A pull-strategy for the appointment scheduling of surgical patients in a hospital-integrated facility

ABSTRACT

This paper addresses the daily appointment scheduling (AS) of patients in a hospital-integrated facility where outpatients and inpatients are treated simultaneously and share critical resources. We propose a lean approach based on the pull-strategy “Constant Work in Process” (ConWIP) to develop robust and easy-to-implement AS rules. Our objective is to reduce patients’ waiting time and maximise the use rate of resources while considering the global surgical process and stochastic service times. The AS rules based on ConWIP are evaluated using a Discrete-Event-Simulation model. Numerical experiments based on a real-life case study are carried out to assess the proposed appointment rules’ performance and compare them to AS rules developed in the literature. The results highlight the robustness of our approach and demonstrate its usefulness in practice.

1. Introduction

The economic benefit is one of the main reasons for promoting alternatives to classical hospitalisation over conventional admission. For instance, cost savings have been the primary incentive for ambulatory surgery development. Hence, for some elective surgeries, over 80% of them are performed on an outpatient basis in several places of the Northern Hemisphere (OECD.Stat, 2020).

In this paper, we focus on ambulatory surgery, also called “day” or “outpatient surgery” (OS). OS is “the practice of admitting into the hospital on the day of surgery carefully selected and prepared patients for a planned, non-emergency surgical procedure and their discharge within hours of that surgery” (Castoro et al., 2007). For many procedures, it appears as an effective and efficient approach, offering several advantages to patients, surgeons, and hospital managers. Besides reducing hospital admission expenditure, OS represents essential benefits in terms of enhanced social and emotional recovery (shorter hospital stays, implying fewer nosocomial infections and earlier mobilisation of the patient). Also, there is a high satisfaction for both surgeons and patients in facilities where the OS has become standard practice. On the one hand, it optimises the use of resources, which means that capital facilities (such as operating and recovery rooms) and staffing (i.e., surgical team and nurses) are used more intensely and effectively (Regions, 2013). Also, patients can be scheduled consecutively with minimal downtime between cases, maximising the use of clinician’s time and minimising the facility completion time or makespan (hereinafter CT). On the other hand, patients overwhelmingly endorse day surgery, with shorter waiting times (hereinafter WT) and less risk of cancellation (IAAS, 2014; Wang et al., 2017).

Appointment scheduling (AS) plays an important role in this context by providing a smooth flow of patients while reducing crowding in waiting rooms. AS is a scheduling method used for managing arrival times of patients. According to (Gupta & Denton, 2008), its objective lies at the intersection of efficiency and timely access to health services to suit all stakeholders’ requirements, including patients, by reducing their WT and health care providers by increasing the use rate of resources. This study addresses the AS of surgical patients at a hospital-integrated OS facility called also Integrated Surgical Suite (ISS). The ISS is based on configurations created for traditional surgery. Outpatients and inpatients go through multiple stages (pre-operation, surgery, and recovery) and share critical resources such as the operating room (OR) and related human resources. AS in such an environment is a challenging task due to the dependencies between the different stages of the ISS, the uncertainty in processing times, and the need to ensure the simultaneous availability of various service providers/resources to deliver the desired services (Gupta & Denton, 2008). In addition, dedicated pre- and post-operative pathways should be established to separate outpatients from inpatients. The co-existence of two clinical flows in the ISS raises the complexity of AS. Given that inpatients are already in the hospital ward, the appointment schedule’s implementation requires the scheduler to identify in each slot, arrival times to the ISS for outpatients, and start times of procedures for inpatients.

An analysis of the literature reveals that most studies delight the AS problem of surgical patients in dedicated facilities to outpatients rather than ISSs. Most of these studies use operational research techniques, namely, i) stochastic programming (Demir et al. 2020; Atighehchian et al., 2019; Batun et al., 2011; Berg et al., 2014; Denton & Gupta, 2003; Denton et al., 2007; Erdogan & Denton, 2013; Gul, 2018; Landa et al., 2016); Liu et al., 2018; Pang et al., 2018; Shehadeh et al., 2019; Srinivas & Ravindran, 2020), ii) mathematical programming with fuzzy sets (Lee & Yih, 2014), iii) analytical methods (Begen & Queyranne, 2011; Begen et al., 2012; Ge et al., 2014, Mak et al. 2014), and iv) hybrid approaches: optimisation-simulation which integrates simulation models into metaheuristic algorithms to evaluate the objective function in each iteration while accounting for randomness (Diaz-Lopez et al. 2018; Ewen & Mönch, 2014; Gul et al., 2011; Saremi et al., 2013, 2015; Zhang & Xie, 2015). Certainly, these methods deliver optimal solutions regarding the studied criteria; however, they present some limitations. On the one hand, methods i), ii), and iii) have difficulty addressing large, complex systems and do not consider all the stages of the global process. Moreover, it mainly relies on simplistic problems considering the single-stage system (OR), which has limited practical values (Papadopoulou, 2013). On the other hand, despite their effectiveness, the implementation of hybrid methods in real cases requires a complex transfer from theory to practice (Fu, 2002; ROADEF, 2011). It also requires the acquisition of a powerful software infrastructure in the care structure, which is not necessarily available in the facility (which is the case of the studied OS facility).

While seeking to address effectiveness and ease-of-implementation, some studies developed AS rules for dedicated facilities for outpatients. Regarding outpatient clinics, Srinivas and Ravindran (2018) and Yang and Cayirli (2020) developed AS rules based on no-show risks (e.g., low-risk patients in the beginning). In fact, variability in arrivals such as walk-ins and no-shows occurs mostly in outpatient clinics, whereas variability in service times affects more AS in surgical settings (Gupta & Denton, 2008). Regarding outpatient surgical facilities, the developed AS rules are based on known scheduling rules (SPT, LPT, Dome scheduling rule) and were compared to the optimisation component results via simulation models (Gul et al., 2011; Peres et al., 2019; Saremi et al., 2013) in order to consider uncertainty in service times. Since these rules are based on a given ordering of surgery durations, their focus was rather on one stage (OR).

Our study is different from the former ones. First, we consider AS in a surgical department and as hypothesis, we assume patient punctuality instead of no-show. Second, we consider the global surgical process (including preparation, surgery, and recovery) and uncertain service times. The need for adopting a multi-stage system for optimising appointment systems in outpatient clinics has been shown by several studies (Srinivas & Khasawneh, 2017; Srinivas & Ravindran, 2017). In a previous work, we considered a multi-stage ISS and tested sequencing rules based on surgery durations as well as on the durations of the three processes of the ISS (e.g., generalised Johnson rule) (Chabouh et al., 2018). We evaluated these rules via a DES model and proved their sensitivity to the distribution of patient types (inpatients vs. outpatients).

This paper fills the existing literature gap by developing a robust, effective, and easy-to-implement AS rules while considering the global surgical process (including preparation, surgery and recovery). The ease-of-implementation refers to the usefulness of the tool by the medical staff in practice without needing a specific competence in scheduling and operation management tools. The effectiveness relates to the quality of the schedules delivered by the tool in terms of CT and WT. Its robustness refers to maintaining the AS tool’s performance when the distribution of patient types and system load varies. Moreover, we tackle the development of AS from a new perspective. We investigate the application of a lean production method “Constant Work in Process or Progress” (ConWIP) for AS in the ISS. ConWIP is a pull system used mainly in manufacturing environments to maintain a constant WIP level that guarantees optimal use of bottleneck resources and reduced WT between processes. Framinan et al. (2003) described ConWIP as a flexible yet robust pull mechanism that can be easily implemented in complex environments since it perceives the studied system as a black box. This method’s particularity stems from its ability to produce a deterministic functioning of the black box (e.g., the ISS) by maintaining a constant WIP. Hence, it reduces the complexity of the problem by reducing the impact of internal stochastic phenomena. In the study by Chabouh et al. (2017, October), we addressed the implementation of ConWIP in a hospital-integrated facility. This manuscript builds on our prior study to develop several AS rules based on ConWIP, evaluates their performance and robustness, and compares them to AS rules developed in the literature. To our knowledge, we are the first to apply ConWIP in an ISS to the AS of inpatients and outpatients.

The remainder of the paper is organised as follows: the next section presents some background on ConWIP while discussing the lean tools applied to surgery departments and justifying the eligibility of ConWIP. Next, we define the problem by describing a case study. We then discuss the methodologies we have applied, including the ConWIP modelling and setting for the ISS, the AS generation using ConWIP, and our simulation model for AS evaluation. We then present and discuss the experimental results. Finally, we summarise the most significant managerial insights.

2. Background on lean in surgical suites and on ConWIP

2.1. Lean in surgical departments

In the healthcare industry, the LEAN approach was investigated in three steps: defining the value from the client’s perspective, mapping the processes and flows, and eliminating waste. The customer group encompasses patients, clinicians, and insurance companies, especially in the public sector (Kollberg et al., 2006). However, patients are identified as prime customers for the healthcare environment since it is established to provide them with the necessary services. Thus, value is mostly identified regarding the patients’ views (Kollberg et al., 2006; Torabi et al., 2018; Wojtys et al., 2009). In the patient-centred view, quality care service that is efficient, appropriate, timely, and safe must be provided by the healthcare system to increase patients’ satisfaction level (Tolga Taner et al., 2007). The value stream mapping (VSM) was used to depict the whole process in a visual map and recognise waste along the whole care value chain. In this way, the focus is on eliminating non-value-added activities to follow lean thinking and make the value chain as lean as possible (Torabi et al., 2018).

In the surgical context, some studies performed VSM (Schwarz et al., 2011), process improvement workshops and 5S techniques (Waldhausen et al., 2010), and lean six sigma (Chang et al., 2018; Gijo & Antony, 2014; Mandahawi et al., 2011; Warner et al., 2013) to improve the surgical process. Paim et al. (2016) and Simon and Canacari (2014) applied a lean approach (on case studies) to improve the surgical scheduling process. These studies point out that patients experience long WT before entering the OR, a major problem in surgical suites. The major causes of this waste are related to the inefficient scheduling process that does not consider the capacity of the surgery department (Paim et al., 2016) and does not personalise the arrival times of patients (Wojtys et al., 2009). According to Chalice (2007), Schwarz et al. (2011), and Torabi et al. (2018), the solution is to switch from a push system to a pull system [i.e., the patients are only transferred inward when the OR gives the go ahead]. In the lean healthcare literature (Brandao De Souza, 2009; Costa & Filho, 2016; D’Andreamatteo et al., 2015), flow control methods including the pull system Kanban were applied only to manage material flows and stocks of a drug, sterile products, etc. used in different phases of the patient care. This method can be used to create signals to pull patient flows into the health system where possible. As soon as a patient needs to be discharged, a new patient must be programmed to take the empty bed (Bowerman & Fillingham, 2007). Kanban’s main limitation is related to its inapplicability in multi-product type environments (which is generally the case in healthcare systems) (Spearman et al., 1990).

ConWIP is a lean production control system used mainly in industrial environments and introduced by Spearman et al. (1990) as a pull alternative to the Kanban system. ConWIP is defined as a generalised or improved form of Kanban that is more flexible and efficient. It shares the benefits of Kanban, such as reduced inventory levels and flow times, while applying to a wider range of manufacturing environments, especially those characterised by product variety and moving demand Courtois et al. (2003) .

2.2. Background on ConWIP

ConWIP refers to any system maintaining the maximum amount of work in a production line constant to avoid the phenomenon of congestion and to maintain a constant lead time. ConWIP differs from pure pull control: pull control is implemented only at system entrance and exit (i.e., the two gates of the system) and push control is applied in all intermediate production stages (Spearman et al., 1990). Thus, the production system for ConWIP is perceived as a “black box” which justifies its ease-of-implementation and its applicability to complex environments. A fixed number of cards is used to maintain a constant WIP level in the system at any time. A job can be released into the system only if there is an available card (Huang et al., 1998). Naturally, a ConWIP system will follow the “bottleneck” rule: there will be just enough WIP to keep the bottleneck system occupied, meaning idle time for other (faster) steps. The ConWIP is described in detail by Hopp and Spearman (1996). Prakash and Feng (2012) distinguished four ConWIP systems; the basic ConWIP system, the hybrid ConWIP system, the multi-product ConWIP system, and the parallel ConWIP system. Our study is interested in the basic ConWIP system, as described by Spearman et al. (1990).

ConWIP has been applied to several manufacturing scenarios, including job-shops, assembly lines, or rework (Framinan et al., 2003). Despite its ability to be applicable in different complex environments, few articles have dealt with ConWIP in non-industrial contexts. To our knowledge, Crop et al. (2015) and Yang et al. (2015) are the main studies that applied ConWIP in the healthcare management field: to optimise the workflow for robotic stereotactic radiotherapy treatments (Crop et al., 2015) and to improve laboratory performance (Yang et al., 2015). However, ConWIP was never applied to manage surgery departments, neither to surgical patients’ AS problem. Our study proposes applying the basic ConWIP system to develop robust and easy-to-implement appointment schedules for inpatients and outpatients. Our approach is applied to an ophthalmology surgery department in a Tunisian hospital described in the next section.

3. The case study

3.1. Description of patient flows and related resources

Hospital-integrated OS facilities deal with elective outpatients and inpatients. We consider an ophthalmology hospital-integrated facility of a Tunisian public hospital.

Figure 1 illustrates patient flow (indicated by arrows) in the studied ISS. We differentiate three paths of patients: outpatient (p = 1), inpatient undergoing general anaesthesia (p = 2), and inpatient undergoing local anaesthesia (p = 3). The patient flow for outpatients and inpatients is given as follows:

Figure 1.

Patient paths in the Tunisian ISS.

1 The outpatient arrives at the ISS based on a preassigned appointment and is sent to the preparation room. The inpatient is already in the ward waiting for the preoperative process to begin.

2 The preoperative stage, preparing the patient for surgery, is performed in the preparation room for outpatients (in the hospitalisation ward for inpatients). A nurse checks the patient’s laboratory values, its pulse and blood pressure, and disinfects his eye(s). The nurse can treat only a single patient at a specific point in time. After this examination, his/her clothes are changed, and he/she is transferred to the surgery area.

3 The surgery area is composed of an anaesthesia room and an OR. The perioperative process consists of anaesthesia and the execution of the surgery for inpatients and outpatients. The anaesthesia is performed by the surgeon accompanied by two anaesthesia technicians in case of local anaesthesia. In the case of general anaesthesia, an anaesthetist should be present.

4 Once the surgery is completed, the outpatient and the inpatient undergoing general anaesthesia are routed to the recovery room. The set of activities immediately after the surgery is called the post-operative process. It is completed, whether in the recovery room or hospitalisation ward, depending on its path and anaesthesia type.

5 The inpatient returns to his ward while an outpatient will be discharged when no difficulties occur. In case of any problem, the patient will be examined by a doctor.

Our field of study is limited to ISS, which involves preoperative, perioperative, and post-operative stages, and does not consider the hospitalisation ward. We consider the following limited resources: preparation beds dedicated to outpatient, surgeons, ORs (the anaesthetists and technicians are available to serve in different ORs), and recovery beds (a predefined quota on the number of beds dedicated to patients p = 1 and p = 2).

3.2. Patient categories

In this case study, we define different categories of patients:

-Patient classes: Outpatients and Inpatients.

-Patient paths p: three paths p as defined in subsection 3.1.

-Patient types j: knowing that for each path of patients, different types of surgeries are considered (which we refer to as the index n), a patient type j = (p,n) is determined based on patient path p and surgery type n.

Three major types of ophthalmologic surgery are performed on the studied ISS. The first two types are related to cataract surgery: phacoemulsification (n =1 with a relative frequency of 45%) and extracapsular (n =2 with a relative frequency of 40%). These surgeries can be done both on an outpatient and inpatient basis. They are performed, respectively, by surgeon type s =1 and type s =2. The third type, strabismus surgery and others (n =3 with a relative frequency of 15%), is performed by surgeon type s =3. This type of surgery is only done on an inpatient basis and can be performed both under local and general anaesthesia. In summary, for the studied ISS, we distinguish six patient types j as depicted in Table 1. Patient categories and their distributions are presented in Table 1 (these probabilities are calculated from historical data of the ISS of the year 2015).

Table 1.

Patient categories and their distribution.

Patient classes		Patient paths		Patient types
Class	Probability of patient class	Path (p)	Probability of patient path	Type j j = (p.n)	Probability of patient type
Outpatient	50%	p =1	50%	j =(1.1)	28%
				j =(1.2)	22%
Inpatient	50%	p =2	47.75%	j =(2.1)	22.5%
				j =(2.2)	20%
				j =(2.3)	5.25%
		p =3	2.25%	j =(3.3)	2.25%

4. Methods

In this paper, we address the daily AS of inpatients and outpatients in an ISS for a known set of patients. We propose a four-step approach based on the ConWIP system:

(1) ConWIP implementation: size properly the number of cards and determine the ISS’s optimal throughput (i.e., number of patients treated per time unit of the bottleneck resource).

(2) Development of appointment rules based on the ConWIP output rate: the appointment rule defines arrival times and the number of patients arriving at each type at each time slot. The optimal throughput will constitute the number of patients (all types confused) arriving at each time unit. We develop several appointment rules considering different priorities of the ConWIP card assignment.

(3) Determination of the best appointment rule based on ConWIP: we evaluate the appointment rules based on ConWIP using a simulation model of the ISS under the ConWIP system in terms of WT and the use rate of bottleneck resources.

(4) Evaluation of the chosen rule: we study the effectiveness of the best appointment rule (s). We compare it to the optimal solution and the best appointment heuristic developed by Chabouh et al. (2018) in terms of WT, CT, and the rate of outpatients discharged after an hour H “RAH”.

The WT measures the patient’s satisfaction. It includes WT1 and WT2. WT1 is a WT between the patient’s arrival and his effective entry into the ISS (i.e., when a ConWIP card is assigned to the patient). WT2 is a WT between the end of the preparation process and the beginning of surgery.

The CT refers to the time at which the last patient leaves the ISS. To compare the CT of an appointment schedule (i) (CT(schedule i)) with the optimal CT (CT*) (i.e., the value of CT yielded after the simulation of the appointment schedule delivered by the MIP developed by Chabouh et al. (2018)), we calculate the gap (Gap(CT)schedule i) given by Equation 1.

$$Gap{(CT)}_{schedulei}=\frac{CT(schedulei)-C{T}^{\ast }}{C{T}^{\ast }}$$ (1)

The latter indicator RAH studies the feasibility of an appointment schedule according to specific constraints related to outpatients. The outpatients must leave the facility before a predetermined time H to ensure that they return home safely. For each appointment schedule i, RAH is calculated using Equation 2.

$$RA{H}_{schedulei}=\frac{\begin{array}{c}Numberofoutpatient\\ dischargedafter{H}_{schedulei}\end{array}}{\begin{array}{c}Totalnumberof\\ outpatientsdischarged\end{array}}$$ (2)

The implementation step has been presented by Chabouh et al. (2017, October). This paper aims to develop and evaluate AS based on the implementation step results. For this purpose, we first recall the methodology for ConWIP implementation and its results in the studied ISS in subsection 4.1 concisely. Second, we detail the algorithm for setting the AS using ConWIP results in subsection 4.2.

4.1. ConWIP implementation

The workflow in the ISS while operating under a basic ConWIP system is illustrated in Figure 2. Upon his arrival, an outpatient takes his place in a dedicated waiting room. Inpatients are already in the hospitalisation ward. Once a ConWIP card is available (or empty), it is assigned to an inpatient or outpatient according to a defined priority (e.g., an outpatient is offered the card first, the cataract surgery patient is prioritised over the others). Once the surgical process is completed, the outpatient (respectively inpatient) is discharged (respectively is transferred to the hospitalisation service) and the card becomes available again and joins the queue of empty cards.

Figure 2.

Workflow in the ISS under a basic ConWIP system.

The implementation of ConWIP requires the determination of the number of cards “m”, which determines the WIP level (Spearman et al., 1990). The card count can be determined at the tactical level but must be adjusted at the operational level. First, in subsection 4.1.1, we determine m theoretically considering the patient types partition and their resource consumptions. Second, to adjust and validate m, a simulation model is used in subsection 4.1.2, considering the stochastic nature of the process in evaluating the performance of the ConWIP system.

4.1.1. Card number determination

Choosing the adequate m involves a compromise between the WIP level and the service level or the production rate. A high m implies a high production rate and a long length of stay (hereinafter LOS) translated by a high WIP level. Conversely, by using a low m, the queues in front of each station will be small and the delay will be short, which degrades the line’s rate. Hence, the number of cards is routed in the Little's Law (Little, 1961), as depicted in Equation (3). This law resulting from the queuing theory establishes a link between the WIP level, the lead time LT (or LOS), and the production rate (or throughput TH). The components of this equation are shown in Table 2.

$$m=TH\times LT$$ (3)

Table 2.

Components of Equation (3) and their specifications.

Term		Signification	Unit
M	Number of cards	Refers to the WIP level we want to keep constant	Number of patients
TH	Throughput	Refersto the output rate of the system which is determined by the capacity of the bottleneck resource	Numberof patients per time unit
LT	Lead time	Refersto the time a job (or a patient) spends in the system	Time units

To determine the TH, we first identify the different resources for each patient type based on patient paths presented in Figure 1. Second, for each resource, we determine the mean throughput. Once calculated, we identify the bottleneck resource, which has the minimum throughput.

To determine the LT, we first calculate for each patient type j, the lead time LT_j corresponding to the sum of the mean processing time D_i,j on all the resources used by patient type j. We then determine the LT as the weighted sum of LT_j.

4.1.2. Adjustment of the number of ConWIP cards

The simulations’ objective is to adjust m, in a stochastic context. We build a DES model of the ISS operating under a basic ConWIP system (hereinafter ConWIP-DES model). The ConWIP-DES model is developed based on the DES model of the ISS presented by Chabouh et al. (2018). The conceptual representation of the ConWIP-DES model and the different data needed for its development are illustrated in Figure 3. The relevant data were collected over 2 months. A total of 70 patients were observed and tracked. Then, distributions were fit for all stages of a patient’s movement through the ISS using the chi-square and Kolmogorov–Smirnov tests using the Arena Input analyser tool.

• Assumptions of the ConWIP-DES model:

Figure 3.

The conceptual representation of the ConWIP DES model based on the DES model developed by Chabouh et al. (2018).

All outpatients arrive at the beginning of the day.

All patients and m cards are created at the beginning of a simulation run.

Arrivals are on time and all patients show up for their scheduled procedure.

The surgeries are considered independent of one another; ORs are identical and parallel, and all the patients are scheduled before the day begins.

Setup times between surgeries (of different types) are not considered.

Preparation and recovery nurses are considered independent resources and operations.

• Simulation parameters

We have steady-state simulations and not terminating ones since the ConWIP system allows us to maintain a constant WIP regardless of the number of patients treated. Given that, in the beginning, the simulation model is empty and idle, and it becomes eventually congested (Kelton et al. 2000), a warm-up period as well as a long replication length, should be defined. The replication length is fixed to 80.000 min and the warm-up period to 200 min (Chabouh et al., 2017, October).

The number of simulation replications was varied. This preliminary experiment shows that 40 runs are needed to obtain a satisfactory half-width of the generated 95% confidence intervals (0.95 CIs) for the selected simulation outputs (e.g., CT).

• The procedure of card number adjustment

It consists of varying m in the ConWIP-DES model and evaluating the average LOS of patients after entering the ISS “LOS-ConWIP” (LT) and the mean output rate of patients in the ISS (TH). The curve of LOS-ConWIP allows determining the range of m. The optimal card number m* is the first value that delivers the maximum output rate. This procedure is illustrated in Figure 4.

Figure 4.

The procedure of adjustment of the number of ConWIP cards.

The application of the implementation step in the Tunisian ISS, the experiments, and the results are detailed in in the study by Chabouh et al. (2017, October). The main results that will be used for the appointment schedules generation (the second step of our approach) are as follows:

-m* = 11 cards ensure that the bottleneck resource (i.e., OR) will be used on average to 97%. The other resources, such as nurses, surgeons, and recovery beds, are also optimised. We note that the OR use rate for each replication is calculated based on the time at which the last entity leaves the ISS and not on the fixed length of the replication.

-The optimal output rate TH* is equal to 4,2 patients per hour.

4.2. Appointment scheduling based on ConWIP

The ConWIP system allows maintaining a constant WIP, without considering WT before the beginning of the treatment. Hence, patients can experience long WT before being assigned a card because their arrivals are not managed using the ConWIP system.

To minimise the patient WT, we developed an algorithm that sets the appointment times and the number of arrivals for each patient category at each time slot.

4.2.1. The appointment setting algorithm based on ConWIP

The algorithm’s principle is simple: the number of patients arriving at each time slot is given by the output rate (TH*) of the ConWIP system. The algorithm is presented in Table 3.

Table 3.

Algorithm for setting patient appointments based on ConWIP.

As inputs, it accepts: the number of patient categories K (we distinguish categories either according to patient types (K =6) or patient classes (K =2)), their distributions, the number of patients to be treated, and the integer part of the output rate TH* (denoted TH*).

We first determine the number of appointments N (or arrival times), which is given by dividing the number of patients by TH*. Second, for each time slot or appointment h in 0.N-1, we determine the category of arriving patients and their number based on the appointment rule (how we call patients to arrive) and patient partitions (category).

We propose three appointment rules to determine the number of patients per category at each appointment: AR1, AR2 and AR3:

• Using AR1, the number of arrivals per patient category at each appointment slot follows patient categories’ distribution, as illustrated in Table 3.

• AR2 consists of arriving all outpatients first.

• AR3 consists of operating all inpatients first.

Hence for AR2 and AR3, and for determining the number of arrivals at each time slot, we do not apply the steps 2.1) and 2.2) of the algorithm, we only apply step 2.3) for the first class of patients (outpatients for AR2; inpatients for AR3) then for the second class of patients (inpatients for AR2; outpatients for AR3). Finally, in order to move to the next appointment slot, the total number of patients called to arrive should be equal to $\mathbf{T}{\mathbf{H}}^{\ast }$TH*. The algorithm ends when all patients are scheduled.

4.2.2. Arrival scenarios to the ISS based on ConWIP

An arrival scenario consists of a way of defining entry times of patients to the ISS. For a given integer TH*, we define several arrival scenarios corresponding to the appointment rules, the considered patient category, and the priority in assigning a ConWIP card. We test several priorities of the card assignment in the ConWIP-DES model:

(1) First-in-first-out (FIFO): the order in which the algorithm defines patients’ arrivals is respected;

(2) Outpatient first: at each appointment, if we have one patient of each type waiting, the outpatients are assigned a ConWIP card first; and

(3) Inpatient first: at each appointment, if we have one patient of each type waiting, the inpatients are assigned the ConWIP card first.

Table 4 defines the eight arrival scenarios based on ConWIP.

Table 4.

The arrival scenarios based on ConWIP for a given integer output rate.

Arrival scenario	Appointment rule	Patient categories	Priority in assigning the ConWIP card
1	AR1: The number of arrivals per patient type/class at each appointment follows the distribution of patient types.	2 classes (K = 2): Outpatient – Inpatient	FIFO
2			Outpatient first
3			Inpatient first
4		6 types (K = 6)	FIFO
5			Outpatient first
6			Inpatient first
7	AR2: Arriving all outpatients first	2 classes	FIFO
8	AR3: Arriving all inpatients first

5. Results and discussion

The numerical experiments are carried out on an Intel@Core i7 processor 2.1 GHz and 8 GB Memory RAM Laptop. The algorithms for setting appointments and DES models are implemented in the MS Visual C++ and Rockwell Arena TM software, respectively. The main objectives of the experiments are to choose the best appointment rule based ConWIP (first experiment), compare it to the current arrival policy and appointment rules developed in literature based on real case dimension (second experiment), and finally analyse its sensitivity to various factors (third experiment). Figure 5 illustrates the three experiments we conducted in this study. We report the results of each experiment over one simulation run for each schedule. The simulation run involves 40 replications.

Figure 5.

Details of the three experiments.

5.1. Evaluation of the ConWIP-based appointment rules

The optimal number of cards is given by TH* = 4.2 patients per hour. Hence, the number of patients that should be called to arrive at each hour can be equal to four or five patients. As we presented in Table 4 (subsection 4.2.2), for each integer TH*, we propose eight arrival scenarios based on ConWIP, which results in 16 schedules in total. The first 8 (respectively last 8) are related to TH* = 4 (respectively 5) patients per hour. The comparison of arrival scenarios is performed in terms of WT1, WT2, and OR use rate.

Scenarios 9 to 16 (related to TH* = 5) patients per hour resulted in long WT (between 700 and 1000 minutes on average) compared to the others (related to TH* = 4) and average use rate of the OR of 94%. The first eight scenarios resulted in much less WT and an OR use rate greater than 60% and less than 70%, which is not a significant variation compared to WT. Thus, we only present results for scenarios 1 to 8 for WT1, WT2, and OR use rate. The results are illustrated in Figure 6. We presented the average values and box plots for performance indicators per arrival scenario. The boxes show 0.95CIs for each metric and whiskers show the best and worst cases for each scenario.

Figure 6.

Comparison of the arrival schedules based on ConWIP in terms of WT1, WT2, and OR use rate for 1000 patients and TH = 4.

Figure 6 shows that the first three scenarios present similar performance and outperform the remaining scenarios, especially in terms of WT. This comparison highlights that the simplest appointment rule (where the number of arrivals at the beginning of each hour is equal to $\mathrm{T}{\mathrm{H}}^{\ast }$⌊TH*⌋, the number of arrivals is determined according to patient classes (outpatient/inpatient), and no priority is defined for assigning a ConWIP card); results in the highest performance in terms of WT and OR use rate.

We notice that there is no significant difference between WT for outpatients and inpatients. For scenario 1, WT for outpatients is slightly less than WT for inpatients (WT1inpatient = 5.438 ± 1.641 min vs. WT1outpatient = 2.730 ± 2.483 min; WT2inpatient = 38.458 ± 2.18 vs. WT2outpatient = 31.418 ± 2.517 min), that is why we consider average WT both classes confused. Finally, if we consider that waiting is β times as costly for an outpatient compared to an inpatient (β ≥ 1), Scenario 1 yields the minimum WT cost compared to all the other arrival scenarios whatever the value of β.

As the patient class is the only criterion that defines who arrives at each appointment, the type of surgery for each class of patients could be chosen arbitrarily or based on the preference of the scheduler/disponibility of the surgeon. Hence, this simple rule offers more flexibility in scheduling patients’ appointments while being efficient in WT and OR use rates.

5.2. Appointment rules comparison

The first experiment allowed us to choose the best appointment rule based on ConWIP (i.e., scenarios 1) hereinafter ConWIP-AS. The aim of this experiment is to test it on a real case dimension and compare it with:

• the current arrival policy used in the Tunisian ISS named “current practice”. It consists of calling all patients to arrive at the beginning of the day

• the simulated optimal solution developed in (Chabouh et al., 2018) for the same problem and,

• the “enhanced SPT” appointment heuristic developed in (Chabouh et al., 2018). This heuristic presented the highest performance in terms of WT, Gap (CT), and RAH (H = 3 pm), among the 12 tested appointment heuristics for 32 patients and 50% of AP.

According to the enhanced SPT, the six patient types are sequenced based on increasing mean of surgery time. The arrivals are generated according to this sequence (i.e., calling all patients in the first type in a sequence to arrive then the second and so on). The number of patients who arrive simultaneously is equal to the number of ORs. If the number of patients of the first type in the sequence is less than the number of ORs, we add the remaining number of the following sequenced type. Finally, appointment times are defined as follows: the first appointment is set at the beginning of the day; subsequent appointments are set to the prior appointment time plus the meantime for the previous patient type’s surgery.

The results of simulated solution delivered by the MIP (mean value ±0.95CI) for 32 patients and 50% of AP are as follows:

CT*≈476 ± 3 min, OR use rate* = 89.6%±0.67%, WT1* = 0 ± 0 min, WT2* = 7 ± 1 min, LOS* = 92 ± 2 min, and RAH* (H = 3 pm) = 0.1%± 0.08%.

The comparison for Gap (CT), WT, LOS (i.e., the time spent between patient arrival and his discharge from the ISS), RAH (H = 3 pm), and OR use rate for 32 patients and AP = 50% are illustrated in Table 5.

Table 5.

Comparison of ConWIP-AS with the current practice and the approach proposed in (Chabouh et al., 2018).

		Arrival scenario
Performance indicator		Current Practice	Enhanced SPT	ConWIP-AS
Gap (CT) (%)	Mean	11.7	3.6	16.4
	± 0.95 CI	± 2.04	± 1.03	± 2.32
OR use rate (%)	Mean	84.4	67.6	76.9
	± 0.95 CI	± 0.66	± 0.8	± 1.5
WT1 (min)	Mean	110.3	0.15	0.01
	± 0.95 CI	± 2.34	±0.69	± 0.01
WT2 (min)	Mean	107	72	18
	± 0.95 CI	± 2.6	± 2.5	±3.6
LOS (min)	Mean	303	157.8	97.5
	± 0.95 CI	± 3.9	± 3.1	± 3.9
RAH (H = 3 pm) (%)	Mean	18.4	0.50	6.50
	± 0.95 CI	± 2.5	± 0.09	± 0.71

From this table, we conclude that:

-The current practice tends to satisfy only health care providers by reducing CT and maximising the OR use rate while dissatisfying patients through long WT, especially outpatients through long LOS and a bigger RAH.

-ConWIP-AS outperforms “Enhanced SPT” in terms of Gap (CT) and WT, especially for WT2 (before surgery), which is justified by the effectiveness of the ConWIP implementation. Furthermore, if we consider that waiting is β times as costly for an outpatient compared to an inpatient (β ≥ 1), ConWIP-AS yields the minimum WT cost whatever the value of β.

Finally, using the ConWIP-DES model and ConWIP-AS, we observed the average WT and LOS of each patient scheduled for the eight appointment slots (from 8 am to 3 pm). LOS is quasi-stable, which is explained by the effect of ConWIP and it increases slightly whenever WT2 increases. WT1 is quasi-null for all patients. However, WT2 increases for patients arriving from 10 am to 1 pm. This is coherent with the load of the ISS at these time slots. In the beginning, the ISS is empty. Thus, WTs are so low. In the middle of the day, the ISS is occupied especially the OR, which explains the rise of WT2 at these times.

5.3. Sensitivity analysis

We analyse the sensitivity of ConWIP-AS, enhanced SPT, and current practice (CP) to two factors: (i) the distribution of patient types and (ii) the system load. First, we vary the AP within a range of ±10% from the current one (50%) while maintaining the number of patients (32) constant “experiment 3.1”. Second, for a working day of 720 min and 50% of AP, we vary the number of patients from 16 to 48 patients “experiment 3.2”. Fig. 7 and Fig. 8 present the results of experiments 3.1 and 3.2, respectively. We presented the average values and box plots for performance indicators per arrival scenario. The boxes show 0.95CIs for each metric and whiskers show the best and worst cases for each scenario.

Figure 7.

Average and 95%CI of performance indicators per AP variation for ConWIP-AS, enhanced SPT, and current practice (CP) (experiment 3.1).

Figure 8.

Average and 95%CI of performance indicators per number of patients and a working day of 720 minutes for ConWIP-AS, enhanced SPT, and current practice (CP) (experiment 3.2).

Both figures confirm the results pointed out in subsection 5.2: the current practice satisfies only the healthcare managers. Using this policy, long WT and LOS have been recorded for all AP and different system loads, which confirms the results obtained for AP = 50% and 32 patients. Enhanced SPT presented the worst performance for the OR use rate. It proved its usefulness only for RAH. Hence, beginning with the shortest durations (i.e., ambulatory cases (Pham & Klinkert, 2008)) increases the number of outpatients treated before a predetermined hour (H). ConWIP-AS is the best if we also consider WT indicators.

Regarding ConWIP-AS:

-First, when AP increases, the OR use rate decreases slightly (Figure 7). However, this variation is not significant (from 80% to 75%). Second, When the system load increases, the OR use rate also increases which is totally logical. ConWIP-AS delivers slightly less OR use rates the current practice but significantly higher than enhanced SPT.

-WT indicators, including LOS, are not sensitive to the AP variation neither to the system load variation, which is explained by the ConWIP effect.

-RAH (H = 3 pm) increases when both AP and system load increase since the number of outpatients increases, which is logical.

In summary, a high decrease of the AP (from 60% to 20%) results in a small variation in all performance criteria, proving the robustness of the ConWIP approach regarding the variation of patient types distribution. The variation of system load affects only the OR use rate and RAH which is related to the increase in the number of patients; however, it maintains its position compared to the other AS rules. The ConWIP approach proved its superiority and robustness especially for WT indicators for both factors.

6. Conclusion

Managing the ISS operations using a basic ConWIP system allowed reducing the length of stay of patients in the ISS and maximising the use of OR and surgeons while considering stochastic process durations. Setting patients’ appointments based on ConWIP consists of calling a number of elective patients to arrive at the beginning of each hour – regardless of their category – equal to the ConWIP output rate. Several appointment schedules based on this principle are developed by considering different patient partitions. The simplest rule based on patient classes: inpatient-outpatient offers more flexibility in scheduling patients’ appointments while being efficient in terms of WT and OR use rate. On the one hand, this rule proved to be easy-to-implement and resulted in a good performance in terms of WT, CT, and OR use rate. On the other hand, varying the distribution of outpatients-inpatients did not affect the ConWIP approach's performance, which justifies its robustness. As this ConWIP-based appointment rule presented promising results for a real-life case study, its real implementation in integrated surgical suites prevails as a first future direction.

This work opens the way to various research perspectives. First, we considered only one source of variability, namely the uncertainty in service times. However, other sources of uncertainty can disrupt the initial AS (Gupta & Denton, 2008). Hence, randomness related to the behaviour of patients such as delay, and no-show as well as the lateness of surgeons, can affect the performance of the developed appointment schedules. It would be interesting to study the behaviour of our methods under these disruptions. Second, the non-consideration of emergencies constituted another fundamental hypothesis of our study. Being in the case of an ISS, the arrival of an urgent case can be quite frequent. Thus, it would be essential to consider urgent cases in the AS. Finally, instead of relying on theoretical distribution for estimating uncertain parameters, predictive models (e.g., machine learning algorithms) can be used to predict patient-specific uncertainty such as no-shows (Simsek et al., 2021), delays (Srinivas, 2020) and service times (Bentayeb et al., 2019) from historical data, and that information can be further leveraged to optimise the AS systems.

Disclosure statement

No potential conflict of interest was reported by the authors.