Transcript simulator

Chapter 15.
Simulation
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
1
Outline



Simulation Process
Monte Carlo Simulation Method
– Process
– Empirical Distribution
– Theoretical Distribution
– Random Number Look Up
Performance Measures and Managerial Decisions
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
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When Optimization is not an option. . .
SIMULATE!
Simulation can be applied to a wide range of
problems in healthcare management and
operations.
In its simplest form, healthcare managers can use
simulation to explore solutions with a model that
duplicates a real process, using a what if approach.
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
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Why Simulate?
To enhance decision making by capturing a
situation that is too complicated to model
mathematically (e.g., queuing problems)
It is simple to use and understand
Wide range of applications and situations in
which it is useful
Software is available that makes simulation
easier and faster
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
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Simulation Process
1. Define the problem and objectives
2. Develop the simulation model
3. Test the model to be sure it reflects the situation
being modeled
4. Develop one or more experiments
5. Run the simulation and evaluate the results
6. Repeat steps 4 and 5 until you are satisfied with the
results.
Chapter 15: Quantitatve
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Management
Yasar A. Ozcan
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Simulation Basics
We need an instrument to randomly
simulate this situation. Let’s call this the
“simulator”.
Imagine a simple “simulator” with two
outcomes.
Chapter 15: Quantitatve
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So how can it help us?
Let’s look at a health care example.
How can we simulate the patient arrivals and
service system response?
System
Waiting Line
Service
Customers
arrivals
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Let’s use this simulator. . .
. . . to simulate patients arrivals in public
health clinic!
If the coin is heads, we will assume that
one patient arrived in a determined time
period (assume 1 hour). If tails, assume
no arrivals.
We must also simulate service patterns.
Assume heads is two hours of service and
tails is 1 hour of service.
Chapter 15: Quantitatve
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Yasar A. Ozcan
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Table 15.1 Simple Simulation Experiment for Public Clinic
Time
Coin toss
for arrival
Arriving
patient
1) 8:00 - 8:59
H
#1
2) 9:00 - 9:59
H
#2
3)10:00 -10:59
H
4)11:00 -11:59
Coin toss
for service
Physician
Departing
patient
H
#1
-
#2
T
#1
#1
#3
#3
T
#2
#2
T
-
-
-
#3
#3
5)12:00 -12:59
H
#4
H
#4
-
6) 1:00 - 1:59
H
#5
#5
H
#4
#4
7) 2:00 - 2:59
T
-
-
-
#5
-
8) 3:00 - 3:59
H
#6
#6
T
#5
#5
Chapter 15: Quantitatve
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Queue
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Calculation of Performance Statistics
System
Waiting Line
Service
Customers
arrivals
Arrivals
Queue (Waiting Line)
Service
Exit
?
?
?
?
Chapter 15: Quantitatve
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Table 15.2 Summary Statistics for Public Clinic Experiment
Patient
Queue
wait time
Service
time
Total time
in system
#1
0
2
2
#2
1
1
2
#3
1
1
2
#4
0
2
2
#5
1
2
3
#6
1
1
2
Total
4
9
13
Chapter 15: Quantitatve
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Performance Measures
Number of Arrivals
Average number waiting
Avg. time in Queue
Service Utilization
Avg. Service Time
Avg. Time in System
Chapter 15: Quantitatve
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But what if we have multiple arrival
patterns?


Can we use a dice or any other shaped
object that could provide random arrival and
service times?
We could use Monte Carlo Simulation and a
Random Number Table!
Chapter 15: Quantitatve
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Yasar A. Ozcan
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MONTE CARLO METHOD



A probabilistic
simulation technique
Used only when a
process has a random
component
Must develop a
probability distribution
that reflects the
random component of
the system being
studied
Chapter 15: Quantitatve
Methods in Health Care
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Yasar A. Ozcan
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MONTE CARLO METHOD
Step 1: Selection of an appropriate probability distribution
Step 2: Determining the correspondence between distribution
and random numbers
Step 3: Obtaining (generating) random numbers and run
simulation
Step 4: Summarizing the results and drawing conclusions
Chapter 15: Quantitatve
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Yasar A. Ozcan
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Empirical Distribution
If managers have no clue pointing to the type of
probability distribution to use, they may use an
empirical distribution, which can be built using the
arrivals log at the clinic.
For example, out of 100 observations, the following
frequencies, shown in table below, were obtained for
arrivals in a busy public health clinic.
Table 15.3 Patient Arrival Frequencies
Chapter 15: Quantitatve
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Management
Number
of arrivals Frequency
0
180
1
400
2
150
3
130
4
90
5 & more
50
Sum
1000
Yasar A. Ozcan
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Table 15.4 Probability Distribution for Patient Arrivals
Number
Cumulative Corresponding
of arrivals Frequency Probability probability random numbers
0
180
.180
.150
1 to 180
1
400
.400
.580
151 to 580
2
150
.150
.730
581 to 730
3
130
.130
.860
731 to 860
4
90
.090
.950
861 to 950
5 & more
50
.050
1.00
951 to 000
Chapter 15: Quantitatve
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Management
Yasar A. Ozcan
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Theoretical Distribution
The second popular method for constructing arrivals is to use
known theoretical statistical distributions that would
describe patient arrival patterns.
From queuing theory, we learned that Poisson distribution
characterizes such arrival patterns. However, in order to use
theoretical distributions, one must have an idea about the
distributional properties for the Poisson distribution, namely
its mean.
In the absence of such information, the expected mean of the
Poisson distribution can also be estimated from the empirical
distribution by summing the products of each number of
arrivals times its corresponding probability (multiplication of
number of arrivals by probabilities).
In the public health clinic example, we get
λ = (0*.18)+(1*.40)+(2*.15)+(3*.13)+(4*.09)+(5*.05) = 1.7
Chapter 15: Quantitatve
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Table 15.5 Cumulative Poisson Probabilities for λ=1.7
Arrivals
x
Cumulative
probability
Corresponding
random numbers
0
.183
1 to183
1
.493
184 to 493
2
.757
494 to 757
3
.907
758 to 907
4
.970
908 to 970
5 & more
1.00
970 to 000
Chapter 15: Quantitatve
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Yasar A. Ozcan
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Table 15.6 Cumulative Poisson Probabilities for
Patients
arrived
0
1
2
3
4 & more
Patients
served
0
1
2
3
4& more
Chapter 15: Quantitatve
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Arrivals: λ=1.7
Cumulative
Corresponding
probability random numbers
.183
1-183
.493
184-493
.757
494-757
.907
758-907
1.000
908 to 000
Service: μ =2.0
Cumulative Corresponding
probability random numbers
.135
1 to135
.406
136 to 406
.677
407 to 677
.857
678 to 857
1.000
858 to 000
Yasar A. Ozcan
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Finding Random Numbers


Numbers must be
both uniformly
distributed and must
not follow any pattern
Always avoid starting
at the same spot on a
random number table
2419
Chapter 15: Quantitatve
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Figure 15.1 Random Numbers*
* Random numbers are generated using Excel
Chapter 15: Quantitatve
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Management
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Table 15.7 Monte Carlo Simulation Experiment for Public Health Clinic
Time
1) 8:00 - 8:59
2) 9:00 - 9:59
3)10:00 -10:59
4)11:00 -11:59
5)12:00 -12:59
6) 1:00 - 1:59
7) 2:00 - 2:59
8) 3:00 - 3:59
Random
numbers
&
(arrivals)
616 (2)
862 (3)
56 (0)
583 (2)
908 (4)
848 (3)
38 (0)
536 (2)
Arriving
patients
#1,#2
#3,#4,#5
#6,#7
#8,#9,#10,#11
#12,#13,#14
#15,#16
Chapter 15: Quantitatve
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Management
Queue
#4,#5
#9,#10,#11
#11,#12,#13,#14
#12,#13,#14
Random numbers
&
(service)
Physician
Departing
Patients
764 (2)
180 (1)
903 (4+)
780 (3)
164 (1)
546 (2)
351 (1)
900 (4+)
#1,#2
#3
#4,#5
#6,#7
#8
#9,#10
#11
#12,#13,#14,#15,#16
#1,#2
#3
#4,#5
#6,#7
#8
#9,#10
#11
#12,#13,#14,#15,#16
Yasar A. Ozcan
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Table 15.8 Summary Statistics for Public Clinic Monte Carlo Simulation Experiment
Queue Service Total time
Patient wait time
time
in system
#1
0
0.5
0.5
#2
0
0.5
0.5
#3
0
1.0
1.0
#4
1
0.5
1.5
#5
1
0.5
1.5
#6
0
0.5
0.5
#7
0
0.5
0.5
#8
0
1.0
1.0
#9
1
0.5
1.5
#10
1
0.5
1.5
#11
2
1.0
3
#12
2
0.2
2.2
#13
2
0.2
2.2
#14
2
0.2
2.2
#15
0
0.2
0.2
#16
0
0.2
0.2
Total
12
8
20
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
24
Performance Measures
Using information from Tables 15.7 and 15.8, we can delineate the
performance measures for this simulation experiment as:
Number of arrivals: There are total of 16 arrivals.
Average number waiting: Of those 16 arriving patients; in 12
instances patients were counted as waiting during the 8 periods, so
the average number waiting is 12/16=.75 patients.
Average time in queue: The average wait time for all patients is
the total open hours, 12 hours ÷ 16 patients = .75 hours or 45
minutes.
Service utilization: For, in this case, utilization of physician
services, the physician was busy for all 8 periods, so the service
utilization is 100%, 8 hours out of the available 8: 8 ÷ 8 = 100%.
Average service time: The average service time is 30 minutes,
calculated by dividing the total service time into number of
patients: 8 ÷ 16 =0.5 hours or 30 minutes.
Average time in system: From Table 15.8, the total time for all
patients in the system is 20 hours. The average time in the system
is 1.25 hours or 1 hour 15 minutes, calculated by dividing 20 hours
by the number of patients: 20 ÷ 16 = 1.25.
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
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Figure 15.2 Excel-Based Simulated Arrivals
Chapter 15: Quantitatve
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Figure 15.3 Excel Program for Simulated Arrivals
Chapter 15: Quantitatve
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Management
Yasar A. Ozcan
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Figure 15.4 Performance-Measure-Based Managerial Decision Making
r1 < rt
r2 < rt
r2 >= rt
r1 >= rt
Marketing and
referral systems
to increase business
volume
Status Quo
Appointment
Scheduling
Increase
Capacity
Busy time during regular hours
r1 = --------------------------------Total regular hours open
Total busy time, including during over time
r2 = -----------------------------------------Total regular hours open
rt = Target utilization rate (e.g., 90%)
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
28
Advantages of Simulation
 Used
for problems difficult to solve
mathematically
 Can experiment with system
behavior without experimenting
with the actual system
 Compresses time
 Valuable tool for training decision
makers
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
29
Limitations
Does not produce an optimum
 Can require considerable effort to
develop a suitable model
 Monte Carlo is only applicable when
situational elements can be described
by random variables

Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
30
The End
Chapter 15: Quantitatve
Methods in Health Care
Management
Yasar A. Ozcan
31