taylor introms10 ppt 14

5/27/2018 Taylor Introms10 Ppt 14

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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 analogousphysical 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 systemsthat 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 forprobabilistic 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 MonteCarlo, Monaco).

Monte Carlo Process



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14-5Table 14.1 Probability Distribution of Demand for Laptop PCs

In the Monte Carlo process, values for a random variable aregenerated by sampling from a probability distribution.

Example: ComputerWorld demand data for laptops selling for$4,300 over a period of 100 weeks.

Monte Carlo Process

Use of Random Numbers (1 of 10)



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The purpose of the Monte Carlo process is to generatethe random variable, demand, by sampling from theprobability distribution P(x).

The partitioned roulette wheel replicates the probabilitydistribution for demand if the values of demand occur ina random manner.

The segment at which the wheel stops indicates demandfor one week.

Monte Carlo Process




7/6514-7Figure 14.1 A Roulette Wheel for Demand

Monte Carlo Process




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Figure 14.2

Numbered Roulette Wheel

Monte Carlo Process


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

number at rim of the wheel.



9/6514-9Table 14.2 Generating Demand from Random Numbers

Monte Carlo Process




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Select number from a random number table:

Table 14.3 Delightfully Random Numbers

Monte Carlo Process




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Repeating selection of random numbers simulatesdemand for a period of time.

Estimated average demand = 31/15 = 2.07 laptop PCsper week.

Estimated average revenue = $133,300/15 = $8,886.67.

Monte Carlo Process




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Monte Carlo Process


Copyright 2010 Pearson Education, Inc. Publishing as Prentice HallTable 14.4


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Average demand could have been calculated analytically:

per weeksPC'1.5

)4)(10(.)3)(10(.)2)(20(.)1)(40(.)0)(20(.)(

:therefore

valuesdemanddifferentofnumberthe

demandofyprobabilit)(ivaluedemand

:where

1)()(

xE

n

xP

x

n

i

xxPxE

i

i

ii

Monte Carlo Process




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The more periods simulated, the more accurate the results.

Simulation results will not equal analytical results unless enoughtrials have been conducted to reach steady state.

Often difficult tovalidateresults of simulation - that true steadystate has been reached and that simulation model truly replicatesreality.

When analytical analysis is not possible, there is no analyticalstandard of comparison thus making validation even more difficult.

Monte Carlo Process




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As simulation models get more complex they become impossibleto perform manually.

In simulation modeling, random numbers are generated by amathematical processinstead of a physical process (such as wheelspinning).

Random numbers are typically generated on the computer using anumerical technique and thus are not true random numbers but

pseudorandom numbers.

Computer Simulation with Excel Spreadsheets

Generating Random Numbers (1 of 2)



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Artificially created random numbers must have the followingcharacteristics:

1. The random numbers must be uniformly

distributed.

2. The numerical technique for generating the numbersmust be efficient.

3. The sequence of random numbers should reflect nopattern.


Generating Random Numbers (2 of 2)



17/6514-17Exhibit 14.1

Simulation with Excel Spreadsheets (1 of 3)



18/6514-18Exhibit 14.2




19/6514-19Exhibit 14.3




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


Decision Making with Simulation (1 of 2)


Exhibit 14.4

C S S d


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Order size of two laptops each week.


Decision Making with Simulation (2 of 2)


Exhibit 14.5

Si l i f Q i S


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Table 14.5 Distribution of Arrival Intervals

Table 14.6 Distribution of Service Times


Burlingham Mills Example (1 of 3)


Si l i f Q i S


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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




Si l i f Q i S


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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, simulationmust run until it replicates normal operating system.

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


C Si l i i h E l


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Exhibit 14.6

Computer Simulation with Excel

Burlingham Mills Example


C i P b bili Di ib i


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minutes2.254x.25,rif:Example

.determinedistime""forxvaluear,number,randomageneratingBy

r4x16x2r

rnumberrandomtheF(x)Let16

2xF(x)

x

02x21

x

0 81dxx81dx

x

08xF(x)

:xofyprobabilitCumulative

(minutes)timexwhere4x0,8xf(x)

:Example

ons.distributicontinuousforusedbemustfunctioncontinuousA

Continuous Probability Distributions


M hi B kd d M i S


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Machine Breakdown and Maintenance System

Simulation (1 of 6)

Bigelow Manufacturing Company must decide if it shouldimplement a machine maintenance program at a cost of $20,000 peryear that would reduce the frequency of breakdowns and thus timefor repair which is $2,000 per day in lost production.

A continuous probability distribution of the time between machinebreakdowns:

f(x) = x/8, 0 x 4 weeks, where x = weeks between

machine breakdownsx = 4*sqrt(ri), value of x for a given value of ri.


M hi B kd d M i S


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Table 14.8

Probability Distribution of Machine Repair Time


Simulation (2 of 6)


M hi B kd d M i t S t


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Table 14.9


Simulation (3 of 6)Revised probability of time between machine breakdowns:

f(x) = x/18, 0 x6 weeks where x = weeks between machinebreakdowns

x = 6*sqrt(ri)


M hi B kd d M i t S t


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Simulation (4 of 6)

Simulation of system without maintenance program

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


M hin Br kd n nd M int n n S t


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Simulation (5 of 6)Simulation of system with maintenance program (total annualrepair cost of $42,000):


M chine Bre kdo n nd M inten nce S stem


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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 peryear.

However, there are potentialproblemscaused by simulatingboth systems onlyonce.

Simulation results could exhibit significant variation since timebetween breakdowns and repair times are probabilistic.

To be sure of accuracy of results, simulations of each systemmust be run many times and 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.


Statistical Analysis of Simulation Results (1 of 2)


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Outcomes of simulation modeling are statisticalmeasuressuch as averages.

Statistical results are typically subjected to additionalstatistical analysisto determine their degree of accuracy.

Confidence limitsare developed for the analysis of thestatistical validity of simulation results.





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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



Simulation Results


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Simulation Results

Statistical Analysis with Excel (1 of 3)

Simulation with maintenance program.


Exhibit 14.9

Simulation Results


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Simulation Results


Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.10

Simulation Results


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Simulation Results



Crystal Ball


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Crystal Ball

Overview

Many realistic simulation problems contain more complexprobability distributionsthan those used in the examples.

However there are several simulation add-insfor Excel that

provide a capability to perform simulation analysis with avariety 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.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall

Crystal Ball


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Recap of Western Clothing Company break-even and profitanalysis:

Price (p) for jeans is $23

variable cost (cv) is $8

Fixed cost (cf) is $10,000

Profit Z = vp - cfvc

break-even volume v = cf/(p - cv)

= 10,000/(23-8)

= 666.7 pairs.

Crystal Ball

Simulation of Profit Analysis Model (1 of 15)


Crystal Ball


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Modifications to demonstrate Crystal Ball Assume volume is now volume demandedand is defined by a

normal probability distributionwith mean of 1,050 andstandard deviation of 410 pairs of jeans.

Price is uncertain and defined by a uniform probabilitydistributionfrom $20 to $26.

Variable cost is not constant but defined by a triangularprobability distribution.

Will determineaverageprofit and profitability with givenprobabilistic variables.

Crystal Ball



Crystal Ball


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Crystal Ball



Crystal Ball


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Crystal Ball



Exhibit 14.12

Crystal Ball


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Crystal Ball



Exhibit 14.13

Crystal Ball


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Crystal Ball



Crystal Ball


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Crystal Ball



Crystal Ball


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Crystal Ball



Crystal Ball


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Crystal Ball



Crystal Ball


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Crystal Ball



Exhibit 14.18

Crystal Ball


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C ystal Ball



Exhibit 14.19

Crystal Ball


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y



Crystal Ball


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y



Crystal Ball


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y



Crystal Ball


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y



Exhibit 14.23

Verification of the Simulation Model (1 of 2)


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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 resultswith actual real-world data.

( )


Verification of the Simulation Model (2 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-stateconditions.

A standard, fool-proof procedure for validation is not available.

Validity of the model rests ultimately on the expertise andexperience of the model developer.

( )


Some Areas of Simulation Application


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Queuing

Inventory Control

Production and Manufacturing

Finance Marketing

Public Service Operations

Environmental and Resource Analysis

pp


Example Problem Solution (1 of 6)


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Willow Creek Emergency Rescue Squad

Minor emergency requires two-person crew

Regular emergency requires a three-person crewMajor emergency requires a five-person crew

p ( )




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Distribution of number of calls per night and emergency type:

Calls Probability

0123456

.05

.12

.15

.25

.22

.15

.061.00

Emergency Type Probability

MinorRegularMajor

.30

.56

.141.00

p ( )


1. Manually simulate 10 nights of calls

2. Determine average number of calls

each night

3. Determine maximum number ofcrew members that might be needed

on any given night.



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Calls ProbabilityCumulativeProbability

Random NumberRange, r1

01

23456

.05

.12

.15.25

.22

.15

.061.00

.05

.17

.32.57

.79

.941.00

1 56 17

18

3233 5758 7980 94

95 99, 00

EmergencyType

ProbabilityCumulativeProbability

Random NumberRange, r1

MinorRegularMajor

.30

.56

.141.00

.30

.861.00

1 3031 86

87 99, 00

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

p ( )




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Step 2: Set Up a Tabular Simulation (use second column of random

numbers in Table 14.3).

p ( )




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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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taylor introms10 ppt 14

Documents

pearson education

monte carlo technique

monte carlo processcopyright

random number table

prentice hallsimulationchapter

analogue simulation

generatethe random variable

values of demand