14-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Simulation Chapter 14.

14-1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Simulation

Chapter 14

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■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 Topics

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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■ 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.

Overview


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■ 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).

Monte Carlo Process


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Table 14.1 Probability Distribution of Demand for Laptop PC’s

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.

Monte Carlo ProcessUse of Random Numbers (1 of 10)


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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.



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Figure 14.1 A Roulette Wheel for Demand



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Figure 14.2Numbered Roulette Wheel


When the wheel is spun, the actual demand for PCs is determined by a number at rim of the wheel.


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Table 14.2 Generating Demand from Random Numbers



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Select number from a random number table:

Table 14.3 Delightfully Random Numbers



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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.



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Table 14.4

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Average demand could have been calculated analytically:

per week sPC' 1.5 )4)(10(.)3)(10(.)2)(20(.)1)(40(.)0)(20(.)(

:therefore

valuesdemanddifferent ofnumber thedemand ofy probabilit )(i valuedemand

:where

1)()(

xE

nxPx

n

ixxPxE

i

i

ii



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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.



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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.

Computer Simulation with Excel SpreadsheetsGenerating Random Numbers (1 of 2)


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Artificially created random numbers must have the following characteristics:

1. The random numbers must be uniformly distributed.

2. The numerical technique for generating the numbers must be efficient.

3. The sequence of random numbers should reflect no pattern.

Computer Simulation with Excel SpreadsheetsGenerating Random Numbers (2 of 2)


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Exhibit 14.1

Simulation with Excel Spreadsheets (1 of 3)


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Exhibit 14.2



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Exhibit 14.3



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Revised ComputerWorld example; order size of one laptop each week.

Computer Simulation with Excel SpreadsheetsDecision Making with Simulation (1 of 2)


Exhibit 14.4

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Order size of two laptops each week.

Computer Simulation with Excel SpreadsheetsDecision Making with Simulation (2 of 2)


Exhibit 14.5

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Table 14.5 Distribution of Arrival Intervals

Table 14.6 Distribution of Service Times

Simulation of a Queuing SystemBurlingham Mills Example (1 of 3)


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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



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Caveats:

■ Results may be viewed with skepticism.

■ Ten trials do not ensure steady-state results.

■ 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.


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Exhibit 14.6

Computer Simulation with ExcelBurlingham Mills Example


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minutes 2 .254 x.25,r if :Example

.determined is time"" for x value a r,number, random a generatingBy r4x

16x2r

r number random the F(x) Let16

2xF(x)

x

02x

21x

0 81dxx

81dx

x

0

8xF(x)

: xofy probabilit Cumulative

(minutes) time x where4x0 ,8xf(x)

:Exampleons.distributi continuous for used be must function continuousA

Continuous Probability Distributions


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Machine Breakdown and Maintenance SystemSimulation (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(ri), value of x for a given value of ri.


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Table 14.8Probability Distribution of Machine Repair Time



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Table 14.9

Machine Breakdown and Maintenance SystemSimulation (3 of 6)Revised probability of time between machine

breakdowns:f(x) = x/18, 0 x6 weeks where x = weeks

between machine breakdowns

x = 6*sqrt(ri)


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Table 14.10

Machine Breakdown and Maintenance SystemSimulation (4 of 6)Simulation of system without

maintenance program (total annual repair cost of $84,000):


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Table 14.11

Machine Breakdown and Maintenance SystemSimulation (5 of 6)Simulation of system with maintenance program

(total annual repair cost of $42,000):


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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.Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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Exhibit 14.7

Machine Breakdown and Maintenance SystemSimulation with Excel (1 of 2)Original machine breakdown example:


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Exhibit 14.8

Machine Breakdown and Maintenance SystemSimulation with Excel (2 of 2)Simulation with maintenance program.


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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.

Statistical Analysis of Simulation Results (1 of 2)


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Formulas for 95% confidence limits:

upper confidence limit

lower confidence limit

where is the mean and s 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.

)/)(.( nsx 961

)/)(.( nsx 961

x

Statistical Analysis of Simulation Results (2 of 2)


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Simulation ResultsStatistical Analysis with Excel (1 of 3)Simulation with maintenance program.


Exhibit 14.9

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Simulation ResultsStatistical Analysis with Excel (2 of 3)


Exhibit 14.10

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Exhibit 14.11

Simulation ResultsStatistical Analysis with Excel (3 of 3)


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Crystal BallOverview

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.


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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.

Crystal BallSimulation of Profit Analysis Model (1 of 15)


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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.



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Exhibit 14.12

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Exhibit 14.13

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Exhibit 14.14

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Exhibit 14.15

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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.16

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Exhibit 14.17

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Exhibit 14.18

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Exhibit 14.19

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Exhibit 14.20

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Exhibit 14.21



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Exhibit 14.22

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Exhibit 14.23

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■ 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.

Verification of the Simulation Model (1 of 2)


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■ 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.

Verification of the Simulation Model (2 of 2)


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■Queuing

■Inventory Control

■Production and Manufacturing

■Finance

■Marketing

■Public Service Operations

■Environmental and Resource Analysis

Some Areas of Simulation Application


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Willow Creek Emergency Rescue Squad

Minor emergency requires two-person crew

Regular emergency requires a three-person crew

Major emergency requires a five-person crew

Example Problem Solution (1 of 6)


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Distribution of number of calls per night and emergency type:

Calls Probability

0 1 2 3 4 5 6

.05

.12

.15

.25

.22

.15

.06 1.00

Emergency Type Probability Minor Regular Major

.30

.56

.14 1.00



1. Manually simulate 10 nights of calls

2. Determine average number of calls each night

3. Determine maximum number of crew members that might be needed on any given night.

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Calls Probability Cumulative Probability

Random Number Range, r1

0 1 2 3 4 5 6

.05

.12

.15

.25

.22

.15

.06 1.00

.05

.17

.32

.57

.79

.94 1.00

1 – 5 6 – 17

18 – 32 33 – 57 58 – 79 80 – 94

95 – 99, 00

Emergency Type

Probability Cumulative Probability

Random Number Range, r1

Minor Regular Major

.30

.56

.14 1.00

.30

.86 1.00

1 – 30 31 – 86

87 – 99, 00

Step 1: Develop random number ranges for the probability distributions.



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Step 2: Set Up a Tabular Simulation (use second column of random numbers in Table 14.3).



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Step 2 continued:



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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 14.



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14-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Simulation Chapter 14.

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