The decrease of ED patient boarding by implementing a stock management policy in hospital admissions

Jaén, Sebastián Article The decrease of ED patient boarding by implementing a stock management policy in hospital admissions Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Jaén, Sebastián (2024) : The decrease of ED patient boarding by implementing a stock management policy in hospital admissions, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 12, pp. 1-11, https://doi.org/10.1016/j.orp.2024.100298 This Version is available at: https://hdl.handle.net/10419/325780 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp The decrease of ED patient boarding by implementing a stock management policy in hospital admissions Sebastián Jaén ALIADO Analytics and Research for Decision Making, Department of Industrial Engineering, Universidad de Antioquia, Calle 67 No. 53-108, Medellín 050010, Colombia ARTICLE INFO Keywords: Inpatient boarding time Hospital overcrowding Bed capacity planning Inpatient flow System dynamics Operations research ABSTRACT The presence of congestion is a common scenario in tertiary-level hospitals worldwide. Current research suggests that an increase in hospital bed capacity is not a long-term solution given that patient demand adapts to added capacity. Recent literature suggests the need for the implementation of a policy of inter-hospital transfers to divert patients to outpatient priority services or home care. This policy has proven to be effective in reducing ED boarding without compromising patient safety. However, determining the required number of patients to be admitted is key. The dynamic nature of hospital bed availability and discharges requires an admission process able to be in synchrony with those variations. A mismatch between patient demand and hospital admissions will result in either ED boarding or idle capacity. The purpose of this paper is to introduce a methodology to support the process of hospital admissions by providing as an input a threshold for the number of patients to be admitted. The methodology is tested using a system dynamics model that replicates one year of operations of a tertiary-level hospital. The simulations reveal the potential of the methodology to decrease the ED inpatient boarding rate as well as ED and hospital length of stay. 1. Introduction The last two decades have witnessed an increasing worldwide demand for attention in emergency departments (EDs) and hospitals [1]. As a consequence, overcrowding is a problem that affects every ED and hospital regardless of their public or private nature [2]. The most pressing effect of this problem is the increase in the number of boarding patients in the ED. That is the number of patients who cannot be moved to inpatient units due to a lack of inpatient bed availability in the hospital wards [3]. The increasing number of boarding patients in the ED is considered one of the most important challenges that hospitals face [4]. In some EDs in the U.S., 22% of the total ED patient census are boarding patients, where 20% of them boarded for at least eight hours or even days [5]. The patient boarding time is crucial due to its correlation with increased ED length of stay (LoS), mortality rates, nosocomial infections, and falls [4,6]. Additionally, it serves as the most common trigger for ambulance diversion [3]. Not only does the ED boarding time impact overall hospital performance, but it is also connected to various societal factors such as patient care safety, efficiency, resource utilization, healthcare costs, patient experience and satisfaction, public health, community impact, and the quality of care and outcomes [1,4, 7]. E-mail address: [email protected]. The ED boarding problem has been associated with various factors such as ‘‘unnecessary ED visits’’, delays in lab and radiology results, inadequate rostering often attributed to understaffing, and the lack of availability of destination ward beds [3,4,7]. Consequently, Boudi et al. [4] suggest that literature on ED has experienced exponential growth in recent years, addressing this phenomenon from various disciplines, including Operational Research/Management Science (OR/MS). The purpose of this research is to address the ED boarding problem by considering a system dynamics-based approach known as stock management policy. This approach refers to the strategy used to control and regulate the levels of stocks or accumulations within a system [8], in this case, ED boarders. Stocks, in system dynamics SD, represent accumulations of entities that change over time due to inflows and outflows. By eliciting the complexity of the system under study (the hospital), this approach has the primary goal of maintaining optimal levels of patients to meet system requirements, avoid overflows, and manage the flows into and out of these stocks effectively [8]. Our research presents findings from simulating the application of a stock management policy that mirrors the operational traits of an actual tertiary hospital. The outcomes indicate promising advantages of this intervention, emphasizing the importance of crafting practical interventions based on these insights. https://doi.org/10.1016/j.orp.2024.100298 Received 30 May 2023; Received in revised form 6 February 2024; Accepted 7 February 2024 Operations Research Perspectives 12 (2024) 100298 2 S. Jaén This study proposes a novel approach to an existing body of literature whose contributions for this problem range from ED-based solutions to hospital-wide initiatives [2]. They consist in improvements in medical procedures, ED layouts and more nurse training [2,9]. Other solutions have as strategy the increase of capacity in rooms, beds, and staff [10]. The increase in the flow of inpatients at the downstream level is another alternative that is achieved by performing one or the combination of the following strategies: improving the classification of inpatients [3,11], better bed management [12], increasing the prediction of demand and the scheduling of discharges [13], and finally, the addition of special units such as acute medical and surgical units [10]. All the aforementioned approaches have an immediate impact on reducing the number of ED boarders. However, these initiatives, as well as those promoting the increase of the flow of inpatients at the downstream level, provide short-term improvements since the demand for beds adapts to added capacity [14,15]. Solutions preventing or diverting the demand of ED or acute hospital services seem to have long-lasting impacts [14,15]. Despite there is not a large body of OR/MS, directly addressing the ED boarding problem, there is a consistent body of research modeling the flow of patients in health care facilities [3]. Studies suggest that ED boarding is due in large part because of the healthcare providers’ incapacity of maintaining patient flow. High ED occupancy and prolonged ED LoS are linked to delays in discharges and low service rates [3,16]. Therefore, OR/MS techniques have significantly contributed to improving the ED performance in several metrics such as ED LoS [3]. The most used methodologies for modeling patient flow are Queuing Theory, Markov models, decision processes, and simulation [3,17]. The use of queuing theory models is among the most basic and simplistic approaches for modeling the patient flow [18]. Cochran and Roche [19,20,21], present several examples intended to increase the capacity of the ED to treat patients. Rodríguez Jáuregui et al. [22], follow the Cochran and Roche approach evaluating capacity and designing staffing policies in an ED. A different approach is introduced by Zonderland et al. [23,24]. Their work introduces the use of queuing theory as a tool for evaluating ED capacity and investigating the best possible strategy for achieving improvements in efficiency. Markov models are a resource for ED and patient flow modeling because health care processes can be represented by a number of interdependent work-stages [3,17]. The growing use of simulation in health care refers to the implementation of either one, or the combination of several of the following techniques: Monte Carlo, discrete event simulation (DES), SD and agent based simulation (ABS) [17,25–29]. Typically, the Monte Carlo simulation has been used as a method for probabilistic sensitivity analysis of analytic formulations (e.g. a Markov Model). In the few studies reported, it is mostly used as a subsidiary technique [28]. DES, on the other hand, is the most commonly employed and widespread modeling methodology for supporting health care decisions to be taken at the operational level [28,30]. This methodology is regarded as more flexible and versatile above other modeling methods due to the following features: DES models are free from both arrival assumptions (Poisson or not) and from time service assumptions (exponential or not) [3]. ABS has been gaining popularity among researchers, surpassing SD and even replacing DES as a tool for realistic health care modeling [28,31]. ABS shares all of the advantages of using DES in terms of eliciting systems complexity. It is claimed that ABS models can also address tactic and strategic level problems outperforming the DES reach [31]. However, the main findings of ABS models focus on operative changes for improving patient throughput time and the other critical performance measures [32]. The improvement of inpatient flow has been specially dealt in the system dynamics (SD) literature [33]. The use of SD is especially suitable for addressing the ED problem given its versatility that allows a more aggregated approach to systems complexity, and in many cases, the modeling process is more efficient than DES and ABS [34]. Also, SD operates under the paradigm of eliciting reality through the interaction of flows and stock, which is particularly suited for modeling patient flows [35,36]. As a seminal research, the work by Lane and Husemann [37], focuses on addressing the flow of acute patients through the UK health care system. Their contribution acknowledges the pertinence of the SD methodology for mapping the flow of patients in a way that the clinicians can both better understand and become an active partner in the process of policy making. Similar applications of system mapping and policy design can be found in works by Wong et al. [38], Vanderby et al. [35], Patrick et al. [39] and Homer et al. [40]. Additionally, Esensoy and Carter [41], use SD modeling to facilitate a better understanding of the system-wide effects of patient flow related interventions. Some SD literature is focused on studying the flow of patients in health care facilities beyond a qualitative approach. For example, Brailsford et al. [42], use SD to simulate patient flows and to identify system bottle-necks in emergency and on-demand health care centers. Furthermore, Vanderby and Carter [36], validate the use of SD for modeling hospital patient flow. Later, Demir et al. [43], use SD for modeling the LoS of a neonatal unit, particularly focusing on patient flow to determine the major drivers of the system. Esensoy and Carter [44], implement a whole-system, strategic perspective, designed to evaluate the direction and magnitude of patient flow. A recent contribution by Grida and Zeid [45] evaluated the impact of hospital resources on inpatient flow and LoS using a SD approach within the framework of the Theory of Constraints. The SD contributions that more closely address the inpatient boarding problem are those by Wong et al. [38,46], Rashwan et al. [15], and Mahmoudian-Dehkordi and Sadat [14]. The simulations performed by Wong et al. [46], showed that by distributing discharges over the course of a week, a decrease in the number of ED boarders is achieved. This work emphasizes the role of the downstream flow on ED boarding. The second work by Wong et al. [38], suggests the use of an admission control policy as a useful strategy to alleviate downstream occupancy rates and boarding times. Rashwan et al. [15], propose a SD methodology to evaluate a comparison between stock (capacity) and flow (demand) interventions to ease the bed blocking problem. Their conclusion suggest that the impact of stock interventions independently, such as increasing post-acute capacity, is time-limited, while the reduction of inflow has more lasting effects. The main findings of SD patient-flow literature has evolved in two distinctive approaches. The first approach remarks the value of the down-stream capacity as a necessary condition for improving inpatient throughput [38,45–48]. The second approach, argues that even if the system reaches the desired level of capacity, it generates a counterintuitive result. Initially, the system shows a short term improvement by reducing pressure in the services. However, the consequence in the long run is an increase in demand that counteracts the impact of the policy over time [15]. Thus, long lasting solutions seem to be placed at the up-stream level where patient demand and admissions are the key drivers [14,15,38]. Several works identify the patient admission process as the main driver in controlling overcrowding [49–51]. Thus, some of the last decade literature has focused specifically on reducing hospital admissions from the ED arguing three main reasons: First, most of the arrivals of patients occur because they do not have access to their primary care doctor or specialist [52]. Second, some patients arrive because they, and their health care providers, have the perception that a comprehensive workup occurs more quickly when a patient is admitted to hospital [52]. Third, there is an increasing evidence that the number of unnecessary hospitalizations range from 5%–82% worldwide [49,53, 54]. As example of the implementation of such approach is provided by the Sunnybrook Health Sciences Center hospital in Toronto [52]. By implementing a patient diverting protocol called ‘‘rapid referral clinic’’ (RRC), which allows ED physicians to refer patients to a general Operations Research Perspectives 12 (2024) 100298 3 S. Jaén internist for urgent outpatient assessment, the hospital reduced admissions, improved ED boarding times, and increased patient flow at the downstream level. In consequence, two type of strategies predominate in the effort of reducing admissions: prevention and inter-hospital transfers or rapid referral protocols [52,55]. The main purpose of the present work is to introduce a methodology to support the inter-hospital transfer process strategy. This is because health care providers need to consider when and how many patients need to be admitted and transferred for this strategy to start to take place and be effective. As an in-and-out flow policy, the rates of patients to be admitted and transferred need to be calibrated in accordance with available capacity and expected inpatient discharges, which vary over time. Inadequate rates of transfers (too low or high) might not produce the desired effect on the number of boarding inpatients, or they will generate unnecessary transfers and idle capacity. The proposed methodology does not intend to constrain clinical criteria for patient admission but provides guidelines to manage inpatient flow and overcrowding in accordance with the hospital’s patient-flow dynamics. Additionally, the methodology incorporates levels of policy adherence that provide flexibility for the proposed methodology to be applied in those cases where medical criteria prioritize patient admission over ED overcrowding. Thus, this paper implements a SD model-methodology which addresses several needs found in the literature: Suggest tactic and strategic changes to support the admission process [36]. Contribute to the limited amount of works addressing the boarding time problem in the ED. Finally, the work addresses the need of a combination of stock (capacity) and flow policies (admissions, inter-hospital transfers, and discharges) that prevent counter-intuitive responses of the system due to adaptations to new capacity [15]. The modeling is tested using the inputs of data from a large third-level. hospital Third-level hospitals, also known as tertiary care hospitals, are healthcare facilities that offer specialized and advanced medical care beyond what is provided in primary and secondary healthcare settings. The selection of a tertiary hospital to simulate the stock management approach considers the unique characteristics of this institution type, given its specialized care that often leads to increased demand, extended lengths of stay, and instances of inpatient boarding. Additionally, it is within these institutions that the negative implications of inpatient boarding are most pronounced [4,7]. 2. Methods and methodology 2.1. Hospital inpatient flow: problem description The patient boarding problem arises when ED doctors decide that a given patient should be hospitalized and its unit bed is not available. Then, the patient has to wait in the ED until a bed is available [3]. To illustrate the problem, in Fig. 1, there is an aggregated and simplified representation of the flow of patients in a hypothetical hospital. The image depicts the most common steps taken upon arrival for almost all patients requiring urgent care. However, there are additional paths of arrival to the hospital, including referrals, transfers from other healthcare facilities, and scheduled non-ambulatory surgeries that necessitate several days of supervised recovery. These pathways are not included in Fig. 1 because patients following these routes typically do not require boarding in the ED due to the scheduled nature of their arrival, which accounts for the availability of beds. After the arrival to the hospital, the patients wait in a waiting area, or pre-triage area which is a designated section within a hospital’s ED where patients initially present themselves before formal triage takes place. This area serves as an initial point of contact for individuals seeking urgent medical attention. The area is usually situated near the entrance or within the ED. In most of the cases, patients are registered in this stage before being examined. After pre-triage and registration, the patient faces an assessment (triage) performed by a physician which determines the pathway of attention. One path is to be discharged after treatment, the other, is to remain in the ED for treatment that incorporates the prescription and arrival of diagnostic aids. The next stage determines if the patient should be discharged, admitted for hospitalization or transferred to a different hospital. The inpatients whose beds are not available are represented as queues of inpatients, and they constitute the boarding inpatients. To be in a given queue depends on the ward of destination in accordance with the inpatient path of treatment. Hospitalization is the next stage. In Fig. 1, hospitalization is represented as levels of inpatients being treated in their respective ward. Wards in a hospital refer to specific units or sections within the healthcare facility that are dedicated to providing care for particular types of patients or medical conditions. The wards depicted in Fig. 1, are representing the most common wards found in a third-level hospital: medical, surgical, intensive care, pediatric, maternity, psychiatric, geriatric, and isolated. Here, inpatients receive the main treatment whose duration depends on the ward LoS. In accordance with the treatment outcome, the inpatient can be transferred to a different ward or discharged. 2.2. Dynamic hypothesis The dynamic hypothesis presented here suggests that the process of patient admissions does not consider available hospital bed capacity, or at least, it is not one of the main drivers to determine admissions. Due to this lack of information, a problematic behavior arises when admissions do not match available bed capacity and inpatients are required to board in the ED (Fig. 2). Under this scenario, depicted by Fig. 2, only when the rates of Inpatients discharge are greater than the rates of Admitted patients, the stock of Boarding inpatients is decreased. When this situation does not occur, the increase in the stock of Boarding inpatients is expected because there exists a difference between the flows of incoming and outgoing patients. 2.3. Model equations The present modeling captures the aforementioned dynamic hypothesis in the context of a tertiary hospital. The choice of such institutions impacts the modeling process in determining the necessary level of aggregation required to elucidate the dynamic interactions among the wards and the ED. Consequently, SD emerges as a suitable and necessary methodology for modeling complex systems like tertiary hospitals [35]. The modeling elicits the hospital using the SD stocks and flows representation (Fig. 3). Double lines indicate that the variable is an array, whose dimension depends upon the number of 𝑈units. The model elicits all of the states in which the inpatient can be in the hospital: Emergency department patients (𝐸), Boarding area patients (𝐵) and Hospitalized patients (𝐻). The hospitalization service 𝐻, is divided into 𝑈units, and thus, the boarding area has 𝑈stocks of inpatients waiting for a bed. The boarding area is normally located inside the ED. However, to simplify the modeling, the large stock of ED is divided into two stocks: the stock 𝐸containing the number of patients admitted, revised, attended and to be discharged, and the stock 𝐵, which accounts for all of the patients waiting for beds to be assigned in the hospitalization units. The level 𝐸has one inflow representing patient arrivals (𝐴), and three outflows, which depict the discharge of patients from the ED (𝑑), transfers to different healthcare facilities (𝑇), and admissions (𝑎), respectively. Admissions (𝑎), refer to patients necessitating hospital stays in the ward due to their medical conditions. The boarding level 𝐵, has an outflow (𝑡) for intra-hospital transfers, indicating the number of patients moved from the ED to the hospital wards. At the hospital 𝐻level, there are two sets of inflows and outflows for patients. The first inflow corresponds to the influx of patients transferred from the ED, previously denoted as outflow 𝑡. Operations Research Perspectives 12 (2024) 100298 4 S. Jaén Fig. 1. Patient flow from the ED to hospitalization. Fig. 2. Effect of inpatient discharge on boarding inpatient. Fig. 3. Stocks and flows diagram of the hospital. The second inflow (𝑔) represents the number of patients arriving from different ward units. Regarding the outflows, the first one (𝑗) represents the number of patients to be transferred to a different ward unit, while the second outflow (𝐷) corresponds to the number of patients to be discharged from their ward. Flows 𝑎,𝑡,𝑔,𝑗and 𝐷, are the respective sum of 𝑎𝑖,𝑡𝑖,𝑔𝑖,𝑗𝑖and 𝐷𝑖flows, where 𝑖= 1,…, 𝑈 units. 𝐸(𝑡)is a stock of patients in the ED, where arrivals 𝐴(𝑡), discharges 𝑑(𝑡), admissions (Admitted patients)𝑎(𝑡), and transfers 𝑇(𝑡)are its flows (Eq. (1)). 𝐸(𝑡) = ∫𝑡 𝑡0[𝐴(𝑡)−(𝑑(𝑡) + 𝑎(𝑡) + 𝑇(𝑡))]𝑑𝑡 +𝐸(0) 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (1) The inflow Arrivals 𝐴(𝑡)(Eq. (2)), is modeled as a GRAPH function retrieving a given number of patients per hour. This function provides time-varying exogenous inputs retrieving the number of patients arriving to the ED per hour. 𝐴(𝑡) = 𝐺𝑅𝐴𝑃 𝐻(𝑡, 𝑡𝑜, 𝛥𝑡, {𝑎𝑡0, 𝑎2,…, 𝑎𝑡})𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 ℎ𝑜𝑢𝑟 (2) The outflow discharges 𝑑(𝑡), is modeled by Eq. (3), where 𝛽1is the proportion of ED patients to be discharged, and 𝜏2is the average ED LoS. Patients discharged in 𝑑(𝑡)are those not requiring hospitalization. 𝑑(𝑡) = 𝐸(𝑡)𝛽1 𝜏2 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (3) Operations Research Perspectives 12 (2024) 100298 5 S. Jaén The outflow of Admitted patients 𝑎(𝑡), (Eq. (4)), is modeled considering 𝛽2as the proportion of ED patients demanding beds in the 𝑈 units. The specific number of patients demanding a bed in an unit 𝑖, 𝑎𝑖(𝑡), corresponds to Eq. (5), where the constant 𝜌𝑖is the proportion of patients demanding a bed in a specific unit 𝑖. 𝑎(𝑡) = 𝐸(𝑡)𝛽2 𝜏2 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (4) 𝑎𝑖(𝑡) = 𝑎(𝑡)𝜌𝑖∨𝑖= 1...𝑈 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (5) 1 = 𝑈 ∑ 𝑖=1 𝜌𝑖(𝑡)𝐴𝑑𝑖𝑚𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 (6) The third outflow 𝑇(𝑡)(Eq. (7)), corresponds to the number of patients to be transferred to a different healthcare facility. These number of transfers are not part of a strategy to reduce boarding patients. They represent the number of patients to have to be transferred to a different health care facilities because the hospital cannot meet the patient needs (i.e. technology, level of care, etc.). 𝑇(𝑡) = 𝐸(𝑡)𝛽3 𝜏2 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (7) 1 = 𝛽1+𝛽2+𝛽3𝐴𝑑𝑖𝑚𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 (8) The stock of boarding inpatients 𝐵(𝑡)(Eq. (9)), corresponds to the sum of the 𝑈stocks (𝐵𝑖(𝑡)Eq. (10)), whose flows correspond to the inflow of Admitted patients 𝑎𝑖(𝑡), and the outflow of Intra-hospital transfer of patients 𝑡𝑖(𝑡)(Eq. (11)). 𝐻∗ 𝑖is the capacity in beds, and 𝐻𝑖(𝑡), the number of occupied beds. The constant 𝜏3corresponds to the average Intra-hospital transfer time from the boarding area in the ED to the units. 𝐵(𝑡) = 𝑈 ∑ 𝑖=1 𝐵𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (9) 𝐵𝑖(𝑡) = ∫𝑡 𝑡𝑜 (𝑎𝑖(𝑡) − 𝑡𝑖(𝑡))𝑑𝑡 +𝐵𝑖(0) 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (10) 𝑡𝑖(𝑡) = 𝑀𝐼𝑁(𝐵𝑖(𝑡), 𝐻∗ 𝑖(𝑡) − 𝐻𝑖(𝑡)) 𝜏3 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (11) The total stock of hospitalized patients 𝐻(𝑡), is defined by the sum of the 𝑈stocks 𝐻𝑖(𝑡)(Eq. (12)). 𝐻𝑖(𝑡)depends on the inflows 𝑡𝑖(𝑡)and 𝑔𝑖(𝑡), modeling transfers from the boarding area and a hospital unit 𝑗, to unit 𝑖. Also 𝐻𝑖(𝑡)depends on the outflows 𝑗𝑖(𝑡)and 𝐷𝑖(𝑡), which determine the number of transfers to a different unit and discharges from the unit 𝑖(Eq. (13)). 𝐻(𝑡) = 𝑈 ∑ 𝑖=1 𝐻𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (12) 𝐻𝑖(𝑡) = ∫𝑡 𝑡𝑜[𝑡𝑖(𝑡) + 𝑔𝑖(𝑡) − 𝐷𝑖(𝑡) − 𝑗𝑖(𝑡)]𝑑𝑡 +𝐻𝑖(0) 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (13) The number of inpatients to be discharged from the unit 𝑖,𝐷𝑖(𝑡) is modeled by Eq. (14). Thus, 𝛽4𝑖corresponds to the proportion of inpatients to be discharged from the unit 𝑖, while 𝜏4𝑖, represents the unit 𝑖LoS. 𝑗𝑖(𝑡)is modeled by Eq. (15), where 𝛽5𝑖, is the proportion of inpatients to be transferred from uni 𝑖to an unit 𝑗. Finally, the flow of transferred inpatients between units 𝑔𝑖(𝑡), is determined by Eq. (16). This is determined by the product of the outflow 𝑗𝑖(𝑡)and the constant 𝛬𝑖𝑗 . This constant is a matrix where each row contains the percentage of inpatients to be transferred from the unit 𝑖to the unit 𝑗. Thus, ∑𝑈 𝑗=1 𝛬𝑖𝑗 = 1, where 𝛬𝑖𝑗 = 0 when 𝑖=𝑗, meaning no inpatient can be transferred to the same unit of origin. 𝐷𝑖(𝑡) = 𝐻𝑖(𝑡)𝛽4𝑖 𝜏4𝑖 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (14) 𝑗𝑖(𝑡) = 𝐻𝑖(𝑡)𝛽5𝑖 𝜏4𝑖 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (15) 𝑔𝑖(𝑡) = 𝑈 ∑ 𝑗=1 𝑗𝑖(𝑡)𝛬𝑖𝑗 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (16) Table 1 Estimated parameters and initial values for levels. Parameter Name Value 𝜌𝑖 Average percentage of patients in the ED demanding a bed in the unit 𝑖0.04 A(t) Arrivals of patients to ED per hour (Averg.) 6.17 𝛽1 Average % of patients to be discharged from ED 36 𝛽2 Average % of patients requiring hospitalization 63 𝛽3 Average % of patients to be transferred to a 1 different hospital 𝜏2Average ED LoS in hour 8.14 𝜏3Average transfer time from 𝐵(𝑡)to 𝐻(𝑡)in hours 1 𝜏4𝑖Average 𝐻𝑖(𝑡)LoS in hours 118 𝛽4𝑖 Average % of inpatients to be 91 discharged from unit 𝑖 𝛽5𝑖 Average of % inpatients to be 9 transferred from unit 𝑖 𝛬𝑖𝑗 Average % of inpatients 3 in ward 𝑖to be transferred to ward 𝑗 Levels Name Initial value 𝐸Patients not boarding in the ED area 55 𝐵Patients boarding in the ED area 50 𝐻Hospitalized inpatients 592 2.4. Simulation reporting Software: Powersim Studio 10 Expert®(10.14.5555.6) 64-bit version. Hardware: OS Windows 10 home. Processor Intel (R) Core(TM) i3-23-10M CPU @ 2.10 GHz 2.10 GHz. RAM 4.00 GB. System type 64-bit Operating System, x64-based processor. Simulation: Euler integration with step 1 h and time horizon 1 year, using Gregorian calendar. 3. Validation 3.1. Source data and analysis The modeling is tested using the inputs of data from a tertiary level hospital. Its capacity accounts for 99 stretchers in the ED, 14 operating rooms, 650 beds allocated in 24 wards or units. The hospital has an annual volume of hospitalized inpatients close to 32,000, with an average LoS of 8.7 days, where 26% of those patients spend on average 36 h boarding in the ED. The data required by the model to replicate the historical behavior of the hospital was extracted from the hospital’s database system. The software retrieved three main databases containing historical information from the 2019 year of operations: •Admissions. Registers each admission to the hospital. •Transfers. Registers the transfer of a patient between units and procedures. •Discharges. Registers the inpatients discharged from wards. After processing the three databases it was possible to determine the parameters of the model presented in Table 1, and the time series of ED and hospital discharges to compare the simulated hospital versus the actual. 3.2. Test of validation The process of validation was guided by the tests suggested by Sterman [8,56]. The validation of the structure was presented in a workshop where the model was discussed among a group of hospital staff members. The validation of the behavior-reproduction test was performed replicating the behavior of two main groups of output variables during a horizon time of 12 months (8640 h) of the hospital operations Operations Research Perspectives 12 (2024) 100298 6 S. Jaén Fig. 4. Simulated and actual ED discharge. Table 2 Summary statistics for assessing simulated fit to actual data. ED Hospital 𝑅20.725 𝑅20.83 MAE 0.887 MAE 1.20 𝑋𝑆Mean 1.468 𝑋𝑆Mean 2.57 𝑋𝐴Mean 1.717 𝑋𝐴Mean 2.11 𝑆𝑆Mean 1.327 𝑆𝑆Mean 4.29 𝑆𝐴Mean 2.103 𝑆𝐴Mean 3.86 MSE 2.199 MSE 5.95 UM 0.028 UM 0.035 US 0.274 US 0.030 UC 0.698 UC 0.935 Sum 1Sum 1 in time steps of one hour. One group of main outputs corresponds to the yearly average occupancy rate of 24 wards and the ED. This test verifies if the stocks of the model and the stocks of the actual system have similar levels of inpatients occupancy. Comparing the simulated data versus the actual, the test performed the following results: MAE: 0.0022821 and MSE: 3.82623E−05, which indicates that the model is able to reproduce the yearly average occupancy rate of 24 wards and the ED. The other group corresponds to the discharge rates of the units (the ED and the 24 hospital wards). To simplify the analysis, the validation compares the sum of the simulated 24 ward and ED discharges, against the sum of the observed 24 ward and ED discharges. In Table 2, the results compare the ability of the model to replicate the ED and hospital discharges. The 𝑅2suggest that the modeling is better at reproducing the behavior of the hospitalization discharges than that of the ED, but they are still in an acceptable range. In addition, according to the Theil’s test [8,56], the large UC values presented in Table 2, indicate that despite the point-by-point values of the simulated series do not match the actual data, this error is unsystematic and is not considered a criteria for rejecting the model Sterman [56]. The Figs. 4 and 5illustrate the fit of the simulated vs the actual outputs of the hospital wards and ED discharges. The figures only compare the first 300 h of 8640, due to the lack of space, illustrating the behavior of simulated and actual outputs. The charts complement the analysis portrayed in Table 2. Given the results of the analysis presented in Table 2, they support the model as an appropriate representation of the hospital behavior under study. The broad validation of the model directly stems from the type of modeling (SD) and the complex nature of the system it represents—a tertiary hospital. Given the intricate dynamics of the tertiary hospital environment, it is important to note that the model provides an aggregated representation of the results. This validation leads us to conclude that despite the complexity and the need for aggregation, the model adeptly describes the aggregated dynamics of both the ED and hospital discharges. 4. Policy analysis 4.1. Policy implementation In this study, to better control the stock of Boarding inpatients, we propose a flow-in alternative methodology as opposed to a flowout methodology which only focuses on decreasing the ward LoS or increasing the stock of beds. The approach presented here includes consideration of the rate of Inpatients discharge and the stock of Available beds as additional inputs needed by the physicians to determine the threshold of admissions. Fig. 6 illustrates how incorporating the variable Available beds to the aforementioned structure (Fig. 2), as an additional input for the calculation of the variable Admitted patients, creates two additional patient admission feedback control loops (R1 and B3). These loops allow the hospital to implement a stock management policy [8] by calculating an inflow rate of Admitted patients, in accordance with both the outflow rate of Inpatients discharge and the stock of Available beds. 4.2. Policy modeling The modeling of a tertiary hospital using an SD approach determines that the policies to be considered (the experiments) operate at the tactical-strategic level. This implies that the results encompass a set of guidelines implemented at the operational level with long-term impacts. The implementation of the proposed methodology demands the introduction of a new variable Available beds, able to determine the number of beds available in each unit 𝑖, to avoid ED overcrowding and idle capacity. In Eq. (17),Available beds is redefined by the addition of two components: expected outflow and adjustment for stock. Expected output is determined by the expected outflow of hospital discharges per hour (Inpatients discharge), 𝐷𝑖(𝑡), (Eq. (14)). The adjustment for stock considers the expression of free capacity 𝐻∗ 𝑖(𝑡) − 𝐻𝑖(𝑡), divided by a term 𝜏1𝑖. This expression, according to the stock management policy defined by Sterman [8], represents a first order material delay of the material in transit (free beds). The term 𝜏1𝑖, established in hours for this model, represents the average delay time of the free beds in transit to be assigned. The value for 𝜏1𝑖controls the desired level of hospital occupancy when bed demand is greater than hospital capacity. The total number of beds available in the hospital 𝑋(𝑡), is given by Eq. (18). The new modeling is illustrated in the stocks and flows diagram in Fig. 7. 𝑋𝑖(𝑡) = 𝐷𝑖(𝑡) + 𝐻∗ 𝑖(𝑡) − 𝐻𝑖(𝑡) 𝜏1𝑖 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (17) 𝑋(𝑡) = 𝑈 ∑ 𝑖=1 𝑋𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (18) As 𝑋𝑖(𝑡)establishes the number of Available beds, the equations determining the number of Admitted patients,𝑎(𝑡), (Eq. (4)), has to be Operations Research Perspectives 12 (2024) 100298 7 S. Jaén Fig. 5. Simulated and actual hospital discharge. Fig. 6. Discharge-Available beds driven admissions policy. Fig. 7. Policy implementation: stocks and flows structure. redefined to determine a limit in the number of admissions (Eq. (19)). If 𝑋𝑖(𝑡)≥𝐸(𝑡)𝛽2𝜌𝑖∕𝜏2, all the patients demanding beds in the unit 𝑖are admitted, otherwise, 𝑋𝑖(𝑡)patients are admitted and 𝑙𝑖(𝑡)patients (Eq. (20)), are those who need to be transferred to a different healthcare facility to decrease ED boarding. Thus, the total number of inter-hospital 𝑇(𝑡)transfers also has to be redefined according with Eq. (22). 𝑎𝑖(𝑡) = 𝑀𝐼𝑁(𝐸(𝑡)𝛽2𝜌𝑖 𝜏2 , 𝑋𝑖(𝑡)) ∨ 𝑖= 1...𝑈 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (19) 𝑙𝑖(𝑡) = 𝑀𝐴𝑋(𝐸(𝑡)𝛽2𝜌𝑖 𝜏2 −𝑎𝑖(𝑡),0) ∨ 𝑖= 1...𝑈 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (20) 𝑙(𝑡) = 𝑈 ∑ 𝑖=1 𝑙𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (21) Operations Research Perspectives 12 (2024) 100298 8 S. Jaén Fig. 8. Boarding inpatients before and after policy. Fig. 9. Inter-hospital transfers. Table 3 Base vs Policy. Indicator Before policy After policy Change Total ED discharges 19,206 19,206 0% Total inter-hospital transfers 533 1440 170% Total hospital discharges 33,246 32,472 −2% Boarding inpatients rate 0.26 0.03 −88% ED occupancy rate 1.99 1.32 −34% Hospital occupancy rate 0.93 0.91 −2% 𝑇(𝑡) = 𝐸(𝑡)𝛽3 𝜏2 +𝑙(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (22) The results of the simulations are presented in Fig. 8 and Table 3. The figure compares the two levels of boarding inpatients in the ED before and after the stock management policy is implemented. The base case shows how the level of inpatient boarding is critical in the second semester of the year reaching values near 60% of the total ED patients. In contrast, after implementing the policy, the figure illustrates how the percent of boarding inpatients decreases from an average of 26% to an average of 3% annually. Table 3 presents how the implementation of this policy impacts the output and level of hospital indicators. The number of total inter-hospital transfers is distributed during the year according to Fig. 9. 4.3. Policy adherence The success of the policy depends on the strict assignment of beds to patients in the ED given the number of beds determined by Eq. (17). However, it may be necessary to allow levels of policy adherence. To Table 4 Levels of policy adherence. 𝐿𝑖Transfers Boarding rate Boarding time (h) 1 906 3.2% 1.5 2 702 3.8% 1.51 3 541 4.5% 1.73 4 412 8% 3.8 5 288 14% 6.27 allow this, Eq. (23) can be modified to incorporate these levels of adherence according with the hospital needs. 𝑎𝑖(𝑡) = 𝑀𝐼𝑁(𝐸(𝑡)𝛽2𝜌𝑖 𝜏2 , 𝑋𝑖(𝑡) ∗ 𝐿𝑖)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡 ℎ𝑜𝑢𝑟 (23) By using the 𝐿𝑖parameter, the equation can increase ‘‘artificially’’ the number of available beds allowing more inpatients to board, avoiding their transfer to another institution. In Table 4, the results of the simulations given different levels of policy adherence. This approach gives the managerial staff the possibility to adjust the policy given the needs or interest of each hospital. 4.4. Statistical screening analysis for policy testing The statistical screening analysis is a technique whose purpose is to identify the key inputs of a model using a sensitivity analysis and the simple correlation coefficient CC [57]. The analysis establishes a range of uncertainty for each input. Then, after performing several simulations it calculates the CC between the inputs and the selected output. The resulting value of the CC ranges from −1 to 1, which indicates the strength of the linear relationship between the input and