries exhibiting more interest in this field compared to developed nations.
Queuing theory, genetic algorithms, simulated annealing, particle swarm optimization, and
other models have been mentioned, with a focus on addressing appointment scheduling and resource
allocation issues in healthcare. Similarly, these models and techniques are widely applied to optimize
resource utilization and enhance efficiency in supply chain management, transportation scheduling,
production planning, customer service, and retail. Compared to scheduling problems in general fields,
healthcare scheduling optimization involves specific models and techniques. This is primarily because
the healthcare domain places a greater emphasis on patient healthcare needs and the specificity of
healthcare resources. For instance, healthcare resource constraint models are designed to address the
unique constraints associated with hospital resources. Disease epidemiology models, on the other
hand, are employed to predict patient visit demands and hospital bed requirements. Additionally,
there are models for patient prioritization, clinical pathway optimization, and healthcare service
capacity planning, ensuring a comprehensive optimization of the entire healthcare service process.
This is also a direction of researching optimization problems in healthcare appointment scheduling,
focusing more on the study of optimization algorithms specifically tailored for the medical field.
Therefore, we can say that, although AS system operation optimization has developed rapidly in
recent years, more focused and practical research is still needed.
Author Contributions: Conceptualization, T.N., B.L. and K.G.; methodology, T.N., B.L. and K.G.;
software, T.N.; validation, T.N.; formal analysis, T.N., B.L. and Q.L.; investigation, K.G.; resources,
L.G., S.F. and L.Y.; data curation, S.F., Q.L., B.G. and L.Y.; writing—original draft preparation, T.N.;
writing—review and editing, T.N. and B.L.; visualization, T.N.; supervision, B.L. and K.G.; project
administration, K.G.; funding acquisition, B.L. All authors have read and agreed to the published
version of the manuscript.
Funding: This work was funded in part by Beijing Social Science Fund Project under Grant 22GLC055
and 21JCB041, the Beijing Municipal Commission of Education under Grant KM202111232007, and
the China Postdoctoral Science Foundation under Grant 2021M693331.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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