chapter 10. simulation an integrated approach to improving quality and efficiency daniel b....

Chapter 10. Simulation

An Integrated Approach to Improving Quality and Efficiency

Daniel B. McLaughlinJulie M. Hays

Healthcare Operations Management

Copyright 2008 Health Administration Press. All rights reserved. 10-2

Chapter 10. Simulation

• Uses of Simulation

• Simulation Process

• Monte Carlo Simulation

• Queueing (Waiting Line) Theory

• Discrete Event Simulation (DES)

• Advanced DES


Simulation

• Process of modeling reality to gain a better understanding of the phenomena or system being studied

• Simulation versus the “real world”- More cost effective- Less dangerous environment- Faster- More practical

• Does not require mathematical models or computers


Types of Simulation

• Performance

• Proof

• Discovery

• Entertainment

• Training

• Education

• Prediction


Simulation Process

• Model development- Define the problem or question- Develop the conceptual model- Collect data- Build computer model

• Model validation

• Simulate and analyze output


Simulation Process

Model Development

• Problem/ question definition

• Develop conceptual model

• Collect data• Build

computer model

Model Validation

• Quantitative comparison

• Expert opinion

Simulation and Analyses

• DOE• Replication• Data

collection, storage, and organization

• Analysis


Monte Carlo Simulation

• Model the output of a system by using input variables that could not be known exactly

• Random variables (those that are uncertain and have a range of possible values) characterized by a probability distribution

• Solution is a distribution of possible outcomes that can be characterized statistically


Simple Monte Carlo ExampleDistribution of Charges

Charges $20.00$ 30.00$ 40.00$ 50.00$ 60.00$ 70.00$ 80.00$ 90.00$

100.00$ 110.00$ 120.00$

Total 360Average 70.00$

302010

Number of Patients (Frequency)

50605040

10203040

0

10

20

30

40

50

60

70

Charges

Nu

mb

er o

f P

atie

nts

(F

req

uen

cy)


Simple Monte Carlo Example

• Fifty percent of the clinic’s patients do not pay for their services, and it is equally likely that they will pay or not pay.

• The payment per patient is modeled by:Probability of payment × Charges/patient = Payment/patient

• A deterministic solution to this problem would be: 0.5 × $70/patient = $35 per patient

00.10.20.30.40.50.6

Pay Do Not Pay

Pro

bab

ilit

y


Simple Monte Carlo ExamplePayment Distribution

Trial #

Coin Flip Payment

Die Total Charges

Patient Payment

1 H 1 7 70.00$ 70.00$ 2 T 0 10 100.00$ -$ 3 H 1 8 80.00$ 80.00$ 4 T 0 8 80.00$ -$ 5 H 1 9 90.00$ 90.00$ 6 T 0 8 80.00$ -$ 7 H 1 7 70.00$ 70.00$ 8 T 0 10 100.00$ -$ 9 H 1 9 90.00$ 90.00$

10 T 0 10 100.00$ -$

$- $60.00 $110.00

Payment

Nu

mb

er

of

Tri

als


Simple Monte Carlo ExampleThe Flaw of Averages

• On average each patient pays $35. However:

- Fifty percent of the patients pay nothing.- A small percentage pay as much as $120.- No individual patient pays $35.

• Monte Carlo simulation can reveal hidden information and a clearer understanding of the risks and rewards of a situation or decision.


VVH Monte Carlo ExampleCAP Payment Distribution

Created with BestFit 4.5, a software product of Palisade Corp., Ithaca, NY; www.palisade.com


VVH Monte Carlo ExampleInput Distributions

Probability Distribution of Cost of Reaching a Score Greater Than 0.90

$10,000 $30,000 $50,000Cost of Reaching a Score Greater Than 0.90

P(X

)

Probability Distribution of Quality Scores

0

0.02

0.04

0.06

0.6 0.7 0.8 0.9Quality Score

P(X

)


VVH Monte Carlo ExampleDeterministic Analysis

Profit = Revenue – Cost

Revenue = (Rev/mon × 12 mon/yr) × Quality

bonus or penalty

= ($250,000/mon × 12 mon/yr) × 0.01

= $30,000/yr

Cost = $30,000/yr

Profit = $30,000/yr – $30,000/yr = $0


VVH Monte Carlo ExampleCAP Pay-for-Performance Simulation Trials

Created with @Risk 4.5, a software product of Palisade Corp., Ithaca, NY; www.palisade.com

@RISK Data ReportData

OutputRevenue/

MonthRevenue/Year

Quality Score

Costs/ Year

Profit =

Revenue – Costs

Iteration/ Cell $B$14 $C$14 $D$14 $G$14 $H$14

1 155,687.16 2,699,013.25$ 0.841 17,032.684 (17,032.68)$ 2 244,965.38 2,903,593.00$ 0.765 15,443.749 (15,443.75)$ 3 257,408.31 2,924,186.25$ 0.785 26,655.609 (26,655.61)$ 4 335,716.84 3,441,799.25$ 0.653 31,370.799 (65,788.80)$ 5 232,497.83 2,857,697.00$ 0.824 46,067.852 (46,067.85)$ 6 249,375.09 3,169,170.50$ 0.839 27,132.934 (27,132.93)$ 7 234,730.83 2,771,886.50$ 0.867 28,037.871 (319.01)$ 8 192,825.16 2,906,499.00$ 0.687 29,651.076 (58,716.07)$ 9 243,230.81 3,045,998.00$ 0.872 44,706.762 (14,246.78)$


VVH Monte Carlo ExampleSimulated Distribution of Profits

-120 -90 -60 -30 0 30 60

5% 90% 5% -66.9894 29.7502

Mean=-19998.71

Distribution for Profit = Revenue - Costs/H14

Val

ues

in 1

0^ -

5

Values in Thousands

0.000

0.500

1.000

1.500

2.000

2.500

Mean=-19998.71

-120 -90 -60 -30 0 30 60

@RISK Student VersionFor Academic Use Only

Created with @Risk 4.5, a

software product of Palisade Corp.,

Ithaca, NY; www.palisade.co

m


VVH Monte Carlo ExampleTornado Graph

Regression Sensitivity for Profit = Revenue -Costs/H14

Std b Coefficients

Revenue/Month 12/T14 .097

Costs/Year/G14-.414

Quality Score/D14 .815

@RISK Student VersionFor Academic Use Only

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1Created with @Risk 4.5, a

software product of Palisade Corp.,

Ithaca, NY; www.palisade.co

m


Simple Queueing System

• Customer population—finite or infinite• Arrival process—often Poisson with mean arrival rate • Queue discipline—first come, first served (FCFS) is one

example• Service process—often exponential with mean service rate

ArrivalCustomer Population

Input Source

Buffer or Queue

Server(s) Exit


Queueing Notation

• A/B/c/D/E- A = Inter-arrival time distribution- B = Service time distribution- c = Number of servers- D = Maximum queue size - E = Size of input population

• M/M/1 queueing system- Poisson arrival distribution- Exponential service time

distribution- Single server

- Infinite possible queue length

- Infinite input population

- Only one queue


Queueing SolutionsM/M/1, <

Capacity utilization

= Percentage of time the server is busy

Average total number of customers in the system =

= Arrival rate × time in the system

arrivals between time mean

time service mean

time service 1/mean

arrivals between time mean1

rate service mean

rate arrival mean

ss WL


Queueing SolutionsM/M/1, <

Average waiting time in the queue

Average time in the system

= Average waiting time in the queue + Average service time

=

Average length of the queue (or average number in the queue)

)(

qW

11qs WW

)(

2

qL


VVH M/M/1 Queue Example

• Goal: Only one patient waiting in line for the MRI

• Data:

- Mean service rate () is four patients/hour and is exponentially distributed

- Arrivals follow a Poisson distribution and the mean arrival rate is three patients/hour ()



If one customer arrives every 20 minutes and it takes 15 minutes to perform the MRI, the MRI will be busy 75 percent of the time.

Capacity utilization of MRI

= Percentage of time MRI is busy

%754

3

%75minutes 20

minutes 15

1

1



Average time waiting in line

Average time in the system

Average total number of patients in the system or

= Arrival rate × Time in the system

= 3 patients/hour × 1 hour

= 3 patients

hours 5704

3

344

3.

)()(

qW

hour 134

11

sW

patients 334

3

sL



• Average number of patients waiting in line =

• VVH needs to decrease the utilization, = /, of the MRI process

• VVH can- Increase the service rate ()- Decrease the arrival rate ()- Do a combination of both

patients 2.254

9

)34(4

3

34

3

4

3

)(

22

qL


Discrete Event Simulation (DES)

Basic Simulation Model

• Entities are the objects that flow through the system.

• Queues hold the entities while they are waiting for service.

• Resources or servers are people, equipment, or space for which entities compete.


Discrete Event Simulation (DES)

Simulation Model Logic

• States are variables that describe the system at a point in time.

• Events are variables that change the state of the system.

• The simulation jumps through time from event to event, and data are collected on the state of the system.


DESRandom Data

1 2 3 4 5 6 7 8 9

0.17 0.37 0.36 0.59 0.14 0.17 0.24 0.06 0.35

0.17 0.54 0.90 1.49 1.63 1.80 2.04 2.10 2.45

0.21 0.56 0.02 0.37 0.34 0.11 1.02 0.01 0.20

Entity NumberExpon (0.33)

Expon (0.25)

Service Time

Inter-arrival Time

Time of Arrival 0.00


DESSimulation Event List

Upcoming EventsStatisticsAttributesVariable

Just Finished Event

En

tity

#

Tim

e

Eve

nt

typ

eL

eng

th o

f q

ueu

eV

aria

ble

Arr

ival

tim

e in

q

ueu

eA

rriv

al t

ime

in

serv

ice

Nu

mb

er c

om

ple

te

wai

ts in

qu

eue

To

tal w

ait

tim

e in

q

ueu

eA

vera

ge

qu

eue

len

gth

Uti

lizat

ion

En

tity

#

Tim

e

Eve

nt

1 0.00 Arr 0 1 0.00 0.00 0 0 0 0 2 0.17 Arr1 0.21 Dep

2 0.17 Arr 1 1 0.17 0 0 0 1.00 1 0.21 Dep3 0.54 Arr

1 0.21 Dep 0 1 0.00 0.00 1 0 0.19 1.00 3 0.54 Arr2 0.77 Dep


DESSimulation Event List

Just Finished Event

Upcoming EventsStatisticsAttributesVariable

En

tity

#

Tim

e

Eve

nt

typ

eL

eng

th o

f q

ueu

eV

aria

ble

Arr

ival

tim

e in

q

ueu

eA

rriv

al t

ime

in

serv

ice

Nu

mb

er c

om

ple

te

wai

ts in

qu

eue

To

tal w

ait

tim

e in

q

ueu

eA

vera

ge

qu

eue

len

gth

Uti

lizat

ion

En

tity

#

Tim

e

Eve

nt

3 0.54 Arr 1 1 0.54 1 0 0.07 1.00 2 0.77 Dep4 0.90 Arr

2 0.8 Dep 0 1 0.17 0.21 2 0.3 0.35 1.00 3 0.79 Dep4 0.90 Arr

3 0.8 Dep 0 0 0.77 3 0.3 0.34 1.00 4 0.90 Arr4 1.27 Dep


DES Arena Screenshot

Arena® screen shots reprinted with permission from Rockwell Automation.


DES—Arena OutputArrival rate = 3 patients/hour; Service rate = 4 patients/hour; 200 hours



DES—Arena Output Arrival rate = 3 patients/hour; Service rate = 4 patients/hour; 200 hours



VVH Simulation• Current situation—on average, 1.5 patients in queue• Goal—1.0 patients in queue• Solution—decrease arrival rate or increase the

service rate• Simulation results:

- Decrease arrival rate to 2.7

- Increase service rate to 4.4

• Actual improvement:- Service rate of 4.2 patients/hour

- Need arrival rate of 2.8 patients/hour


DES—Arena Output Arrival rate = 2.8 patients/hour; Service rate = 4.2 patients/hour; 10 hours



Simulation

• Simulation is a powerful tool for modeling processes and systems to evaluate choices and opportunities.

• Simulation can be used in conjunction with other initiatives such as Lean and Six Sigma to enable continuous improvement of systems and processes.

chapter 10. simulation an integrated approach to improving quality and efficiency daniel b....

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