Simulation

Chapter 14 14-1

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Chapter Topics 14-2

■ 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

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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.

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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).

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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 14.1 Probability Distribution of Demand for Laptop PC’s 14-5

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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.

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Monte Carlo Process

Use of Random Numbers (3 of 10)

Figure 14.1 A Roulette Wheel for Demand 14-7

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Monte Carlo Process

Use of Random Numbers (4 of 10)

When the wheel is spun, the actual demand for PCs is determined by a

number at rim of the wheel.

Figure 14.2 14-8

Numbered Roulette Wheel

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Monte Carlo Process

Use of Random Numbers (5 of 10)

Table 14.2 Generating Demand from Random Numbers 14-9

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Monte Carlo Process

Use of Random Numbers (6 of 10)

Select number from a random number table:

Table 14.3 Delightfully Random Numbers 14-10

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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.

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Monte Carlo Process

Use of Random Numbers (8 of 10)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Table 14.4 14-12

Monte Carlo Process

Use of Random Numbers (9 of 10)

Average demand could have been calculated analytically:

E(x) = n

∑ 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

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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.

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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.

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Computer Simulation with Excel Spreadsheets

Generating Random Numbers (2 of 2)

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.

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Simulation with Excel Spreadsheets (1 of 3)

Exhibit 14.1

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Simulation with Excel Spreadsheets (2 of 3)

Exhibit 14.2

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Simulation with Excel Spreadsheets (3 of 3)

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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.

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14-20

Computer Simulation with Excel Spreadsheets

Decision Making with Simulation (2 of 2)

Order size of two laptops each week.

Exhibit 14.5 14-21

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Simulation of a Queuing System

Burlingham Mills Example (1 of 3)

Table 14.5 Distribution of Arrival Intervals

Table 14.6 Distribution of Service Times

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Simulation of a Queuing System

Burlingham Mills Example (2 of 3)

Average waiting time = 12.5days/10 batches 14-23

= 1.25 days per batch

Average time in the system = 24.5 days/10 batches

= 2.45 days per batch

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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.

■ 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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Computer Simulation with Excel

Burlingham Mills Example

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14-25

Continuous Probability Distributions

A continuous function must be used for continuous distributions.

Example :

f(x) = x , 0 ≤ x ≤ 4 where x = time (minutes)

8

Cumulative probability of x:

F(x) = 0x∫ 8x dx = 810x∫ x dx = 1 1 x2 x

8 2 0

F(x) = x2

16

Let F(x) = the random number r

r = x2

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

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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(ri), value of x for a given value of ri.

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Machine Breakdown and Maintenance System

Simulation (2 of 6)

Table 14.8 14-28

Probability Distribution of Machine Repair Time

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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(ri)

Table 14.9

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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):

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Table 14.10 14-30

Machine Breakdown and Maintenance System

Simulation (5 of 6)

Simulation of system with maintenance program (total annual repair

cost of $42,000):

Table 14.11

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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.

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Machine Breakdown and Maintenance System

Simulation with Excel (1 of 2)

Original machine breakdown example:

Exhibit 14.7 14-33

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Machine Breakdown and Maintenance System

Simulation with Excel (2 of 2)

Simulation with maintenance program.

Exhibit 14.8 14-34

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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.

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Statistical Analysis of Simulation Results (2 of 2)

Formulas for 95% confidence limits:

upper confidence limit = x+(1.96)(s/ n)

lower confidence limit = x−(1.96)(s/ n)

where x 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.

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Simulation Results

Statistical Analysis with Excel (1 of 3)

Simulation with maintenance program.

Exhibit 14.9 14-37

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Simulation Results

Statistical Analysis with Excel (2 of 3)

Exhibit 14.10

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Simulation Results

Statistical Analysis with Excel (3 of 3)

Exhibit 14.11

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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.

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Crystal Ball

Simulation of Profit Analysis Model (1 of 15)

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.

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Crystal Ball

Simulation of Profit Analysis Model (2 of 15)

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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Crystal Ball

Simulation of Profit Analysis Model (3 of 15)

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Crystal Ball

Simulation of Profit Analysis Model (4 of 15)

Exhibit 14.12

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Crystal Ball

Simulation of Profit Analysis Model (5 of 15)

Exhibit 14.13 14-45

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Crystal Ball

Simulation of Profit Analysis Model (6 of 15)

Exhibit 14.14

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Crystal Ball

Simulation of Profit Analysis Model (7 of 15)

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14-47

Crystal Ball

Simulation of Profit Analysis Model (8 of 15)

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Crystal Ball

Simulation of Profit Analysis Model (9 of 15)

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14-49

Crystal Ball

Simulation of Profit Analysis Model (10 of 15)

Exhibit 14.18

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