Transcript presentation

Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 4
Simulation
Chapter 4 - Simulation
1
Chapter Topics
The Monte Carlo Process
Computer Simulation with Excel Spreadsheets
Simulation of a Queuing System
Continuous Probability Distributions
Statistical Analysis of Simulation Results
Crystal Ball
Verification of the Simulation Model
Areas of Simulation Application
Chapter 4 - Simulation
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Overview
Analogue simulation replaces a physical system with an
analogous physical system that is easier to manipulate.
In computer mathematical simulation a system is replaced
with a mathematical model that is analyzed with the
computer.
Simulation offers a means of analyzing very complex
systems that cannot be analyzed using the other
management science techniques in the text.
Chapter 4 - Simulation
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Homework Reduction
From problems 5, 9, 13, 15, 19, 21, 23:
select four, and complete only those four.
Then, do:
Either one of the problems from among 25, 27, 29:
(Qrystal Ball program — so, you must describe how you used the
program, cite the results, and explain what the results indicate.)
Or do problem 17 (by hand or using Excel)
Simulate five trials of ten random turns around the corner
Chapter 4 - Simulation
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Monte Carlo Process
A large proportion of the applications of simulations are for
probabilistic models.
The Monte Carlo technique is defined as a technique for
selecting numbers randomly from a probability distribution
for use in a trial (computer run) of a simulation model.
The basic principle behind the process is the same as in
the operation of gambling devices in casinos (such as those
in Monte Carlo, Monaco).
Gambling devices produce numbered results from welldefined populations.
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (1 of 10)
In the Monte Carlo process, values for a random variable
are generated by sampling from a probability distribution.
Example: ComputerWorld demand data for laptops selling
for $4,300 over a period of 100 weeks.
Table 4.1
Probability Distribution of Demand for Laptop PC’s
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (2 of 10)
The purpose of the Monte Carlo process is to generate the
random variable, demand, by sampling from the probability
distribution P(x).
The partitioned roulette wheel replicates the probability
distribution for demand if the values of demand occur in a
random manner.
The segment at which the wheel stops indicates demand
for one week.
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (3 of 10)
Figure 4.1
A Roulette Wheel for Demand
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (4 of 10)
When wheel is spun actual demand for PC’s is determined
by a number at rim of the wheel.
Figure 4.2
Numbered Roulette Wheel
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (5 of 10)
Process of spinning a wheel can be replicated using
random numbers alone.
Transfer random numbers for each demand value from
roulette wheel to a table.
Table 4.2
Generating Demand from Random Numbers
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (6 of 10)
Select number from a random number table:
Table 4.3
Random Number Table
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (7 of 10)
Repeating selection of random numbers simulates demand
for a period of time.
Estimated average demand = 31/15 = 2.07 laptop PCs per
week.
Estimated average revenue = $133,300/15 = $8,886.67
($133,300 = $4,300  31).
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (8 of 10)
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (9 of 10)
Average demand could have been calculated analytically:
n
E( x)   P( xi) xi
i1
where:
xi  demand value i
P( xi)  probability of demand
n  the number of different demand values
therefore:
E( x)  (.20)(0)  (.40)(1)  (.20)(2)  (.10)(3)  (.10)(4)
 1.5 PC's per week
Chapter 4 - Simulation
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Monte Carlo Process
Use of Random Numbers (10 of 10)
The more periods simulated, the more accurate the results.
Simulation results will not equal analytical results unless
enough trials have been conducted to reach steady state.
Often difficult to validate results of simulation - that true
steady state has been reached and that simulation model
truly replicates reality.
When analytical analysis is not possible, there is no
analytical standard of comparison thus making validation
even more difficult.
Chapter 4 - Simulation
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Computer Simulation with Excel Spreadsheets
Generating Random Numbers (1 of 2)
As simulation models get more complex they become
impossible to perform manually.
In simulation modeling, random numbers are generated by
a mathematical process instead of a physical process (such
as wheel spinning).
Random numbers are typically generated on the computer
using a numerical technique and thus are not true random
numbers but pseudorandom numbers.
Chapter 4 - Simulation
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Computer Simulation with Excel Spreadsheets
Generating Random Numbers (2 of 2)
Artificially created random numbers must have the following
characteristics:
The random numbers must be uniformly distributed.
The numerical technique for generating the numbers
must be efficient.
The sequence of random numbers should reflect no
(discernible) pattern.
Chapter 4 - Simulation
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Simulation with Excel Spreadsheets (1 of 3)
Exhibit 4.1
Chapter 4 - Simulation
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Simulation with Excel Spreadsheets (2 of 3)
“Lookup”
Chapter 4 - Simulation
Exhibit 4.2
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Simulation with Excel Spreadsheets (3 of 3)
Exhibit 4.3
Chapter 4 - Simulation
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Computer Simulation with Excel Spreadsheets
Decision Making with Simulation (1 of 2)
Revised ComputerWorld example; order size of one laptop
each week.
Exhibit 4.4
=1+MAX(G6-H6,0)
Order one laptop
each week
Chapter 4 - Simulation
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Computer Simulation with Excel Spreadsheets
Decision Making with Simulation (2 of 2)
Order size of two laptops each week.
=2+MAX(G6-H6,0)
Order two laptops
each week
Exhibit 4.5
Chapter 4 - Simulation
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Simulation of a Queuing System
Burlingham Mills Example (1 of 3)
Table 4.5
Distribution of Arrival Intervals
Table 4.6
Distribution of Service Times
Chapter 4 - Simulation
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Simulation of a Queuing System
Burlingham Mills Example (2 of 3)
Average waiting time = 12.5days/10 batches
= 1.25 days per batch
Average time in the system = 24.5 days/10 batches
= 2.45 days per batch
Chapter 4 - Simulation
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Simulation of a Queuing System
Burlingham Mills Example (3 of 3)
Caveats:
Results may be viewed with skepticism.
Ten trials do not ensure steady-state results.
In fact, the statistical error for N data-points is sqrt(N),
so the relative statistical error is ~1/sqrt(N).
Starting conditions can affect simulation results.
If no batches are in the system at start, simulation
must run until it replicates normal operating system.
If system starts with items already in the system,
simulation must begin with items in the system.
Chapter 4 - Simulation
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Computer Simulation with Excel
Burlingham Mills Example
Exhibit 4.6
Chapter 4 - Simulation
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Continuous Probability Distributions
A continuous function must be used for continuous distributions.
Example :
1/2
x
f(x) , 0  x  4 where x  time (minutes)
8
Cumulative probability of x :
xx
x
1
x
1
1
F(x)  dx   x dx   x2 
80
82 0
08
2
x
0
4
F(x)
16
Let F(x)  the random number r
2
x
r
16
x 4 r
By generating a random number,r, a value x for "time" is determined.
Example : if r  .25, x  4 .25  2 minutes
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation (1 of 6)
Bigelow Manufacturing Company must decide if it should
implement a machine maintenance program at a cost of
$20,000 per year that would reduce the frequency of
breakdowns and thus time for repair which is $2,000 per
day in lost production.
A continuous probability distribution of the time between
machine breakdowns:
f(x) = x/8, 0  x  4 weeks, where x = weeks between
machine breakdowns
x = 4*sqrt(r1), value of x for a given value of r1.
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation (2 of 6)
Table 4.8
Probability Distribution of Machine Repair Time
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation (3 of 6)
Revised probability of time between machine breakdowns:
f(x) = x/18, 0  x6 weeks where x = weeks between
machine breakdowns
x = 6*sqrt(r1)
Table 4.9
Revised Probability Distribution of Machine Repair Time with the Maintenance Program
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation (4 of 6)
Simulation of system without maintenance program (total
annual repair cost of $84,000):
x = 4*sqrt(r1)
Table 4.10
Simulation of Machine
Breakdowns and Repair
Times
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation (5 of 6)
Simulation of system with maintenance program (total
annual repair cost of $42,000):
x = 6*sqrt(r1)
Table 4.11
Simulation of Machine
Breakdowns and Repair
with the Maintenance
Program
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation (6 of 6)
Results and caveats:
Implement maintenance program since cost savings
appear to be $42,000 per year and maintenance program
will cost $20,000 per year.
However, there are potential problems caused by
simulating both systems only once.
Simulation results could exhibit significant variation since
time between breakdowns and repair times are
probabilistic.
To be sure of accuracy of results, simulations of each
system must be run many times and average results
computed.
Efficient computer simulation required to do this.
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation with Excel (1 of 2)
Original machine breakdown example:
Exhibit 4.7
Chapter 4 - Simulation
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Machine Breakdown and Maintenance System
Simulation with Excel (2 of 2)
Simulation with maintenance program.
Exhibit 4.8
Chapter 4 - Simulation
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Statistical Analysis of Simulation Results (1 of 2)
Outcomes of simulation modeling are statistical measures
such as averages.
Statistical results are typically subjected to additional
statistical analysis to determine their degree of accuracy.
Confidence limits are developed for the analysis of the
statistical validity of simulation results.
Chapter 4 - Simulation
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Statistical Analysis of Simulation Results (2 of 2)
Formulas for 95% confidence limits:
upper confidence limit x(1.96)( / n)
lower confidence limit  x(1.96)( / n)

where x is the mean and  the standard deviation

from a sample of size n from any population.
We can be 95% confident that the true population mean will
be between the upper confidence limit and lower
confidence limit.
Chapter 4 - Simulation
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Simulation Results
Statistical Analysis with Excel (1 of 3)
Simulation with maintenance program.
Exhibit 4.9
Chapter 4 - Simulation
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Simulation Results
Statistical Analysis with Excel (2 of 3)
Exhibit 4.10
Chapter 4 - Simulation
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Simulation Results
Statistical Analysis with Excel (3 of 3)
Exhibit 4.11
Chapter 4 - Simulation
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Crystal Ball
Overview
Many realistic simulation problems contain more complex
probability distributions than those used in the examples.
However there are several simulation add-ins for Excel that
provide a capability to perform simulation analysis with a
variety of probability distributions in a spreadsheet format.
Crystal Ball, published by Decisioneering, is one of these.
Crystal Ball is a risk analysis and forecasting program that
uses Monte Carlo simulation to provide a statistical range of
results.
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (1 of 17)
Recap of Western Clothing Company break-even and profit
analysis:
Price (p) for jeans is $23; variable cost (cv) is $8; fixed
cost (cf ) is $10,000.
Profit Z = vp - cf - vc; break-even volume v = cf/(p - cv)
= 10,000/(23-8) = 666.7 pairs.
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (2 of 17)
Modifications to demonstrate Crystal Ball:
Assume volume is now volume demanded and is
defined by a normal probability distribution with mean
of 1,050 and standard deviation of 410 pairs of jeans.
Price is uncertain and defined by a uniform probability
distribution from $20 to $26.
Variable cost is not constant but defined by a triangular
probability distribution.
Will determine average profit and profitability with given
probabilistic variables.
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (3 of 17)
Exhibit 4.12
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (4 of 17)
Exhibit 4.13
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (5 of 17)
Exhibit 4.14
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (6 of 17)
Exhibit 4.15
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (7 of 17)
Exhibit 4.16
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (8 of 17)
Exhibit 4.17
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (9 of 17)
Exhibit 4.18
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (10 of 17)
Exhibit 4.19
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (11 of 17)
Exhibit 4.20
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (12 of 17)
Exhibit 4.21
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (13 of 17)
Exhibit 4.22
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (14 of 17)
Exhibit 4.23
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (15 of 17)
Exhibit 4.24
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (16 of 17)
Exhibit 4.25
Chapter 4 - Simulation
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Crystal Ball
Simulation of Profit Analysis Model (17 of 17)
Exhibit 4.26
Chapter 4 - Simulation
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Verification of the Simulation Model (1 of 2)
Analyst wants to be certain that model is internally correct
and that all operations are logical and mathematically
correct.
Testing procedures for validity:
Run a small number of trials of the model and compare
with manually derived solutions.
Divide the model into parts and run parts separately to
reduce complexity of checking.
Simplify mathematical relationships (if possible) for
easier testing.
Compare results with actual real-world data.
Chapter 4 - Simulation
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Verification of the Simulation Model (2 of 2)
Analyst must determine if model starting conditions are
correct (system empty, etc).
Must determine how long model should run to insure
steady-state conditions.
A standard, fool-proof procedure for validation is not
available.
Validity of the model rests ultimately on the expertise and
experience of the model developer.
Chapter 4 - Simulation
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Some Areas of Simulation Application
Queuing
• Inventory Control
• Production and Manufacturing
• Finance
• Marketing
• Public Service Operations
• Environmental and Resource Analysis
Chapter 4 - Simulation
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Example Problem Solution (1 of 6)
Data
Willow Creek Emergency Rescue Squad
Minor emergency requires two-person crew, regular, a
three-person crew, and major emergency, a fiveperson crew.
Chapter 4 - Simulation
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Example Problem Solution (2 of 6)
Distribution of number of calls per night and emergency
type:
Calls
0
1
2
3
4
5
6
Probability
.05
.12
.15
.25
.22
.15
.06
1.00
Emergency Type
Minor
Regular
Major
Probability
.30
.56
.14
1.00
Required: Manually simulate 10 nights of calls; determine
average number of calls each night and maximum number
of crew members that might be needed on any given night.
Chapter 4 - Simulation
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Example Problem Solution (3 of 6)
Solution Step 1: Develop random number ranges for the
probability distributions.
Calls
Probability
0
1
2
3
4
5
6
.05
.12
.15
.25
.22
.15
.06
1.00
Emergency
Type
Minor
Regular
Major
Chapter 4 - Simulation
Cumulative
Probability
.05
.17
.32
.57
.79
.94
1.00
Probability
.30
.56
.14
1.00
Random Number
Range, r1
1–5
6 – 17
18 – 32
33 – 57
58 – 79
80 – 94
95 – 99, 00
Cumulative
Probability
.30
.86
1.00
Random Number
Range, r1
1 – 30
31 – 86
87 – 99, 00
64
Example Problem Solution (4 of 6)
Step 2: Set Up a Tabular Simulation (use second column of
random numbers in Table 4.3).
Chapter 4 - Simulation
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Example Problem Solution (5 of 6)
Step 2 continued:
Chapter 4 - Simulation
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Example Problem Solution (6 of 6)
Step 3: Compute Results:
average number of minor emergency calls per night
= 10/10 =1.0
average number of regular emergency calls per night
= 14/10 = 1.4
average number of major emergency calls per night
= 3/10 = 0.30
If calls of all types occurred on same night, maximum
number of squad members required would be 4.
Chapter 4 - Simulation
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Chapter 4 - Simulation
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