Hindawi Advances in Operations Research Volume 2021, Article ID 7120291, 11 pages https://doi.org/10.1155/2021/7120291 Research Article An Optimization Model to Address Overcrowding in Emergency Departments Using Patient Transfer 1 1,2 1,3 Zeynab Oveysi , Ronald G. McGarvey , and Kangwon Seo Department of Industrial and Manufacturing Systems Engineering, E3437 Laﬀerre Hall, University of Missouri, Columbia 65211, Missouri, USA Truman School of Public Aﬀairs, E3437 Laﬀerre Hall, University of Missouri, Columbia 65211, Missouri, USA Department of Statistics, E3437 Laﬀerre Hall, University of Missouri, Columbia 65211, Missouri, USA Correspondence should be addressed to Ronald G. McGarvey; mcgarveyr@missouri.edu Received 22 April 2021; Revised 7 July 2021; Accepted 29 July 2021; Published 23 August 2021 Academic Editor: Panagiotis P. Repoussis Copyright © 2021 Zeynab Oveysi et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Overcrowding of emergency departments (EDs) is a problem that aﬀected many hospitals especially during the response to emergency situations such as pandemics or disasters. Transferring nonemergency patients is one approach that can be utilized to address ED overcrowding. We propose a novel mixed-integer nonlinear programming (MINLP) model that explicitly considers queueing eﬀects to address overcrowding in a network of EDs, via a combination of two decisions: modifying service capacity to EDs and transferring patients between EDs. Computational testing is performed using a Design of Experiments to determine the sensitivity of the MINLP solutions to changes in the various input parameters. Additional computational testing examines the eﬀect of ED size on the number of transfers occurring in the system, identifying an eﬃcient frontier for the tradeoﬀ between system cost (measured as a function of the service capacity and the number of patient transfers) and the systemwide average expected waiting time. Taken together, these results suggest that our optimization model can identify a range of eﬃcient al- ternatives for healthcare systems designing a network of EDs across multiple hospitals. which leave them unsatisﬁed [1, 8, 9]. According to the 1. Introduction National Hospital Ambulatory Medical Care Survey in 2017 Overcrowding in hospital emergency departments (EDs) is in the United States, the average waiting time at emergency recognized as a serious problem in many countries around departments for a patient to visit a physician, physician the world [1] and aﬀected many hospitals especially during assistant, or nurse is about 40 minutes and around 17 the response to emergency situations such as pandemics or percent of patients waited more than an hour [10]. In fact, up disasters [2]. Overcrowding occurs when the arrival rate of to 10% of patients can become frustrated from long waiting patients exceeds the ED’s available capacity [3]. One con- times and may leave the ED without treatment [11], which increases such patients’ risk of death or hospital readmission tributing factor to overcrowding in the USA is a reduced supply of EDs; from 1995 to 2016, although the number of within the next seven days [12]. ED staﬀ frustration is ED visits increased by 51 percent, the number of EDs de- recognized as one of the negative eﬀects of overcrowding on creased by 12 percent [4]. Other key factors which cause healthcare providers [13]. overcrowding in the USA are the aging population, limited One potential solution to overcome the overcrowding access to medical care from other providers, the safety net, problem and have quick and high-quality services in EDs is seasonal illness, surgical scheduling, and high utilization of adding more resources to EDs [14]. However, such an ap- ED for nonemergency care [5,6]. Overcrowding has some proach is limited not only by operating budget constraints negative outcomes for both patients and service providers but also by limitations on available personnel and the size of [7]. Patients may face prolonged pain and long waiting times the ED facility [14]. 2 Advances in Operations Research Introducing an incentive policy to accident and emer- interval” is the time interval utilized for analysis, during gency departments by the UK government in 2000 was which the number of staﬀ is constant (a representative time might be one or two hours in an ED) [15]. Izady and another attempt to reduce patient waiting time in such departments [15]. (is policy requires 98% of patients to be Worthington [15] applied their method in a generic ED and discharged, transferred, or admitted to inpatient care within showed that signiﬁcant improvement with respect to this 4 hours of their arrival [15]. Large penalties were imposed for target can be made even without an increase in total staﬀ failing to meet the 98% target and some hospital managers hours. Sinreich et al. [22] introduced two iterative heuristic even lost their jobs for this reason [16]. Gruber et al. showed algorithms, which combined simulation and optimization that this policy reduced patient waiting time by 19 minutes models for scheduling the work shifts of the ED medical and it also decreased mortality by 14% [17]. staﬀ. (ese authors’ algorithm shifted the resource capacity Transferring patients has also been discussed in some from low-demand hours to peak demand hours, and as a studies as an option to help address overcrowding [3, 18, 19] result, there was a signiﬁcant reduction in patient waiting and was utilized in some areas facing large numbers of time as well as the peak utilization values of the ED medical staﬀ. patients due to emergency situations, such as New York City [2]. Nezamoddini and Khasawneh [3] proposed a mathe- Some studies examined the eﬀects of transferring pa- matical model to quantify the eﬀect of transferring patients tients between hospitals in a multihospital setting [3, 18]. between hospitals on patients’ waiting time in a multihos- Nezamoddini and Khasawneh [3] found that transferring pital system. patients between hospitals can be an eﬀective way to reduce In this study, we propose a novel mathematical model to patients’ waiting time. (ey used the concept of a capacitated capture the eﬀects of transferring patients between hospitals network to model a multihospital system and allowed the on patients’ waiting time. Similar to Nezamoddini and nonemergency patients to be transferred between hospitals Khasawneh [3], the objective of our model is to determine subject to capacity constraints on the maximum number of the number of servers in each ED and the rates of patient transfers allowed per unit time. Soni [18] developed rule- transfer between EDs, in such a way that the cost of the based patient transfer protocols and tested the protocols in a multihospital patient ﬂow simulation model and found that system is minimized. However, unlike [3], who did not explicitly account for queueing eﬀects, our model includes eﬀective patient transfer protocols can optimize the patient concepts from queueing theory (QT) to account for delays in ﬂow in a hospital system. patients’ receiving service due to overcrowding. Regarding the solution techniques utilized, some re- (e remainder of the paper is organized as follows. searchers have used simulation models to capture the Section 2 presents a literature review on research examining complexity and dynamic nature of processes in EDs. Cabrera ED overcrowding. Section 3 provides our mathematical et al. [14] used an agent-based simulation to model EDs. modeling approach. Section 4 presents the results of nu- (ey concluded that although their simulation experiments merical testing and sensitivity analysis. Section 5 provides a helped to generate a better understanding of the problem, conclusion and suggestions for future work. they were time-consuming even for a small problem. Hung and Kissoon [19] used discrete-event simulation (DES) to evaluate the eﬀect of using an Observation Unit (OU) and 2. Literature Review patient transfer to other inpatient units on overcrowding in a pediatric emergency department (PED). (ey considered Overcrowding in EDs has been recognized as a problem for many years. Various solutions and methods have been four scenarios representing combinations of regular PED operations with and without a ﬁve-bed OU and transfer applied to improve patient ﬂow in EDs. In many operations research studies examining the overcrowding problem in mandate. (ey concluded that a combination of an OU and EDs, the main question was how many resources should be patient transfer mandate improved the waiting time com- allocated to each queue in an ED or to each hospital in a pared to PED with neither an OU nor a transfer mandate. multihospital system, to reduce the patients’ waiting times. Moreover, their results showed that the simulated OU without transfer mandate had an occupancy rate of 73.1%; Some researchers have found that an optimized manpower allocation can reduce the patients’ waiting time in ED by up this rate dropped to 48.1% by applying the transfer policy, indicating a signiﬁcant improvement in the occupancy rate to 20% [3, 20]. Daldoul et al. [5] proposed a stochastic mixed-integer linear programming (MILP) model to opti- of OU. Gul et al. [23] also used DES to analyze the eﬀect of the patient surge in EDs after an earthquake. (ey ﬁrst used mize the number of staﬀ and beds in each queue (six queues for six main activities) in an ED to minimize patients’ Artiﬁcial Neural Networks (ANNs) to estimate earthquake waiting time. El-Rifai et al. [21] also proposed a stochastic causalities and generate inputs for the DES model. (en, the MIP model to ﬁnd the optimal number of personnel for each DES model used the ANN outputs to simulate a network of shift to minimize patients’ waiting time. Izady and Wor- EDs and generate performance outputs for the corre- thington [15] proposed a heuristic algorithm that combined sponding EDs. After constructing the simulation model, a queueing and simulation models to determine the required Design of Experiments (DOE) was conducted to assess the eﬀects of diﬀerent factors on the LoS in the ED and the number of each type of medical staﬀ during each “staﬃng interval” to meet a 4-hour sojourn time target (98% of utilization of ED resources. To show their framework, Gul et al. [23] used a network with ﬁve EDs located in one of the patients must be discharged, transferred, or admitted to inpatient care within 4 hours of arrival), where a “staﬃng regions with the highest estimated injury rate after an Advances in Operations Research 3 earthquake in Istanbul, Turkey. (e results from their study λ y 1,k 1 can be helpful for planning for the expected earthquake in Istanbul. ED 1 Some researchers have combined QT concepts and 1,2 3,1 simulation to analyze patient ﬂow [24, 25]. Alavi-Mog- y 1,3 2,1 haddam et al. [26] showed that by using QT analysis (with 2,3 discrete-event simulation to model and validate patient ﬂow ED 3 ED 2 metrics), one can identify solutions that improve patients’ 3,2 ﬂow and reduce waiting times in EDs. Hu et al. [7] compared 3,k 2,k the use of QT with discrete-event simulation in modeling Figure 1: (ree-ED network. EDs. (ey reported that QT models had lesser data re- quirements and computational cost, due to QT models’ tendency to simplify the problems, while simulation models (v) μ : service rate at ED i captured more details in systems but were more sensitive to (vi) λ : arrival rate, from outside of the system, of ik changes of parameters. (us, they suggested that a combi- patient type k at ED i nation of both was the ideal approach to model such (vii) θ : travel time required to transfer a patient from problems. ii ED i to ED i (viii) π : maximum number of patients allowed to be 3. Modeling transferred from ED i (ix) η : total budget available for servers at ED i Our model attempts to reduce the negative impacts of ED (x) ϕ : expected waiting time in queue for an ED mn overcrowding in a multihospital system by making optimal having ζ servers operating at a utilization rate κ m n allocation decisions in two areas: (1) the number of servers at each hospital’s ED and (2) the rate of nonemergency patient Decision variables transfers between hospitals. To capture the nonlinear (i) v : expected number of patients in a queue queueing eﬀects, we utilize an approach based on that of awaiting service at ED i [27], which allows for an MILP model to represent each ED (ii) r : expected number of patient type k in a queue ik as an M/M/C queueing system. Our research extends the awaiting service at ED i model of [27] in that it allows for each queueing system (ED) (iii) w : expected time in queue per patient treated at to potentially transfer some patients to other EDs, which ED i requires that we utilize a mixed-integer nonlinear pro- (iv) z : expected waiting time in system (time in queue gramming (MINLP) model. Figure 1 presents such a no- plus time in transfer) per patient treated at ED i tional three-ED system, showing the arrivals of patients into (v) p : maximum utilization allowed at ED i the system and transfers of patients between EDs. (vi) s : number of servers at ED i (e sets and indices, data parameters, and decision (vii) y : eﬀective arrival rate of patients into ED i variables used in the MINLP model are as follows: (viii) f : rate at which nonemergency patients are ii Sets and indices transferred from EDi to EDi; note f � 0 by ii assumption (i) I: set of EDs, indexed by i (ii) M: set of values considered for a number of 1 if ED i operates with ζ servers at utilization rate κ m n (ix) x � mni servers, indexed by m 0 otherwise (iii) N: set of values considered for server utilization, Note that this model makes the following assumptions: indexed by n (iv) K: set of patient types, indexed by k, where k � 1 (i) (e patient interarrival times follow an exponential denotes emergency, k � 2 denotes nonemergency distribution (v) T: set of time periods, indexed by t (ii) All patients who enter the system from the outside Data parameters must be treated at some ED (iii) Due to the potential risks of transferring emergency (i) ζ : number of servers associated with the set patients such as heart rate changes, increased in- element m tracranial pressure, and respiratory rate changes (ii) κ : server utilization associated with the set ele- [28], it is assumed that they are admitted imme- ment n diately after arrival to the ED. However, non- (iii) α : cost per unit service capacity at ED i emergency patients can be transferred between EDs (iv) c : cost per patient transferred ED i to ED i ii (v) β : waiting penalty cost, per unit time spent (iv) (e service time per patient follows an expo- nential distribution, and to simplify the model, it waiting (time in queue plus time in transfer), for patients treated at ED i is not diﬀerentiated by patient type. However, it can be diﬀerentiated by patient type for future (vi) δ : queueing penalty cost, per patient type k in a ik queue, at ED i research 4 Advances in Operations Research (v) Each patient departs the system following service. f ≤ λ , ∀i. i2 ii (11) Objective function Min α s + c f + β z + δ r . Constraint (11) allows only nonemergency patients to i i i i ik ik ii ii (1) i i i i i be transferred between EDs. Objective function (1) minimizes the total system cost, f ≤ π , ∀i, ∀i. (12) ii deﬁned as the sum of each ED’s service capacity cost, the cost of transferring patients between EDs, along Constraint (12) permits at most π patients to be with penalty costs associated with the average waiting transferred from ED i to ED i . time per patient at each ED, and the average number of patients waiting in a queue at each ED. α s ≤ η , ∀i. (13) i i i Constraints Constraint (13) limits the total cost for servers at ED i to x � 1, ∀i, mni (2) not exceed η . m n i f θ + w + λ − f w i k ik i i ii ii i ii ζ x � s , ∀i, z � , ∀i. m mni i (3) m n i (14) κ x � p , ∀i. n mni i (4) m n Constraint (14) computes the expected waiting time in the system (time in queue plus time in transfer) per Constraints (2)–(4) assign a unique number of servers patient treated at ED i. and utilization levels to each ED. v , r , w , z , p , s , y , f ≥ 0, ∀i, ∀i,∀k, i ik i i i i i ii w � ϕ x , ∀i. (15) i mn mni (5) x ∈ 0, 1 , ∀m, ∀n, ∀i. m n { } mni Constraint (5) calculates the expected waiting time in queue for patients at ED i, based on the M/M/C 4. Experimental Results queueing system with ζ servers and utilization level of κ . Note that this can be computed a priori for all pairs 4.1. Example Problem. Consider the following example (ζ , κ ) utilizing the standard M/M/C formulae. m n problem, similar in many respects to that presented in [3]. It is assumed that there are three emergency departments in ⎣ ⎦ ⎡ ⎤ ζ κ x μ ≥ y , ∀i. (6) m n mni i i the system, each having identical arrival rates of 5 and 5.5 m n emergency and nonemergency patients per hour, respec- tively. Each unit of service capacity costs $30 per hour. Constraint (6) ensures that the utilization level at ED i Transferring one patient between any pair of EDs costs $10 does not exceed p . and takes 0.25 hours. (e service rate at each ED is 0.5 patients per hour. A penalty cost of $2 per hour is assumed y � λ + f − f , ∀i. i ik ii ii (7) for patient waiting time in the system (time in queue plus i i time in transfer). Penalty costs are also incurred based on the Constraint (7) computes the eﬀective arrival rate into average number of patients waiting in a queue at each ED, at a cost of $5 and $2 per emergency and nonemergency pa- EDi, comprised of both patients arriving into ED i from outside of the system, and the net patients transferred tient, respectively. Table 1 presents the sets of utilization into ED i. values and the number of servers considered for each ED. (e available budget for servers at each ED is assumed to be v � y w , ∀i, (8) i i i $1700, which is greater than the expense incurred if the maximum number of servers (56, from Table 1) was selected. i1 r � ∗ v , ∀i, (9) i1 i 4.2. Computational Results. (e mathematical model pre- y − λ sented in Section 3 was coded in the GAMS 27.2.0 modeling i i1 r � ∗ v , ∀i. (10) i2 i y environment and solved using the MINLP solver SCIP 27.2.0. (e optimal solution has an objective function value Constraints (8)–(10) compute the total number of of $2,195. No patients are transferred between EDs in this patients in the queue and then disaggregate this into the optimal solution. Table 2 presents the optimal values for the number of emergency patients and nonemergency decision variables p , s , w , z , and r at each ED; note that i i i i ik patients in the queue, respectively. these values are identical at each ED in this solution. Advances in Operations Research 5 Table 1: Sets of utilization values and number of servers considered Table 3: DOE design factors and their levels. for each ED. Levels Factors Units ζ κ m n −1 +1 20 0.60 α $/server hour 0 60 24 0.64 c $/patient 0 20 1i 28 0.68 β $/hour 0 4 32 0.72 δ $/patient 0 10 36 0.76 δ $/patient 0 4 40 0.80 μ Patient(s)/hour 0.36 0.65 44 0.84 λ Patient(s)/hour 1 9 48 0.88 λ Patient(s)/hour 1 10 52 0.92 θ Hour 0 0.5 1i 56 0.96 η $/hour 1920 3360 presents a table of signiﬁcant factors for each response Table 2: Optimal variable values for each ED. containing the level of signiﬁcance and the direction of eﬀects, ED i p s w z r r i i i i i1 i2 along with plots of signiﬁcant two-way interactions (see 1 0.88 24 0.307 0.307 1.535 1.689 Supplementary Materials, Figures D.1–D.8). 2 0.88 24 0.307 0.307 1.535 1.689 (e remainder of this section presents a detailed ex- 3 0.88 24 0.307 0.307 1.535 1.689 amination of two responses of particular importance to ED overcrowding: z (expected waiting time in queue plus time in a transfer per patient treated at ED1) and f + f 21 31 (number of patients transferred to ED1). 4.3. Sensitivity Analyses. To determine the eﬀects of the various input parameters on the optimal solutions obtained by our MINLP, a Design of Experiments (DOE) was con- ducted; all statistical analyses were performed utilizing 4.3.1. Sensitivity Analysis for z . Consider the response z , 1 1 Minitab 17. In this DOE, input parameters were varied at the expected waiting time in queue plus time in a transfer per only one emergency department (denoted ED1), and all patient treated at ED1. (e stepwise regression procedure parameters at the other two EDs remained unchanged from described above returned the regression model (in uncoded their previously tested baseline values, with one exception: units) presented in equation (17); this regression model had values η and η were set equal to $1400, such that at the an adjusted R-squared value of 71%. Table 4 presents sta- 2 3 assumed value of α � α � $30/ server hour, up to 44 tistics on this (coded) regression model’s coeﬃcients. 2 3 servers would be feasible at each of ED2 and ED3. In total, According to this analysis, there are seven main eﬀects and ten input parameters were examined in this DOE, with a nine interaction terms signiﬁcant at the p � 0.05 level 10−3 resolution V fractional factorial design (2 ) utilized for (factors c and λ , while not signiﬁcant individually, are V 12 1i screening, using a single replicate for each point and zero included to retain a hierarchical model, since they appear in center points. Table 3 presents the high and low levels tested statistically signiﬁcant interaction terms). (ree of these for each input parameter in this DOE for ED1 (the values for main eﬀects, α , θ , and μ , are signiﬁcant at the p � 0.001 1 1 1i the other two EDs correspond to the center point of the level, indicating that the expected waiting time plus time in values in Table 3). For each of these 128 experiments, the the transfer is impacted considerably by changes to the cost MINLP model was solved using GAMS/SCIP to obtain the per unit service capacity and the travel time between EDs optimal values for all decision variables. Appendices A and B (with time in system increasing as each of these parameters present the designs and responses, respectively, for these 128 increases) and to the service rate (with time in system de- experiments. creasing as this parameter increases). Figure D.5 in the (e following responses were tracked with respect to Supplementary Materials presents interaction plots for the ED1: s , r , r , w , z , p , and the number of patients nine signiﬁcant interaction terms. Observe that three in- 1 11 12 1 1 1 transferred from and to ED1 (f + f and f + f , re- teraction terms are signiﬁcant at the p � 0.001 level, namely, 12 13 21 31 spectively). (e regression model speciﬁcation considered all α ∗ μ , θ ∗ λ , and θ ∗ λ . (e latter two of these in- 1 1 11 12 1i 1i potential ﬁrst and two factor interaction terms. (e regression teraction terms somewhat mediate the eﬀects of the travel model selection was performed using a stepwise procedure, time on the expected time in the system; on average, the with the p value threshold to enter and depart the model set reduced level of the arrival rate of patients from outside of equal to 0.05, with the necessary ﬁrst-order terms retained to the system accelerates the increase of the expected time in produce a hierarchical model. Appendices A, B, and C (see the system when the travel time increases. (is would only Supplementary Materials) present the fractional factorial be reasonable if this increased arrival rate of patients from designs, table of coded coeﬃcients, and signiﬁcant main outside of the system is impacting the likelihood of patient eﬀects and interaction terms for all responses. Appendix D transfers between EDs, which will be examined next. 6 Advances in Operations Research Table 4: Coded regression model coeﬃcients. Term Eﬀect Coef. SE coef. T-value p value VIF Constant — 0.18861 0.00766 24.61 ≤0.001 — c 0.02225 0.01113 0.00766 1.45 0.149 1.00 1i α 0.14602 0.07301 0.00766 9.53 ≤0.001 1.00 β −0.03066 −0.01533 0.00766 −2.00 0.048 1.00 δ −0.04873 −0.02437 0.00766 −3.18 0.002 1.00 δ −0.04056 −0.02028 0.00766 −2.65 0.009 1.00 θ 0.12642 0.06321 0.00766 8.25 ≤0.001 1.00 1i μ −0.09382 −0.04691 0.00766 −6.12 ≤0.001 1.00 λ −0.04142 −0.02071 0.00766 −2.70 0.008 1.00 λ 0.00740 0.00370 0.00766 0.48 0.630 1.00 c ∗ δ 0.03588 0.01794 0.00766 2.34 0.021 1.00 1i α ∗ θ −0.04605 −0.02303 0.00766 −3.01 0.003 1.00 1 1i α ∗ μ −0.05341 −0.02670 0.00766 −3.48 0.001 1.00 1 1 α ∗ λ −0.03084 −0.01542 0.00766 −2.01 0.047 1.00 1 11 α ∗ λ 0.03736 0.01868 0.00766 2.44 0.016 1.00 1 12 β ∗ δ 0.03866 0.01933 0.00766 2.52 0.013 1.00 1 11 θ ∗ λ −0.06884 −0.03442 0.00766 −4.49 ≤0.001 1.00 1i θ ∗ λ −0.09034 −0.04517 0.00766 −5.89 ≤0.001 1.00 1i λ ∗ λ 0.03320 0.01660 0.00766 2.17 0.032 1.00 11 12 z � 0.1379 − 0.00068c + 0.00618α − 0.01733β − 0.00874δ − 0.01911δ + 0.7379θ − 0.1394μ 1 1 1 11 12 1 1i 1i + 0.00221λ + 0.00210λ + 0.000897c ∗ δ − 0.00307α ∗ θ − 0.00614α ∗ μ − 0.000128α ∗ λ 11 12 12 1 1 1 1 11 1i 1i (16) + 0.000138α ∗ λ + 0.001933β ∗ δ − 0.03442θ ∗ λ − 0.04015θ ∗ λ 1 12 1 11 1i 11 1i 12 + 0.000922λ ∗ λ . 11 12 4.3.2. Sensitivity Analysis for f + f . Consider the re- terms. Observe that the interaction terms α ∗ λ and 21 31 1 11 sponse f + f , the number of patients transferred into α ∗ λ all magnify the main eﬀects of these individual 21 31 1 12 ED1. (e stepwise regression procedure described above terms, with even greater decreases in the number of patients returned the regression model presented in equation (17); transferred into ED1 when either pair of these parameters this regression model had an adjusted R-squared value of are jointly increased. In aggregate, an increase in the arrival 78%. Table 5 presents statistics on this (coded) regression rate of emergency or nonemergency patients into ED1 from model’s coeﬃcients. According to this analysis, there are outside the system is associated with a decreased number of four main eﬀects and six interaction terms signiﬁcant at the patients transferred into ED1, which partly explains the p � 0.05 level. Each main eﬀect is signiﬁcant at the p � 0.001 interaction eﬀect discussed in the previous section, in which level, indicating that the number of patients transferred into the reduced level of the arrival rate of patients from outside ED1 is impacted considerably by changes to the cost per unit of the system accelerates the increase of the expected time in service capacity and the arrival rate of both emergency and the system when the travel time increases. Recall that z , the nonemergency patients from outside of the system (with the expected waiting time in queue plus time in a transfer per number of transferred patients decreasing as each of these patient treated at ED1, does not account for the time in the parameters increases) and to the service rate (with the system spent by patients transferred from ED1 to other EDs; number of transferred patients increasing as this parameter the only transfer time that it accounts for is that of patients increases). Figure D.8 in the Supplementary Materials transferred into ED1. presents interaction plots for the six signiﬁcant interaction f + f � 5.367 − 0.0036α − 0.22μ − 0.4009λ − 0.3273λ − 0.0427α ∗ μ 21 31 1 1 11 12 1 1 (17) − 0.001712α ∗ λ − 0.001811α ∗ λ + 0.422μ ∗ λ + 0.315μ ∗ λ + 0.01032λ ∗ λ . 1 11 1 12 1 11 1 12 11 12 4.4. Sensitivity Analyses on ED Size. To assess the eﬀect of ED diﬀerent sizes. (e large ED has arrival rates of 10 and 11 size on the number of transfers occurring in the system, a emergency and nonemergency patients per time unit, re- spectively. (e medium ED has arrival rates of 5 and 5.5 sensitivity analysis was performed examining three EDs of Advances in Operations Research 7 Table 5: Coded regression model coeﬃcients. Term Eﬀect Coef. SE coef. T-value p value VIF Constant — 2.3642 0.0849 27.85 ≤0.001 α −2.6216 −1.3108 0.0849 −15.44 ≤0.001 1.00 μ 0.6784 0.3392 0.0849 4.00 ≤0.001 1.00 λ −1.4609 −0.7305 0.0849 −8.60 ≤0.001 1.00 λ −1.5391 −0.7695 0.0849 −9.06 ≤0.001 1.00 α ∗ μ −0.3716 −0.1858 0.0849 −2.19 0.031 1.00 1 1 α ∗ λ −0.4109 −0.2055 0.0849 −2.42 0.017 1.00 1 11 α ∗ λ −0.4891 −0.2445 0.0849 −2.88 0.005 1.00 1 12 μ ∗ λ 0.4891 0.2445 0.0849 2.88 0.005 1.00 1 11 μ ∗ λ 0.4109 0.2055 0.0849 2.42 0.017 1.00 1 12 λ ∗ λ 0.3716 0.1858 0.0849 2.19 0.031 1.00 11 12 emergency and nonemergency patients per time unit, re- Table 6: Objective values. spectively. (e small ED has arrival rates of 3.75 and 4.125 emergency and nonemergency patients per time unit, re- Solution # Objective value spectively. (e optimization model is modiﬁed slightly here, 1 1020.24 to include only constraints (2)–(7), (11), and (14). (e ob- 2 970.10 jective function is modiﬁed as represented in equation (18), 3 960.00 deleting the ﬁnal two summation penalty terms from ob- 4 940.00 5 921.87 jective (1). Rather than associating a ﬁnancial penalty with 6 920.00 delay times, we introduce a new constraint (19) which 7 910.00 imposes an upper bound, denoted by σ, on the systemwide 8 900.00 average expected waiting time in queue plus time in transfer, 9 890.68 which can be computed as y z / λ . We varied this i i i i k ik 10 881.53 upper bound σ across a range of values, from a minimum 11 880.00 value of 0.0284 to a maximum value of 1.3913 (the sys- 12 873.87 temwide average for the minimum cost solution if constraint 13 871.15 (19) is not considered). In total, 26 diﬀerent solutions were 14 870.00 identiﬁed, constituting an eﬃcient frontier for the tradeoﬀ 15 863.91 between objective function (18) and the left-hand side of 16 860.05 17 854.91 constraint (19). All parameters were assumed to take the 18 852.97 baseline values from Section 4.1 with two exceptions: we 19 851.16 assume that the cost per unit service capacity at each ED is 20 845.39 equal to 10 times the cost per patient transferred between 21 842.60 EDs, say, $10 and $1, respectively. (e potential numbers of 22 840.05 servers considered at each ED were also modiﬁed from the 23 837.79 values presented in Table 1; for this sensitivity analysis, ζ m 24 833.03 was varied to include all integer values between 2 and 60. 25 830.05 Table 6 presents the sensitivity analysis’ objective values. As 26 830.00 it can be seen, the objective value decreases as the upper bound value increases. In fact, it implies that as the average expected waiting time in queue plus time in transfer in the (ese results demonstrate how the optimization model system becomes more ﬂexible, a fewer number of servers and utilizes a variety of strategies to achieve a constrained sys- fewer patient transfers are required in EDs. (erefore, the temwide average expected waiting time at minimum cost. associated costs (equation (18)) decrease. (e following ﬁgures present the sensitivity analysis’ expected waiting time Consider, for example, solutions 22 and 23. (ey achieve relatively similar performance, with respective objective in queue plus time in a transfer per patient treated (Figure 2), function values of 840.06 and 837.79 and respective sys- number of servers (Figure 3), ED utilization (Figure 4), and temwide average expected waiting times of 0.7988 and percent of nonemergency patients transferred (Figure 5). 0.8733. (e utilization at each ED is essentially unchanged Min α s + c f , i i ii ii (18) across solutions 22 and 23, with 96%, 92%, and 92% utili- i i i zation, respectively, at the large, medium, and small ED in each solution. However, the underlying structure has f θ + w + λ − f w y z i i k ik i i i i i ii ii i ii changed signiﬁcantly, with solution 22 utilizing 44, 23, and � ≤ σ. λ λ 17 servers, respectively, at the large, medium, and small EDs, i k ik i k ik and very little patient transfer (1% of the nonemergency (19) patients transferred from the small ED to each of the 8 Advances in Operations Research 2.5 2.0 1.5 1.0 0.5 0.0 123456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 solution # small large medium systemwide average Figure 2: Expected waiting time in queue plus time in a transfer per patient treated. 123456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 solution # small large medium sum Figure 3: Number of servers. medium and large EDs). By contrast, solution 23 utilizes 60, Across all 26 solutions identiﬁed, the optimization model 14, and 9 servers, respectively, at the large, medium, and utilized patient transfer extensively for nonemergency patients small EDs (one fewer server, in total, than does solution 22), arriving at the small ED; on average, 25.3% and 4.4% of such but extensive patient transfer (91% and 74% of the non- patients were transferred to the large and medium EDs, re- emergency patients from the small and medium EDs, re- spectively. (e patient transfer was utilized less frequently for spectively, are transferred to the large ED). nonemergency patients arriving at the medium ED; on average, z_i s_i Advances in Operations Research 9 100% 95% 90% 85% 80% 75% 70% 1 23456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 solution # small medium large Figure 4: ED utilization. 1 2 3 4 5 6 7 8 9 101112131415 16 17 18 19 20 21 22 23 24 25 26 solution # f_small,large f_medium,small f_small,medium f_large,medium f_medium,large f_large,small Figure 5: Percent of nonemergency patients transferred. % non-emergency patients transferred rho_i 10 Advances in Operations Research medium, and small). (e MINLP was modiﬁed slightly here; 6.5% and 0.1% of such patients were transferred to the large and small EDs, respectively. (ere were no instances across all 26 rather than including a ﬁnancial penalty for delay times in the objective, we introduce a new constraint imposing an solutions in which nonemergency patients were transferred from the large ED to another ED. upper bound on the systemwide average expected waiting time in queue plus time in the transfer. Computational testing varied this upper bound across a range of values, 5. Conclusions and Future Work identifying an eﬃcient frontier for the tradeoﬀ between the Overcrowding in hospital emergency departments (EDs) is a modiﬁed objective function and the systemwide average problem that aﬀected many hospitals especially during the expected waiting time. (is optimization model utilizes a response to emergency situations such as pandemics or variety of strategies to achieve a constrained systemwide disasters. In this study, we propose a novel optimization average expected waiting time at minimum cost, balancing changes to the numbers of servers at each ED with patient model to address overcrowding in a network of EDs via a combination of two decisions: modifying service capacity to transfers across EDs. Across all points identiﬁed on the eﬃcient frontier, the MINLP utilizes patient transfer ex- EDs and transferring patients between EDs. (is model is similar to that presented in [3]; however, whereas the au- tensively for nonemergency patients arriving at the small thors in [3] did not account for queueing eﬀects, our model ED, somewhat infrequently for arrivals to the medium ED, includes queueing considerations in a MINLP, capitalizing and in no instances for arrivals to the large ED. Taken to- on the closed-nature form of M/M/C queueing eﬀects, gether, these results suggest that our optimization model can similar to the approach utilized in [27]. identify a range of eﬃcient alternatives for healthcare sys- Computational testing was performed, using a Design of tems designing a network of EDs across multiple hospitals. Moreover, the model can be helpful to have more balanced Experiments to determine the eﬀects of changes to the various input parameters for a single ED (denoted ED1) on EDs with respect to the number of patients and patient waiting time in a network of EDs in case of emergency the optimal solutions obtained by our MINLP. Regarding the expected waiting time in queue plus time in a transfer per situations such as natural disasters. Future work could extend this analysis by considering patient treated, the most signiﬁcant main eﬀects indicated that this response is impacted considerably by changes to the queueing systems other than M/M/C to represent the sto- cost per unit service capacity and the travel time between chastic nature of patient arrivals and service times at EDs. EDs (with time in the system increasing as each of these Further, while this analysis models steady-state perfor- parameters increases) and to the service rate (with time in mance, which is useful for network design, an extension to system decreasing as this parameter increases), with inter- transient system performance in nonsteady-state would action terms somewhat mediating the eﬀects of the travel allow for similar models to be used in a real-time dispatching environment. Finally, a more nuanced diﬀerentiation be- time on the expected time in system; on average, the reduced level of the arrival rate of patients from outside of the system tween patient types, which are modeled as being either emergency or nonemergency patients in this research, could accelerates the increase of the expected time in the system when the travel time increases. (is would only be rea- allow for such an MINLP approach to be used to allocate special types of ED service (e.g., pandemic virus testing). sonable if this increased arrival rate of patients from outside of the system is impacting the likelihood of patient transfers between EDs. Examining this further, we ﬁnd that for the Data Availability number of patients transferred into ED1, the most signiﬁcant main eﬀects indicated that this response is aﬀected signif- Supplementary materials, including data, will be posted at icantly by changes to the cost per unit service capacity and the University of Missouri’s data repository https://mospace. the arrival rate of both emergency and nonemergency pa- umsystem.edu/xmlui/. tients from outside of the system (with the number of transferred patients decreasing as each of these parameters Conflicts of Interest increases) and to the service rate (with the number of transferred patients increasing as this parameter increases), (e authors declare that they have no conﬂicts of interest. with interaction terms between the cost and each arrival rate magnifying the main eﬀects of each these individual terms. Supplementary Materials In aggregate, an increase in the arrival rate of emergency or nonemergency patients into ED1 from outside the system is Appendix A: a table of fractional factorial designs. Appendix associated with a decreased number of patients transferred B: a table of responses for fractional factorial designs. Ap- into ED1, which partly explains the aforementioned inter- pendix C: detailed statistical model outputs for responses. action eﬀect, in which expected time in the system is found Appendix D: detailed statistical model outputs for responses. to increase with increases in the travel time between EDs (Supplementary Materials) only when the arrival rate of patients from outside of the system into ED1 is at its reduced level. References Additional computational testing examined the eﬀect of ED size on the number of transfers occurring in the system, [1] R. W. Derlet and J. R. 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Advances in Operations Research – Hindawi Publishing Corporation
Published: Aug 23, 2021
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