Transcript Workshop-DM

Probability Distributions in Decision Making &
Monte Carlo Simulation Workshop
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Outline/Coverage

Terms for reference
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Introduction

The Uniform Distribution
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The Triangular Distribution
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The Normal Distribution
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Monte Carlo Simulation Overview
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Monte Carlo Simulation Practice Examples
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Carefully read and study each slide
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Some Terms used in the workshop

x is a “random variable”, a symbol representing an outcome that can take on a
value. For example, x may be used to represent outcomes that could occur in say
flipping two coins. So, x could take on the values/outcomes of “Head Head”,
“Head Tails”, “Tails Head”, or “Tails Tails”. A different example would be purchasing
a part from a supplier where the price could range from $100 to $200 depending
on market conditions. In this case, x could take on a number of values such as
$141.25 or $162.89 and so on.

f(x), f represents frequency and f(x) represents frequency of a particular outcome,
x. If we were flipping a single coin 10 times, f(Heads) might be anywhere from one
to ten “Heads” outcomes.

p(x), p represent probability or likelihood and p(x) represents probability of a
particular outcome, x. In the case of flipping a single coin, p(Heads)=0.50.

e(x), e represents expected value of a particular outcome, x. For example, if we
were to flip a single coin 100 times, we would expect 50 Heads on average, thus
e(Heads)=50. We use probabilities to obtain expected values. In this example, we
have 100 flips and the probability of “Heads” is 0.50, so 100 X 0.50 = 50, e(Heads).

mean is simply the average. It is the sum of the values divided by the number of
values. The mean is one measure of “central tendency”, used to help describe
where the center of a set of data set is located.
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Introduction
A Probability Distribution is a graphical representation of possible values a
random variable (x) may assume along with the probability of the variable taking
on these values.
The simplest probability distribution is the Uniform Distribution. The X-axis
(horizontal axis) is the list of possible values the variable can assume. The Y-axis
(vertical axis) represents the probability of the variable taking on a certain
value. In a uniform distribution, the probabilities for the values the random
variable are all equal.
For example, flipping a coin can be represented using a uniform distribution.
The possible outcomes or values are Heads and Tails. The probability of flipping
a Head is 0.5 and the probability of flipping a Tail is 0.5, the same. Notice in the
table below the probabilities are the same or uniform, 0.50.
outcome, x
p(x)
Heads
0.50
Tails
0.50
This is a probability distribution shown
as a table.
This is a probability distribution shown
as a graph.
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Introduction
Now, consider rolling a single die. Use the table below to fill in random
variable values or possible outcomes along with their associated probabilities.
x
p(x)
This is a probability distribution shown as a table.
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Introduction
Using the table from the prior slide, build the uniform distribution for the
single die roll:
p(x)
likelihood
x
outcome
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Triangular distribution
The triangular distribution is very popular in business because it is easy to
understand and use effectively. It only requires the specification of three
values, its minimum, most likely, and maximum.
The mean of a triangular distribution is simply (min + most likely + max) / 3.
For example, we are not not sure how many hours it will take to complete a
task. The shortest time is estimated to be 12 hours, but the task could take as
long as 22 hours. Most likely, the task will complete in 14 hours, but could take
any amount of time between 12 and 22 hours.
The task time is modeled below using the triangular distribution.
likelihood
mean task time is (12+14+22) / 3 = 16
12
14
22
task time (hours)
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The normal distribution
Another very popular distribution is the normal distribution. Like the
triangular distribution, the normal distribution is a continuous distribution
meaning the random variable can take on any value in its range. However, the
normal distribution is always symmetric, with its mean always at the center
value. Values on each side of the center are equally likely.
For example, below is a normal distribution of class grades with a mean of 80.
A score around 80 is the most likely. Scores around 70 and 90 are less likely
than a score around 80, but are equally likely to each other.
likelihood
The mean of a normal distribution is its center, in this case 80.
mean class grade is 80
class test grades
70
80
90
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MC Simulation
Monte Carlo simulation is a technique in which many values of a random variable are
generated from a distribution. When a value is obtained from a distribution it is called a trial.
Two examples are shown below. The first shows Monte Carlo simulation trail results from a
normal distribution, the second shows MC trial results from a uniform distribution.
Example 1:
class test grades (x)
likelihood
MC Simulation
81.2
68.7
91
82.5
79
82.5
81.6
77.68
MC Simulation
Trial Results
57.8
class test grades (x)
70
80
90
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MC Simulation
Supplier part price (x)
Example 2:
MC Simulation
148
153
160.2
137
MC Simulation
Trial Results
likelihood
152.4
Supplier part price (x)
$100
$150
$200
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The uniform distribution
Two values are needed to specify a uniform distribution, a small and a large.
The uniform distribution is a continuous distribution and can take on any value
between its smallest and largest, all will equal likelihood.
The Uniform Distribution is shown below and its mean is simply the middle,
(small value + large value) / 2.
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New but related topic,
Monte Carlo Simulation
Monte Carlo simulation generates the outcomes from a probability distribution, in
this case the uniform distribution which has 2 outcomes with equal likelihoods. The
following Excel spreadsheet is an example of the Monte Carlo technique generating
20 outcomes for flipping a coin.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Outcome (x)
T
T
H
H
T
T
T
H
T
H
H
H
T
H
H
H
H
H
T
T
=IF(RAND()<0.5,"H","T")
Note: A random number generator (shown above) is used in
Monte Carlo simulation to obtain values for the random variable
or outcome. No need to worry about this Excel function!
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MC Simulation
Taking the coin flip example one step further, we can easily generate 1000 trials and
outcomes, count the Head and Tail outcomes, and plot the Monte Carlo simulation
results in a distribution. Note: some rows (trials) are not shown (hidden) for clarity.
491
509
f(x)
MC Simulation with 1000 trials.
Results; 491 Tails and 509 Heads.
f(x) stands for frequency, f(Tails)=491,
f(Heads)=509 in this run of 1000 trials.
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MC Simulation
expected frequency, e(x)
Now, considering the graph on the prior slide, how would the graph look if you
replaced f(x) with e(x), the expected number of each outcome in 1000 flips or trials?
Complete the graph below:
600
Coin flip Histogram
expected number, e(x) of Heads and Tails
500
400
300
200
100
0
Number of T
Number of H
Outcome (random variable), x
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MC Simulation
Now, considering the graph on the prior slide, how would the graph look if you
placed probabilities of outcomes on the vertical axis? Complete the graph below:
probability, p(x)
1
Coin flip Histogram
probability distribution, p(x)
0.75
0.5
0.25
0
Number of T
Number of H
Tails
Heads
Outcome (random variable), x
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MC Simulation
Go to the following website and download the example MC simulator. Instructions
below:
1. Goto the website: www.uta.edu/faculty/weltman
2. Click on the OPMA 3306 link
3. Download the file Excel Workshop Simulator
4. Open the file in Excel
Download this one
This one is for review, later!
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MC Simulation
Considering the rolling of a single die example, use the simulator spreadsheet to fill in
the following table and draw the frequency distribution below. f(x) stands for relative
frequency and is used for counting the frequency (or totaling up the number of times
each outcome, x occurs).
x
f(x)
f(x)
x
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MC Simulation
Considering the rolling of a single die example, in 600 rolls how many 1’s, 2’s, 3’s, 4’s, 5’s,
and 6’s would you expect using statistics? Fill in third column, e(x), in the table below:
x
f(x)
e(x)
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MC Simulation
Considering the rolling of a single die example, in 600 rolls how many 1’s, 2’s, 3’s, 4’s, 5’s,
and 6’s would you expect using statistics? Fill in the fourth column, p(x), in the table
below and the graph the probability distribution. Why are the columns f(x) and
e(x) different?
x
f(x)
e(x)
p(x)
p(x)
x
Note: f(x) comes from the MC simulation results.
e(x) comes from statistical probabilities.
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MC Simulation
Now, let’s use MC simulation for a slighted more involved decision (Profit Worksheet)
using multiple distributions to model the decision variables. Let’s try to see if MC
Simulation can help us arrive at a good decision based on risks and unknowns.
The selling price for a part we sell is fixed at $159. Depending on market conditions
and supplier, the part costs us between $80 and $100, and the cost is equally likely
anywhere between $80 and $100. Administrative cost is most likely to be $700,000
but could be as low as $300,000 or as high as $800,000. Demand for the part follows
a normal distribution with mean of 15,000 units and a standard deviation of 4,500.
The formula for profit is [revenue] – [costs] as shown in the formula below:
Profit = [rQ]-[AC+cQ], where r is the revenue per unit, AC is the administrative costs,
c is the supplier cost per unit, and Q is the demand or quantity sold. Thus,
Using the means of each distribution (the uniform, the triangular, and the normal),
calculate the profit below,
Mean part cost=
Note: You do not
use the MC
simulator to obtain
these values.
Mean admin cost=
Mean demand, Q=
Profitmean=
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MC Simulation
Next, use the MC simulator (Profit Worksheet) with 500 trials, fill in the yellow boxes
and obtain the (from the simulation) the average/mean profit, max and min profit, and
probability of a loss with this product. First, fill in the yellow boxes in the spreadsheet.
Once the data is entered (in the yellow boxes), refresh the spreadsheet several times. Study the spreadsheet
trials and results information to try to understand what is going on. Observe each column; parts cost, admin
cost, demand, and profit.
What does the MC simulation provide that is different from the mean profit calculation done on the prior slide?
Should we introduce this product? Why or why not?
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