Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2004 Thomson/South-Western
Slide 1
Chapter 15
Simulation







Advantages and Disadvantages of Using
Simulation
Modeling
Random Variables and Pseudo-Random Numbers
Time Increments
Simulation Languages
Validation and Statistical Considerations
Examples
© 2004 Thomson/South-Western
Slide 2
Simulation

Simulation is one of the most frequently employed
management science techniques.

It is typically used to model random processes
that are too complex to be solved by analytical
methods.
© 2004 Thomson/South-Western
Slide 3
Advantages of Simulation

Among the advantages of simulation is the ability to
gain insights into the model solution which may be
impossible to attain through other techniques.

Also, once the simulation has been developed, it
provides a convenient experimental laboratory to
perform “what if” and sensitivity analysis.
© 2004 Thomson/South-Western
Slide 4
Disadvantages of Simulation

A large amount of time may be required to develop the
simulation.

There is no guarantee that the solution obtained will
actually be optimal.

Simulation is, in effect, a trial and error method of
comparing different policy inputs.

It does not determine whether some input which was
not considered could have provided a better solution
for the model.
© 2004 Thomson/South-Western
Slide 5
Simulation Modeling

One begins a simulation by developing a
mathematical statement of the problem.

The model should be realistic yet solvable within
the speed and storage constraints of the computer
system being used.

Input values for the model as well as probability
estimates for the random variables must then be
determined.
© 2004 Thomson/South-Western
Slide 6
Random Variables

Random variable values are utilized in the model
through a technique known as Monte Carlo simulation.

The mapping is done in such a way that the
likelihood that a particular number is chosen is the
same as the probability that the corresponding value
of the random variable occurs.
© 2004 Thomson/South-Western
Slide 7
Pseudo-Random Numbers

Because a computer program generates random
numbers for the mapping according to some formula,
the numbers are not truly generated in a random
fashion.

However, using standard statistical tests, the
numbers can be shown to appear to be drawn from a
random process.

These numbers are called pseudo-random numbers.
© 2004 Thomson/South-Western
Slide 8
Time Increments

In a fixed-time simulation model, time periods are
incremented by a fixed amount. For each time
period a different set of data from the input
sequence is used to calculate the effects on the
model.

In a next-event simulation model, time periods are
not fixed but are determined by the data values
from the input sequence.
© 2004 Thomson/South-Western
Slide 9
Simulation Programs




The computer program that performs the simulation
is called a simulator.
Flowcharts can be useful in writing such a program.
While this program can be written in any general
purpose language (e.g. BASIC, FORTRAN, C++, etc.)
special languages which reduce the amount of code
which must be written to perform the simulation
have been developed.
Special simulation languages include SIMSCRIPT, SPSS,
DYNAMO, and SLAM.
© 2004 Thomson/South-Western
Slide 10
Model Verification/Validation

Verification/validation of both the model and the
method used by the computer to carry out the
calculations is extremely important.

Models which do not reflect real world behavior
cannot be expected to generate meaningful results.

Likewise, errors in programming can result in
nonsensical results.
© 2004 Thomson/South-Western
Slide 11
Model Verification/Validation

Validation is generally done by having an expert
review the model and the computer code for errors.

Ideally, the simulation should be run using actual
past data.

Predictions from the simulation model should be
compared with historical results.
© 2004 Thomson/South-Western
Slide 12
Experimental Design




Experimental design is an important consideration in
the simulation process.
Issues such as the length of time of the simulation and
the treatment of initial data outputs from the model
must be addressed prior to collecting and analyzing
output data.
Normally one is interested in results for the steady
state (long run) operation of the system being
modeled.
The initial data inputs to the simulation generally
represent a start-up period for the process and it may
be important that the data outputs for this start-up
period be neglected for predicting this long run
behavior.
© 2004 Thomson/South-Western
Slide 13
Experimental Design

For each policy under consideration by the decision
maker, the simulation is run by considering a long
sequence of input data values (given by a pseudorandom number generator).

Whenever possible, different policies should be
compared by using the same sequence of input data.
© 2004 Thomson/South-Western
Slide 14
Example: Dynogen, Inc.
The price change of shares of Dynogen, Inc.
has been observed over the past 50 trades. The
frequency distribution is as follows:
Price Change
-3/8
-1/4
-1/8
0
+1/8
+1/4
+3/8
+1/2
© 2004 Thomson/South-Western
Number of Trades
4
2
8
20
10
3
2
1
Total = 50
Slide 15
Example: Dynogen, Inc.

Relative Frequency Distribution and
Random Number Mapping
Price Change Relative Frequency Random Numbers
-3/8
.08
00 - 07
-1/4
.04
08 - 11
-1/8
.16
12 - 27
0
.40
28 - 67
+1/8
.20 to random
68 - 87
numbers in
Assign this
+1/4
.06 this range
88 - 93
+3/8
.04
94 - 97
+1/2
.02
98 - 99
Total 1.00
© 2004 Thomson/South-Western
Slide 16
Example: Dynogen, Inc.
If the current price per share of Dynogen is
23, use random numbers to simulate the price per
share over the next 10 trades.
Use the following stream of random numbers:
21, 84, 07, 30, 94, 57, 57, 19, 84, 84
© 2004 Thomson/South-Western
Slide 17
Example: Dynogen, Inc.

Simulation Worksheet
Trade
Random
Number Number
1
21
2
84
3
07
4
30
5
94
6
57
7
57
8
19
9
84
10
84
© 2004 Thomson/South-Western
Price
Change
-1/8
+1/8
-3/8
0
+3/8
0
0
-1/8
+1/8
+1/8
Stock
Price
22 7/8
23
22 5/8
22 5/8
23
23
23
22 7/8
23
23 1/8
Slide 18
Example: Dynogen, Inc.

Spreadsheet for Stock Price Simulation
A
B
1
Lower
Upper
2 Random Random
3 Number Number
4
0.00
0.08
5
0.08
0.12
6
0.12
0.28
7
0.28
0.68
8
0.68
0.88
9
0.88
0.94
10
0.94
0.98
11
0.98
1.00
12
C
Price
Change
-0.375
-0.250
-0.125
0.000
0.125
0.250
0.375
0.500
The look-up table named,
say, “StockChng”.
© 2004 Thomson/South-Western
D
Trade
Number
1
2
3
4
5
6
7
8
9
10
E
Price
Change
0.125
0.375
0.000
0.000
0.000
0.000
0.125
0.125
0.000
0.125
F
Stock
Price
23.125
23.500
23.500
23.500
23.500
23.500
23.625
23.750
23.750
23.875
A random number is compared to
the table, to choose a price change
Slide 19
Example: Dynogen, Inc.
“StockChng”

Spreadsheet for Stock Price Simulation
A
B
1
Lower
Upper
2 Random Random
3 Number Number
4
0.00
0.08
5
0.08
0.12
6
0.12
0.28
7
0.28
0.68
8
0.68
0.88
9
0.88
0.94
10
0.94
0.98
11
0.98
1.00
12
C
Price
Change
-0.375
-0.250
-0.125
0.000
0.125
0.250
0.375
0.500
D
Trade
Number
1
2
3
4
5
6
7
8
9
10
E
Price
Change
0.125
0.375
0.000
0.000
0.000
0.000
0.125
0.125
0.000
0.125
F
Stock
Price
23.125
23.500
23.500
23.500
23.500
23.500
23.625
23.750
23.750
23.875
The formula, “=VLOOKUP(INT(100*RAND()),StockChng,3)”, in
this cell had RAND() return a number between 0.68 and 0.88.
© 2004 Thomson/South-Western
Slide 20
Example: Dynogen, Inc.

Theoretical Results and Observed Results
Based on the probability distribution, the expected
price change per trade can be calculated by:
(.08)(-3/8) + (.04)(-1/4) + (.16)(-1/8) + (.40)(0)
+ (.20)(1/8) + (.06)(1/4) + (.04)(3/8) + (.02)(1/2) = +.005
The expected price change for 10 trades is
(10)(.005) = .05. Hence, the expected stock price after 10
trades is 23 + .05 = 23.05.
Compare this ending price with the spreadsheet
simulation and “manual” simulation results on the
previous slides: 23.875 and 23 1/8.
© 2004 Thomson/South-Western
Slide 21
Example: Mark Off’s Process
Mark Off is a specialist at repairing large metalcutting machines that use laser
technology. His repair
territory consists of the cities
of Austin, San Antonio, and
Houston. His day-to-day
repair assignment locations
can be modeled as a Markov
process. The transition matrix is shown
on the next slide.
© 2004 Thomson/South-Western
Slide 22
Example: Mark Off’s Process
This Day’s
Location
Austin
San Antonio
Houston
Next Day's Location
Austin San Antonio Houston
.60
.20
.15
.15
.75
.05
.25
.05
.80
If Mark’s in Austin today, chances are
60% that he’ll be in Austin tomorrow.
If Mark’s in Houston today, chances are
5% that he’ll be in San Antonio tomorrow.
© 2004 Thomson/South-Western
Slide 23
Example: Mark Off’s Process

Random Number Mappings
Currently in
Currently in
Currently in
Austin
San Antonio
Houston
Next-Day Random Next-Day Random Next-Day Random
Location Numbers Location Numbers Location Numbers
Austin
00 - 59
Austin
00 - 19
Austin
00 - 14
San Ant. 60 - 74
San Ant. 20 - 94
San Ant.
15 - 19
Houston 75 - 99
Houston
95 - 99
Houston 20 - 99
© 2004 Thomson/South-Western
Slide 24
Example: Mark Off’s Process
Assume Mark is currently in Houston.
Simulate where Mark will be over the next 16 days.
What percentage of time will Mark be in each of the
three cities?
Use the following random numbers:
93, 63, 26, 16, 21, 26, 70, 55, 72, 89, 49, 64, 91, 02, 52, 69
© 2004 Thomson/South-Western
Slide 25
Example: Mark Off’s Process

Simulation Worktable
Starting in Houston
Random N.Day's
Random
Day Number Location
Day Number
1
93
Houston
9
72
2
63
Houston
10
89
3
26
Houston
11
49
4
16
San Ant.
12
64
5
21
San Ant.
13
91
6
26
San Ant.
14
02
7
70
San Ant.
15
52
8
55
San Ant.
16
69
N.Day's
Location
San Ant.
San Ant.
San Ant.
San Ant.
San Ant.
Austin
Austin
San Ant.
In Houston, 20 ≤ r ≤ 99  Houston
© 2004 Thomson/South-Western
Slide 26
Example: Mark Off’s Process
Repeat the simulation with Mark currently
in Austin. Use the following random numbers:
13, 08, 60, 13, 68, 40, 40, 27, 23, 64, 36, 56, 25, 88, 18, 74
Compare the percentages with those found with
Mark starting in Houston.
© 2004 Thomson/South-Western
Slide 27
Example: Mark Off’s Process

Simulation Worksheet
Starting in Austin
Random
Day's
Random
Day Number Location
Day Number
1
13
Austin
9
23
2
08
Austin
10
64
3
60
San Ant.
11
36
4
13
Austin
12
56
5
68
San Ant.
13
25
6
40
San Ant.
14
88
7
40
San Ant.
15
18
8
27
San Ant.
16
74
© 2004 Thomson/South-Western
Day's
Location
San Ant.
San Ant.
San Ant.
San Ant.
San Ant.
San Ant.
Austin
San Ant.
Slide 28
Example: Mark Off’s Process

Simulation Summary
Starting in Houston
Austin
= 2/16 = 12.50%
San Antonio = 11/16 = 68.75%
Houston
= 3/16 = 18.75%
Starting in Austin
Austin
= 4/16 = 25%
San Antonio = 12/16 = 75%
Houston
= 0/16 = 0%
© 2004 Thomson/South-Western
Slide 29
Example: Mark Off’s Process

Partial Spreadsheet w/Variable Look-up Table
A
B
1
2
Aus.
3 LRN URN
6 0.00 0.60
7 0.60 0.75
8 0.75 1.00
9
C
D
E
F
G
Current Location
S.A.
NDL LRN URN NDL LRN
Aus. 0.00 0.20 Aus. 0.00
S.A. 0.20 0.95 S.A. 0.15
Hou. 0.95 1.00 Hou. 0.20
H
I
Hou.
URN
0.15
0.20
1.00
NDL
Aus.
S.A.
Hou.
URN = Upper Random Number
LRN = Lower Random Number
NDL = Next-Day Location
© 2004 Thomson/South-Western
Slide 30
Example: Mark Off’s Process

Partial Spreadsheet with Simulation Table
11
12
13
14
15
16
17
18
19
A
B
Current
Location
Aus.
S.A.
S.A.
S.A.
S.A.
S.A.
S.A.
C
D
E
F
Random Next-Day
Number Location
0.64
S.A.
0.33
S.A.
0.87
S.A.
0.61
S.A.
0.30
S.A.
0.45
S.A.
0.71
S.A.
=IF(A13=$A$2,VLOOKUP(C13,$A$6:$C$8,3),
IF(A13=$D$2,VLOOKUP(C13,$D$6:$F$8,3),
VLOOKUP(C13,$G$6:$I$8,3)))
© 2004 Thomson/South-Western
Slide 31
Example: Wayne International Airport
Wayne International Airport primarily serves
domestic air traffic. Occasionally, however,
a chartered plane from abroad will arrive
with passengers bound for Wayne's two
great amusement parks, Algorithmland
and Giffith's Cherry Preserve.
Whenever an international plane
arrives at the airport the two customs
inspectors on duty set up operations to
process the passengers.
© 2004 Thomson/South-Western
Slide 32
Example: Wayne International Airport
Incoming passengers must first have their
passports and visas checked. This is handled by
one inspector. The time required to check
a passenger's passports and visas can be
described by the probability distribution
on the next slide.
© 2004 Thomson/South-Western
Slide 33
Example: Wayne International Airport
Time Required to
Check a Passenger's
Passport and Visa
20 seconds
40 seconds
60 seconds
80 seconds
© 2004 Thomson/South-Western
Probability
.20
.40
.30
.10
Slide 34
Example: Wayne International Airport
After having their passports and visas checked,
the passengers next proceed to the second customs
official who does baggage inspections. Passengers form
a single waiting line with the official inspecting
baggage on a first come, first served basis. The time
required for baggage inspection follows the probability
distribution shown on the next slide.
© 2004 Thomson/South-Western
Slide 35
Example: Wayne International Airport
Time Required For
Baggage Inspection
No Time
1 minute
2 minutes
3 minutes
© 2004 Thomson/South-Western
Probability
.25
.60
.10
.05
Slide 36
Example: Wayne International Airport

Random Number Mapping
Time Required to
Check a Passenger's
Passport and Visa
20 seconds
40 seconds
60 seconds
80 seconds
© 2004 Thomson/South-Western
Probability
.20
.40
.30
.10
Random
Numbers
00 - 19
20 - 59
60 - 89
90 - 99
Slide 37
Example: Wayne International Airport

Random Number Mapping
Time Required For
Baggage Inspection
No Time
1 minute
2 minutes
3 minutes
© 2004 Thomson/South-Western
Probability
.25
.60
.10
.05
Random
Numbers
00 - 24
25 - 84
85 - 94
95 - 99
Slide 38
Example: Wayne International Airport

Next-Event Simulation Records
For each passenger the following information
must be recorded:
• When his service begins at the passport control
inspection
• The length of time for this service
• When his service begins at the baggage inspection
• The length of time for this service
© 2004 Thomson/South-Western
Slide 39
Example: Wayne International Airport

Time Relationships
Time a passenger begins service
by the passport inspector
= (Time the previous passenger started passport
service)
+ (Time of previous passenger's passport service)
© 2004 Thomson/South-Western
Slide 40
Example: Wayne International Airport

Time Relationships
Time a passenger begins service
by the baggage inspector
( If passenger does not wait in line for baggage
inspection)
= (Time passenger completes service
with the passport control inspector)
(If the passenger does wait in line for baggage
inspection)
= (Time previous passenger completes
service with the baggage inspector)
© 2004 Thomson/South-Western
Slide 41
Example: Wayne International Airport

Time Relationships
Time a customer completes service
at the baggage inspector
= (Time customer begins service with baggage
inspector) + (Time required for baggage inspection)
© 2004 Thomson/South-Western
Slide 42
Example: Wayne International Airport
A chartered plane from abroad lands at
Wayne Airport with 80 passengers. Simulate the
processing of the first 10 passengers through customs.
Use the following random numbers:
For passport control:
93, 63, 26, 16, 21, 26, 70, 55, 72, 89
For baggage inspection:
13, 08, 60, 13, 68, 40, 40, 27, 23, 64
© 2004 Thomson/South-Western
Slide 43
Example: Wayne International Airport

Simulation Worksheet (partial)
Passport Control
Baggage Inspections
Pass. Time Rand. Service Time Time Rand. Service Time
Num Begin Num. Time End Begin Num. Time End
1
2
3
4
5
0:00
1:20
2:20
3:00
3:20
93
63
26
16
21
1:20
1:00
:40
:20
:40
1:20
2:20
3:00
3:20
4:00
1:20
2:20
3:00
4:00
4:00
13
08
60
13
68
0:00
0:00
1:00
0:00
1:00
1:20
2:20
4:00
4:00
5:00
Must wait 40 seconds.
© 2004 Thomson/South-Western
Slide 44
Example: Wayne International Airport

Simulation Worksheet (continued)
Passport Control
Baggage Inspections
Pass. Time Rand. Service Time Time Rand. Service Time
Num Begin Num. Time End Begin Num. Time End
6
7
8
9
10
4:00
4:40
5:40
6:20
7:20
26
70
55
72
89
:40
1:00
:40
1:00
1:00
© 2004 Thomson/South-Western
4:40
5:40
6:20
7:20
8:20
5:00
6:00
7:00
8:00
8:20
40
40
27
23
64
1:00
1:00
1:00
0:00
1:00
6:00
7:00
8:00
8:00
9:20
Slide 45
Example: Wayne International Airport

Explanation
For example, passenger 1 begins being served by
the passport control inspector immediately. His service
time is 1:20 (80 seconds) at which time he goes
immediately to the baggage inspector who waves him
through without inspection.
Passenger 2 begins service with passport inspector
1:20 minutes (80 seconds) after arriving there (as this is
when passenger 1 is finished) and requires 1:00 minute
(60 seconds) for passport inspection. He is waved
through baggage inspection as well.
This process continues in this manner.
© 2004 Thomson/South-Western
Slide 46
Example: Wayne International Airport

Question
How long will it take for the first 10 passengers
to clear customs?

Answer
Passenger 10 clears customs after 9 minutes and
20 seconds.
© 2004 Thomson/South-Western
Slide 47
Example: Wayne International Airport

Question
What is the average length of time a customer
waits before having his bags inspected after he clears
passport control? How is this estimate biased?
© 2004 Thomson/South-Western
Slide 48
Example: Wayne International Airport

Answer
For each passenger calculate his waiting time:
(Baggage Inspection Begins) - (Passport Control
Ends)
= 0+0+0+40+0+20+20+40+40+0 = 120 seconds.
120/10 = 12 seconds per passenger
This is a biased estimate because we assume that
the simulation began with the system empty. Thus,
the results tend to underestimate the average waiting
time.
© 2004 Thomson/South-Western
Slide 49
General algorithm
Initial model set up
Probabilistic
inputs
Controllable
inputs
Model
Controllable
inputs
Random generation
of probab. inputs
Model processing
Next
random
number
Recording results
N
End?
Y
Concluding analysis
© 2004 Thomson/South-Western
Slide 50
Covered in the text:

Risk analysis
• Simulated probabilistic inputs:
costs & demands

Inventory simulation
• Simulated probabilistic input: demand

Waiting line simulation
• Simulated probabilistic input:
inter-arrival time & service time
© 2004 Thomson/South-Western
Slide 51
Comments

What-if analysis (as a warm-up)
• Establish limit values of all probabilistic inputs
• Select the worst-case and best-case scenario
• Choose appropriate decisions for each
• Calculate bottom-line for both marginal cases
© 2004 Thomson/South-Western
Slide 52
Comments (cont’d)


Random data generation
• Look-up table vs. probab. distribution function
• Finite vs. discrete vs. continuous
• Symmetric vs. asymmetric (one-sided)
Excel functions
• =IF(T1=N1, VLOOKUP( RAND(), U1:W4, 4),
VLOOKUP( RAND(), X1:Z4, 4))
• T1 = Table choice, N1 = Table name, for:
U1:W4 & X1:Z4 = 3  4 look-up tables (min,max,val)
• = A1 + RAND()*(B1 – A1) : uniform p.d.
• A1 = minimum, B1 = maximum
• = NORMINV( RAND(), M1, S1) : normal p.d.
• M1 = mean, S1 = standard deviation
© 2004 Thomson/South-Western
Slide 53
End of Chapter 15
© 2004 Thomson/South-Western
Slide 54