Forecasting hospital bed availability using computer simulation and neural networks

Rochester Institute of Technology

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2004

Forecasting hospital bed availability using

computer simulation and neural networks

Matthew J. Daniels

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Rochester

Institute

of

Technology

FORECASTING HOSPITAL BED AVAILABILITY USING

COMPUTER

SIMULATION

AND NEURAL NETWORKS

A Thesis

Submitted in

partial

fulfillment

ofthe requirements

for

the

degree

of

Master

of

Science in

Industrial

Engineering

in

the

Department ofIndustrial

& Systems

Engineering

Kate Gleason College

of

Engineering

By

Matthew

J.

Daniels

B.S.,

Industrial

Engineering,

2003

Thesis/Dissertation Author Permission Statement

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DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING

KATE GLEASON COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

CERTIFICATE OF APPROVAL

M.S. DEGREE THESIS

The M.S. Degree Thesis of Matthew J. Daniels has been examined and approved by the thesis

committee as satisfactory for the thesis requirement for the Master of Science degree

Approved by:

Michael E. Kuhl

Dr. Michael E. Kuhl, Advisor

Moises Sudit

Dr. Moises Sudit

Elisabeth Hager

Abstract

The success of hospitals in _treating patients and _{staying in business} relies on their

efficientuse of resources. Inparticular, theutilization ofhospital beds is a critical_concern, since

over-crowding will result in delays or transfers of _patients, and under-utilization will result in

lost opportunityto treat patients and generate profit. To this end, hospital decision makers must

havereliableforecasts of patientdemandandbed availability. The objectiveofthis thesis was to

create a general methodtoforecast the _availabilityofhospital beds inthe short_{term, up}to2 days

into the future. _{Specifically,} this thesis employed a computer simulation model ofthe hospital

and a time-dependent neural network to learn from the simulated model and forecast the

availability ofbeds. The computer simulation model was found to be well suited to the task of

describing a general hospital system and_{creating training}data fora neural network. The neural

network was found to provide accurate performance in predicting bed availability in the short

term. The network incorporated the effect of time _explicitly to capture the _{non-stationary}

behavior of hospital systems. These findings have a number of implications that will be

Table

of

Contents

1. Introduction 1

2. Problem Statement 6

3.Literature Review 8

3.1 Long-Term

Forecasting

in Health Care 8

3.2 Short-Term

Forecasting

in Health Care 12

Time-seriesmethodforshort-termbedutilization

forecasting

12

Neural networksformeanE.D. waittime

forecasting

13

3.3 Other Methods for Short-Term BedUtilization

Forecasting

14

Regression for

forecasting

15

Expert systemsfor

forecasting

16

Computersimulation for_forecasting 17

3.4 Neural Networks 18

Structure ofthebrain 19

Structure of artificial neural networks 20

Trainingneural networks 23

Neural networks applicationsin_forecasting 26

Drawbacksof neural networks 28

3.5 ComputerSimulation 28

Definition of computer simulation 28

Applicationsof computer simulationin health care 30

Drawbacks of computer simulation and neural networks as a _{complementary}

technique 33

4. Methods 36

4.1 System Description 36

Hospital size 36

Hospital layout and patient flow 37

Patientarrivals and volume 38

4.2 Computer Simulation Model ofthe Generic Hospital 39

Modeling

software 40

Patient types 40

Patient arrivals 41

Resources 42

Model outputs 43

Verification and validation 43

Experimentto determineeffect of patient mix 45

4.3 Neural Network for Bed_{AvailabilityForecasting} 46

Network inputs 47

Networkarchitecture 49

Network _training 51

Building

forecasttolerance intervals 52

Testing

theneural networks 53

Neural network for predicting forecasterror 53

5. Conclusions andRecommendations for Future Research 57

5.1 Conclusions 57

5.2 Recommendations for Future Research 57

References 59

Appendices 62

Appendix I: Generic Hospital Patient Characteristics 62

Appendix II: Distribution of_{Beds in Generic Hospital Simulation Model} 62

Appendix III: Generic Hospital Model Verification Scenarios 63

Appendix IV: Attached Computer Files 64

List

of

Figures

Figure 1: Arepresentative neuron in a neural network 20

Figure 2: Neural transfer functions: _{(a) non-squashing linearfunction, (b) log} sigmoid_squashing

function,and _(c) hyperbolictangentsigmoid _{squashing function} 22

Figure 3: A representative layer in a neural network. Outputs from previous layer are

represented

by

boxes at _left;outputsfrom current layerare represented_byboxes at right..23

Figure 4: Flow diagram showingthe flowof patientsthrough thegenerichospital 37

Figure 5: _{(a) Emergency} arrivals and _{(b) non-emergency} arrivals to the generic hospital model.

Note the difference in scale (maximum emergency arrivals in one hour is 4, versus 13 for

scheduled_patients) 42

Figure 6: comparisonofmeans for basemodel (here _{"sm4_l")} and alternative model 46

Figure 7: Boxplotsofbed _availability foreachhourofthe week, over300weeks 48

List

of

Tables

Table 1: Validation results forgeneric hospitalsimulation model 44

Table 2: Test results for various network sizes, for

determining

network hidden layer size.

Resultsare after200passes of_training on each network 50

Table 3:

Training

resultsforthe expected value network 52

Table 4: testresults fromtrainedexpected valuenetwork,using noveltest data 53

Table5:

Training

results forthe six-hourexpected value _network, after400passes 55

Table 6: Testresultsforthesix-hour expected value _network, after400passes 55

Table 1.1:Characteristics of patienttypesusedin modelingthe generichospital system 62

Table 1.2: Characteristics of patient types used in modeling the Case

Study

hospital system.

Error! Bookmarknotdefined.

Table II. 1: Distributionofbeds in generichospital system 62

1.

Introduction

Health care _{is possibly} the most important

industry

in the United States for the

application of process improvementtechniques because ofits sheer _{size, its}current performance

with respect_{to efficiency,}and its central placein oursociety.

The enormous size of the US health care system is plainly evidenced _by the dollars it

consumes, the number of patients it serves, and the number ofpeople it employs. In terms of

health care _{costs, total} expendituresand average charges perhospital stay are two measures that

illustrate the size of the US health care system: national health care expenditures were $1.4

trillion in 2001 (Centers for Medicare and Medicaid _{Services, 2003),} and the national average

charges per hospital discharge in 2002 were over _{$17,000, up from} $11,000 five years earlier

(Agency for Healthcare Research and _Quality, 2004). In terms ofthe number of people served

by

health _{care, the} 2.46 million patients admittedto hospitals inNew Yorkstate in 2002 and the

more than 37 million patients nationwide in the same year attest to the size of this _industry

(Agency for Healthcare Research and _Quality, 2004). Finally, in terms of employment in this

sector, the Bureau ofLabor Statistics ₍₂₀₀₄₎ reports that 14,188,000 people were employed in

health services as of

July

2004

-that _{is, nearly 11%} ofUS employment. Health care is _truly a

system ofgigantic proportions in theUnited States.

Beyond the size ofthe system, its widely reported inefficient practices contribute to the

need to make systematic improvements. Alarmingly, health care costs are _rising at a rate of

approximately 9%per year (Centers for Medicareand MedicaidServices, 2003). One culprit in

of5,000 new compounds tested in the _{laboratory, only 5 actually} makeit to clinicaltrials... and

only one is approved for patient use. Elapsed time can be 12 to 15 years. _Hence, the average

cost of a new medicine is around $500

million"

("Drug-price program

notes,"

2000). As a

further example, US health care providers are notorious for

being

behind the times with respect

to information technology: the Joseph H. Kanter

Family

Foundation ₍₂₀₀₂₎ reports that savings

on the order of$80billion per year could be realized

by

_{making better}use of the most current

information _technology available.

Any

system of this _{size, especially} with the

inefficiency

indicated intheUS healthcare_system, is an importanttarget_{for study} andimprovement.

In addition to the size and

inefficiency

of health care _{systems, however,} the nature of

theirfunction in our _societyprovides yet another motivation towork toward theirimprovement.

Asmentioned above,_they provide alarge source ofemployment; and _theyare not _onlyused_by a

large volume of people in _{this society, they}are used

by

peoplein every demographic and _every

segment ofit. Further, these services are not mere conveniences or luxuries: the wellness of all

people _depends, at some pointintheir_lives, ontheiraccesstoand adequate service fromahealth

care provider. _So, in summary, the US health care system is a critical subject forimprovement

by

scientific methods.

Indeed, many improvement effortshavebeenundertaken overtheyears. One significant,

though not _completely successful, effort toward cost containment has been the transition from

fee-for service payment to managed care. _{The existing fee-for-service} system paid a provider

according to _anyservices rendered to a patient. Managed care, in contrast, is a system that pays

consumed in _treating that patient.

By

not _providing additional financial incentive to use extra

resources on individual _patients, it is intended as a means to deter overuse of health care

resources. Under this system, the _{only way} to generate more revenue is

by

_treating more

patients, and therefore an important result of managed care has been a significant decrease in

patients'

lengths of_{stay in hospitals.}

Given this incentive to treat as _many patients as possible to maximize _profit, it has

become imperative to _closely manage the utilization ofhospital beds. Several tools are used_by

hospitals to manage theirhospital bed utilization. One important example is utilization _review,

where case managers track and evaluate each patient's use of resources. Medical _technology

-better procedures, tools, and medicines- is

a second example of animportant tool forutilization

management. A final example is the _scheduling of patients and _{staff; this} critical toolis usedin

hospitals to_tryto controltheflow of patients.

The optimization of bed utilization, despite the efforts of utilization review and other

techniques, is difficult to achieve. The high _variability in demand for services and

patients'

lengths of _stay prohibits simple _forecasting of demand. Without this _ability to forecast the

hospital's capacity status into the _future, one of two negative situations will arise. In the first

scenario, the hospital reserves a number of beds for patients that _may come in through an

Emergency Department. If these beds go unfilled, the hospital loses out on potential revenue.

The second situation is that in whichthe hospital fills (or schedulesto _fill) as _{many beds} as are

available, with little regard to _emergency patients at a future time. This creates a situation of

scheduled_patients, must be turned _away and re-scheduled. _{This, in turn,} creates dissatisfaction

on thepart ofpatients and resultsin_{increasingwaiting lists for}scheduled surgicalprocedures. It

is not difficult forahospital to maximizebedutilization withoutrespectto the negative scenarios

above. _However, it is difficult

-and _{necessary for}the success ofthe hospital system

-to

time-dependently optimize bed utilization. _Therefore, intelligent methods of

forecasting

bed

availabilityare crucial to_{maintaining high} utilization while not

delaying

or_{turning away}patients

in need.

One important measure for optimizing bed utilization and the hospital's success is the

number of available beds at points in the near future. If many beds are held for potential

emergency patients and _only a few patients show _{up, then} beds sit _empty that could have been

used for scheduled procedures. On the other _hand, if not enough beds are held for potential

emergency patients and _many patients do _{arrive, then} either scheduled patients must be sent

away and re-scheduled, or else _emergency patients will have to be sent for treatment at a

different hospital.

This proposal will describe the approach to a new solution this important problem. _By

forecasting

future bed availability more _accurately than current methods allow, hospitals will

realize improvements in staffing accuracy and reductions in patients turned _away upon arrival.

Further, waiting lists forscheduled procedures will diminish as patients are treated on theirfirst

visit rather than _being re-scheduled. The results ofthis improvement will be

immediately

felt.

Betterinformation about bed availability will allow _schedulingof specialized nurses aroundthe

knowing busy

and idle times in advance. _Moreover, consider the impact on hospital revenue of

optimized bed utilization. As cited _previously, the national average charges per hospital

discharge in 2002 were $17,000 _(Agency for Healthcare Research and _Quality, 2004).

Maintaining the same levels of service to _emergency patients while

being

able to accommodate

justten moredischarges peryear,forexample,would generateadditionalrevenue ontheorder of

$170,000per year.

This research addressesthe identified need

by developing

a general methodto _accurately

and _{quickly forecast bed availability up} to 48 hours in advance. The general method uses a

computer simulation model with a neural network

forecasting

_system, in a two-phase approach,

toprovide an expected value and associatedtolerance intervalforeach prediction. Thecomputer

simulation model provides an accurate picture of the processes and interactions in the real

system. As such, it can also provide volumes of usable _training data forthe neural network. In

the second phase ofthe method, the neural network

forecasting

system uses the simulated data

for training, after which it can _quickly generate predictions, and it can _continually update its

learning

basedon _newlyreceived actual valuestoremain accurate andrelevant.

This paperis organized as follows: Section 2 containsthe Problem Statement. Section 3

reviews literaturerelated to_forecasting, computersimulation, and neural networks, aswell as the

application of each in health care. Section 4 details the general method for creation of the

forecasting

system for the case of _predicting one time _{step into} the _future, and the various

experiments performed aredescribed. Conclusions andrecommendationsfor future workfollow

2. Problem

Statement

The problem addressed_by thisresearch is thatof_accurately

forecasting

the_availabilityof

hospital beds into the immediate future. The problem of_{managing bed} utilization is important:

overcrowded hospitals mean _{delaying, re-scheduling,} or _transferring to other hospitals the

treatment of _{many patients; meanwhile,} under-utilized beds mean a squandered _opportunity to

treatpatients and generate revenue.

This research will _apply a novel approach to this

forecasting

problem through the use of

two phases. _First, a typical hospital system will be modeled through the use of computer

simulation. This simulation model will allow the _study and _{understanding} ofthe system

being

modeled. Specifically, factors most

influencing

bed availability and utilization willbeexamined.

The model will create an _{understanding} of arrival patterns of _{patients, treatment lengths,} and

current resource utilization in the system. This model will be verified and validated, to ensure

that _{it accurately} represents_realityand can provide reliableoutputs.

Second, the simulation model will be used to _apply a neural _{network, to} increase the

efficiency ofthe solution on an _{everyday basis. This} neural network phase will provide two

key

benefits in

forecasting

_{bed availability} and, therefore, in optimizing bed utilization. The first

benefit is increased speed over the use ofthe simulation model for everyday

forecasting

needs.

While a simulation model of this _{complexity may} take at least several minutes to achieve a

forecast result, because it requires sufficient run time to build an acceptable confidence interval

of the estimate, a neural network can achieve a result much more quickly, without _sacrificing

adapt themselves to updated actual _data; that is, as conditions change in the system, the neural

network will be able to "learn" these changes and continue to produce accurate results. Its

effectiveness will notbe limitedto a static situation ofthe system parameters.

The type of result that will be produced

by

the

forecasting

system is an estimate ofthe

3.

Literature

Review

Two characteristics of the real world make

forecasting

an important subject for study:

one is _randomness, and the otheris scarcity. With certainknowledge offuture demand, or with

unlimited resources to meet future _demand, success could be guaranteed. _However, since there

is _variability in demand for resources, and since resources to meet that demand are _limited,

successis based on theoptimal use ofresourcesto meetdemands. _Thus,

forecasting

is critical to

success in the real world: optimal forecasts provide the _{opportunity for} optimal usage of

resources. As MacNiece _{(1961) succinctly} writes, without

forecasting

"both short- _and

long-range_planningrests on foundationsmuch lesssubstantialthan

sand"

(p. 109).

This thesisaddresses theproblem of short-term

forecasting

ofhospital bed availability; as

such, this literature review will discuss how this problem has been addressed and what methods

are available to arrive at a better solution. _{Traditionally,}

forecasting

in health care _usually

considers long-term forecasts of resource _utilization, rather than hour-to-hour forecasts. These

methods are shown to be inadequate for use in the thesis problem. Other methods that are

available for

forecasting

short-term bed utilization include Expert Systems and artificial neural

networks. While neural networks require a large amount of _data, and the determination of

network structure and inputs can be problematic, they are shown to _{be ideal for addressing} this

problem when complemented

by

computer simulation models.

3.1 Long-Term

Forecasting

in Health Care

Mostoftheliterature_dealingwith _forecastingin health care systemsis based onthe

and staff needs over

long

_{times, say} over a period of months to years. In this _section, some

works inthelong-term

forecasting

ofhealthcare systemsarereviewed.

Cote and Tucker ₍₂₀₀₁₎ provide an excellent _summary as a _starting point for the

practitionerof strategic

forecasting

ofhospital demand: "Four common methods of

forecasting

are percent _{adjustment, 12-month movingaverage, trendline,} and seasonalized forecast" (p. 54).

The first two _methods, percent adjustment and 12-month _{moving average,} are simplistic

forecasting

techniques that assume no process changes. While this _{may be} a valid assumption

forsome _{systems, the} authorspoint outthat thesemethods donot incorporate anyofthe seasonal

effects that are inherent in monthly patient demand throughout a year. Where there are trends

and spikesintheprocess that_{significantly}affect themeasure

being

_forecasted, neither ofthe two

methods will catch_up to_changing actualdemand. Trend lines arethe third

forecasting

technique

described as common _by Cote and _Tucker, for predicting hospital demand in the medium and

long

term. This method is a simple case of a linearregression: a line is fitted to the observed

datapoints in order to extract the expected trend. While this method is more sophisticated than

theprevious two methods, theeffect of_{seasonality in}thedatais still not considered.

The final techniquein this paperdoes accountfor seasonal trends _explicitly and produce

more realistic results than the other three techniques. The seasonalized forecast transforms the

observed values into "deseasonalized" values _by comparing each season's averageto the overall

average (forexample, to saythat month xis 10% above the average of all months).

Using

this

information torelate allthe datapointsbackto a commonbaseline, atrendlineis _calculated,and

month's seasonal factor. While this final method is often more appropriate for this task, based

on the nature of the _system, the authors issue a final caution with respect to all four methods:

"Healthcare financial managers know that change is perhaps the _only constant for the future.

Therefore, knowledge of both internal and external environments must be factored into final

forecasts"

toincreasetheirpractical usefulness andvalue(Cote and_{Tucker, 2001,}p. 57).

Another paper on health care utilization

forecasting

in the

long

term is a contribution

from MacStravic (2004). The author again emphasizes the

inadequacy

of simple models or

measures in addressing complex problems: "I once

'proved'

via an exponential trend line

forecast, that _{in only 25years, every}person _living in the State ofVirginiawould be a registered

nurse.... Of course, no trend _line, however large or small its goodness of _fit, should be chosen

for use in

forecasting

without _{understanding} and _having confidence in the continued and

consistent action of whatever dynamics caused thepast

trend"

(p. 11). The main _deficiencyof

these simple models is _precisely that _inability to account for (and react _to) the dynamics ofthe

real health care world. This is the arena where more complex models are appropriate for

forecasting.

In long-range utilization prediction in health care, models of increased _complexity

usuallyoccuras extensions ofthe simple modelsdescribedabove. Woodruffsguidelinesonbed

need _{planning (2002)}incorporate additional factorsbesides time-lagged values ofdemand itself.

The recommendation is to include community-level measures, such as population change and

market share, along with hospital-level variables like past discharge and _{length-of-stay} data and

at estimates for the average

daily

census of patients and the peak census (when the hospital is

mostfull). From this_estimate, Woodruffrecommends use ofthe normal_{probability distribution}

todetermine a safe interval intowhich the needed numberofbeds shouldfall. Even forcreation

of a design-phase or strategic need _forecast, the author acknowledges that the dynamics and

variabilityofhealthcare systems mustbetakeninto account:

"Predicting

the _variability offuture

demand is _probably the most critical element of a sound _{bed need-planning}

model"

(Woodruff,

2002,p. 2). Here, then, a more in-depth assignment ofthe causes of patientdemand fluctuation

isattempted, inorderto _accuratelycapturelong-range bedneed.

Continuing toward _{higher complexity is} a final example from Myers and Green (2004).

The authors incorporate the "incremental impact of medical

advances"

in their two-phase

approach to _forecasting utilization (p. 35). The first phase, again, is a traditional forecast as

described in the

foregoing

papers. The

authors'

approach here attempts to add the effect of

medical _technology as one of the variables influencing utilization. The idea is, with medical

advances come reduced patient lengths of _stay and reduced chance that the same patient will

return for the same procedure. This same ideacould be generalized to _anyof a vast number of

effects on health care utilization, including community immunization rates, climactic change,

standard of

living

inthe hospital'sservicearea, and _manymore.

In summary, utilization forecasting in health care is often aimed at predicting utilization

levels far into the future (months as opposed to _days), as this impacts the hospital's

construction/staffing priorities as well as its bottom line. For its importance, many of the

commonlyusedmodels for thesepredictions aretoo simplistictoaccuratelycapture changesthat

can affect utilization. _{The sensitivity} of hospitals to patient length of _stay, for example, is

highlighted in Myers and Green's paper (2004): "For a large system such as [the case _study

hospital], a

0.1-day

changein LOS resultsin a 10- to_{20-bed swing in} required

capacity" (p.37).

Therefore, the models that have developed to address this _{complexity have} added additional

factors to these_simple, basicmethods.

3.2 Short-Term

Forecasting

in Health

Care

While a number of instances are found that examine long-range utilization of hospital

resources, fewer entries in the literature address short-term _{(i.e., hour-to-hour)}

forecasting

in

health care situations. The methods and scope found in the short-term utilization

forecasting

literature differ significantly from the _relatively simple methods discussed above. In particular,

the applications of

forecasting

seem to focus on single hospital units, most _popularly the

Emergency Department(ED). Further, the methods thatare used are more complex and capable

of_handlingthe _{high variability} and _{complexity inherent in hospitals.} Two examples_employing

statistical methods and artificial neural networks arediscussed in this section.

3.21 Time-series methods for short-term bed utilization

forecasting

The statistical methods used in _{Jones, Joy,} and Pearson's 2002 paper _"Forecasting

demand of _emergency

care"

are _notably different from the methods used in longer-range

forecasting. The authors use an Auto-Regressive Inductive_MovingAverage _(ARIMA)model to

predict the

daily

number of occupied beds in a hospital due to _emergency admissions. ARIMA

isa statisticalmodel thatassumes that observed values of a process arethe resultof a

"shock"

to

anoverall process average,usingthe form

P q

Xt =

Y]

_(f>iXt-i

+

_8, +

Y]

OiSf-i

(Jones' Jy> andPearson> 2002,p. 297)

Inthe above _{equation, cp} and 9 are model _parameters, and e is an error term. The shock

term can act as a type of seasonal modeling. This seasonal component is enhanced in the

authors'

work

by

use oftheSeasonal _ARIMA, orS_ARIMA, modelto addressrecognized_"strong

seasonality"

in the data (p. 298). This extended model adds two more terms to capture the

history

of observed values and shocks inpast seasons. This methodhas promise as a

forecasting

method in this type ofapplication: the authors achieve a root mean squared error of_{only 3% in}

their forecasts of the mean bed usage for emergency admissions. _{Nevertheless,} this technique

uses parameters that are fixed once _they are _determined; as _{such, these} models are

likely

not

suitablefor

highly

dynamic and _{non-stationary} systems.

3.22 Neural networks for mean E.D. wait time

forecasting

Artificial neural networks (or _simply neural _networks) are another sophisticated

forecasting

method thathas been appliedto this specific problem_{byKilmer, Smith,} and Shuman

(1997). Aneural network (see full discussion of neural networksin Section 3.4 ofthis literature

review) is a model thatuses numerous, simple parallel processorsto handletasks such as pattern

recognition, classification, machine _control, and _others, as well as for forecasting. The weights

between the processors are changedbased on feedback aboutthe network's estimation errors, so

that through_training (ratherthan programming), thenetwork

"learns"

the_relationship ofinterest.

As a_{result, the} network is extremely flexible to change as the system changes, andonce _trained,

forecast values can be generated almost instantly. The authors in this case use the neural

networkas ameta-model extension of a computersimulation model: the model creates data from

the model system, which the network then

"learns."

The application, here again, is an

EmergencyDepartment, andthefocus oftheworkison _estimatingmean patienttime intheE.D.

The authors begin

by building

a simulation model of the _{E.D., where patients} arrive and are

treated

by

various people and resources in the system. (See Section 3.5 ofthis literaturereview

for a discussion of computer simulation). Once the model is _built, the authors build parallel

networks to estimate the mean and variance of

patients'

lengths of _{stay in} the E.D. These

networks are shown

by

the authorstoperform atleastas well asthe simulation model. _However,

theinputs used inthe model

-theintensive care unit _{bed waiting time,} general/surgicalunit bed

waitingtime, labservice _time, and

X-ray

servicetime

-are notinformation thatwouldbe readily available in a real-world _situation, and indeed if this information were _known, a sophisticated

model-network solution might not be necessary. _{Further, only}a single hospital unit is modeled

in the

authors'

work. _Still, this _methodology allows an _outstanding richness and

flexibility

of

modeling, and it isconsidered a superiorapproach toothertechniques thus far discussed.

Research in health care

forecasting

has been progressing for some _{time, focused mostly}

on strategic _{capacity planning in hospitals.} Where short-term bed availability

forecasting

is involved, the research appears still to_{be in early} stages; however, the richness ofthe _technology

in use and the results obtained thus far are encouragements that these problems can be treated

effectively. Because health care systems are complex, variable, and non-stationary, sophisticated

methods like the meta-model computer simulation and neural network approach are most

appropriate for addressingthe

forecasting

ofbed availability inthe shortterm.

3.3 Other Methods for Short-Term Bed Utilization

Forecasting

As highlighted in the above sections of this literature review, any method that will

must have a combination of several features: it must produce quick results for real-time

forecasting; it must account for_{hourly, daily,} and other cyclical changes in the system; it must

adapt as the system changesover_time; anditmustbe capable of_capturingenoughof ahospital's

complexity and inter-relatedness to _adequately model measures based on the whole system's performance. Three explanatory methods are discussed in this section: _regression, expert

systems, andcomputer simulation.

3.31 Regression for

forecasting

Regression is a general term _encompassing a number of _{techniques, among} which the

common theme is that an output result can be explained

by

some combination of one or more

input variables. Regression techniques _{vary greatly in} sophistication. The basic form of

regression is linear _regression, which supposes that an output of interest arises as a linear

function of one other variable. While this technique is simple, multiple linearregression is more

sophisticated, modeling an output as a function of multiple _{variables; further,} non-linear regression techniques allow such a model to transcend the linearrestrictions of simpler forms of

regression.

Whilethese techniques areuseful in manycases, and indeeda multiple regression model

may be abletocapture systembehavior verywell, the problem with these techniques is that _they

are static. Foreach technique,parameters ofthemodel mustbeestimatedthat formalizethe _way

inwhich inputs combine to create an output. Once these parameters are _{determined, however,}

they arefixed. Therefore,thoughregressionis a usefultechnique thatcan explain alotofsystem

behavior, ratherthanjust followatime series ofoutputs,it doesnot meet the needs ofthe current

thesis problem.

3.32 Expert systems for

forecasting

Expert systems are a good example of the potential _{for using human} expertise for

problemsolving. Anexpert systemis a computerprogramthattransforms experts'

opinions into

specific rules, so that the system is "sufficient to perform as a skillful and cost-effective

consultant"

(Bramer, 1982). The systemtakesin system_inputs, andbased on the values ofthose

inputs and the rules encoded in the _system, it generates a decision or predicted value. _Thus, a

goodexpert system should make the same decisions that a humanexpert would _make, basedon

the sameinput information.

Expert systems have already been created to address _{issues in many fields,}

including

knowledge engineering, medicine, chemistry, and others. One example of an expert system

applicationin the clinicalhealth care areais the HERMES system in the work of_{Bonfa, Maioli,}

Sarti, Milandri, and Dal Monte (1993). In this _application, an expert system is built to

determine,fromvarious_inputs, theprognosis of patients with potential liverdiseases. Anumber

of experts are interviewed to glean from them the process and rules _by which prognoses are

successfullymade. Theirprocess continued with _{encoding into} a_system, refinement ofthe rules

(upon review with the experts), and

finally

implementation. Systems like this are capable of

filling

"the need to compare and evaluate forms of _reasoning and to manage elements which cannot be

rigidly

schematized"

(p. 240). A drawback ofthis method is that _it, like regression

techniques, is a static system once its rules have been specified. _Further, from a practical

standpoint, thetimerequired to

develop

such a system(given the_interviewing and rule _encoding requirements) mayprohibitits implementation in areasonabletimeframe. Themodel discussed

bed utilization dynamics can be expected to change _{significantly,}

rendering such a system

obsolete

by

thetime it is finished.

3.33Computer simulation for

forecasting

A third

forecasting

method that seeks toexplain output_behavior, ratherthan just follow

its trend through _{time, is computer simulation. Simulation is a}

preferred method for

learning

about _complex, inter-related _systems, and as _such, a valid and accurate computer simulation

modelis capableof_producingreliable predictionsoffuture performance ofthose systems.

While discussion of computer simulation is deferred to section 3.5 of this literature

review, an example application ofcomputer simulation shows _{its ability}to capture the behavior

of whole systems. _{Bagust, Place,} and Posnett _{(1999) study hospital bed} usage and the effect of

emergency admissions on the _system, through the use ofcomputer simulation.

By

_modeling a

whole hospital and _enabling the _tracking of various measures of_interest, the authors are able to

reachsignificant conclusions _regarding the _availabilityand usage ofbeds: _{specifically, that}when

thehospital in this _study was more than 85% _occupied, therisk of_runningout ofavailable beds

became significant, with long-term impact. "Even a_relativelylow risk offailure can disrupt the

operation of ahospital for a considerable time: at 85% mean _occupancy, a hospital that runs out

ofbeds for four days in a year _{may be disrupted for up} to eight weeks in

total"

(p. 157). This

example demonstrates the type of larger-context analysis that can make simulation models

powerful for forecasting. Simulation as a

forecasting

method, however, is not without its

drawbacks. The drawback associatedwithcomputer simulationis that, like thestatistical models

above, it must be manually adjusted to some degree as the real system changes. _Further,

simulation models are not practical as a real-time _forecasting method _because, in order to

generate precise confidence intervals for the output measure of _interest, the model must be run

fora significant amount of simulatedtime.

From the review ofliterature thus far, it is apparent that _{simple, static methods} will not

suffice to modelthe _complexityand dynamics ofshort-term hospital bedavailability. _However,

neural networks(discussed in the

following

_section)and computer simulation (in section _3.5) are

effective _{complementary technologies,} as was showninthe work of_{Kilmer, Smith,} and Shuman

(1997), thatcanlead to _accurate,flexible forecasts.

3.4Neural Networks

An artificial neural network is a model that _{is conceptually} similar to the human brain.

Specifically, a neural network employs parallel distributed processingto perform _{computations,}

andit is composed of

"neurons."

In the_brain, neurons receive inputs from the outside world or

from other _neurons, and _they fire an output signal if the inputs are sufficient to overcome a

required activation level. The neurons are connected _by synapses whose strengths change

dynamically;

learning

takes place _by the reinforcement or _weakening of _{these connections,} so

thatsignals areprocessed correctly. Aneural networkfunctions inmuchthe same way: inputs to

each artificial neuron are received _along weighted

"synapses;"

the output ofthe neurondepends

ontheweighted sum oftheseinputs. The

following

section ofthis literaturereviewdiscussesthe

structure andfunction of neuralnetworks, which are usefulfor many tasks. These tasksinclude

prediction, function approximation, speech recognition, pattern _{classification,} and machine

control. _{Further, forecasting} applications of neural networks are discussed, and

finally

drawbacks to the use ofneural networks arediscussed thatwill motivate the use ofatwo-phase

3.41 Structure of the brain

The human _{brain, weighing in} at a mere 3.25 pounds (Blinkov and _{Glezer, 1968),} can

outperformthe most advanced computersmade

by

_{humans in performing many}tasks. Thegame

of chess is justone example ofman's _ability to out-think machinein many arenas. _Recognizing

patterns (such as a particular person's _speaking voice, or _{face), making} estimations based on

incomplete and flawed _data, and the adaptation of a learned relationship based on changed

circumstancesareotherexamples ofareaswhere humansexcel.

The basic unit ofthebrain's function is the neuron. The neuron's _elementarystructureis

as follows: it has branches (called _dendrites) to receive input signals from other neurons. The

junctions between neurons, where the signal propagation takes place, are called synapses. A

neuron has a cell _body with a specialized area where the input signals combine (called the axon

hillock). If the combination of the input signals received within a small window of time is

sufficientlystrong, then anoutput signalis _{fired along}the outputbranch (calledthe axon), where

itfans out andisgiven as an inputto other neurons.

The neurons are interconnected as a network, and as networks go the brain _{is extremely}

large: _{"The multiplicity}of neurons andinterconnectionsin ahumanbrain farexceeds thatof_any

artificial neural network. It is estimated that the brain contains on the order of

(10)"

neurons,

comparable to the number of stars in our galaxy, and

(10)'4 to

(10)'6

synaptic interconnections

among these"

(Chester, 1993). _Using the more conservative synaptic estimate of 100 trillion

synapses, this represents on the order of 1000 average connections per neuron. Consider a

hypothetical demonstration of the parallel _computing power of the brain: If a single neuron

processes asingle signal inone millisecond_(certainlynot fast

by

standards oftoday's _PCs), then

after that one _millisecond, that same signal is passed to 1000 neurons. Ifwe assume in this

example that the brain's neurons are all connected and _passing signals to each other, then that

signal has reached one million neurons after _onlytwo _{milliseconds,}

illustrating

the powerofthe

braintocompute andthe reasonit has inspiredthe developmentofartificial neural networks.

3.42 Structure of artificial neural networks

As mentioned _above, neural networks are structured as a simplified model ofthe brain.

As with the _brain, each "neuron" of a neural network has weighted _inputs, a simple_processing

function, and outputs (shown in Figure 1). These neurons are connected into _layers, which in

turn form whole networks. Networks are trained to "learn" appropriate _weightings, so that the

weightsneed notbespecified and can adapt continuously.

Input units toa neural network act as the artificial _analog to an external stimulus to the

human brain (a pin prick or muscle contraction, for example). In a mathematical sense, the

inputs are those factors that are thought to have an effect on the output ofthe network. In the

representative neuronin Figure _1,theinputunits arerepresented_{byx_, x2, x?,}x4, and x5.

ho=i|v

The neuron's weights in Figure 1 are represented

by

_w0 through w5. Whereas in a

biological neuron, signals_{may be} of

differing

_strengths, inartificial neuronsboththe inputsignal

and the weight _{connecting it} to the neuron _may change. _Indeed, it is through the adjustment of

the weights in a network that optimal performance is achieved. This is done through _training,

discussed below. In this _{training process, the} weight adjustments will be based on the errors

committed

by

the network. To begin, weights are _{initialized, usually}either to zero or to small

random values.

The weight _wo in theFigure above andthe node_Xo are representative ofthe use of abias

node in the neuron. In similar fashion to the y-intercept of a linear equation, a bias allows an

additional degree of

flexibility

in modeling, so that the _{relationship learned by}the networkis not

restrained topass through an origin ifall the input values are zero. The bias "input" _{is usually}

+1,withtheweight w0 learningin similarfashionto the other neuronsinthe network.

An activation function (or

"transfer"

function or

"squashing"

function)is the actiontaken

inside the "body" of the artificial neuron to determine the neural output. The input to this

function is some combination of the weighted input signals, usually as a straightforward

weighted sum. This yields, as a general activation,

Yj

=f(Z(Xj*Wjj)+b). Here,

Yj

is theneuron's

output,Z(Xj*Wji) istheweighted sum oftheinputsand their weights, andb isthebias value.

While any number of different effects can be achieved through the use of activation

functions, three are most common. The value _may be simply passed on as is (with no

transformation), it may be limited tocertain discretevalues (aswith a_{stepfunction),} orit may be

squashedtosome valueinthe range _[0,1] or_[-1,1] (withthe use of asigmoid function). Sigmoid

functions have the _{desirable property (seen in Figure 2}

(b) and _{(c) below)} of _{asymptotically}

approachingadesired limitas the input's magnitude_increases to_infinity, while

becoming

_nearly

linearwhen the input is between 0and 1 (forthe

log

sigmoidfunction in _(b)) or-1 and 1 (forthe

hyperbolic tangent sigmoid function in (c)). The further importance of the types offunctions

shown in the Figure is that _they are _{differentiable,} which becomes important to the error

back-propagation_training algorithmdiscussed below.

(a) (b) (c)

Figure 2: Neural transferfunctions: _{(a)non-squashing}linear_{function, (b) log}sigmoid_squashing

function,and _(c)hyperbolic tangent sigmoid _{squashing function.}

The activity in a single artificial neuron is finished when it passes its output signal

(resulting

from the activation _function) toall the neurons to whichit is connected. Atthis point,

thenext levelof network organization

-the layer

-comesintoplay. Alayerof a neural network

(see Figure 3 _below) consists ofthose neurons _processing the same set of input signals at the

same time. A single brain neuron _may be connected to hundreds of other neurons in all

directions; in contrast, the most common neural network structure, called the feed-forward

architecture, only has weights _{going from} one layer to the next. That _is, neurons aren't

connected to themselves or other neurons in the same _layer, and their outputs are not fed

backward in the network. There are, however, infinite ways of_arranging neural _networks, and

Figure 3: Arepresentativelayer in a neural network. Outputs fromprevious layerare

represented

by

boxes at_left;outputs fromcurrent layerare represented _byboxes at right.

A neural _{network, then, is} a collection of neurons - from

one node _up to _many nodes in

many layers

-arranged as described above. The layers between the inputs and the output_layer,

called hidden layers, are what enable neural networks to model such a _variety of different

systemswith accuracy. In fact,giventhe correct architecturefora neural network andtheproper

training, itcanbeshown toapproximate _{any function}to _anydesireddegreeof accuracy.

3.43

Training

neural networks

The usefulnessof neural networks is _dependent, inthefirstplace, on_choosing thecorrect

network architecture and inputs. Once these decisions are made, however, it is _training that

makes networks useful and enables them to be adaptive asconditions change in the system _they

arebuiltto model.

With respectto neural _{networks, training simply}means the adjustment ofthe weights in

the network, with the goal of_reducing the total prediction error the network commits. That is,

for any configuration of_weights, and given an input vector_{to the network, the} output result will

be different and will be closer _to, or farther _from, the target value

depending

on that

configuration. While the focus of much neural network research has focused on different

methods of_training neural _networks, errorback-propagation _(BP) is the method thathas become

most common since its introduction nearly 20 years ago

by

_{Rumelhart, Hinton,} and Williams

(1986). This method minimizes the total error of the network across a _training data set, by

adjustingthe setting of each network weight_{proportionally} to its contribution to the error. _First,

an input pattern is fed the network in order to generate a network output. _Second, the error is

calculated _using a known desiredoutput. _Then, this erroris propagated backwards through the

network,

beginning

with the output neuron and _{working backwards.} At each step, the weights

are changed

by

an amount that depends on the input to that neuron and the neuron's

learning

rate. The

learning

rateis a parameterthancanbe adjustedtocontrolthespeed oftraining.

The number of_training input vectors that are needed for successful training, as well as

thenumberoftimes the set of vectors will needto be seen

by

the network, will _vary

largely

with

the structure ofthe network and the _difficulty of the problem. The number of _training input

patterns can _{be roughly} determined _using a rule ofthumb from Swingler (1996). The author's

guideline is to choose_N, thenumber ofinputpatterns, as N > _W/, where W is the total number

of network_weights, and eis thedesired level of errorto reach. For a network with 25 _weights,

Considering

that such a network wouldbe small compared to_practically applied networks, this

guideline illustratestheneed of neuralnetworksforextensivedata.

After obtaining the appropriate amount of _{training data,} the next _step must be to

pre-process those data for best use in the network. The most common _{pre-processing involves}

removing outliers and _scaling the data.

Scaling

is done so that the network's various inputs do

not

individually

influence the network output too _much, andthemost common _{scaling is into}an

interval similarto(orthesame _as)the outputinterval thatis desired.

A network will see each input pattern numerous times in order to learn the appropriate

weight settings to minimize error over the whole data set. Each instance of_presenting all the

input patternsis one pass or epoch. Thenumberof_{training passes,} in conjunction with the size

ofthenetworkandthenumbers of_trainingpatternsineach_pass,mustbe controlledto ensurethe

best possible training. Iftoo _many passes are performed, the network will _eventually fit every

datapointofthat set_closely, at the sacrifice of not_beingable togeneralize tonew information in

the future. However, too few passes (or too little data or too small a _network) will result in

insufficient _training to modelthe _training data or _{any future data.} Commonly, networks will be

trained to reach a certain performance goal, so that _{(for example)} the average error or the

maximumerroris belowa certain threshold.

Lastly, once a network has been sufficiently trained, its further usage _{may be in} one of

two modes that will have a significant impact on its adaptability. In _essence, the modeler must

choose whether the network will continue to learn from new data or _simply use its current

learning

goingforward. Ifthenetworkis notset_up toincorporate sometypeof_feedback,then a

networkin simulation mode is astatic network

-thatis,no

learning

will occur. This is afurther

consideration for practitioners when _establishing networks that will be intended to work over

timeinthereal world.

3.44 Neural networks applications in

forecasting

An important application of neural networks is in forecasting. Because neural networks

can, when_{properly built} and_trained, mimic _{virtually any input-output relationship, they} are well

suited to this task of _predicting a system's future performance. _Further, the _ability of these

networks to continue

fine-tuning

their

learning

as new data are assimilated is a

key

asset in

environments that are complex and dynamic. _Below, two

forecasting

applications of neural

networks are discussedasillustrationsofthe benefits of_using neural networksforthese tasks.

Afirst example is provided

by

Hsu and _{Yang (1991),} who use neural networks for

short-termelectric load forecasting. Thisproblem addressedin thispaper

-the short-term _forecasting

of a resource utilization measure- bears important

similaritiesto the currentthesis problem, and

the authors illustrate that these types of problems can be successfully addressed _using neural

networks. The utilization measures in question, peak load and _{valley load,} are each forecasted

by a separate networkinthe

authors'

work. Eachnetwork'sinputsare temperature data fromthe

current _time, as well as temperature data and electric load information from past observations.

Further,the days are split into typesofdays: "Sundays and _holidays; Mondays and the

day

after

a_{holiday; Saturdays;} and normal days"

(p. 416). Thus, the time-dependenttrends in this system

are atleast _partiallyaccounted for inthe design ofthe neural network and selection ofits inputs.

less for each of the 24 hours in the prediction _horizon, with average error around 0.5%.

By

incorporating

past information and _explanatory variables into a neural network structure, the

authors are able to achieve _very close performance on this problem and declare "accurate

forecasting

of

hourly

loads canbe achieved

by

the neural networkin a_veryefficient

manner" (p.

418).

Walczak, Pofahl, andScorpio ₍₂₀₀₃₎ provide a second

key

exampleof a neural network

applicationin _{forecasting, by}

determining

_emergencypatient needsbased on "informationthatis

available within the first 10 min (without the use of invasive _tests) ofthe patient's

arrival"

(p.

446). Here again, two separate networks are used. _However, in _{this work, the two} networks

address two different types of patients (pediatric trauma patients and pancreatitis _patients) that

require _{qualitatively different} types of _care, rather than to model two different measures of

interest. The authors use a single network(foreach patient_type) topredictboththe _severityof a

patient and that patient's expected length of_stay (LOS): "The _likely correlation between

injury

severity and LOS indicates that if characteristic variables of _{injury severity} can be _identified,

they can also be used to model LOS"

(p. 446). The inputs for each model include information

like type of_injury, blood pressure, and otherinformation _readily available to the potential users

ofsuch a system. The resultingpediatric trauma LOS prediction model is able to predict LOS

within one

day

for 52% ofpatients, in a population where LOS ranged from zero to 146 days.

Improvement _{in predicting} the LOS allows better planning in the hospital, and it also has an

important effect on bed utilization (with higher utilization _resulting from more

longer-staying

patients, for example). _Clearly, neural networks are a capable _technology for

filling

the

forecasting

needs ofhealth care systems.

Drawbacks of neural networks

While neural networks have been shown to be excellent tools for short-term utilization

forecasting, there are important limitations in their ease of implementation. _First, neural

networks require an extensive amount of data for training. This is because of the _way that

networks are trained

-the weights are changed

by

_(usually) small amounts with each

presentationofinputs. Asreferenced above, Swingler(1996) recommendsuse of a guideline for

determining training data needs: based on an acceptable level of _{error, s,} and the number of

weights in the network, W, the number of_training patterns should be N > W/ e. That _is, ifthe

acceptable erroris 0.05, thenat least 20 input patterns shouldbe available for everyweightinthe

network. Put simply, as the network grows in size, the number of input patterns needed for

trainingincreases rapidly.

In addition to the data needs for _training neural networks, a second pitfall is in the

selection of network inputsand architecture. An element oftrial-and-error is apparent in the vast

array of neural architectures that have been chosen to model different problems, and even to

model _very similar problems. Computersimulation, as used in Kilmer, Smith, Shuman (1997),

offers a convenient _way to address the concerns of _training data and input selection. As

mentioned previously, simulation models capture the richness and randomness of processes in

detail and allow a treatment of whole systems. The next section explores computer simulation

andhighlights itsusefulness asacomplement to neural networksforforecasting.

3.5 Computer Simulation

3.51 Definition of computer simulation

Computer simulation is a tool used to analyze and optimize real-world systems. In

simulation models provide a simplified process model of the whole real-world system. From

these _models, analysis can be made on a host of measures

(including

examples like resource

utilization,

entities'

time in the system, and lengths of queues that build in front of various resources).

Entities are the objects _{(or people)} that flow through the process and are acted upon. In

terms ofhealth care,patients are the _primary entities in anysimulation model. _{Isken, Ward,} and

McKee ₍₁₉₉₉₎ and Cahill and Render ₍₁₉₉₉₎ both corroborate this _choice, and _indeed, the

patients are (for most_{hospital modeling situations)}the main people or objects ofinterest moving

through the system. Entities in simulation models are assigned various _attributes,which are used

to create various behaviors. _Using the example of hospital patients, attributes can _specify a

patient's _diagnosis, the length oftime a certain type of patient remains in a given hospital unit,

the _severityofthe _diagnosis, or_anyother patient characteristics. _Further, thearrivals of entities

to the system can be controlled to _any volume ordegree of randomness _desired, with statistical

distributions used to generate random arrivals. Arrivals can also be modeled on schedules, so

that(asin _hospitals)we_may showmore elective patients _arrivinginthe daytime

during

theweek

and more _emergency patients _arriving at night and on the weekends. In short, the modeler

controlsthedegreetowhich themodel reflects the actual system.

The machines, people, or other things _acting on entities are modeled as resources in

simulationmodels. In manymodels, resources are fixed as to their_{location; however,} there_may

also be types of _moving resources (such as transporters _moving parts from station to station).

Importantly

to the present thesis, hospital beds are _commonly modeled in the literature as

resources. _{Isken, Ward,} and McKee _(1999), Cahill and Render _(1999), Harper and Shahani

(2002), and others _employ thisconvention in their respective works. _{Doctors, nurses,} and other

hospital staff would also _{commonly be} modeled as resources in hospital simulations. The

utilization of these resources is then _easily tracked through the simulation software, so that its

results can beuseful for analysis. As with _entity attributes and _arrivals, resource characteristics

canbe controlled to _{many different levels} ofdesired control, allowing the modeling of resource

efficiencies, down times (or work _breaks), and schedules of resource capacities. _Furthermore,

resource queues are an essential part of simulation models that

help

determine the performance

of the system and the length of time entities will wait to be served _by a resource. The

combination of_{entities, resources,} and queuesin a simulation model allows a wealth ofdifferent

systemstobemodeled to _{virtually any desired degree} of accuracy.

3.52 Applications of computer simulation in health care

The applications of such a rich _technology as simulation are, of course, many and

extremely varied. In general,however, one _{may say} that within the context of system analysis,

and there are two main categories: optimization _(asking the question, "How can the system be

improved?") and system design _(asking "How would a given system operate if it were built?").

Design models allow the modelerto see systemsthat donot yet _exist, while optimization models

focus on thecurrent system and_changing variables within it. The common theme, again, is that

the analysis ofthese systems (whether_theyexistyet or _{not) may}be conducted at a _{substantially}

lower cost and risk than _implementing the same changes in the actual system (or

building

the

A first example of simulation _{modeling for} optimization is the work of Harper and

Shahani (2002). The authors build a_{comprehensive,} whole-hospital model and _apply the model

to_{study bed capacity planning (at}a _{monthly level). Their}modeluses patients as entities _moving

through the hospital from when _they enter until _they are discharged. Arrival distributions and

length of _{stay (LOS)} distributions are used to model the randomness of the hospital patient

processes, and pools of beds are modeled as resources that the patients attempt to use upon

entering the system. This model includes logic to remove patients from the system if a bed

cannot be seized in a reasonable _{time, mimicking} the actual practice in full hospitals of

transferringemergencypatients and_{re-scheduling} elective patients.