Introduction to management science 10e by bernard taylor chapter 14

■ The Monte Carlo Process■ Computer Simulation with Excel Spreadsheets ■ Simulation of a Queuing System ■ Continuous Probability Distributions ■ Statistical Analysis of Simulation Resul

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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

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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

variable are generated by sampling from a

probability distribution.

selling for $4,300 over a period of 100 weeks.

Use 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

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Figure 14.1 A Roulette Wheel for Demand

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Figure 14.2 Numbered 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:

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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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Average demand could have been calculated

analytically:

per week s

PC' 1.5

) 4 )(

10 (.

) 3 )(

10 (.

) 2 )(

20 (.

) 1 )(

40 (.

) 0 )(

20 (.

) (

: therefore

values demand

different of

number the

demand of

y probabilit )

( demand value i

: where

1 ( )

) (

i i

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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

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 As simulation models get more complex they

generated by a mathematical process instead of

a physical process (such as wheel spinning).

computer using a numerical technique and thus are

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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

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Exhibit 14.2

Simulation with Excel Spreadsheets (2 of 3)

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

Computer Simulation with Excel

Spreadsheets

Decision Making with Simulation (1

of 2)

Exhibit 14.4

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Order size of two laptops each week.

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Table 14.5 Distribution of

Arrival Intervals

Table 14.6 Distribution of

Simulation of a Queuing System

Burlingham 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

replicates normal operating system.

■ If system starts with items already in the system, simulation must begin with

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Exhibit 14.6

Burlingham Mills Example

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minutes 2

.25 4

x 25,

r if : Example

determined is

time"

"

for

x value a

r, number, random

a generating

By

r 4

x 16

x2

r

r number random

the F(x)

Let 16

2 x F(x)

x 0

2 x 2

1

x

0 8

1 dx

x

8 1 dx

x

0 8

x F(x)

: x of

y probabilit Cumulative

(minutes) time

x where 4

x 0

,

8 x f(x)

: Example

ons.

distributi continuous

for used be

must function

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Machine Breakdown and

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(r i ), value of x for a given value of r i

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Table 14.8 Probability Distribution of Machine Repair Time

Maintenance System

Simulation (2 of 6)

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Table 14.9

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Simulation (4 of 6) Simulation of system without

maintenance program (total annual

repair cost of $84,000):

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Table

Simulation (5 of 6) Simulation of system with maintenance program

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

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Simulation (6 of 6)

Results and caveats:

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

average results computed.

■ Efficient computer simulation required to do this.

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Exhibit 14.7

Simulation with Excel (1 of 2) Original machine breakdown example:

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Simulation 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

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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

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Simulation Results

Statistical Analysis with Excel (1 of

3) Simulation with maintenance program.

Exhibit 14.9

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Simulation Results

Statistical Analysis with Excel (2 of

3)

Exhibit 14.10

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40

14-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

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Recap of Western Clothing Company

break-even and profit analysis:

Price (p) for jeans is $23 variable cost (c v ) is $8

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Modifications to demonstrate Crystal Ball

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.

Crystal Ball

Simulation of Profit Analysis Model

(2 of 15)

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Crystal Ball

(6 of 15)

Exhibit 14.14

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Crystal Ball

(8 of 15)

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Crystal Ball

(10 of 15)

Exhibit 14.18

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Crystal Ball

(12 of 15)

Exhibit 14.20

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Crystal Ball

(14 of 15)

Exhibit 14.22

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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

Verification of the Simulation Model (1 of 2)

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■ Analyst must determine if model starting conditions

are correct (system empty, etc).

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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■ 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 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, r

.05 17 32 57 79 94 1.00

.30 86 1.00

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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

If calls of all types occurred on same night,

maximum number of squad members required

would be 14.

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