Chapter 4 - Simulation 1 Chapter 4 Simulation Introduction to Management Science 8th Edition by Bernard W. Taylor III.

Chapter 4 - Simulation 1

Chapter 4

Simulation

Introduction to Management Science

8th Edition

by

Bernard W. Taylor III


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


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


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


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 well-defined populations.

Monte Carlo Process


Table 4.1Probability 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)


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.



Figure 4.1A Roulette Wheel for Demand



Figure 4.2Numbered Roulette Wheel


When wheel is spun actual demand for PC’s is determined by a number at rim of the wheel.


Table 4.2Generating Demand from Random Numbers


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.


Select number from a random number table:

Table 4.3Random Number Table



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).





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



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.



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)


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.

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


Exhibit 4.1

Simulation with Excel Spreadsheets (1 of 3)


Exhibit 4.2


“Lookup”


Exhibit 4.3



Exhibit 4.4

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

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

=1+MAX(G6-H6,0)

Order one laptopeach week


Exhibit 4.5

Order size of two laptops each week.

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

=2+MAX(G6-H6,0)

Order two laptopseach week


Table 4.5Distribution of Arrival Intervals

Table 4.6Distribution of Service Times

Simulation of a Queuing SystemBurlingham Mills Example (1 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




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.


Exhibit 4.6

Computer Simulation with ExcelBurlingham Mills Example


€

A continuous function must be used for continuous distributions.Example:

f(x)= x8

, 0≤ x≤ 4 where x = time (minutes)

Cumulative probability of x:

F(x)= x8

0

x∫ dx= 1

8x dx= 1

80

x∫ 1

2x2 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟0

x

F(x)= x216

Let F(x) = the random number r

r = x216

x= 4 rBy generating a random number,r, a value x for "time" is determined.

Example: if r = .25, x= 4 .25 = 2 minutes

Continuous Probability Distributions

0 4

1/2


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(r1), value of x for a given value of r1.


Table 4.8Probability Distribution of Machine Repair Time



Table 4.9Revised Probability Distribution of Machine Repair Time with the Maintenance Program


Revised probability of time between machine breakdowns:

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

x = 6*sqrt(r1)


Table 4.10Simulation of Machine

Breakdowns and Repair Times


Simulation of system without maintenance program (total annual repair cost of $84,000):

x = 4*sqrt(r1)


Table 4.11Simulation of Machine

Breakdowns and Repair with the Maintenance

Program


Simulation of system with maintenance program (total annual repair cost of $42,000):

x = 6*sqrt(r1)



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.


Exhibit 4.7

Machine Breakdown and Maintenance SystemSimulation with Excel (1 of 2)

Original machine breakdown example:


Exhibit 4.8

Machine Breakdown and Maintenance SystemSimulation with Excel (2 of 2)

Simulation with maintenance program.


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)


Formulas for 95% confidence limits:

upper confidence limit

lower confidence limit

where 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.

€

=x+(1.96)(σ / n)

€

=x−(1.96)(σ / n)

x

Statistical Analysis of Simulation Results (2 of 2)


Exhibit 4.9

Simulation ResultsStatistical Analysis with Excel (1 of 3)

Simulation with maintenance program.


Exhibit 4.10



Exhibit 4.11



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.


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 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.



Exhibit 4.12



Exhibit 4.13



Exhibit 4.14



Exhibit 4.15



Exhibit 4.16



Exhibit 4.17



Exhibit 4.18



Exhibit 4.19



Exhibit 4.20


Chapter 4 - Simulation 53Exhibit 4.21



Exhibit 4.22



Exhibit 4.23



Exhibit 4.24



Exhibit 4.25



Exhibit 4.26



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)


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)


Queuing

• Inventory Control

• Production and Manufacturing

• Finance

• Marketing

• Public Service Operations

• Environmental and Resource Analysis

Some Areas of Simulation Application


Data

Willow Creek Emergency Rescue Squad

Minor emergency requires two-person crew, regular, a three-person crew, and major emergency, a five-person crew.

Example Problem Solution (1 of 6)


Distribution of number of calls per night and emergency type:

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.

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



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

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



Step 2: Set Up a Tabular Simulation (use second column of random numbers in Table 4.3).



Step 2 continued:



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 1 Chapter 4 Simulation Introduction to Management Science 8th Edition by Bernard W. Taylor III.

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