A simulation optimization approach for resource allocation in an emergency department

The emergency department (ED) is a primary health care unit and one of the main entrances to the hospital system where appropriate, timely and good performance can save lives. Lack of sufficient resources, such as beds and qualified health care professionals, are major stumbling blocks to providing timely and suitable services; but resources availability and moving towards the ideal situation without attention to budget restrictions is neither practical nor achievable. In this study, simulation optimization is used to finding the best configuration in ED resources (e.g., Bed, Nurse, and GP) that affects a patient’s length of stay, subject to budget constraints. Simulation is used to analyze the system and estimate target function an optimization model is then solved under different budget constraints. By considering the current budget, the new configuration of 20 inpatient beds, 3 nurses and 1 GP, with 554.4 minutes of a patient’s length of stay shows 8.1% length of stay (LOS) improvement. Whilst with a maximum 35.5 budget units allocation of 20 inpatient beds, 4 nurses and 3 GPs a 9.5% decrease in LOS is proposed.


INTRODUCTION
A healthcare system is one of the most important services with rapid growth demands in both developed and less developed countries.The main focus of a health care system is on the patients who have different expectations, in different times and situations [Faezipour and Ferreira, 2013].Indeed, in recent years changing patterns of disease, progress in medical knowledge and technology, public awareness of modern medical facilities, along with an increasing elderly population have resulted in greater demand for medical services, as well as more pressure on the health care budget.In this way, attempts to prevent the rising costs of healthcare by better distribution of scarce resources and by applying cost-containment strategies are major challenges for healthcare authorities [Ferreira et al., 2011].
The emergency department (ED) as a primary health care unit and one of the main entrances to the hospital system, therefore necessitates considerable attention.Proper, correct and timely function in an ED can save many lives and decrease various adverse effects.This requires an understanding and the scrutiny of the current situation in this sector.In many cases, lack of resources such as beds, qualified health care professionals and nurses in care prevent timely and suitable services.This can also lessen health care quality, increase potential medical errors, add to long waiting times and increase a patient's length of stay (LOS), which is followed by deleterious effects on the patient, as well as unintended higher costs for service providers.Hence, an equilibrium between supply and demand can provide obvious improvement in service quality and bring higher patient satisfaction in the emergency department [Xu et al., 2013].
Optimum resource allocation in the healthcare sector and improving the quality of communal health is important and has always been considered as a controversial issue [Asante and Zwi, 2009].Because of scarce and costly resources, as well as sensitive, yet challenging conditions, a large body of literature has noted the allocation of health care resources.Laker et al, [2014], by using discrete-event simulation, demonstrated that adding some flexibility into bed allocation between low-and high-acuity can provide substantial reductions in overall patient waiting and efficiency in ED.Holm et al, [2013] analyzed bed allocation in hospital wards to minimize overcrowding by developing a novel, generic discrete event simulation (DES) model.The number of beds in a hospital and the impact on hospital operations, length of stay, as well as waiting lists were studied by Reyes- Santı ´as et al, [2013].Schmidt et al, [2013] focused on improving bed assignment by considering adaptable length of stay estimation, which results in a 30% reduction of the dismissal rate.Bekker and Koeleman [2011]addressed the impact of resource variability in determining capacity requirements and admission quota for planned admissions to regulate the pattern of bed occupancy.Griffiths et al, [2010], determined the number of beds that were needed in the critical care unit (CCU) with discrete event simulation.
Optimal allocation of human resources is one appropriate strategy to meet demand and provide suitable service with high quality, to increase patient and personal satisfaction.Lim et al, [2013], modelled physicians and their delegates in the ED as interacting pseudo-agents in a discrete event simulation and compare it with the traditional approaches.They found with this approach physician utilization increase from 23% to 41% and delegate utilization increase from 56% to 71%.Izady and Worthington [2012] demonstrated how queuing models with simulation can be used to reduce overcrowding in ED by modifying the staffing profiles.Brenner et al, [2010] identified bottlenecks and evaluation of human resources and equipment in ED using simulation.
In the ED, with several dynamic variables and random features, such as unscheduled admissions, irregular patient arrival times as well as multiple types of patients who require various care processes, resource allocation seems a more important issue than in other wards.Shimada et al, [2012], determined the resource utilization of a tertiary care, Japanese emergency department before and after the 2011 Great East Japan earthquake and tsunami to help planning the allocation of national resources for disasters.Ahmad et al, [2012] in their paper presented a computer simulation model to assess the use of resource utilization in the ED of a public hospital in Malaysia.By using this model administrators could monitor the patient flow in ED, determine possible areas for improvement and ensure the best resource allocation.Weng et al, [2011], stated that the purpose of their research was to find an optimal allocation of resources in the ED through simulation.They used Simul8 and the results showed overall performance in an ED can increase by 8% through allocating new human resources.Cochran and Broyles et al, [2010], proposed making strategic decisions for future ED capacity, based on patient safety (rather than congestion measures) in their study.Ng et al, [2010], compared priorities and the differences between resource utilization in four-levels of Taiwan triage and five-levels of Canadian triage and acuity scale among patients.The results showed that hospitalization rates, length of stay and medical resource consumption are different between the two systems, and Canadian triage is more efficient in predicting patient acuity and resource utilization.Holm and Dahl [2010] focused on the expected increase in patient volume and its significant impact on patient flow in the ED, and asked hospital managers the following question: "What is the lowest number of additional resources that would be needed in the ED as a result of the increase in patient volume, that would not compromise the patient flow?"The results of this study show that increasing the number of nurses from eight to nine and physicians from eight to twelve would be sufficient to meet the requirements of the hospital's ED.Using simulation optimization, Ahmed and Alkhamis, [2009], presented a decision support tool for performance evaluation an ED of a public hospital in Kuwait.Their main purpose was to determine the optimum number of doctors, nurses and laboratory technicians needed to maximize patient throughout and to reduce patient waiting time, considering budget constraints.Finding a correlation between Canadian pediatric emergency triage and acuity scale to ED resource utilization and how it could be the basis for allocating ED resources, was investigated by Ma et al, [2008].
This paper describes a case study undertaken at a public and training hospital in the city of Bushehr, Iran.Since the ED performance is assessed by the average patient's length of stay (6 hours), managers intended to configure a suitable combination of available resources in this ward in order to respond to demands properly.In this study, the main problem is the patient's long waiting time and length of stay in the emergency department, due to several issues such as lack of adequate resources.However, providing resources and moving towards the ideal situation without attention to budget restrictions is neither practical nor achievable.Hence, the proposed approach compares different possible configurations of resources by considering their cost in the optimization model to minimize the patient's length of stay.The paper is organized as follows: Section 2 is devoted to research methodology; results and discussion are presented in Section 3 and Section 4, respectively; and finally, the paper is concluded in Section 5.

RESEARCH METHODOLOGY 2.1. System description
Persian Gulf hospital is a teaching hospital of Bushehr Medical Sciences University, with 280 beds.The hospital emergency department is open 24 hours a day, in morning, afternoon and night shifts.Medical services per shift are provided by 2 GPs, 1 specialist and 8 nurses (3 nurses in the hospital and 5 nurses for other services) and on call specialists.
The most crowded hours are between 14:00 and 23:00.According to the ED manager, the patient flow process in an emergency department is depicted in Figure 1.Patients arrive by ambulance or on foot.Then, based on patient condition and Emergency Severity Index (ESI), patients are classified into five levels in triage: . Patients in level one are in critical condition and should be transferred to CPR, immediately. .Level two are called high-risk patients and have severe pain and they may require CPR in some cases. .Level three are patients that require two or more health care services, such as a blood/urine tests, ECG, or serum. .Level four are patients that require one health care service.
. Patient in level five receive a prescription or consultation and leave the department after that.
Patients may tolerate different waiting times depends on their ESI and the availability of different resources, such as medial staff and empty sickbeds to receive related services.However, patients with ESI 1 and 2, because of their critical condition, must receive immediate lifesaving interventions.In fact, every minute of delay in their treatment process could reduce their chance of survival.
All patients, after receiving the order and prescription from an ED clinician follow their treatments in different areas (e.g.Laboratory, radiology, etc.) and in the last step may be discharged or admitted to the hospital.

Key performance indicators
There are many different performance measures within healthcare.Depending on the purpose of study, one can select key performance indicators (KPIs) to evaluate the ED performance.In this research, after visiting and interviewing the emergency department manager, three main key performances have been identified: . Resource utilization (%), that is total busy time of resources compared with total working time.
. Average patient waiting time, defined as the mean duration that a patient spends in the ED waiting room or in a queue to receive treatment.It is desirable to minimize this measurement. .Patient's total average length of stay in ED, the total time a patient spends in ED.It is the period of time from patient arrival until his/her exit from ED.It is desirable to minimize this measurement.

Data collection
To make a simulation model, data was collected directly from the ED by using a triage database, as well as tracing patients in different time intervals.In some cases, when direct tracing and random sampling was not practically possible, department staff were interviewed to collect accurate data.In this case, a combination of open interview and multiple choice questionnaires have been used.All data has been cross-checked to minimize data collection errors.

Simulation and optimization
Today, new methods, for example computer simulation provide unlimited opportunities for managers to achieve the highest quality in different systems, such as healthcare.It is an effective tool to support decisions on staffing requirements, planning, resource allocation and evaluation of processes in hospitals.Healthcare managers can apply simulation for assessing current performance, predicting the impact of operational changes and examining the tradeoffs between system variables safely at lower cost [Ferreira et al, 2011].Furthermore, discrete event simulation is an especially well-suited tool to tackle problems in emergency departments, where resources are scarce and patients arrive at irregular times [Abo-Hamad and Arisha, 2013].However, discrete-event simulation models have some drawbacks, such as being heavily reliant on adequate data, time consuming and an unfair representation of model outputs.
On the other hand, an optimization technique in healthcare systems may also require too many unrealistic assumptions about the process, because it cannot be used to study the details of a complex system like a medical clinic; therefore the solution may be invalid and unrealistic.In this way, to implement the advantages of both techniques and reduce the disadvantages, operations researchers have attempted to combine simulation with other operations research techniques.Simulation optimization is the practice of combining a simulation model with an optimization algorithm to obtain the maximum performance of the simulated model, incorporating the stochastic behavior of the system [Ahmed and Alkhamis, 2009].Simulation optimization attempts to determine the best values of input parameters, given an output criterion.
In this study a discrete event simulation model is implemented in Arena Simulation Software.The simulation model consists of complex care processes, including triage, diagnosis of disease by general practitioners, and patient flow within the department.In order to build a conceptual model, the core business processes of the ED were mapped into a flowchart.Based on the conceptual model, building blocks of the model were integrated into the Arena.The model is calibrated to represent the emergency conditions of patients and the medical decisions that have been made by doctors.
In order to evaluate the performance of the ED under different operational circumstances, a number of scenarios were developed.The ED operates 24 hours everyday, therefore, the replication time horizon is set to 365 days a year.In addition, to achieve credible output, 10 replications were run.Key performance indicators, including length of stay, average waiting time (AWT), and resource utilization in the system.
Consequently, target function aimed at minimization inpatient LOS has been estimated in a Statistical Package for the Social Sciences (SPSS) by using simulation results.The presented model was solved using Lingo software in order to assess patient's length of stay, based on the number of ED resources level, considering their corresponding cost.

Goodness of fit
In modeling discrete event systems, it is usually assumed that input arrival data follows a Poisson distribution and the distribution of interarrival times is an exponential distribution [Banks et al, 2004].
By analyzing patients arrival pattern, we have found that patients interarrival time follows an exponential distribution, with a mean of 8.1 minutes.To check whether the exponential distribution correctly captures the arrival pattern of incoming patients, a Kolmogorov-Smirnov statistical test, with a ¼ 0.05, was used.Since p-value is larger than 0.05, statistically no significant difference was detected between real data and the theoretical exponential distribution, with l ¼ 8.1.Therefore, the null hypothesis of fitting interarrival data with exponential distribution function is accepted.More details can be found in Table 1.
For other specific procedures in ED care process, the identified probability distribution functions are summarized in Table 2, in which all distributions at each stage were analyzed by using Kolmogorov-Smirnov goodness of fit test, with a 0.05 significance level.

Simulation model validation
Three methods such as face validation test, comparison testing, and hypothesis testing can be used for evaluating the result of simulation model [Abo-Hamad and Arisha, 2013].
Face validity of the model was analyzed by interviewing the ED manager and nurses who are knowledgeable about the system.In fact, they became involved in model construction from conceptualization to implementation, to ensure that processes, patient flow and collected data are reliable.After making some revisions they acknowledged that the constructed model appears reasonable.
In the next step, by using a comparison test, the final results of the simulation model and real data were compared, no significant difference was observed.Differences between final results of the simulation model and real data were also assessed by Student's t-tests, with a ¼ 0.05 in SPSS software.Since, Sig is more than 0.05 the null hypothesis (H 0 : m 1 ¼ m 2 , H 1 : m 1 -m 2 ) is accepted and the simulation model was considered to be valid (Table 3).
However, within the chaotic, busy ED environment and with gathering data at specific time intervals (e.g.lack of data in night shifts) it is reasonable to expect some data will always remain unavailable or too costly to acquire, and therefore result in some differences between simulation results and the real system.

Optimization model description
Discrete event simulation can provide estimates of performance measures for various alternatives, but is not an optimization tool [Jacobson et al, 2006].Moreover, optimization in a stochastic system, in which a function object or constraints have no analytical form, can only be evaluated through simulation.Hence, a combination of simulation with optimization is proposed for implementing the advantages of both techniques in many cases [Ahmed and Alkhamis, 2009].
The optimization problem considered in this paper aims to minimize patient's total length of stay subject to a budget constraint.When a good and reliable estimate of target function is obtained, constraints will be added to define an optimization model: Cðx 1 ; x 2 ; . . .; Where (1) is the target function in order to minimize patient's total length of stay in ED.Constraint (2) represents the cost of operating configuration ðx 1 ; x 2 ; . . .; x n Þ, B is the budget available to operate the service and l i and u i are the lower and upper bounds of the number of servers in service i, respectively.We assume x i as decision variables of the problem that represent the number of main resources used in ED.

Estimation of the objective function
After verification and validation, simulation model was run under different, feasible configurations of the system.Scenarios to obtain a reliable 'set' from which relevant analytical relations among parameters of system (e.g.ED inpatient beds, nurses and GPs) and measures of performance (Total LOS) could be estimated.In fact, for every configuration x ¼ ðx 1 ; x 2 ; . . .; x n Þ a value of length of stay (y) is obtained and functional relation y ¼ f(x) can be estimated from this set.For obtaining a reliable set, only feasible configurations are accepted (it is not possible to have more than 20 ED inpatient beds, 5 nurses and 3 GPs in ED, according to interviews with hospital managers) and non-realistic configurations are omitted (Table 4).Functional relation y ¼ f(x) is estimated between these three selected resources in the model (e.g.ED inpatient beds, nurses and GPs) and patient's total length of stay in ED using SPSS software.In fact, in this model y represents LOS criterion, which has a regression linear function to estimate the relation between the LOS and the configuration ðx 1 ; x 2 ; . . .; x n Þ.According to Table 5, as R is close to one, it is acceptable the relation between dependent and independent variables is linear.Furthermore, this value of R squared, that is equal to 0.92, indicates a good fit of the data in the model.
Before expanding the target function the null hypothesis, H 0 : a 1 ¼ 0, H 1 : a 1 -0, i ¼ 0, 1, 2, 3 were examined with SPSS, because of the Sig value, which is more than 0.05, H 1 is acceptable.Based on regression analysis in SPSS software, initial values of model parameters a 0 ; a 1 ; . . .; a n were gained (Table 6).In this model, a 0 is a constant parameter that includes other values, except x 1 , x 2 , . . ., x n that will affect LOS.
The following, 0:5208 x 1 þ 2:5 x 2 þ 5 x 3 # 24:7 represents the cost constraint of operating the current configuration, with 14 inpatient beds, 2 GPs and 3 nurses, obtained from hospital financial department.In addition, thanks to administrative policy and budget limitation, the number of resources from the current situation should not exceed 20 inpatient beds, 3 GPs and 5 nurses.Therefore, we will have 14 # x 1 # 20; 3 # x 2 # 5; 1 # x 3 # 3 and x 1 , x 2, x 3

Optimization problem results
In Table 7, the results of optimization problems with respect to the optimal value of inpatient LOS and the related costs are presented.For example, under 21 budget units an optimal configuration will be 16 inpatient beds, 3 nurses and 1 GP, with 576 minutes as LOS.In addition, in some cases increasing the budget has no effect in configuration or length of stay, such as 28.5, 29, 29.5 and 30 budget units.

Comparing simulation and optimization results
For further validation, simulation and optimization results in 14 different configurations were compared (Table 8).The results from optimization are reasonably close to the simulation outputs, this indicates the estimated length of stay in the model is acceptable and validated.The best allocation corresponding to closest values for the length of stay given by the two methods is 19 inpatient beds, 3 nurses and 3 GPs in 32.5 budget units, with 0.5 minutes in LOS.

DISCUSSION
This study is focused on improving healthcare efficiency using operations research tools.In order to improve resource allocation in ED, and thereby improve hospital operation efficiency, a simulation optimization framework for distributed resource allocation in ED with considering their corresponding cost is proposed.Initial results from simulation show average lengths of stay (LOS) for inpatients and outpatients are 603.6 and 78.53 minutes, respectively.Furthermore, outpatients' average waiting time to visit a doctor is 4.91 minutes and average waiting time for inpatient to be hospitalized in the ED is 59.6 minutes.In addition, the main resource utilization in the ED is related to inpatient beds, GPs and nurses, these are the primary bottlenecks in patient flow.They play key roles in the average patient's waiting time and patient's average total length of stay in the ED (Table 9).
In the second stage, an optimization model has been defined by considering a number of experiments in the simulation model.As optimization results in the Table 7 indicate, in some cases increasing the budget has no effect in configurations.With a maximum budget of 35.5 units, allocating 20 inpatient beds, 4 nurses and 3 GPs, results in a minimum length of stay of 545.9 minutes, with 9.5% improvement.However, budgeting over 35.5 units has no effect on the length of stay reduction and therefore was not considered.Furthermore, in the current resource level of 14 inpatient beds, 2 GPs and 3 nurses in the simulation model, an average length of stay is 603.6 minutes, which costs 24.7 units money measurement.Interestingly, at the same cost, 24.5 units in the optimization model, the new configuration will improve the length of stay (by 8.1%) through focusing on the significant role of inpatient beds.At this level the best value of length of stay was obtained at 554.4 minutes, this is the optimum stay amongst the other budgets.
Additional validation of the optimal configurations has been done in the last stage.Figure 2 shows a comparison between simulation and optimization outputs.In configuration 19 inpatient beds, 3 nurses and 2 GPs and 19 inpatient beds, 3 nurses and 3 GPs, simulation and optimization outputs are very close, the agreement between these two figures is substantial, which confirms validation.This study has many limitations, mainly because of the unwillingness and lack of cooperation from healthcare authorities in using operation research techniques in this sector.We faced other limitations in gathering accurate data.For instance, tracing patients in different times without computerized and smart devices is not precise.In addition, lack of access to relevant information in emergency departments with other hospital wards caused restrictions in parts of the simulation model.Furthermore, we present a simple model and use a linear function, while implementing a non-linear function leads to more realistic results.

CONCLUSION
Globally, it is the most challenging time for the healthcare industry.Growing demand for services, ageing population, rising costs, disease burden, poor access and inequitable care have created a crisis and put heavy pressure on the limited resources in healthcare [Brailsford and Vissers, 2011].In order to cope with this situation, healthcare managers and decision makers require innovative methods to allocate scare resources in an efficient way to be able to provide appropriate, as well as accessible services with high quality.
In this study simulation optimization was used to determine the best configuration in ED resources (inpatient beds, nurses and GPs) that affect a patient's length of stay subject to budget constraints.It will help to bridge the gap in the literature, which have focused on budget constraints in resource allocation issues.We also use optimization for revalidation of the model.In this case, inpatient beds are one of the main bottlenecks limiting throughput in ED.Budgeting to add this resource results in a significant reduction in length of stay, in comparison to adding nurses and GPs.The results show, with a current budget of 20 inpatient beds, 3 nurses in hospital and 1 GP, the best length of stay is estimated at 554.4 minutes, while with maximum budget, 35.5, 20 inpatient beds, 4 nurses and 3 GPs, the length of stay is 545.9 minutes.A manager can evaluate the length of stay with budget to decide whether there should be an increase in budget for special level of performance, or whether the patient should spend more time in the system and accept this level of service because of limitations in budget.
For further research it seems static resources and other factors might limit patient throughput (e.g.boarding of admitted patients, patient transport times, laboratory/radiology turnaround time, lag times awaiting patient transport to radiology or wards, etc.) provides more realistic results about system improvements that could not be applied because of our limitations in this hospital.

Table 1 .
K-S test for inter-arrival time.

Table 2 .
Service time distributions at each stage of the process.

Table 3 .
Model validation by using T-test.

Table 4 .
LOS under different possible configurations.
* The current situation.
a Dependent Variable: LOS input.

Table 8 .
Comparing simulation and optimization results.

Table 7 .
Optimization output with different budget.

Table 9 .
Resource utilization in current system.
Figure 2. Comparison chart of simulation and optimization outputs for LOS.