Simulation Modelling Practice and Theory 61 (2016) 14–27

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Simulation Modelling Practice and Theory

journal homepage: www.elsevier .com/locate /s impat

Patient referral mechanisms by using simulation optimization

http://dx.doi.org/10.1016/j.simpat.2015.11.0041569-190X/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +886 3265 4428; fax: +886 3265 4499.E-mail address: kanghungyang@cycu.edu.tw (K.-H. Yang).

Ping-Shun Chen a, Kang-Hung Yang a,⇑, Rex Aurelius C. Robielos b, Rozel Aireen C. Cancino b,Lea Angelica M. Dizon b

aDepartment of Industrial and Systems Engineering, Chung Yuan Christian University, Chung Li District, Taoyuan City 320, Taiwan, ROCb School of Industrial Engineering and Engineering Management, Mapúa Institute of Technology, Intramuros, Manila 1002, Philippines

a r t i c l e i n f o

Article history:Received 17 August 2015Received in revised form 13 November 2015Accepted 24 November 2015

Keywords:Magnetic resonance imagingCollaborationPatient-referring mechanismSimulation optimization

a b s t r a c t

Building patient referral mechanisms between two hospitals is the focus of this research,which considers different kinds of magnetic resonance imaging (MRI) services offered bythe case hospitals. The simulation optimization approach is the main tool for analysis,along with the formulation of a mathematical model and a simulation framework to con-ceptualize a collaborative MRI patient-referring mechanism. The objective of the study is toobtain the best feasible number of referral outpatients to minimize patients’ averagewaiting time or to maximize the revenues of both case hospitals. The results can serveas guidelines for hospital collaboration.

� 2015 Elsevier B.V. All rights reserved.

1. Introduction

According to [1], collaborations are essential among researchers and practitioners, policymakers, business leaders, andcommunity advocates. It can also be perceived as significant in the medical field, whose professionals are concerned withthe health of patients who require reliable and fast service [2]. Hospital collaboration has been studied for decades andhas become a critical topic for hospital management. Collaborative activities among allied hospitals achieve objectives thatimprove healthcare quality and enhance patient satisfaction [3–5]. For example, sharing medical resources (physicians, beds,or machines) or reengineering the hospital processes might reduce patients’ cycle time and waiting time [6].

The National Health Insurance Administration (NHIA) is concerned with the health insurance of Taiwan’s citizens. Themajor factors affecting the organization’s annual losses and earnings come from the premiums, insurance payments, baddebts, and other incomes, such as those coming from the public welfare. In 1995, NHIA implemented the National HealthInsurance System (NHIS), whereby payments are transferred to the global budget. The global budget system controls thetotal NHIS payments in order to remain on budget. Each hospital needs to negotiate annually with NHIA for the annual totalbudget and insurance payment for each kind of treatment. Therefore, hospitals can calculate their annual allowable quota. Ifhospitals treat more patients than their annual allowable quota, the hospital still obtains the fixed annual total budget fromNHIA for its insurance payment. Yet the global budget system created potential inefficiencies for hospital operations andalleviated hospitals’ willingness (such as unwillingness to purchase new medical equipment or to increase the staff’s workhours) to promote quick services for patients. Consequently, patients in overcrowded hospitals wait longer time for theirtreatment. However, some hospitals with fewer patients have abundant medical resources, leading to the medical resourcesstanding idle or being unused and to inefficiencies for hospital operations. From the viewpoint of hospital management, a

P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27 15

hospital must promote its service to survive and utilize its resources in an efficient way. In order to increase competitiveadvantages under the global budget system, collaboration can be used to give hospitals competitive capacities and servicequality.

The annual budget of the NHIS has been in a deficit since 1995. In order to make up the deficit, NHIA increased the insur-ance premiums. The original insurance premiums of the NHIS remained at 4.25% for the insured people from 1995/03 to2002/08. NHIA adjusted the insurance premiums rate to 4.55% in 2002/09, to 5.17% in 2010/04, and to 4.91% in 2013/01.Finally, the accumulated financial report of the NHIS was changed from budget deficits to budget surpluses in 2012/02.Although the global budget increased about 2% to 6% annually between 2009 and 2015, hospitals are still under strong finan-cial pressure to obtain more insurance payments from the NHIS. Hospital managers therefore needed to improve their oper-ational effectiveness and efficiency, such as collaboration with adjacent hospitals, to prevent hospitals’ resources from notbeing used.

Taiwan has seen increasing demand for healthcare treatment [6]. However, a systematic and efficient collaborationframework of hospitals is now being established. The referral system has been successful in hospital collaborations inTaiwan, but needs improvement. The primary objective of the study is to improve Taiwan’s healthcare system using the sim-ulation optimization method for the mutual benefit of patients and hospitals.

In general, patients are quite concerned with access to and the reputation of their hospital. With this kind of setup,patients will visit only the best-known hospitals in their vicinity. This mentality creates long waits at prominent hospitals.Conversely, short waiting lines are found in suburban hospitals. Hence, promoting collaboration among hospitals throughreferrals will reduce the long queues and waiting times for patients. At the same time, machines located at each hospital willnot stand idle. This study determines the most feasible numbers of fixed and unfixed referrals of magnetic resonance imag-ing (MRI) patients per day between the two case hospitals. The objectives are to minimize eligible patients’ average waitingtime or maximize hospitals’ revenues.

The remainder of this article is organized as follows. Section 2 reviews the literature on simulation and simulation opti-mization in healthcare applications. Section 3 introduces the background of the study hospital. Section 4 describes thedetails of the simulation model and related parameters. In Section 5, two optimizationmathematical models are constructed.The simulation optimization procedures are then developed to solve the proposed model. Section 6 describes two scenariosto compare the fixed and unfixed patient referral mechanisms. Two sensitivity analyses are also performed, and the resultsand discussions of the patient referral mechanisms are summarized. Finally, Section 7 concludes the paper and suggestsfuture directions for research.

2. Literature review

Kobayashi et al. [7] built an electronic geographical information system (GIS) in order to improve the ability of clinics andhospitals to collaborate. They concluded that some information from paper maps used as a guide to search for physiciansmight be outdated. Consequently, the researchers used GIS with the use of Google Map API to refer patients to other clinicsin a fast and informative manner.

With the same goal, Ramis et al. [8] explored Medical Imaging Centers (MIC). The researchers used a simulator imple-mented in Flexsim GP to shorten waiting times. With the aid of four hospitals in Chile, the researchers identified all possibleflows, resources, schedules, and exams. After comparing seven setups, they were able to decrease patients’ total waiting timeby 35% without changing personnel, but allocating common functions. Consequently, the productivity of a center can beincreased by 54%, supposing an infinite demand.

Ahmed and Alkhamis [9] used the same approach, but with the incorporation of optimization. The researchers integratedsimulation with optimization to produce a decision support system that would help decision makers in the operation of anemergency department unit located in Kuwait. In this paper, budget constraints were observed in the determination of theoptimal number of staff members to maximize patients’ throughput while reducing patients’ waiting time.

Rub et al. [10] addressed the importance and usefulness of modeling tools for analyzing complex processes in the health-care sector. Modeling tools identify process bottlenecks, invalidated data, and cross-functional boundary problems. Modeloutputs can be provided on the basis of future scenarios. The study applied Role Activity Diagramming modeling techniqueto analyze the collaboration of cancer centers and cancer registries for improving administrative quality of cancer treatmentand enriching collection of information about cancer cases.

Griffiths et al. [11] studied the optimization of the number of beds to minimize the cancelation of elective surgery withthe use of discrete event simulation, which maintained an acceptable level occupied hospital beds. Furthermore, ’what-if’scenarios were used in the increasing number of beds and ring-fencing beds, reducing the length of stay due to the delayeddischarge and schedule changes for elective surgery. The results showed that, if the hospital’s original 24 beds were used, itwould not adhere to the bed-occupancy guidelines. Therefore, additional five beds were required for the hospital. With these,there would be no elective surgeries would be canceled. Moreover, different configurations were tested to solve the bedblocking and ring-fencing bed problems for elective patients and unplanned admissions.

Korst et al. [12] supported the claim that it is significant to evaluate the hospital’s capability to collaborate first becausedata sharing is a challenging step to take. Factor analysis was conducted to determine the most important factor in foresee-ing success: 1. the presence of a supportive hospital executive; 2. the extent to which a hospital values data; 3. the presence

16 P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27

of leaders’ vision for how the collaborative approach advances the hospital’s strategic goals; 4. the hospital’s use of the col-laborative data to track quality outcomes; and 5. the staff’s recognition of a strong mandate for collaborative participation.

Zhang et al. [13] utilized demographic and survival analysis, discrete event simulation, and optimization for setting long-term care capacity planning in order to achieve certain target wait time service levels. The researchers developed a decisionsupport system that would be of help to real-world conditions. Based on the simulation results of long-term bed capacityplanning, this approach was found to be preferable to the fixed ratio approach, stationary, independent, period by period(SIPP), and modified offered load (MOL) as it lengthened the service times. The fixed ratio approach meant that if the ratiowas 0.075, 75 beds would be needed for every 1000 population aged over 75 years old. Within the same year (2012),Masselink et al. [14] scrutinized the effect of pharmacy policies on patients’ waiting time in the Chemotherapy Day Unitof the Netherlands Cancer Institute – Antoni van Leeuwenhoek Hospital. Using a case study and simulation model, patients’waiting time and wastage costs were evaluated to define the trade-off. Thus, this study established a new policy at the careercenter that is expected to decrease the waiting time by half while increasing the pharmacy’s costs by only 1% to 2%.

Al-Araidah et al. [15] examined outpatient clinics at a local hospital in Tehran, Iran. The researchers used a discrete-event-simulation model with the total visit time and service times at stations as data. Statistical comparison was also used to con-firm the model with the performance in the actual system. The outcomes showed that many improvement scenarios can beapplied to have a decrease in waiting time up to 29% and decrease in total visit time up to 19% without buying new resources.

Aeenparast et al. [16] also applied a scenario comparison in which the researchers studied a model that would reduce theoutpatients’ waiting time through system simulation. The simulation models were categorized from Scenarios 1 to 10, withdifferent waiting times. Patients’ waiting times were divided into three levels, depending on their physician. The result of thestudy showed that the mix of increasing work time of the resident physicians, increasing work time of the senior staff physi-cians, and changing the start of patients’ admission was the best alternative for reducing patients’ waiting time by about73.09%.

Gupta et al. [17] applied simulation optimization with Arena 13.0 to analyze the drive-throughmass vaccination problem.The distinctive feature of the simulation optimization allows decision makers to investigate interacting control variables byoptimizing criteria and performance measures under uncertain simulation scenarios. This approach has been applied suc-cessfully in Louisville, Kentucky (United States) in 2009.

Chen and Juan [18] studied a three-hospital patient-referral problem, which consisted of two large hospitals and a smallhospital. They considered the patients’ shortest waiting time as the objective function and added a budget constraint on hos-pital decisions for allowing staff overtime. In order to simplify the proposed problem, they assumed that the service time ofall computed tomography (CT) scans was the same distribution, the upper bound for the number of the daily referringpatients was 10 for all decision variables, and used the policy that referred the fixed daily patients among three hospitals.The optimal solution was obtained by Arena software and compared to the practical results. Further, they performed twosensitivity analyses, such as reducing the value of the maximal average patients’ waiting time and increasing the numberof annual CT patients, in order to explore the managerial implications for hospital managers. The main differences betweenChen and Juan [18] and this research were that this study relaxed the simplification part of Chen and Juan’s [18] research andmade their model more realistic, considering two kinds of objective functions, assuming that the service time of each MRIscan has its own distribution, assuming that the upper bound for the number of the daily referring patients was based on themaximal number of the daily arrival patients in a month, and considering the policies that referred to the fixed and unfixeddaily patients among two hospitals.

Gillespie et al. [19] compared the deterministic model and the stochastic model of the Orthopedic Integrated ClinicalAssessment and Treatment Service (OICATS). The researchers specifically studied the deficiency in the number of staff mem-bers allocated if the uncertainty in the planning stage is disregarded. If this happened, then queues will start to build up, andthe patients will experience a decrease in service quality. In the deterministic model, it was assumed that no variationoccurred in the OICATS department, where the results exhibited that 5625 patients could be treated. In contrast, in thatstochastic simulation with some realistic variations, the OICATS staff can only treat up to 3734 patients. The results showedthat variation had an effect when queuing is an essential property of the system.

Most researchers have used simulation models to understand the behavior of a hospital system and to evaluate the var-ious strategies for the operation of the system. Different types of the simulation models, such as discrete event, deterministic,and stochastic, have been developed and utilized. Optimization was also taken into consideration in some research studies todetermine the best option for minimizing waiting time and the total cost or maximizing revenues. The objective of mostresearch studies is to attain a certain waiting-time level for patients. Various facets in the medical field have been tackledin the prior years, including collaboration among hospitals, MIC, elective surgery, long-term capacity care planning, andchemotherapy units. However, this study will focus on hospital collaboration to develop a patient-referral mechanism toimprove waiting time. Nevertheless, existing research will contribute to the development of the study.

3. System description

The two case hospitals belong to one organization and have a collaborative relationship. One hospital director is in chargeof the operating procedures for both Hospitals A and B. The hospital director wants to shorten patients’ average wait, whichis a key performance indicator in the global budget system.

P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27 17

The system considered in the study involves only the MRI departments for the case hospitals (i.e., Hospitals A and B),which offer MRI services. For both hospitals, these MRI services are classified into six types depending on the body partto be examined, such as brain, chest, liver, spine, limb, and artery. The case hospitals offer these kinds of MRI services foroutpatients, inpatients, health check patients, and emergency patients. The physician decides whether or not the patientrequires an enhancement injection undergoing the MRI service. Hospitals A and B both have one MRI machine. The relatedparameters of MRI services and MRI patients will be described in Section 4.2.

4. Model simulation

4.1. Simulation framework

Based on the descriptions in Section 3, a simulation framework was established. Patient arrival patterns were determinedusing probability distribution with real data provided by case hospitals. Outpatient rules are defined for each hospital.Optimization models were embedded within the simulation model to determine the best number of referral patients.Fig. 1 shows an overview of the simulation flow occurring between the two collaborative hospitals from the time the patiententers the hospital, whether or not the patient is transferred, receives MRI service, and records data until the patient leavesthe original hospital.

The simulation models include the following details:

1. Four types of patients receive MRI services: outpatients, inpatients, emergency patients, and health check patients.2. The priority level of patients from highest to lowest is emergency patients, inpatients, health check patients, and

outpatients.3. Only outpatients can be referred to other hospitals as the transfer of inpatients and emergency patients can be a critical

condition. In addition, health check patients require only simple check-ups that consume minimal time; thus, referralsare not needed in such cases.

4. Only brain, spine, and limb MRI services are considered; these three MRI services account for more than 92% of all MRIpatients in both Hospitals A and B.

5. MRI services are offered 8 h per day in both hospitals.6. The service times for each type of MRI service for both hospitals are assumed to have the same distribution.7. MRI machines are assumed to be in good working conditions during the operating hours; the maintenance of MRI machi-

nes in the case hospitals is performed during off hours.8. The referral transportation time from Hospital A to Hospital B is approximately 50 min; the same is true from Hospital B

to Hospital A.

4.2. MRI patients’ inter-arrival time and MRI service time

The inter-arrival time of patients is important for estimating the time between two consecutive patients entering the sys-tem, considering all types of patients (i.e., outpatients, inpatients, health check patients, and emergency patients). However,service time is essential to determine the average service time of each kind of MRI services.

Using the Input Analyzer Tool,1 inter-arrival times were data-fitted to ensure that they passed chi-square and Kolmogorov–Smirnov tests, wherein the p-value should be greater than 0.05. Figs. 2 and 3 show the distributions of MRI patients’ inter-arrival times for Hospitals A and B, which were used to create the patient entities entering in the model. Figs. 4–6 indicatethe distributions of MRI service times for brain, spine, and limb services used for both case hospitals.

4.3. Parameters of simulation models

In order to obtain steady-state data, simulation model parameters were computed, including the warm-up period, repli-cation length, and replication number. The value for a warm-up period was set to 180 days, 60 for replication length, and 80for replication number. The coefficient of variation (C:V .), which is the standard deviation divided by the mean, should besmall. This research assumed that the value of C:V . should be less than 0.15 based on the empirical rule. For the replicationnumber (n) in Eq. (1), the half-width value (the half of the confidence interval, h) is also used as the basis in Eq. (2) andshould fall within an acceptable range. In Eq. (1), n = required replication number; n0 = initial replication number;h = required half width of the confidence interval; and h0 = initial half width of the confidence interval. In Eq. (2), a = levelof the confidence interval and s = standard deviation of the replication data.

1 Inp

n � n0 � h20

h2 ð1Þ

h ¼ tn�1;1�a2� sffiffiffi

np ð2Þ

ut Analyzer Tool is software developed by the Rockwell Company that can determine inter-arrival and service times of real data.

Hospital A Hospital B

A patient arrives at Hospital A

A patient arrives at Hospital B

Outpatient refer to Hospital B?

Outpatient refer to Hospital A?

Within quota of Hospital B?

Within quota of Hospital A?

MRI at Hospital A MRI at Hospital B

Track patient data in Hospital A

Track patient data in Hospital B

The patient leaves Hospital A

The patient leaves Hospital B

Y Y

N N

NY Y

N

Optimization model

Fig. 1. Flowchart model between two collaborative hospitals.

18 P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27

5. Optimization model

5.1. Mathematical models

Revised mathematical models from Chen and Juan [18] were used to define the objective functions and constraints. Thefollowing indices, parameters, objectives, and constraints come from the detailed models.

Histogram Beta

MRI patients' inter-arrival time (in minutes)160140120100806040200

0.150.140.130.120.11

0.10.090.080.070.060.050.040.030.020.01

0

Expression in ARENA of MRI patients’ inter-arrival time

Prob

abili

ty D

ensi

ty F

unct

ion

of H

ospi

tal A

×

Fig. 2. Formula based on Beta distribution for Hospital A.

Histogram Weibull

MRI patients' inter-arrival time (in minutes) 5004003002001000

0.44

0.4

0.36

0.32

0.28

0.24

0.2

0.16

0.12

0.08

0.04

0

Expression in ARENA of MRI patients’ inter-arrival time

Prob

abili

tyD

ensi

ty F

unct

ion

of H

ospi

talB

Fig. 3. Formula based on Weibull distribution for Hospital B.

P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27 19

Indices:

i and j represent the hospital index. In this paper, i ¼ 1;2; j ¼ 1;2; i– j.k represents the patient index, where k ¼ 1;2; . . . ;Ni. Ni is the original maximum number of patients in Hospital i.‘ represents the MRI service. ‘ ¼ 1 represents the Brain MRI service; ‘ ¼ 2 represents the Spine MRI service; and ‘ ¼ 3represents the Limb MRI service.

Parameters:

Oik‘ represents the revenue for patient k for ‘th MRI service in Hospital i.

Aik‘ represents the appointment’s start time of patient k for ‘th MRI service in Hospital i.

Pik‘ represents the examination time of patient k for ‘th MRI service in Hospital i.

Qi represents the maximum waiting time at Hospital i.Uij represents the upper bound of the number of MRI patients referred each day from Hospital i to Hospital j.Lij represents the lower bound of the number of MRI patients referred each day from Hospital i to Hospital j.Ci represents the close time of MRI each day at Hospital i.

Histogram Gamma

MRI service time (in minutes)200150100500

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

Expression in ARENA of the brain MRI service time

Prob

abili

ty D

ensi

ty F

unct

ion

of th

e br

ain

serv

ice

Fig. 4. Formula based on Gamma distribution for the brain MRI service.

Histogram Gamma

MRI service time (in minutes)100806040200

0.30.280.260.240.22

0.20.180.160.140.12

0.10.080.060.040.02

0

Expression in ARENA of the spine MRI service time

Prob

abili

ty D

ensi

ty F

unct

ion

of th

e sp

ine

serv

ice

Fig. 5. Formula based on Gamma distribution for the spine MRI service.

20 P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27

Ti represents the maximum overtime at Hospital i.S represents the predetermined time by Hospitals i and j so that the hospitals can serve as many patients as possible.

Decision variables:

n represents the number of days.wi

k‘ represents the waiting time of patient k for ‘th MRI service in Hospital i.

f ik‘ represents the finish time of patient k for ‘th MRI service in Hospital i.x‘ij represents the number of ‘th MRI patients referred each day from Hospital i to Hospital j.Xij represents the number of MRI patients referred each day from Hospital i to Hospital j. This decision variable is deter-

mined before a simulation experiment. Xij ¼P3

‘¼1x‘ij.

Optimization Model 1:

z ¼ E minXN1þX21

k¼1

X3‘¼1

w1k‘ þ

XN2þX12

k¼1

X3‘¼1

w2k‘

!ð3Þ

Histogram Weibull

MRI service time (in minutes)200150100500

0.64

0.56

0.48

0.4

0.32

0.24

0.16

0.08

0

Expression in ARENA of the limb MRI service time

Prob

abili

ty D

ensi

ty F

unct

ion

of th

e lim

b se

rvic

e

Fig. 6. Formula based on Weibull distribution for the limb MRI service.

P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27 21

Subject to

wik‘ ¼ f iðk�1Þ‘ þ Pi

k‘ � Aik‘ ð4Þ

wik‘ 6 Qi ð5Þ

Lij 6 Xij 6 Uij ð6Þf iðk�1Þ‘ þ Pi

k‘ 6 nðCi þ TiÞ ð7Þ

Optimization Model 2:

z ¼ E maxXN1þX21�X12

k¼1

X3‘¼1

O1k‘ þ

XN2þX12�X21

k¼1

X3‘¼1

O2k‘

!ð8Þ

Subject to

wik‘ ¼ f iðk�1Þ‘ þ Pi

k‘ � Aik‘ ð9Þ

wik‘ 6 Qi ð10Þ

Lij 6 Xij 6 Uij ð11Þf iðk�1Þ‘ þ Pi

k‘ 6 S ð12Þ

Twomodels were established with different objectives: to minimize the MRI patients’ average waiting time at Hospitals 1

and 2, as shown in Eq. (3), or to maximize the revenues of Hospitals 1 and 2, as shown in Eq. (8). The two models have thesame constraints except for constraints (7) and (12) because of the different scenario settings. In optimization model 2, afixed predetermined time is required as the base for comparing the revenues of two case hospitals without consideringmaximum overtime. However, predetermined time is not fixed for optimization model 1 because the objective functionof optimization model 1 is in time unit whiles the objective function of optimization model 2 is in money units (New TaiwanDollar, NTD).

Constraints (4) and (9) determine patients’ waiting time. Constraints (5) and (10) represent the patients’ average waitingtime in different kinds of MRI services at Hospital i, which must be less than or equal to the maximum allowed waiting time,Qi. Constraints (6) and (11) represent the number (Xij) of MRI patients referred each day from Hospital i to Hospital j, whichmust be between the lower bound (Lij) and upper bound (Uij). Based on the interviews of the hospital manager, the upperbound is set to the maximal number of the daily arrival MRI patients in a month. Constraint (7) ensures that all patients haveMRI services. Constraint (12) provides a time base for estimating total revenues of different scenario settings.

5.2. Simulation optimization

The researchers used Arena OptQuest software and followed the simulation optimization algorithm formulated byKlassen and Yoogalingam [20]. First, for the initialization step, candidate solutions were generated and the diverse popula-tion identified. Second, in the simulation process, the researchers proceeded with 80 replications to generate more stable and

22 P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27

better solutions in every run of the simulation. Third, the optimization process combined the existing feasible solution (par-ents) to create new solutions (offspring) and replace the worst parent with the best offspring. To avoid revisiting the preced-ing inferior solutions, the offspring was modified with tabu memory functions. Finally, the researchers used a user-specifiednumber of heuristic iterations, such as 80 iterations, as the stopping criterion. Each iteration contains 80 replications.

The feasible solutions of Arena OptQuest software were generated based on iteration searching (the scatter searchmethod) with 80 replications for each iteration. The objective value of each iteration was the average objective value ofthe 80 replications. The optimal solution was the best average objective value of the 80 iterations.

5.3. Verification and validation of simulation models

After building the simulation model, the researchers used verification and validation to scrutinize the logic and process ofthe model. The built models were checked to determine if the logical flow of the entities and processes were correct. Forexample, when an outpatient arrives at Hospital A, a decision is made whether or not to transfer that outpatient to HospitalB. Afterwards, the simulation model will check if the outpatient is within the number of allowed referred patients. If thereferral number is below the referring quota, the patient could be referred to Hospital B with an approximate delay of50 min for the transportation time; otherwise, the patient referral will not take place.

After the MRI service, the patient will be transferred back to Hospital A for records with a delay of approximately 50 minfor the transportation time. Finally, the patient will leave the original hospital. The verification process will be successfulonly if all the logical flows are appropriate for the desired outcome of the system.

In order to make sure that the result of the simulation model will execute a system comparable to the actual condition,the researchers validated the service time of each kind of MRI processes, average cycle time of all types of patients, and thetotal number of patients who received MRI services and left Hospitals A and B for two months by running 80 replications.This validation process was done to test if a significant difference exists between the actual and simulated systems. Given thedata for patients’ average waiting time, the values of the mean and standard deviation were used with the help of the pairedt-test of statistical comparison test using the 95% confidence interval.

6. Results and discussions

6.1. First objective: minimize patients’ average waiting time

Hospital A began its collaboration with Hospital B in 2011. Hospital A is located in a larger city with 230,202 residents;Hospital B is located in a smaller city with 59,657 residents. Due to the geographical factor, patients’ arrivals at Hospital Aoutnumber patients’ arrivals at Hospital B. Therefore, patients of Hospital A experience long waiting times. Thus, a patientreferral mechanism will be implemented to accelerate the MRI service. Referral system models of the fixed number and theunfixed number of referral outpatients per day were built. Therefore, the researchers’ first objective is to minimize patients’average waiting time in Hospitals A and B.

Scenario 1: Fixed number of referral outpatients per dayUsing the OptQuest Tool, the researchers defined the maximum number of referral outpatients as a control, and defined

the waiting times of the three considered MRI services and patients’ average waiting time as responses. Moreover, the wait-ing times in the brain, spine, and limb MRI services were set to be less than or equal to 21 days due to hospital’s criteria.Finally, the objective function is to minimize patients’ average waiting time in Hospitals A and B. After simulating the data,the researchers obtained an optimal solution that referred to 3 referral outpatients per day with a minimized waiting timefor patients equal to 0.53 days.

Scenario 2: Unfixed number of referral outpatients per dayDue to the fluctuating number of patients entering Hospital A, having a fixed number of referral outpatients might not be

sufficient for addressing patients’ long waiting time problems in the long run. Therefore, the researchers also consideredbuilding a model based on Scenario 1, but with an unfixed number of referral outpatients per day for a span of one month.

The model for Scenario 2 is similar to that of Scenario 1, except for a few modifications, such as adding the maximumnumber of referral outpatients per day, which are variables, for the next 29 days. Initial values of these variables were setto 0 to determine the appropriate number of referral outpatients per day.

Using the OptQuest tool, the responses, constraints, and objective functions for this scenario were similar in Scenario 1.However, additional responses were included, such as the maximum number of referral outpatients for the second day untilthe last day of the month. Afterwards, the researchers obtained an optimal solution with a minimized waiting time forpatients equal to 0.48 days

6.2. Second objective: maximize hospitals’ revenues

The researchers considered the hospitals’ financial needs to render better quality services to patients. From this view-point, another objective was defined: to maximize hospitals’ revenues. In this case, the needs of both the patient and thehospital are satisfied. Therefore, both case hospitals’ revenues will be studied further.

P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27 23

Scenario 1: Fixed number of referral outpatients per dayIn this model, the same logic of the system was considered, but additional variables for the calculation of revenues were

included, such as the revenues, injection points, non-injection points, and equivalent points. These variables were addedbecause these can affect the revenues of the hospital. The injection point for one patient is 11,500 points compared to6500 points for non-injection patients. To convert these points to a monetary value, they were multiplied to the equivalentpoint, 0.9568. The equivalent point is predetermined by hospitals and government agencies every year.

With the OptQuest tool, the researchers defined the maximum number of referral outpatients as a control, and definedthe waiting times of the three considered MRI services and total revenues for Hospitals A and B as responses. Furthermore,the waiting times in the brain, spine, and limb MRI services were set to be less than or equal to 21 days, based on the hos-pitals’ criteria. Finally, the objective function was to maximize the revenues of both Hospitals A and B.

In considering the revenues of Hospital A, the researchers obtained the optimal solution with one number of referral out-patients per day with a maximized revenue of 6,071,664 NTD for Hospital A and 3,157,533 NTD for Hospital B. The computedcosts were subsequently subtracted from the revenues of both Hospitals A and B. These are the labor and transportationcosts of transferring an outpatient to another hospital. The distance between the two hospitals is 45.2 km. After the laborand transportation costs are deducted from the revenues, the remaining revenues for Hospitals A and B are 6,061,849NTD, and 3,147,718 NTD, respectively. The depreciation cost of the MRI machine and associated cost are not consideredin this research. In general, the profits of Hospitals A and B will be much less than the remaining revenues.

Scenario 2: Unfixed number of referral outpatients per dayUsing the OptQuest tool, the constraints and objective function for this scenario are the same as in Scenario 1. In addition,

there were more responses, such as the maximum number of referral outpatients for the second day until the last day of themonth. In considering the revenues of Hospitals A and B, the researchers obtained the optimal solution with the number ofreferral outpatients per day with a maximized revenue of 2,450,025 NTD for Hospital A and 2,271,577 NTD for Hospital B.Successively, the computed costs were subtracted from the revenues of both Hospitals A and B. These are the labor and trans-portation costs of transferring an outpatient to another hospital. The remaining revenues for Hospitals A and B are 2,445,117NTD and 2,266,669 NTD, respectively.

Table 1 summarizes the number of daily referral outpatients in the first 10 days with different objective functions undercase settings. For the scenario of fixed number of referral outpatients per day, the number of daily referral outpatients con-sidering minimizing patients’ average waiting time is larger than that of considering maximizing hospitals’ revenues. Thereason comes from extra cost of referral outpatients. This implies if Hospital A takes patients’ position rather themselves,it will improve its service level comparing to its original operations. The tendency of the second scenario of unfixed numberof referral outpatients per day is the same as the first scenario. In general, the number of referral outpatients per day withconsidering the first objective is larger than that with considering the second objective.

6.3. Sensitivity analysis

In order to investigate the effect of two key parameters (revenues and patients’ average waiting time) by adding a newconstraint on revenues or patients’ average waiting time, this study conducted two sensitivity analyses. The details aredescribed in the following subsections.

6.3.1. First objective: minimize patients’ average waiting timeScenario 1: Fixed number of referral outpatients per dayIn minimizing patients’ average waiting time, revenues were reduced at most by 5%, 10%, 15%, and 20%. These values are

the revenues that the hospital director is willing to sacrifice in accordance with the hospital’s criterion of less than or equal to21 days for patients’ average waiting time. Thus, if the hospital director is willing to sacrifice revenues under 10%, the opti-mal number of outpatients to be referred per day is 1, that is, cases of 5% and 10% revenues reduction make the director dothe same decision, which results in a 6.49-day patients’ average waiting time. By reducing the revenues by less than or equalto 15% to 20%, the optimal number of outpatients to be referred per day are 2 and 3, respectively. Moreover, patients’ averagewaiting times are 0.62 days and 0.55 days respectively when reduced to less than or equal to 15% and 20% of the revenues.Fig. 7 shows results of Scenario 1 of the first objective.

Scenario 2: Unfixed number of referral outpatients per dayFig. 8 shows results of Scenario 2 of the first objective. If the hospital director considers the unfixed patient referral mech-

anism and sacrifices the revenues by at most 5%, 10%, 15%, and 20%, patients’ average waiting times are 1.27, 1.27, 0.52, and0.49 days, respectively, with corresponding revenues of 2,910,097 NTD, 2,910,097 NTD, 2,628,224 NTD, and 2,421,620 NTD.

6.3.2. Second objective: maximize hospitals’ revenuesScenario 1: Fixed number of referral outpatients per dayIn the maximized revenues in Hospitals A and B, patients’ average waiting time was reduced to less than or equal to 18,

15, 12, and 9 days. Fig. 9 shows results that indicate the impact of reducing patients’ average waiting time for revenues. If thehospital director is willing to prioritize patients’ average waiting time by reducing it to less than or equal to 18, 15, or12 days, the optimal number of outpatients to be referred per day is 1, resulting in a value of 6.49 days with 6,071,664

Table 1Number of referral outpatients per day in the first 10 days.

Day 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

1st obj. Scenario 1 3 3 3 3 3 3 3 3 3 3Scenario 2 7 4 5 6 6 5 2 7 6 6

2nd obj. Scenario 1 1 1 1 1 1 1 1 1 1 1Scenario 2 6 2 6 6 3 5 2 4 6 2

0

1

2

3

4

5

6

7

0

1

2

3

4

0% 5% 10% 15% 20% 25%

Act

ural

ave

rage

wai

ting

time

unde

r ref

erra

l pol

icy

Num

ber o

f out

patie

nts

Percentage of reduced revenues (under the referral policy) comparing to the original revenues (no referral policy)

Optimal number of outpatients to be referred per day

Patients' average waiting times

Fig. 7. Simulation optimized cases for Hospital A with the first objective for Scenario 1.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.0

1.0

2.0

3.0

4.0

0% 5% 10% 15% 20% 25%

Act

ual a

vera

ge w

aitin

g tim

e un

der t

he re

ferr

al

polic

y

Rev

enue

(10e

6)

Percentage of reduced revenues (under the referral policy) comparing to the original revenues (no referral policy)

Revenue Patients' average waiting times

Fig. 8. Simulation optimized cases for Hospital A with the first objective for Scenario 2.

24 P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27

NTD in revenue. By reducing patients’ average waiting time to less than or equal to 9 days, the optimal number of outpa-tients to be referred per day is 3, resulting in a value of 0.55 days with 4,936,275 NTD in revenue.

Scenario 2: Unfixed number of referral outpatients per dayFig. 10 indicates the results of Scenario 2 of the second objective. If the hospital director considers the unfixed patient

referral mechanism and prioritizes patients’ average waiting times by reducing them to less than or equal to 18, 15, 12,

0

1

2

3

4

5

6

7

0

1

2

3

4

8 9 10 11 12 13 14 15 16 17 18 19

Act

ual a

vera

ge w

aitin

g tim

e un

der r

efer

ral p

olic

y

Num

ber o

f out

patie

nts

Pre-determined patients' waiting time under the referral policy

Optimal number of outpatients to be referred per dayPatients' average waiting times

Fig. 9. Simulation optimized cases for Hospital A with the second objective for Scenario 1.

0

1

2

3

4

5

6

1

2

3

4

8 9 10 11 12 13 14 15 16 17 18 19 Act

ual a

vera

ge w

aitin

g tim

e un

der r

efer

ral p

olic

y

Rev

enue

(10e

6)

Pre-determined patients' waiting time under the referral policy

Revenue Patients' average waiting times

Fig. 10. Simulation optimized cases for Hospital A with the second objective for Scenario 2.

P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27 25

and 9 days, the corresponding revenues are 3,004,292 NTD, 3,004,292 NTD, 3,004,292 NTD, and 2,993,874 NTD. The reduc-tion of patients’ average waiting times to less than or equal to 18, 15, and 12 days yielded the same results. The result indi-cates that if the hospital is willing to sacrifice some revenues to make patients’ waiting times be reduced by 3 days, in reality,it might have advantages that the actual reduction time can reach 9 days, that is, the average patients’ waiting time for thepre-determined criterion is 12 days.

7. Conclusions and future research

The first academic contributions of this research is that it proposed the more rigid mathematical formulations, whichrelaxed the simplification part of Chen and Juan’s [18] research. The proposed models of this research were more realisticfor the further academic study. Second, for simulation methodology, combining stochastic parameters (random patients’arrival and random patients’ service time) with constraints, the ’what-if’ simulation scenario would be required in eithera trial-and-error way to search for an optimal solution. Take the unfixed referral policy as an example, 30 decision variables,

26 P.-S. Chen et al. / Simulation Modelling Practice and Theory 61 (2016) 14–27

such as the number of the daily referring patients, were integer variables, ranging from 0 to the maximal number (i.e., Npatients) of the daily arrival patients in a month. The problem is similar to the multi-stage stochastic problem (cf. Dyerand Stougie [21]). For each day (stage), Hospital A has to determine the number of referral outpatient (decision) under uncer-tain environment parameters. This implies the problem of this study is a NP-hard problem. This led to long computation timefor exhaustive searching. However, by simulation optimization, the convergence could be quickly obtained within 80 iter-ations and 80 replications, which were total 6400 runs. The simulation time was not proportional to the value of N. There-fore, the simulation optimization method indeed shortened a lot of computing time compared to the traditional simulationmethod.

From the viewpoint of hospital management, this study determined a collaborative patient referral mechanism betweentwo case hospitals with different objectives: minimizing patients’ average waiting time and maximizing the revenues in bothhospitals. The researchers used scenarios for each objective by determining a fixed and unfixed optimal number of outpa-tients to be referred per day between the two hospitals.

After evaluating the two objectives with different scenarios, it was observed in the first objective that there is a drasticdecrease occurred in patients’ average waiting time along with the revenues. In maximizing revenues, there will also be adecrease in patients’ average waiting time, but only a small decrease in revenues. These factors could help the hospital direc-tor make decisions in guiding, leading, and rendering a quality service.

Finally, the researchers validated that simulation optimization could be a helpful tool in the healthcare sector withoutchanging any processes and entities in a system. With the use of Arena software, the researchers were able to conduct anal-yses and achieve the purpose of their study.

For further development, the researchers recommend that future researchers examining a similar topic with a deeper dis-cussion of hospital collaboration. Other considerations, such as referral not only of MRI outpatients but also other types ofpatients, can be tackled. In this paper, the researchers focused on minimizing patients’ average waiting time or maximizingthe revenues for Hospitals A and B. On the other side, the objective function can be changed by simultaneously minimizingpatients’ average waiting time and maximizing revenues.

Due to service time constraints, future researchers might also consider another type of hospital services in the imagingcenter, such as X-ray, CT, or ultrasound tests. A collaboration involving more than two hospitals could have a more conve-nient service. Other service industries can also apply collaboration to fully comprehend its importance.

Acknowledgements

The authors are grateful to anonymous referees for their very valuable comments, which have significantly enhanced thiswork. This research is supported by the Ministry of Science and Technology under contract No. MOST 104-2221-E-033-025-.

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