Robust Overbooking for No-Shows and Cancellations in Healthcare

Abstract

Any country’s healthcare system is vital for its progress, quality of life, and long-term viability. During the pandemic, many developed countries encountered challenges of differing degrees in the administration of their healthcare systems. The overloading of healthcare services is common, leading to prolonged waiting times for medical services. Thus, the wastage of hospital resources must be taken seriously. In this paper, we examine the problem of no-shows and cancellations in outpatient clinics. By examining the literature and drawing from practical industry experience, we uncover the operational procedures of these clinics. We then suggest a robust optimization strategy for overbooking, incorporating both a conventional overbooking model and a resilient system approach. The proposed model seeks to address the substantial uncertainties in parameters encountered during the pandemic. Taking into account risk aversion, we develop an optimal overbooking policy that considers the associated costs. The primary contribution lies in introducing an alternative approach to manage the uncertainty of no-shows and cancellations through the utilization of an overbooking technique.

1. Introduction

The COVID-19 pandemic in recent years provided evidence of the vulnerability of healthcare systems all over the world [1]. Unprecedented challenges are evident in different areas, such as resource management, including outpatient doctors, nurses, and ambulances [2]. Even during the pre-pandemic period, traditional healthcare systems struggled with demand uncertainties as they typically had to plan their resources months in advance. Doctors needed to be recruited ahead of time, and medicines and medical equipment had to be deployed well before patients required treatment. Although many countries have invested heavily in their respective healthcare systems, their efforts seem to be hampered by forecasting mistakes and uncertainties [3]. Hence, a more robust system is needed to accommodate those factors. Furthermore, healthcare significantly contributes to sustainable economic growth, alongside other essential government spending areas such as technology and infrastructure [4,5].

In this paper, we aim to demonstrate how robust optimization can solve the healthcare problem with the data obtained in outpatient clinics. Resource planning is crucial for outpatients. Typically, managers must decide how many doctors should be available for outpatient clinics and how many nurses should be assigned to assist. They also determine the number of rooms available for treatment. The overall efficiency and profit of clinics largely depends on how accurately managers match patient appointments with the number of doctors assigned [6,7,8]. However, patients often change their plans at the last minute or cancel their appointments [9,10], creating challenges for managers in dealing with appointment demand uncertainties, which can result in idle doctors or untreated patients. The traditional solution is overbooking, commonly used in industries with a high number of no-shows and replenishable products, such as the aviation industry. Nevertheless, existing overbooking methods with deterministic approaches impose restrictions on the assumptions about parameter distributions [11,12,13,14]. Therefore, we propose a robust optimization approach to manage no-shows and cancellations in healthcare systems and to deal with uncertain data assumptions and reliance.

2. Background

The primary issues of the operational design and patient appointment systems for ambulatory care center services were broken down into two main categories: major resource factors and modeling the problems. Gupta and Denton [15] delved into critical the aspects concerning operational design and patient appointment systems at ambulatory care centers. They explored complex variables and diverse processes by which to formulate an optimization model aimed at reducing costs associated with waiting times, tardiness, and idle periods. Ultimately, by integrating these factors comprehensively, their model aims to maximize revenue and profitability. Three distinct arrival patterns were analyzed, namely, periodic process, unit process, and single batch process, which define how patients arrive within specific timeframes. This adaptable model can be applied to groups, clusters, or individual cases. Our study expanded upon previous work by Gupta and Denton, incorporating additional considerations such as the cost of errors, which is particularly pertinent in the context of Hong Kong.

Stanciu, Vargas, and May [16] addressed revenue management challenges in operating theaters by adapting Lobaba’s EMSRb algorithm. Their primary objective was to optimize patient scheduling for different surgical services within specific timeframes, considering procedure types and patient reimbursements. Unlike previous models, their approach incorporates penalties to quantify the costs associated with surgical errors, which can affect service quality and prolong medical care. These penalties are assessed in both monetary and non-monetary terms, enhancing the model’s comprehensiveness.

Ratcliffe, Gilland, and Marucheck [17] likened revenue management in outpatient appointment clinics to practices in the airline industry, particularly in managing issues like overbooking and no-shows. They emphasize that inefficiencies arise when appointments are missed by either doctors or patients, leading to lost revenue due to the underutilized time of medical staff. From a practical standpoint, researchers can derive a deterministic formula for optimizing booking requests based on specific booking data. Removing constraints allows for the redefinition of upper and lower limits and the revaluation of optimal booking strategies. Their sensitivity analysis explores ten different policies to understand how model variables affect expected profit.

Additionally, insights from the American Health Information Management Association on ambulatory care centers, including hospital-based and community-based ambulatory care, provide valuable perspectives for extending models to incorporate features specific to hospital-based settings. Roski and Gregory [18] discussed various numerical metrics for measuring healthcare performance, focusing on care effectiveness, healthcare options, service accessibility, and patient satisfaction. They prioritize care efficiency as a key metric, recognizing its multifaceted impact. Service quality and patient satisfaction serve as secondary metrics for evaluating effective healthcare management.

3. Literature Review

Overbooking is widely employed in the hotel and aviation industries, where resources are perishable, such as reservations and appointment bookings [19,20]. Appointment overbooking is a method of mitigating the impact of no-shows by reducing idle time and increasing the efficiency and utilization rate of the system [21]. Overbooking appointments involves scheduling more patients in a session than the number of available service providers [22,23,24]. An excessive number of appointments can compensate for no-shows. Muthuraman and Lawley [25] utilized a stochastic model to maximize expected profit with overbooking and unimodal no-show rates. Amnon [26] provided advice on managing overbooking in endoscopy units. Consequently, patient access times and provider productivity could be improved. Kazim et al. [27] determined the optimal overbooking strategy via machine learning (ML) using a greedy algorithm. Kuiper et al. [14] used phase-type distributions to approximate the distribution of patient service times for overbooking decisions. Han et al. [28] first used artificial neural networks (ANNs) for predictive calculation of a patient’s probability of absence and different appointment durations before booking optimization. However, if the overbooking policy is implemented inefficiently, patient waiting times and system overtime may also increase. To determine the optimal overbooking policy and achieve the intended objectives, Liu and Ziya [29] used the single-server queuing model, and Kim and Giachetti [30] adopted the stochastic model. LaGanga and Lawrence [31] illustrated numerical examples of overbooking strategies that lead to improvements in the performance of different service settings and cost structures in hospitals. Kolisch and Sickinger [32] and Zeng et al. [33] provided evidence of the efficiency of using overbooking for advanced policy systems. Liu and Ziya [29] stated that the overbooking strategy is the best among other strategies when the number of patients is comparatively low. Kros et al. [22] demonstrated that large facilities tend to benefit from overbooking when no-show rates are high.

Among all the uncertain factors, patient no-shows and cancellations are the most significant issues. Typically, patients cancel their appointments just before the scheduled time, resulting in last-minute cancellations. If patients cancel their appointments sufficiently in advance, the hospital can reallocate the vacancy to new patients. However, according to Liu, Ziya, and Kulkarni [34], from the patient’s perspective, they might suffer if they later find that the service is needed. Consequently, many patients choose not to show up or cancel at the last moment. Parizi and Ghate [35] considered multi-class and multi-resource scenarios, while Schuetz and Kolisch [36] discussed the demand for different customer services and products. Most papers treat late cancellations as no-shows because rebooking is required if the cancellation occurs late, necessitating the rearrangement of resources. Bellini et al. [37] indicated that delays in treatment caused by no-shows may affect the patient’s own health. Wang and Gupta [38] demonstrated that the probability of no-shows depends on time, service, and patient type, in addition to homogeneous types. Samorani and LaGanga [39] showed that appointments can also be affected by weather conditions. Bheemidi et al. [40] demonstrated that physical distance to the hospital affects whether patients no-show. Cayirli and Veral [41], as well as Gupta and Denton [15], revealed that shorter appointment intervals and overbooking can mitigate the impact of no-shows.

We categorized the existing research according to specific topics (see

Table 1. Summary of the relevant literature.

No.	Topics	Sources	Findings/Methodologies
1	Factors affecting no-show	[15,38,39,40,41]	Weather, physical distance, waiting time, appointment intervals, etc.
2	Impact of no-show	[34,35,36,37]	Waste of resources, reprogramming, harm to patient health, etc.
3	Effectiveness of overbooking	[22,23,24,25,26,29,31,32]	Maximize expected profits, reduce waiting times, optimize performance, etc.
4	Optimal overbooking strategy	[14,27,28,29,30]	Single-server queuing model, stochastic model, phase-type distributions, ML, ANNs, etc.

) and found that no-shows and cancellations are not uncommon occurrences in healthcare services. Factors such as weather conditions and appointment intervals can influence whether patients attend their scheduled appointments. Patient no-shows not only impact patients’ health but also lead to the underutilization and wastage of hospital resources or incur additional costs due to rescheduling. The concept of overbooking has been extensively validated for its effectiveness in addressing the issue of no-shows, demonstrating its potential to maximize expected profits and enhance efficiency. However, many existing overbooking studies rely on assumptions regarding parameter distributions, which, if inaccurate, compromise the validity of subsequent optimization results. In recent years, some studies have attempted to utilize technologies such as ML and ANNs to address no-shows and cancellations. These advanced technologies depend on substantial data and technical capabilities, rendering them impractical for many hospitals at the operational level. This paper introduces a robust optimization model for healthcare overbooking strategies to mitigate no-shows and vacancies. It emphasizes balancing profitability with risk aversion by determining optimal overbooking thresholds based on cost considerations. Future research directions include integrating real-time data analytics, refining predictive models, optimizing department-specific overbooking thresholds, and conducting comprehensive cost–benefit analyses to enhance operational efficiency and patient care standards.

4. Proposed Model

4.1. Basic Model

Following a thorough examination of operational processes at 27 hospitals in Hong Kong and Mainland China, the most crucial factors were identified and distilled to represent each hospital’s situation accurately and streamline the complex scenario. The model is structured as follows: when considering K types of healthcare services in an ambulatory care center or K departments, the planned time frame consists of a single session with consecutive time slots divided into equal intervals based on center requirements. The appointment-scheduling challenge revolves around two critical decisions:

• Determining the number of consultation rooms to open on any given day;
• Assigning appointments to each consultation room.

We introduce the following variables and indices to develop the model.

Indexes:

i: Index of starting time points i, where i = 0, 1, …, T – 1;

j: Index of ending time point j, where j= 1, 2, …, T;

T: Total points within a scheduling system in the context of healthcare operations;

k: Index of healthcare department k, where k = 1, 2, …, K;

K: Total number of numbers of distinct healthcare departments;

${\mathrm{C}}^{\mathrm{k}}$: Capacity of health care department $k$;

${\mathrm{R}}_{\mathrm{i},\mathrm{j},\mathrm{k}}$: Revenue from an appointment generated from time i to j within department k;

${\mathrm{x}}_{\mathrm{i},\mathrm{j},\mathrm{k}}$: Number of accepted appointments from time i to j within department k;

${\mathrm{U}}_{\mathrm{i},\mathrm{j},\mathrm{k}}$: The number of uncertain demand of appointments from time i to j within department k;

${\mathrm{P}}_{\mathrm{i},\mathrm{j},\mathrm{k}}$: The probability of showing up for the accepted appointment from time i to j within department k;

$O^{t,k}$: Number of overbooked appointments at time $t$ within department k.

Initially, at time 0, it is assumed that there are no appointments, while all scheduled appointments are expected to be completed by the end time T. To manage overbooking effectively when starting with no appointments and needing to complete all by time $T$, we should initially schedule more appointments than the number of available slots, based on anticipated no-shows and cancellations. This requires the use of historical data to estimate the right amount of overbooking and adjusting schedules dynamically to handle any deviations. Clear patient communication and contingency plans for managing unexpected delays are crucial to ensuring that care is delivered efficiently and on time.

In practice, the hospital operates from 9 a.m. to 6 p.m., denoted as time 0 to T in this study. Another assumption is that each accepted appointment spans at least one time slot, such as from time 1 to time 2. The network can be represented by nodes that signify the beginning and end times of each appointment, as depicted in Figure 1.

For a particular time slot t and t ϵ {1, 2, …, T − 1} in our planning horizon, the formula below indicates the utilization rate of department k, k ϵ {1, 2, …, K} at time t, which embraces the existence of no-shows:

$$\sum _{i=0}^t\sum _{j=t+1}^T{P}_{i,j,k}{x}_{i,j,k}-\sum _{i=0}^{t-1}{P}_{i,j,k}{x}_{i,j,k}+\sum _{j=i+1}^T{P}_{i,j,k}{x}_{i,j,k}\le C^k+O^{t,k}.$$

(1)

For all $t, k$, the capacity of department that has been utilized is represented in the first term. The capacity which was occupied previously and is vacant again is shown in the second term. Finally, for the latest accepted appointments which will continue, at least one should be represented by the third term. Due to the problem of no-shows after the appointments are made in healthcare, the overbooking policy is introduced to manage the appointment scheduling. For each specialty department k with constraints on their capacities, decision makers in hospitals may choose to invite more patients and accept more appointments than the available appointment session to mitigate the effect of no-shows. Therefore, the constraints will be modified as follows:

$$\sum _{j=i+1}^T{P}_{0,j,k}{x}_{0,j,k}\le C^k+O^{0,k}, \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} k.$$

(2)

The revenue produced by fulfilling the consultation sessions is shown below:

$$\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\sum _{k=1}^K\sum _{i=0}^T\sum _{j=t+1}^T{R}_{i,j,k}{P}_{i,j,k}{x}_{i,j,k},$$

(3)

$$\mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o} \sum _{i=0}^{t-t}\sum _{j=t+1}^T{P}_{i,j,k}{x}_{i,j,k}-\sum _{i=0}^{t-1}{P}_{i,j,k}{x}_{i,j,k}+\sum _{j=i+1}^T{P}_{i,j,k}{x}_{i,j,k}\le C^k+O^{t,k},$$

$$\sum _{j=i+1}^T{P}_{i,j,k}{x}_{0,j,k}\le C^k+O^{0,k},$$

$$x_{i,j,k}\le U_{i, j,k},$$

$$x_{i,j,k}\ge 0, \mathrm{for} \mathrm{all} i,j,k.$$

The equations above seem to outline a linear programming problem (LPP). However, the parameters $p_{i,j,k}$ and $U_{i,j,k}$ are uncertain within the initial time frame. Since complete elimination of uncertainty is challenging in this complex scenario, our approach involves acknowledging and integrating it into our model. To address uncertain parameters, we employ linear interval programming based on interval analysis. This method is designed to manage no-shows for appointments by incorporating overbooking.

4.2. Preliminary Foundation

Lai et al. [42] presented the theoretic principle and framework of linear interval programming. First of all, assuming Ω is a set of real number, an ordered pair is defined in a bracket as an interval:

$$a=\left[ \underset{\_}{a}, \overline{a} \right]=\left\{ x :\underset{\_}{a}\le x \le \overline{a} , x ϵ \mathrm{\Omega }\right\},$$

(4)

where $\underset{\_}{a}$ refers to the lower bound and $\overline{a}$ refers to the upper bound of interval a, respectively.

The interval arithmetic is basically an extended version of ordinary arithmetic. The definitions of the mathematical operations between a pair of intervals are shown as below.

Definition 1 

(Alefeld and Herzberger [43]). Let ° ∈ {+, −, ×, ÷} be a binary operation on Q. If two intervals, a and b, are involved, then

$$a\circ b=\left\{ x\circ y :x\in a , y \in b\right\}.$$

(5)

Note that (5) defines a binary operation on the set of intervals. For division, it is assumed that $0\in b$. Based on this definition, the operations between pairs of intervals are as follows:

$$a+b=\left[ \underset{\_}{a}+\underset{\_}{b} ,\overline{a}+\overline{b} \right],$$

(6)

$$a-b=\left[ \underset{\_}{a}- \overline{b},\overline{a}-\underset{\_}{b} \right],$$

(7)

$$ka=\left\{\begin{array}{c}\left[k\underset{\_}{a},k\overline{a}\right], k\ge 0\\ \left[k\overline{a},k\underset{\_}{a}\right] k<0,\end{array}\right.$$

(8)

where $k$ is any real number.

4.3. The Proposed Approach

With intervals involved given in Alefeld and Herzberger [43], the linear programming problem (ILP) will be converted:

$$\begin{gathered} {\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}}_{{\le }_1} Z\left(x\right)=\sum _{j=1}^n\left[{\underset{\_}{r}}_j , {\overline{r}}_j\right]x_j, \\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o} \sum _j^n \left[{\underset{\_}{a}}_{ij} , {\overline{a}}_{ij}\right]x_j{\le }_2 \left[{\underset{\_}{b}}_j, {\overline{b}}_j\right], i=1,\dots ,m, \\ {x}_j\ge 0, j=1,\dots ,n. \end{gathered}$$

(9)

To maximize the return under uncertainty constraints, which are represented as interval numbers, the objective of the interval linear programming is changed from the ordinary one. Interval linear programming (ILP) is a mathematical optimization technique that addresses decision-making under uncertainty, where parameters are described by intervals rather than precise values. This approach is particularly useful in situations where data are uncertain, imprecise, or subject to variability. The objective is to maximize or minimize a linear combination of decision variables $x_j$, where each variable is multiplied by an interval coefficient $\left[{\underset{\_}{\mathrm{r}}}_{\mathrm{j}},{\overline{\mathrm{r}}}_{\mathrm{j}}\right]$.

In simple terms, the expected value of any uncertain variable typically lies at the midpoint of its interval. When presenting uncertain returns within an interval, the lower bound represents the pessimistic return, while the upper bound signifies the optimistic return. The ILP discussed earlier can be likened to a hybrid model combining elements of expected value modeling with uncertainty and a pessimistic decision-making approach. The objective function to maximize $\mathrm{Z}\left(\mathrm{x}\right)={\sum }_{\mathrm{j}=1}^{\mathrm{n}}\left[{\underset{\_}{\mathrm{r}}}_{\mathrm{j}},{\overline{\mathrm{r}}}_{\mathrm{j}}\right]{\mathrm{x}}_{\mathrm{j}}$ can be regarded as a variant version of max–min problem. The constraint requirements ${\sum }_{\mathrm{j}=1}^{\mathrm{n}}\left[{\underset{\_}{\mathrm{a}}}_{\mathrm{i}\mathrm{j}},{\overline{\mathrm{a}}}_{\mathrm{i}\mathrm{j}}\right]{\mathrm{x}}_{\mathrm{j}}{\le }_2\left[{\underset{\_}{\mathrm{b}}}_{\mathrm{i}},{\overline{\mathrm{b}}}_{\mathrm{i}}\right],\mathrm{i}=1,\dots ,\mathrm{m}$ specify the fact that the feasible solutions to ILP indicate the cost factors in the worst-case scenario. The average costs are less than or equal to the maximum possible value and the expected average value of the resources with uncertainties, respectively.

Definition 2 

(Lai et al. [42]). Let $x$ be defined as a feasible solution to a non-inferior solution to ILP if, and only if, no other feasible solution $x^{\prime }$ is present such that

$$z\left(x\right){\le }_{1}Z.$$

(10)

Based on the Definition 2, the non-inferior solution to ILP can be derived in the following bi-objective programming:

$$\left(ILP\right)\left\{\begin{array}{c}max\left\{{\displaystyle \sum _{j=1}^n}{\underset{\_}{r}}_j{x}_j\right.,{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{\underset{\_}{r}}_j+{\overline{r}}_j}{2}} x_j\\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^n}{{\overline{a}}_i ,jx}_j \le {\overline{b}}_i ,\\ {\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{\underset{\_}{a}}_{i,j}+{\overline{a}}_{i,j}}{2}} x_j\le {\displaystyle \frac{{\underset{\_}{b}}_i+{\overline{b}}_i}{2}},\\ x_j\ge 0.\end{array}\right.\end{array} \right.$$

(11)

To tackle multiple-objective decision making (MODM) programming, a linear combination of objective functions is suggested by Chankong and Haimes [44]. Hence, the objective function of BIL can be rewritten as follows:

$$\lambda \sum _{j=1}^n{\underset{\_}{r}}_j{x}_j+\left(1-\lambda \right)\sum _{j=1}^n{\displaystyle \frac{{\underset{\_}{r}}_j+{\overline{r}}_j}{2}} x_j= \sum _{j=1}^n{\displaystyle \frac{{(\underset{\_}{r}}_j+{\overline{r}}_j)-\lambda {(\underset{\_}{r}}_j-{\overline{r}}_j)}{2}} x_j.$$

(12)

Eventually, the ILP solution can be generated from the parametric linear programming (PLP) model as follows:

$$\left(PLP\left(\lambda \right)\right)\left\{\begin{array}{c}\max \left\{{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{(\underset{\_}{r}}_j+{\overline{r}}_j)-\lambda {(\underset{\_}{r}}_j-{\overline{r}}_j)}{2}} \left.x_j\right\}\right.\\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^n}{{\overline{a}}_i ,jx}_j \le {\overline{b}}_i,\\ {\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{\underset{\_}{a}}_{i,j}+{\overline{a}}_{i,j}}{2}} x_j\le {\displaystyle \frac{{\underset{\_}{b}}_i+{\overline{b}}_i}{2}},\\ x_j\ge 0.\end{array}\right.\end{array} \right.$$

(13)

From (12), $\lambda \in \left(\mathrm{0,1}\right)$ indicates the risk appetite for decision makers. The aforementioned equations have some strength for calculation. The non-inferior solution is comparatively easy to be calculated by the suggested interval linear programming (λ) corresponding to various values of $\lambda \in \left(\mathrm{0,1}\right)$. The distinguished characteristic of the suggested ILP (λ) is that the model has already been in linear programming form; hence, it is able to be solved efficiently via simple different linear model packages [45], while $\lambda \in \left(\mathrm{0,1}\right)$ can be determined.

4.4. Robust Optimization Model

The suggested ILP (λ) can be used to tackle the appointment overbooking problem with no-shows in an ambulatory care center. The uncertainty is depicted by the parameters $p_{i,j,k}$ and $U_{i,j,k}$ in an objective function and constraints model. Eventually, the proposed model by which to manage no-shows with overbooking is as follows:

$$\mathrm{M}\mathrm{a}\mathrm{x} \sum _{k=1}^K \sum _{i=0}^{T-1} \sum _{j=t+1}^T{\displaystyle \frac{R_{i,j,k} \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}}-C_o\sum _{k=1}^K\sum _{t=1}^{T-1}{ \left[{C^k+O}^{t,k }-A \right]}^{+},$$

where

$$\begin{gathered} \mathrm{A}=\sum _{i=0}^{t-1} \sum _{j=t+1}^T{\displaystyle \frac{ \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}}-\sum _{i=0}^{t-1}{\displaystyle \frac{ \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}} \\ +\sum _{j=i+1}^T{\displaystyle \frac{ \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}} \end{gathered}$$

$$s.t.\sum _{i=0}^{t-1} \sum _{j=t+1}^T{\overline{P}}_{i,j,k}{x}_{i,j,k}- \sum _{i=0}^{t-1}{\overline{P}}_{i,j,k}{x}_{i,j,k}+ \sum _{j=i+1}^T{\overline{P}}_{i,j,k}{x}_{i,j,k}\le C^k+O^{t,k} ,$$

$$\begin{array}{c}{\displaystyle \sum _{i=0}^{t-1}} {\displaystyle \sum _{j=t+1}^T}{\displaystyle \frac{{\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k}}{2}}{x}_{i,j,k}-{\displaystyle \sum _{i=0}^{t-1}}{\displaystyle \frac{{\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k}}{2}}{x}_{i,j,k} \\ +{\displaystyle \sum _{j=i+1}^T}{\displaystyle \frac{{\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k}}{2}}{x}_{i,j,k}\le C^k+O^{t,k }\end{array}$$

$$x_{i,j,k}\le {\overline{U}}_{i,j,k}$$

$$x_{i,j,k}\le {\displaystyle \frac{{\underset{\_}{U}}_{i,j,k}+{\overline{U}}_{i,j,k}}{2}}{x}_{i,j,k},$$

$$x_{i,j,k}\ge 0 , for all i, j, k.$$

The constraint of ${\displaystyle \frac{{\underset{\_}{U}}_{i,j,k}+{\overline{U}}_{i,j,k}}{2}}\le {\overline{U}}_{i,j,k}$ is always true. This means the constraint of $x_{i,j,k}\le {\overline{U}}_{i,j,k}$ can be removed to simplify the proposed model as follows:

$$\mathrm{M}\mathrm{a}\mathrm{x} \sum _{k=1}^K \sum _{i=0}^{T-1} \sum _{j=t+1}^T{\displaystyle \frac{R_{i,j,k} \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}}-C_o\sum _{k=1}^K\sum _{t=1}^{T-1}{ \left[{C^k+O}^{t,k }-A \right]}^{+},$$

where

$$\begin{gathered} \mathrm{A}=\sum _{i=0}^{t-1} \sum _{j=t+1}^T{\displaystyle \frac{ \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}}- \sum _{i=0}^{t-1}{\displaystyle \frac{ \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}} \\ + \sum _{j=i+1}^T{\displaystyle \frac{ \left[\right({\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k})-\lambda ({\underset{\_}{P}}_{i,j,k}-{\overline{P}}_{i,j,k}\left)\right]x_{i,j,k}}{2}} \\ s.t.\sum _{i=0}^{t-1} \sum _{j=t+1}^T{\overline{P}}_{i,j,k}{x}_{i,j,k}- \sum _{i=0}^{t-1}{\overline{P}}_{i,j,k}{x}_{i,j,k}+ \sum _{j=i+1}^T{\overline{P}}_{i,j,k}{x}_{i,j,k}\le C^k+O^{t,k} , \\ \sum _{i=0}^{t-1} \sum _{j=t+1}^T{\displaystyle \frac{{\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k}}{2}}{x}_{i,j,k}- \sum _{i=0}^{t-1}{\displaystyle \frac{{\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k}}{2}}{x}_{i,j,k} \\ + \sum _{j=i+1}^T{\displaystyle \frac{{\underset{\_}{P}}_{i,j,k}+{\overline{P}}_{i,j,k}}{2}}{x}_{i,j,k}\le C^k+O^{t,k} \\ {x}_{i,j,k}\le {\displaystyle \frac{{\underset{\_}{U}}_{i,j,k}+{\overline{U}}_{i,j,k}}{2}}, \\ {x}_{i,j,k}\ge 0 , for all i, j, k. \end{gathered}$$

(14)

In the next section, the usefulness and effectiveness of the proposed linear interval programming model to manage no-shows with overbooking for appointments in an ambulatory care center will be shown.

5. Result and Discussion

During this session, we will illustrate the utility of the model. To facilitate this demonstration, certain assumptions need to be established.

• The number of departments and their capacities cannot be expanded quickly.
• The revenue generated from each appointment is likely fixed in accordance with the policy of the Hong Kong Hospital Authority.
• The probability of patients attending their scheduled appointments remains constant.
• Demand is consistently high and exceeds capacity, making overbooking advantageous.

To simplify the numerical example illustration, six sessions are set to be available per day; i.e., T = 6. The length of each session can be any value and is defined by the decision-makers. Three departments will be involved. The demands for the lower bound and the upper bound are shown in

Table 2. The lower bound of the demand for different periods.

From/To	1	2	3	4	5	6
0	60	21	28	30	25	35
1		30	28	35	25	21
2			25	35	25	21
3				25	21	25
4					25	21
5						25

and

Table 3. The upper bound of the demand for different periods.

From/To	1	2	3	4	5	6
0	120	42	55	60	50	70
1		60	55	70	50	42
2			50	70	50	42
3				50	42	50
4					50	42
5						50

, respectively.

For each period, a show-up probability (equal to 1 minus the no-show probability) can be found in Table 3 or

Table 4. Parameters for overbooking.

Denotation	${\mathit{R}}_{\mathit{i},\mathit{j},\mathit{k}}$	${\underset{\_}{\mathit{p}}}_{\mathit{i},\mathit{j},\mathit{k}}$	${\overline{\mathit{p}}}_{\mathit{i},\mathit{j},\mathit{k}}$	${\mathit{C}}^{\mathit{k}}$	${\mathit{C}}_{\mathit{o}}$	${\mathit{O}}^{\mathit{t},\mathit{k}}$
Value	100	0.8	0.95	500	10	10

. For simplicity, hospital managers decide to use constant values of 0.95 and 0.8 for the upper bound and lower bound of the show-up rate (5% and 20% for no-show rates, respectively). The revenue for each period is set at 100. These three parameters can be adjusted according to the precise information provided by the hospital managers.

Therefore, other parameters which are made constant are shown in Table 4.

First, the risk parameter λ is examined. This parameter informs decision-makers of the expected profit after a designated period, given a certain value of λ. It also influences the configuration of the assignment between patients and doctors. The expected profit aids in capacity planning and financial management of the hospital. Note that the results of the following graphs are based on the parameters mentioned above; the patterns remain similar when the parameters vary.

Figure 2 illustrates that when decision-makers have higher confidence in estimating uncertain parameters such as upper and lower bounds of demand and no-show probabilities, profitability increases consistently, irrespective of how demand is allocated. The level of confidence among decision-makers positively correlates with expected profits.

The overbooking cost exerts influence over both the upper and lower bounds of profit, the allocation of patient-doctor assignments, and the extent of overbooking utilized. In Figure 3, when overbooking is cost-free, it is fully leveraged to meet the demand for consultation services, resulting in maximum profit equivalent to the revenue portion of the objective function, unaffected by overbooking costs. However, as the overbooking cost increases, profitability decreases significantly. Initially, patient–doctor assignment configurations remain fully utilized until the overbooking cost reaches 10% of revenue. Beyond this point, adjustments in assignment configurations commence, leading to a reduction in overbooking levels, as discussed in Figure 4. When the overbooking cost reaches 15% of revenue, the expense becomes prohibitive, prompting managers to consider not meeting all demands, which subsequently reduces profit. Overbooking units decrease to less than 5% when the cost reaches 25% of revenue. Moreover, decision-makers willing to accept higher risks demonstrate wider upper and lower bounds of profit, suggesting that they anticipate greater profitability across various scenarios.

In Figure 4, the level of overbooking, measured by the ratio of total overbooked units to total capacity, reaches its peak when the overbooking cost ranges from 0% to approximately 10% of total revenue. During this range, overbooking is maximized to meet all demand requirements. If there is increased uncertainty in demand or a wider range of demand fluctuations, the overbooking level will naturally rise accordingly. Ideally, the optimal overbooking level will tend to stabilize around 1.6, effectively accommodating demand fluctuations. It is noteworthy that decision-makers who are more confident may delay the decline in their overbooking levels, implying that they continue with full overbooking, even at higher costs.

As the overbooking cost approaches 10%, the overbooking level begins to decline, resulting in the inability to accommodate all potential patients. Changes in scheduling configurations occur until the overbooking cost reaches between 15% and 18%, contingent upon the confidence level of decision-makers. Those with higher confidence levels can maintain a higher overbooking level at the same cost. At a 15% overbooking cost, less-confident decision-makers tend to minimize overbooking, whereas more-confident ones typically maintain levels between 10% and 20% until costs escalate to 18%. The specific shapes of these curves may vary with different parameters, such as no-show rates, but the overall patterns remain consistent.

Figure 5 depicts the correlation between profit and the overbooking ratio, considering an overbooking cost set at 14% of revenue and a risk level of 0.5. The no-show rate ranges from 5% to 20%. Under these conditions, revenue increases steadily up to an overbooking level of 40%. This level allows additional patients to fill vacant consultation slots left by no-shows, representing an optimal strategy by which to maximize hospital revenue. However, beyond the 40% threshold, further increasing the overbooking level leads to diminishing returns. The blue line on the graph indicates that maintaining an overbooking level of around 40% is advisable. If hospital managers opt for higher overbooking levels to accommodate all potential patients, the associated costs escalate significantly, resulting in reduced profit margins. Once the optimal overbooking level is determined, managers can finalize the overbooking schedule and allocate the number of consultations for each period, as detailed in

Table 5. Overbooking level at the cost of 14% of the revenue.

Overbooking	t = 1	t = 2	t = 3	t = 4	t = 5
Specialty 1	200	198	196	200	198
Specialty 2	199	200	198	200	200
Specialty 3	200	200	200	197	199

and

Table 6. Assignment of patient under the scheme of overbooking at 0.4 of the total capacity.

From/To, Specialty 1	1	2	3	4	5	6
0	90	31	41	45	37	51
1		16	12	18	20	31
2			14	27	16	31
3				15	30	37
4					17	31
5						37
Specialty 2
0	40	39	44	49	66	49
1		9	30	19	10	48
2			8	26	4	41
3				7	25	45
4					7	52
5						56
Specialty 3
0	39	37	41	46	63	50
1		5	45	15	17	47
2			3	34	14	39
3				10	23	42
4					8	50
5						54

.

It is worth noting that decision-makers may prioritize scheduling patients who require longer appointments as these tend to generate higher revenue when patients attend. Adjusting the scheduling to achieve a more balanced distribution across different periods can help maximize revenue for private hospitals. Meanwhile, for public hospitals, optimizing scheduling can minimize wasted vacancies caused by patient no-shows and cancellations.

6. Management Insights

Based on the preceding analysis, hospital managers must carefully determine two critical parameters prior to establishing appointment assignment policies: the level of risk aversion and the cost associated with overbooking. Risk aversion can be likened to the confidence level of managers in their decisions. Figure 2 illustrates that profitability correlates directly with the degree of risk aversion among hospital managers. Opting for a more cautious approach to risk and its implications on overbooking, managers may choose a smaller value of λ, which corresponds to lower expected profits. It is essential for managers to align their decisions with both short-term operational goals and long-term sustainability strategies, including investments and specialty choices. Once the risk aversion level is defined, managers need to meticulously assess the cost of overbooking. This cost can be tangible, such as overtime pay for medical staff, increased utility expenses, or compensation for unattended patients. Alternatively, it can be intangible, encompassing reputational damage to the hospital. Figure 3 demonstrates that as the cost of overbooking rises, profits decline due to the incurred expenses from wasted appointment slots. Beyond a certain threshold, managers should scale back overbooking levels, causing profits to plateau. Minimal overbooking levels mitigate the impact of overbooking costs on profitability, though higher compensation for associated issues can further reduce expected profits.

Armed with insights into potential profits considering risk factors and overbooking costs, managers can decide on the appropriate overbooking level. Figure 4 demonstrates that managers opt for higher overbooking levels when costs are perceived to be low. However, there exists an optimal point beyond which increasing overbooking levels becomes counterproductive, necessitating careful allocation of resources. Conversely, when overbooking costs are prohibitively high, managers should minimize overbooking levels. Different parameter sets lead to varying decisions across different cost scenarios. It is important to note that despite low overbooking costs, hospital policies typically advise against excessive overbooking. In such cases, managers must judiciously evaluate whether to maintain optimal overbooking levels. Figure 5 illustrates that peak profitability is achieved at the optimal overbooking level. Further, the decision flow for overbooking is described in Figure 6. Deviating from this optimal level results in decreased profitability due to unmet demand and penalties associated with overbooking. Subsequently, managers should determine overbooking levels for each period, as detailed in Table 4, and allocate patients accordingly, as shown in Table 5. Additional parameters can be incorporated to cater to specific hospital requirements, culminating in a comprehensive action plan

7. Superiority and Inferiority of the Proposed Optimization Strategy

The proposed robust optimization strategy for overbooking outpatient clinics offers several superior advantages over conventional models. By integrating a conventional overbooking model with a resilient system approach, this strategy addresses the critical uncertainties exacerbated by pandemics and other large-scale disruptions. The incorporation of risk aversion into the model allows for a nuanced balance between potential benefits and the associated risks, such as patient dissatisfaction and reputational damage. This adaptability enables the system to handle unpredictable no-shows and cancellations more effectively, leading to improved resource utilization and reduced waiting times. In contrast, traditional overbooking models often rely heavily on historical data and static predictions, which can fail to account for unexpected fluctuations and dynamic changes in patient behavior, particularly during crises.

However, the proposed strategy is not without its limitations. The complexity of integrating both conventional and resilient approaches may lead to higher implementation costs and require more sophisticated management and training. The need for ongoing adjustments and fine-tuning based on real-time data can also be resource-intensive, potentially straining the administrative capacities of some outpatient clinics. Additionally, while the model improves overall efficiency, it may not fully eliminate the inherent challenges of no-shows and cancellations, especially in settings where patient behavior is highly unpredictable. Despite these limitations, the robustness and adaptability of the proposed strategy provide a significant improvement over traditional methods, making it a valuable tool for managing the uncertainties and inefficiencies inherent in outpatient clinic operations.

8. Conclusions

This paper introduces a robust optimization model designed to tackle issues of no-shows and vacancies in healthcare operations through the implementation of overbooking strategies. Drawing from the existing literature and practical insights, we establish a foundational model commonly utilized in overbooking scenarios. Unlike deterministic approaches, our model incorporates robust parameters, which prove advantageous particularly as the probability of no-shows fluctuates. Our model suggests that profitability can be enhanced with increased overbooking yet reaches a peak beyond which further overbooking diminishes returns. This prompts the calculation of an optimal maximum overbooked appointments threshold. The decision-making process regarding overbooking begins with an assessment of its associated costs. When costs are high, our findings advocate for cautious overbooking levels, potentially exploring alternative no-show mitigation strategies. Conversely, lower costs endorse more aggressive overbooking strategies, encouraging widespread adoption among decision-makers. For moderate costs, a balanced overbooking approach is recommended. This paper underscores the significance of risk aversion in determining optimal strategies, particularly in uncertain times such as pandemics. By presenting this approach, our contribution aims to enhance operational efficiency in healthcare settings, supporting robust decision-making amidst evolving challenges. Looking ahead, there are several promising avenues for advancing the robust optimization model proposed to address no-shows and vacancies in healthcare through overbooking strategies. One key direction involves the integration of real-time data analytics, which could allow for dynamic adjustments in overbooking levels based on up-to-date information such as historical attendance trends and external factors influencing patient behavior. Furthermore, advancements in machine learning and predictive modeling hold potential to significantly enhance the accuracy of forecasting no-show probabilities, enabling continuous improvement and adaptation of the model. Another critical area for future development includes refining the methods by which to optimize overbooking thresholds tailored to different departments and specific operational contexts, ensuring effective utilization of resources while minimizing risks. Additionally, conducting comprehensive cost–benefit analyses will be essential to evaluate the financial impact of overbooking strategies in conjunction with factors like patient satisfaction and staff workload. Simulation techniques can further support these efforts by simulating various scenarios to test the robustness of the model under different conditions, including periods of high demand or unexpected disruptions like pandemics. Ultimately, ongoing advancements in adaptive decision-support systems and ethical considerations in scheduling will play crucial roles in further refining and implementing these strategies across healthcare settings, ultimately enhancing operational efficiency while maintaining high standards of patient care.