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DECISION MODELINGDECISION MODELINGWITH WITH

MICROSOFT EXCELMICROSOFT EXCEL

Chapter 9

Copyright 2001Prentice Hall

Monte Carlo SimulationMonte Carlo Simulation

Part 1

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Simulation allows you to quickly and inexpensively Simulation allows you to quickly and inexpensively acquire knowledge concerning a problem that is usually acquire knowledge concerning a problem that is usually gained through experience (which is often costly and gained through experience (which is often costly and time consuming).time consuming).

IntroductionIntroduction

An experimental device (An experimental device (simulatorsimulator) will “act like” ) will “act like” ((simulatesimulate) the system of interest in a quick, cost-) the system of interest in a quick, cost-effective manner.effective manner.

Goal:Goal: To create an environment in which To create an environment in which information about alternative actions can information about alternative actions can be obtained through experimentation. be obtained through experimentation.

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SIMULATION vs. OPTIMIZATIONSIMULATION vs. OPTIMIZATION

In an In an optimization modeloptimization model, the values of the decision variables are , the values of the decision variables are outputsoutputs..

In a In a simulation modelsimulation model, the values of the decision variables are , the values of the decision variables are inputsinputs. The model evaluates the objective function for a particular . The model evaluates the objective function for a particular set of values.set of values.

The result of the model is a set of values for the decision variables The result of the model is a set of values for the decision variables that will maximize (or minimize) the value of the objective function.that will maximize (or minimize) the value of the objective function.

The result of the model is a measure of the The result of the model is a measure of the qualityquality of a suggested of a suggested solution and the variability in various performance measures due to solution and the variability in various performance measures due to randomness in the inputs.randomness in the inputs.

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Simulation and Random VariablesSimulation and Random Variables

Simulation models are often used to analyze a Simulation models are often used to analyze a decision under decision under riskrisk. Under risk, the behavior of one or more factors is not . Under risk, the behavior of one or more factors is not known with certainty. For example:known with certainty. For example:

demand for a product during the next monthdemand for a product during the next month

the return on an investmentthe return on an investment

the number of trucks that will arrive to be unloaded the number of trucks that will arrive to be unloaded

The factor that is not known with certainty is called the The factor that is not known with certainty is called the random variablerandom variable. .

The behavior of the random variable can be described by a The behavior of the random variable can be described by a probability distributionprobability distribution. .

MONTE CARLO METHOD:MONTE CARLO METHOD:

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When should simulation be used?When should simulation be used?

Simulation is one of the most frequently used tools of Simulation is one of the most frequently used tools of quantitative analysis today because:quantitative analysis today because:

1.1. Analytical models may be difficult or impossible toAnalytical models may be difficult or impossible to obtain, depending on complicating factors. obtain, depending on complicating factors.

2.2. Analytical models typically predict only average orAnalytical models typically predict only average or “steady-state” (long-run) behavior. “steady-state” (long-run) behavior.

3.3. Simulation can be performed with a variety ofSimulation can be performed with a variety of software on a PC or workstation. The level of software on a PC or workstation. The level of computing and mathematical skill required to computing and mathematical skill required to design and run a simulator has been substantially design and run a simulator has been substantially reduced. reduced.

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Types of Simulation Models

• Continuous– Based on mathematical equations.– Used for simulating continuous values for all

points in time.– Example: The amount of time a person spends in

a queue.

• Discrete– Used for simulating specific values or specific

points.– Example: Number of people in a queue.

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Simulation Methodology:

• Obtain probabilities from historical data, knowledge of underlying probability distribution, or expert opinion of (subjective) probabilities; Assign random number ranges to probabilities.

• Obtain random numbers from software packages, printed random number tables (or from your instructor); Use random numbers to “simulate” events.

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Advantages of Simulation

• Often leads to a better understanding of the real system.

• Years of experience in the real system can be compressed into seconds or minutes.

• Simulation does not disrupt ongoing activities of the real system.

• Simulation is far more general than mathematical models.

• Simulation can be used as a game for training experience.

• Simulation provides a more realistic replication of a system than mathematical analysis.

• Simulation can be used to analyze transient conditions, whereas mathematical techniques usually cannot.

• Many standard packaged models, covering a wide range of topics, are available commercially.

• Simulation answers what-if questions.

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Disadvantages of Simulation

• There is no guarantee that the model will, in fact, provide good answers.

• There is no way to prove reliability.

• Building a simulation model can take a great deal of time.

• Simulation may be less accurate than mathematical analysis because it is randomly based.

• A significant amount of computer time may be needed to run complex models.

• The technique of simulation still lacks a standardized approach.

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Simulation Applications

• Machine Breakdown problems

• Queuing problems

• Inventory problems

• Portfolio allocations

• Training

• Many other applications

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To generate a random variable, To generate a random variable, draw a random sampledraw a random sample from a given probability distribution.from a given probability distribution.

Generating Random VariablesGenerating Random Variables

Two broad categories of random variables:Two broad categories of random variables:

Can assume only certain specific values Can assume only certain specific values (e.g., integers)(e.g., integers)

Can take on any fractional value (an infinite Can take on any fractional value (an infinite number of values)number of values)

ContinuousContinuous

DiscreteDiscrete

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The game spinner below is an example of a physical device used to The game spinner below is an example of a physical device used to generate demand in a given model. generate demand in a given model.

Once spun, the spinner is equally likely to point to any point on the Once spun, the spinner is equally likely to point to any point on the circumference of the circle. circumference of the circle.

If the areas of the If the areas of the sectors are made to sectors are made to correspond to the correspond to the probabilities of different probabilities of different demands, the spinner demands, the spinner can be used to simulate can be used to simulate demand. Each spin demand. Each spin represents a trial.represents a trial.

12 (10.0%)12 (10.0%)

13 (10.0%)13 (10.0%) 8 (10.0%)8 (10.0%)

11 (20.0%)11 (20.0%)

10 (30.0%)10 (30.0%)

9 (20.0%)9 (20.0%)

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Although easy to understand, the spinner method of Although easy to understand, the spinner method of generating random numbers would be difficult to use if generating random numbers would be difficult to use if thousands of trials are necessary. Therefore, random thousands of trials are necessary. Therefore, random number generators have been developed in number generators have been developed in spreadsheets. spreadsheets.

Using a Random Number Using a Random Number Generator in a SpreadsheetGenerator in a Spreadsheet

To generate demand for a given model, first assign a To generate demand for a given model, first assign a range of random numbers to each possible demand.range of random numbers to each possible demand.

To do this correctly, the proportion of total numbers To do this correctly, the proportion of total numbers assigned to a demand must equal the probability of that assigned to a demand must equal the probability of that demand.demand.

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For example, using the interval from 0 to 1, make the For example, using the interval from 0 to 1, make the following assignment:following assignment:

10% of the 10% of the interval is interval is

assigned to 13assigned to 13

30% of the 30% of the interval is interval is

assigned to 10assigned to 10

20% of the 20% of the interval is interval is

assigned to 11assigned to 11

The probability of The probability of drawing a number drawing a number in the range of in the range of .90 to .99999 is 1 .90 to .99999 is 1 out of 10 or 0.1 out of 10 or 0.1 (10%).(10%).

This method is useful for generating discrete random This method is useful for generating discrete random variables.variables.

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To generate a To generate a discrete random variablediscrete random variable with the RAND() function in a with the RAND() function in a spreadsheet, two things are needed:spreadsheet, two things are needed:

A GENERALIZED METHOD:A GENERALIZED METHOD:

1. The ability to generate discrete uniform random 1. The ability to generate discrete uniform random variables variables

2. The distribution of the discrete random variable 2. The distribution of the discrete random variable to be generated to be generated

To generate a continuous random variable, two things are needed:To generate a continuous random variable, two things are needed:

1. The ability to generate continuous uniform1. The ability to generate continuous uniform random variables on the interval 0 to 1 random variables on the interval 0 to 1

2. The distribution (in the form of the 2. The distribution (in the form of the cumulativecumulative distribution function distribution function) of the random variable to) of the random variable to be generated be generated

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Continuous Uniform Random Variables. It is important to distinguish Continuous Uniform Random Variables. It is important to distinguish between between UU (the uniform random variable on the interval 0 to 1) and (the uniform random variable on the interval 0 to 1) and uu (a (a specific realization of that random variable). specific realization of that random variable).

.25.25

.5.5

.75.75

00

The game The game spinner can be spinner can be

used to used to generate generate

values of values of UU. .

Every point on the circumference of the circle corresponds to a number Every point on the circumference of the circle corresponds to a number between 0 and 1. between 0 and 1.

However, it is However, it is impractical for a impractical for a continuous continuous distribution since distribution since the exact point the exact point must be read must be read (e.g., .4999999999).(e.g., .4999999999).

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The Cumulative Distribution Function (CDF). Consider a The Cumulative Distribution Function (CDF). Consider a random variable, random variable, DD, the demand. The CDF for , the demand. The CDF for DD [called F(x)] [called F(x)] is then defined as the probability that is then defined as the probability that DD takes on a value takes on a value << x. x.

F(x) = Prob{D F(x) = Prob{D << x} x}

Knowing the probability distribution for Knowing the probability distribution for DD, the CDF for key , the CDF for key values of values of DD is: is:

With a continuous distribution, the probability that any With a continuous distribution, the probability that any specific value occurs is 0. Therefore, continuous random specific value occurs is 0. Therefore, continuous random variables do not have probability distributions. They are variables do not have probability distributions. They are defined by the density function and the CDF. defined by the density function and the CDF.

X 8 9 10 11 12 13X 8 9 10 11 12 13F(x) 0.1 0.3 0.6 0.8 0.9 1.0F(x) 0.1 0.3 0.6 0.8 0.9 1.0

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0

0.2

0.4

0.6

0.8

1

1.2

7 8 9 10 11 12 13 14

ProbabilityProbability

xx

F(F(xx))

Here is a graph of the CDF. To generate a discrete Here is a graph of the CDF. To generate a discrete demand using the graph:demand using the graph:

Step 2: Read the Step 2: Read the particular value particular value of the random of the random quantity, quantity, dd, on , on

this axisthis axis

Step 1: Locate Step 1: Locate the particular the particular value of value of UU on on

this axisthis axis

uu

dd

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THE GENERAL METHOD THE GENERAL METHOD APPLIED TO CONTINUOUS DISTRIBUTIONS: APPLIED TO CONTINUOUS DISTRIBUTIONS: The two-step process for generating a continuous random variable The two-step process for generating a continuous random variable WW is is shown below:shown below:

00.1

0.20.30.40.5

0.60.70.80.9

11.1

0 1 2 3 4 5 6 7 8 9 10xx

ProbabilityProbability

F(F(xx)=Prob{W)=Prob{W<<x}x}

The cumulative The cumulative distribution distribution

function of function of WWuu

As before, first As before, first locate the value locate the value

((uu) of the random ) of the random variable variable UU

ww

F(F(ww))

Then, read the Then, read the particular value of particular value of

the random the random quantity, quantity, WW, on , on

this axisthis axis

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Data Collection and Random Number Interval Example

Suppose you timed 20 athletes running the 100-yard dash and tallied the information into the four time intervals below.

Seconds 0-5.996-6.997-7.998 or more

Tallies Frequency41042

You then count the tallies and make a frequency distribution.

%20502010

Then convert the frequencies into percentages.

Finally, use the percentages to develop the random number intervals.

RN Intervals01-2021-7071-9091-100

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Converting Known Probabilities to Cumulative Distribution Frequencies for Discrete Events

1) Determine the number of discrete events and assign each event a probability.

2) Ensure that the sum of all probabilities add to 1. No individual probability is zero or negative.

3) Review all probabilities to determine the level of precision needed. For example, if three possible outcomes has probabilities of 30%, 20%, and 50% the required precision is two decimals.

4) Create a random number table.

5) Pick a random number, per the table. Find which row it falls in. The associated event is considered to have occurred.

Event Probability Random Number Cumulative ProbabilityA 0.123 001 - 123 0.123B 0.014 124 - 137 0.137C 0.652 139 - 789 0.789D 0.211 790 -1000 1.000

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Converting Known Probabilities to Cumulative Distribution Frequencies for Discrete Events - PRACTICE

Event Probability Random NumberA 0.150 001 - 150B 0.063 151 - 213C 0.308 214 - 521D 0.124 522 - 645E 0.087 646 - 732F 0.2 733 - 932G 0.068 933 - 1000

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Machine Breakdown Example

Suppose that a given production machine has a 4% chance of breakdown the day after it is serviced. Each day it runs without service the chance of breakdown increases by 3%. The P(Breakdown) can be described as = 0.01 + 0.03x where x = the number of days since service.

Day X P(F) RN Breaks

1 1 0.04 0.41 02 2 0.07 0.25 03 3 0.1 0.67 04 4 0.13 0.16 05 5 0.16 0.41 06 6 0.19 0.01 17 1 0.04 0.69 08 2 0.07 0.29 09 3 0.1 0.11 010 4 0.13 0.32 0

1) Calculate X, P(F) and Breaks

2) What are the total number of breakdowns?

3) What was the most days without a breakdown?

4) What was the average number of days without a breakdown?

5) Did this machine ever breakdown two days in a row?

Assume service was performed at day zero.

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Queuing Example

Suppose a bank has two loan officers. On average each officer can process 12 clients per hour. On average 24 clients arrive each hour. The true distributions of arrivals and hourly processing abilities are:

Prob RN Cum Freq Number Prob RN Cum Freq Number

0.20 001 - 020 0.20 23 0.13 001 - 013 0.13 100.30 021 - 050 0.50 24 0.25 014 - 038 0.38 110.42 051 - 092 0.92 25 0.30 039 - 068 0.68 120.08 093 - 100 1.00 26 0.32 069 - 100 1.00 13

Service RatesArrival Rates

1) Assuming the beginning queue is 0, develop a model to simulate a 10 hour day.2) What is the average number of people in the system?3) What is the average number of people in queue at the end of each hour?4) What is the average number of people served each hour?5) What was the maximum ending queue?6) What was the most loans processed in a single hour?7) How many hours had zero people in the ending queue ?

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Queuing Example

HourBegin Queue Arrive RN Arrivals

In System Serv1 RN

Server One

Capacity Serv2 RN

Server Two

CapacityTotal

CapacityNumber Served

Ending Queue

1 0 0.87 25 25 0.53 12 0.60 12 24 24 12 1 0.04 23 24 0.82 13 0.52 12 25 24 03 0 0.24 24 24 0.39 12 0.03 10 22 22 24 2 0.83 25 27 0.36 11 0.14 11 22 22 55 5 0.14 23 28 0.94 13 0.08 10 23 23 56 5 0.40 24 29 0.94 13 0.90 13 26 26 37 3 0.76 25 28 0.92 13 0.85 13 26 26 28 2 0.16 23 25 0.02 10 0.69 13 23 23 29 2 0.38 24 26 0.43 12 0.53 12 24 24 210 2 0.91 25 27 0.17 11 0.19 11 22 22 5

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Important Distributions

• Uniform - All possible values are equally likely. Typically uses the =RAND() function in excel to return a continuos number evenly distributed between 0 and 1.

• Normal - Values are distributed around a mean with a given standard deviation. This is the classic bell curve. In Excel the formula is =NORMINV(RAND(),mean, StDev)

• Exponential - Typical of arrivals in queuing theory where there is a mean arrival rate. The Excel formula is = -mean * LN (1-rand())

• Poisson - Also used for arrival rates in queuing theory. Though possible to formulate in Excel this is generally used in add-in packages such as @Risk or Crystal Ball.

• Triangular - Use a triangular distribution when the most likely outcome, the minimum outcome, and the maximum outcome are the only things known about the distribution. This can be implemented with Crystal Ball

• Discrete - Used when something either occurs or does not. For example 2.5 people can not arrive. Either 2 or 3 people show up. These distributions can be uniform, normal, exponential or custom.

• Custom - This is a user supplied distribution.

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Generating from the Normal Distribution. The normal distribution plays Generating from the Normal Distribution. The normal distribution plays an important role in many simulation and analytic models. Normality is an important role in many simulation and analytic models. Normality is often assumed.often assumed.

Consider drawing a random demand from a normal distribution with a Consider drawing a random demand from a normal distribution with a mean (mean () of 1000 and a standard deviation () of 1000 and a standard deviation () of 100. ) of 100.

If Z is a unit normal random variable (normally distributed with a mean of If Z is a unit normal random variable (normally distributed with a mean of 0 and a standard deviation of 1) then 0 and a standard deviation of 1) then + Z + Z is a normal random variable is a normal random variable with mean with mean and standard deviation and standard deviation . .

So, we can draw from a unit normal distribution. Excel has a built-in So, we can draw from a unit normal distribution. Excel has a built-in function that can do this:function that can do this:

= NORMINV( RAND() , 1000, 100)= NORMINV( RAND() , 1000, 100)

Excel will automatically return a normally distributed random number with Excel will automatically return a normally distributed random number with mean 1000 and std. dev. 100.mean 1000 and std. dev. 100.

Generating the Normal Distribution

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Generating from the Exponential Distribution. The Generating from the Exponential Distribution. The exponential distribution is often used to model the time exponential distribution is often used to model the time between arrivals in a queuing model. Its CDF is given between arrivals in a queuing model. Its CDF is given by:by:

F(x) = Prob{W F(x) = Prob{W << w} = 1 – e w} = 1 – e--ww

Where 1/Where 1/ is the mean of the random variable is the mean of the random variable WW..

Therefore, we want to solve the following equation for Therefore, we want to solve the following equation for ww::

u = 1 – eu = 1 – e--ww

The solution is: w = -1/The solution is: w = -1/ ln(1- u) ln(1- u)

Generating the Exponential Distribution

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Now, to draw a sample from an exponential distribution Now, to draw a sample from an exponential distribution with a mean of 20 (1/with a mean of 20 (1/) using this equation:) using this equation:

1. First generate a continuous, uniform random1. First generate a continuous, uniform random number with RAND() (for example, .75). number with RAND() (for example, .75).

2.2. Apply the formula: w = -1/Apply the formula: w = -1/ ln(1- u) ln(1- u)

= -20 (ln(1- .75))= -20 (ln(1- .75))

= -20 (-1.386)= -20 (-1.386)

= 27.72= 27.72

In a spreadsheet cell, simply enter:In a spreadsheet cell, simply enter:

= -20 *LN(1 – RAND() )= -20 *LN(1 – RAND() )

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Suppose you want to model a discrete uniform distribution of demand Suppose you want to model a discrete uniform distribution of demand where the values of 8 through 12 all have the same probability of where the values of 8 through 12 all have the same probability of occurring (uniform, equally likely). occurring (uniform, equally likely).

The spreadsheet has a function, =RAND(), that returns a random number The spreadsheet has a function, =RAND(), that returns a random number between 0 and 1. However, this will result in a continuous uniform between 0 and 1. However, this will result in a continuous uniform distribution.distribution.

To create a discrete uniform distribution, use the INT() function. For To create a discrete uniform distribution, use the INT() function. For example:example:

In general, if you want a discrete, uniform distribution of integer values In general, if you want a discrete, uniform distribution of integer values between between xx and and yy, use the formula:, use the formula:

INT(INT(xx + ( + (yy – – xx + 1)*RAND() ) + 1)*RAND() )

Discrete Distributions

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Simulations can be performed with spreadsheets alone. Simulations can be performed with spreadsheets alone. However, add-in software packages can enhance the However, add-in software packages can enhance the capabilities of Excel.capabilities of Excel.

Simulating with a SpreadsheetSimulating with a Spreadsheet

Two Excel add-in packages that will be used are Crystal Two Excel add-in packages that will be used are Crystal Ball and @Risk. Ball and @Risk.

These add-ins offer additional random distributions and These add-ins offer additional random distributions and easy commands to set up and run many more iterations easy commands to set up and run many more iterations than could be run in Excel. than could be run in Excel.

In addition, they automatically gather statistical and In addition, they automatically gather statistical and graphical summaries of the results.graphical summaries of the results.

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DISCRETE EVENT SIMULATIONDISCRETE EVENT SIMULATIONINTRODUCTIONINTRODUCTION

Typically, Typically, simulation simulation refers to a specific refers to a specific class of dynamic models involving the class of dynamic models involving the detailed observation of a complex detailed observation of a complex probabilistic system over time. probabilistic system over time. Continuous simulationContinuous simulation models involve models involve aggregate variables that change more or aggregate variables that change more or less continuously with time (such as those less continuously with time (such as those models in Chapter 9).models in Chapter 9).Discrete event simulationDiscrete event simulation is applied is applied whenever individual items are tracked and whenever individual items are tracked and in which abrupt or nonsmooth changes in in which abrupt or nonsmooth changes in the timing of events is the norm.the timing of events is the norm.

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Discrete event simulation models can be Discrete event simulation models can be used to delve into the fine details of used to delve into the fine details of complex systems with many interactions.complex systems with many interactions.

However, discrete event simulation models However, discrete event simulation models can become extremely complex to build and can become extremely complex to build and analyze.analyze.Discrete event simulation models are most Discrete event simulation models are most commonly used to create detailed commonly used to create detailed operational systems representing demands operational systems representing demands among activities requiring scarce resources among activities requiring scarce resources over time.over time.

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A single server queuing situation will be A single server queuing situation will be used to illustrate a discrete event used to illustrate a discrete event simulation. simulation.

EXAMPLE 1EXAMPLE 1A SIMULATION MODEL IN EXCELA SIMULATION MODEL IN EXCEL

We will illustrate the building of such We will illustrate the building of such models using a non-spreadsheet simulation models using a non-spreadsheet simulation package called Extend.package called Extend.

Extend can be used for both continuous Extend can be used for both continuous simulation and discrete event simulation.simulation and discrete event simulation.We will investigate the relative efficiency of We will investigate the relative efficiency of an airport car rental business (Hervis) by an airport car rental business (Hervis) by modeling the movement of customers modeling the movement of customers through the rental office.through the rental office.

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Customers arrive in an airport van and queue up, Customers arrive in an airport van and queue up, first come – first serve, for service at the rental first come – first serve, for service at the rental counter. counter.

Eventually, a given customer moves to the head of Eventually, a given customer moves to the head of the queue and, when the rental clerk is free, will the queue and, when the rental clerk is free, will receive service that consists ofreceive service that consists of

Finally, customers depart, collect their rental car Finally, customers depart, collect their rental car and drive away.and drive away.

filling out the rental formsfilling out the rental forms

choosing the rental optionschoosing the rental options

receiving some driving instructionsreceiving some driving instructions

etc.etc.

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Consider the following data giving the relative Consider the following data giving the relative frequency distribution of customer arrivals in any frequency distribution of customer arrivals in any given 5-minute block of time throughout a typical given 5-minute block of time throughout a typical 24-hour day.24-hour day.

Let’s simulate a week of 24-hour operations and Let’s simulate a week of 24-hour operations and segment time into 5-minute blocks. We assume segment time into 5-minute blocks. We assume that it will always take that it will always take exactlyexactly 5 minutes to serve a 5 minutes to serve a waiting customer.waiting customer.

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Because 7*24*60/5 = 2016 time blocks (exceeding Because 7*24*60/5 = 2016 time blocks (exceeding the 256-column limitation in Excel), the time blocks the 256-column limitation in Excel), the time blocks will be modeled in rows.will be modeled in rows.

To build the Hervis model in Excel, first label the To build the Hervis model in Excel, first label the time blocks by using the time blocks by using the Fill SeriesFill Series option from the option from the EditEdit menu. menu.

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Next, create a probability distribution table:Next, create a probability distribution table:

Cell K8 computes the expected number of customer Cell K8 computes the expected number of customer arrivals.arrivals.

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The computer will be driven by randomly drawn The computer will be driven by randomly drawn probabilities, which in turn will determine the probabilities, which in turn will determine the number of customers arriving in any 5-minute time number of customers arriving in any 5-minute time block.block.

The RAND() function will draw a random fraction The RAND() function will draw a random fraction greater than or equal to 0 and less than 1, one for greater than or equal to 0 and less than 1, one for each of the 2016 time blocks.each of the 2016 time blocks.

Set the Set the Tools – Options Tools – Options – Calculation – Calculation option to option to ManualManual to avoid to avoid drawing new random drawing new random numbers each time the numbers each time the worksheet is changed.worksheet is changed.

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Place the =RAND() formula in cell B5 and copy it Place the =RAND() formula in cell B5 and copy it down to cell B2020 to create all 2016 random down to cell B2020 to create all 2016 random fractions.fractions.

Next, create a modified probability table based on Next, create a modified probability table based on the =RAND() formulas (the inputs).the =RAND() formulas (the inputs).

=K4=K4=M5 + K5=M5 + K5=M6 + K6=M6 + K6

=VLOOKUP(B5,$M$4:$N$7,2)=VLOOKUP(B5,$M$4:$N$7,2)

The number of customer arrivals, as outputs, are The number of customer arrivals, as outputs, are computed by =VLOOKUP() formulas.computed by =VLOOKUP() formulas.

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Column D formulas compute the number of Column D formulas compute the number of customers in line and being served.customers in line and being served.

=C5 + F4=C5 + F4=MIN(D5,$C$2)=MIN(D5,$C$2)=D5 + E5=D5 + E5

=MAX(D$5:D$2020)=MAX(D$5:D$2020)=SUM(D$5:D$2020)=SUM(D$5:D$2020)=AVERAGE(D$5:D$2020)=AVERAGE(D$5:D$2020)

Column E formulas compute the number of customers Column E formulas compute the number of customers receiving rental counter service as being the minimum of the receiving rental counter service as being the minimum of the number in line and the number of rental counter stations in number in line and the number of rental counter stations in C2.C2.

Column F Column F computes the computes the number of number of customers customers remaining remaining unserved at unserved at the end of the the end of the 5-minute 5-minute block.block.

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The initial results show that on the average, ½ The initial results show that on the average, ½ customer arrives in each 5-min. block (or 1 customer arrives in each 5-min. block (or 1 customer every 10 minutes). It always takes only customer every 10 minutes). It always takes only one 5-min. interval to serve a customer, thus, the one 5-min. interval to serve a customer, thus, the rental counter is busy only ½ the time on average. rental counter is busy only ½ the time on average.

However, during peak times, you could have 9 However, during peak times, you could have 9 customers awaiting service, leading to a delay of customers awaiting service, leading to a delay of 45 minutes for the 945 minutes for the 9thth customer. customer.

To alleviate this wait, we could add another rental To alleviate this wait, we could add another rental station. station. We can use this model to evaluate the effect of We can use this model to evaluate the effect of adding another rental station, thus allowing up to adding another rental station, thus allowing up to two waiting customers to receiver service two waiting customers to receiver service simultaneously. simultaneously.

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Below is the revised model for 2 rental Below is the revised model for 2 rental stations.stations.

The revised statistics indicate that the The revised statistics indicate that the maximum number of customers delayed maximum number of customers delayed falls from an average of .499 to an average falls from an average of .499 to an average of .025.of .025.

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A CAPITAL BUDGETING EXAMPLE: A CAPITAL BUDGETING EXAMPLE: ADDING A NEW PRODUCT LINEADDING A NEW PRODUCT LINE

Airbus Industry is considering adding a new jet airplane Airbus Industry is considering adding a new jet airplane (model A3XX) to its product line. The following financial (model A3XX) to its product line. The following financial information is available:information is available:

Startup CostsStartup Costs $150,000$150,000Sales PriceSales Price $ 35,000$ 35,000Fixed Costs (per year)Fixed Costs (per year) $ 15,000$ 15,000Variable Costs (per year)Variable Costs (per year) 75% of revenues75% of revenues

Tax depreciation on the new equipment would be $10,000 per Tax depreciation on the new equipment would be $10,000 per year over the 4 year expected product life.year over the 4 year expected product life.Salvage value of the equipment at the end of the 4 years is Salvage value of the equipment at the end of the 4 years is estimated to be 0.estimated to be 0.Airbus’ cost of capital is 10% and tax rate is 34%.Airbus’ cost of capital is 10% and tax rate is 34%.

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If demand is known, then a spreadsheet can be used to If demand is known, then a spreadsheet can be used to calculate the calculate the net present valuenet present value (NPV). For example, (NPV). For example, assume that the demand for A3XXs is 10 units for each assume that the demand for A3XXs is 10 units for each of the next 4 years:of the next 4 years:

=C9*$B$3=$B$4

=C10*$D$2=$B$5

=C10-SUM(C11:C13)=$D$4*C14=C14 – C15=C16 + C13

=-$B$2

=NPV($D$3,C17:F17)+B17

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It is unlikely that demand will be the same every year. A more realistic It is unlikely that demand will be the same every year. A more realistic model would be one in which demand each year is a sequence of random model would be one in which demand each year is a sequence of random variables.variables.

THE MODEL WITH RANDOM DEMANDTHE MODEL WITH RANDOM DEMAND

This model of demand is appropriate when there is a constant base level This model of demand is appropriate when there is a constant base level of demand that is subject to random fluctuations from year to year. of demand that is subject to random fluctuations from year to year.

Sampling Demand with a Spreadsheet:Sampling Demand with a Spreadsheet: Assume initially that the demand Assume initially that the demand in a year will be either 8, 9, 10, 11, or 12 units with each value being in a year will be either 8, 9, 10, 11, or 12 units with each value being equally likely to occur.equally likely to occur.This is an example of a This is an example of a discrete uniform distributiondiscrete uniform distribution..

Now, use the formula =INT(Now, use the formula =INT(88 + + 55*RAND() ) to sample from a discrete *RAND() ) to sample from a discrete uniform distribution on the integers 8, 9, 10, 11, 12 .uniform distribution on the integers 8, 9, 10, 11, 12 .

Multiple trials can be performed by pressing the recalculation key for the Multiple trials can be performed by pressing the recalculation key for the spreadsheet (e.g., F9).spreadsheet (e.g., F9).

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=INT(8+5*RAND() )

Using this formula results in random demands. Using this formula results in random demands.

Hitting the F9 key would result in a different sample of demands, and Hitting the F9 key would result in a different sample of demands, and possibly a different NPV. possibly a different NPV. The demands are random variables, therefore, the NPV is also a random The demands are random variables, therefore, the NPV is also a random variable.variable.

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Two questions need to be answered about the NPV Two questions need to be answered about the NPV distribution:distribution:

EVALUATING THE PROPOSALEVALUATING THE PROPOSAL

1. What is the 1. What is the meanmean or expected value of the NPV? or expected value of the NPV?

2. What is the probability that the NPV assumes a 2. What is the probability that the NPV assumes a negative value (making the proposal to add the negative value (making the proposal to add the A3XX less attractive)? A3XX less attractive)?

To answer these questions, a simulation model must be To answer these questions, a simulation model must be built. To run the simulation automatically and capture built. To run the simulation automatically and capture the resulting NPV in a separate spreadsheet, use the the resulting NPV in a separate spreadsheet, use the Data TableData Table command. command.

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Start with a blank worksheet by clicking on the Start with a blank worksheet by clicking on the InsertInsert menu and select menu and select WorksheetWorksheet

Next, rename this blank worksheet Next, rename this blank worksheet 100 Iterations100 Iterations

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Type the starting value (1) in cell A2 and hit Type the starting value (1) in cell A2 and hit EnterEnter, then , then return to cell A2. return to cell A2.

Click the Click the EditEdit menu and menu and choose choose Fill – SeriesFill – Series..

In the resulting dialog, In the resulting dialog, select select Series in Series in ColumnsColumns and enter a and enter a stop value of 100. stop value of 100. Click Click OK OK to fill series. to fill series.

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Add column titles and the following formula to cell B2.Add column titles and the following formula to cell B2.

Now select the range A2:B101 and click Now select the range A2:B101 and click DataData – – TableTable..

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In the resulting dialog, enter C1 for the column input cell In the resulting dialog, enter C1 for the column input cell and click and click OKOK..

Note that since a random Note that since a random number generator is used in number generator is used in the formula, you may get the formula, you may get different values than these.different values than these.

Excel will recalculate the Excel will recalculate the values and store the values and store the resulting NPV in the resulting NPV in the adjacent cells in column adjacent cells in column B. B.

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Now, to turn the formulas into actual values upon which we Now, to turn the formulas into actual values upon which we can focus, first select the range of cells B2:B101, then click can focus, first select the range of cells B2:B101, then click on the on the Edit – CopyEdit – Copy menu. menu.

Next, click on the Next, click on the Edit – Paste Edit – Paste SpecialSpecial menu option and in the menu option and in the resulting dialog, choose resulting dialog, choose ValuesValues..

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To get a summary of the 100 iterations, use Excel’s built-in To get a summary of the 100 iterations, use Excel’s built-in data analysis tool. Click on data analysis tool. Click on Tools – Data AnalysisTools – Data Analysis. .

If you do not have this option, click on the If you do not have this option, click on the Add-inAdd-in option on option on the the ToolsTools menu and in the resulting dialog, click on menu and in the resulting dialog, click on Analysis Analysis ToolPakToolPak.. After clicking After clicking OKOK, the , the Data Data

AnalysisAnalysis dialog will open. dialog will open.

Select the Select the Descriptive Descriptive Statistics Statistics option and click option and click OKOK..

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In the resulting dialog, choose the In the resulting dialog, choose the Input RangeInput Range to to include the 100 iterations. include the 100 iterations.

Now click Now click on on Output Output RangeRange and and enter the enter the cell where cell where the output the output will be will be placed. placed.

In addition, In addition, select select Summary Summary StatisticsStatistics and click and click OKOK. .

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The resulting analysis gives the estimated mean NPV The resulting analysis gives the estimated mean NPV and standard deviation.and standard deviation.

Downside Risk and Upside Risk:Downside Risk and Upside Risk: To get a better idea about the range of possible NPVs that To get a better idea about the range of possible NPVs that could occur, look at the minimum and maximum NPVs.could occur, look at the minimum and maximum NPVs.

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Distribution of Outcomes:Distribution of Outcomes: Now we ask the question:Now we ask the question:How likely will these extreme outcomes occur?How likely will these extreme outcomes occur?

To answer this, examine the shape of the distribution of the To answer this, examine the shape of the distribution of the NPV by creating a histogram. NPV by creating a histogram.

Click on Click on Tools – Data AnalysisTools – Data Analysis and choose and choose HistogramHistogram. .

In the resulting dialog, set In the resulting dialog, set the input range and choose the input range and choose to save the results in a to save the results in a worksheet called worksheet called NPV NPV DistributionDistribution. .

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In the resulting analysis, the In the resulting analysis, the FrequencyFrequency (column B) indicates (column B) indicates the number of trials that fell into the bins (categories) defined the number of trials that fell into the bins (categories) defined by column A. by column A.

The cumulative % column indicates the cumulative The cumulative % column indicates the cumulative percentage of observations that fall into each category or bin.percentage of observations that fall into each category or bin.

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The histogram gives a visual representation of the The histogram gives a visual representation of the distribution of NPVs. Note that it is somewhat bell shaped. distribution of NPVs. Note that it is somewhat bell shaped.

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The next questions to ask are:The next questions to ask are:

How Reliable is the Simulation?How Reliable is the Simulation?

1. What is the 1. What is the meanmean or expected value of the NPV? or expected value of the NPV?

2. What is the probability that the NPV assumes a 2. What is the probability that the NPV assumes a negative value (making the proposal to add the negative value (making the proposal to add the A3XX less attractive)? A3XX less attractive)?

In this trial, the mean is $12,100.In this trial, the mean is $12,100.

In this trial, the probability is >15%.In this trial, the probability is >15%.

1. How much confidence do we have in the answers 1. How much confidence do we have in the answers from the first trial? from the first trial?

2. Would we be more confident if we ran more trials?2. Would we be more confident if we ran more trials?

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For a 95% confidence interval of the Mean, the formula For a 95% confidence interval of the Mean, the formula is:is:

estimated mean estimated mean ++ 1.96(standard deviation) 1.96(standard deviation)

In this case, the standard deviation is the standard error (the In this case, the standard deviation is the standard error (the standard deviation divided by the square root of the number standard deviation divided by the square root of the number of trials). of trials).

Based on this trial, the upper and lower confidence limits are:Based on this trial, the upper and lower confidence limits are:=$E$4-1.96*$E$8/SQRT($E$16)=$E$4-1.96*$E$8/SQRT($E$16)

=$E$4+1.96*$E$8/SQRT($E$16)=$E$4+1.96*$E$8/SQRT($E$16)

So, we have 95% confidence that the true mean NPV is So, we have 95% confidence that the true mean NPV is somewhere between $9,679 and $14,521.somewhere between $9,679 and $14,521.

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In summary,In summary,

1. Increasing the number of trials is apt to give a 1. Increasing the number of trials is apt to give a better estimate of the expected return. However, better estimate of the expected return. However, there can still be a difference between the there can still be a difference between the simulated simulated averageaverage and the true expected return. and the true expected return.

2. Simulations can provide useful information on the2. Simulations can provide useful information on the distribution results. distribution results.

3. Simulation results are sensitive to assumptions 3. Simulation results are sensitive to assumptions affecting the input parameters. affecting the input parameters.

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Key Learning's

• Monte Carlo method• Random variables• Distributions• Cumulative distribution functions• Machine breakdown models• Queuing models