Roget Pinky's Problem. Roget Pinky, a talented and wealth

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The Choice Phase
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Decision Analysis The Three Components

• A set of alternative actions
• We may chose whichever we please.
• A set of possible states of nature
• One will be correct, but we dont know in
  advance.
• A set of outcomes and a value for each
• Each is a combination of an alternative action
  and a state of nature.
• Value can be monetary or otherwise.

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Three Levels of KnowledgeDecision Situation
Categories

• Certainty
• Only one possible state of nature
• Ignorance
• Several possible states of nature
• Risk
• Several possible states of nature with an
  estimate of the probability of each

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States of Knowledge

• Certainty
• DM knows with certainty what the state of nature
  will be.
• Ignorance
• DM Knows all possible states of nature, but does
  not know probability of occurrence.
• Risk
• DM Knows all possible states of nature, and can
  assign probability of occurrence.

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Decision Making Under Ignorance

• LaPlace-Bayes
• Select alternative with best average payoff.
• Maximax
• Select alternative which will provide highest
  payoff if things turn out for the best.
• Maximin
• Select alternative which will provide highest
  payoff if things turn out for the worst.
• Minimax Regret
• Select alternative that will minimize the maximum
  regret.

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Roget Pinkys Problem
Roget Pinky, a talented and wealthy businessman,
has committed to promote an IndyCar race at Road
Alpharetta next March. Roget would have preferred
a date later in the Spring, but this was the best
date available considering Road Alpharetta's and
the IndyCar Series' schedules. He estimates that
it will cost him 2,000,000 to put on the race,
plus an average variable cost per spectator of
10. On a warm, dry day, he estimates that he
will draw 62,500 spectators the first year. Of
course, if it is cold and wet, he won't do as
well he figures he might get 25,000 hard core
fans. Cold and dry would improve on that by 10.
Since rain races can be very dramatic, if it is
wet but warm he can probably draw 30 more fans
than on a cold wet day. Including tickets and
his cut of concessions and souvenirs, he figures
he will bring in 75 from the average spectator.
Parking at Road Alpharetta is plentiful and free,
so that won't bring in any revenue.
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Of course, not even Roget Pinky can control the
weather. Next March any of the 4 states of
nature might happen. There are some things Roget
might do to alter the scenario. MARTA
(Metropolitan Alpharetta Random Transit
Authority) has offered him a transportation deal
that is hard to refuse. For a mere 500,000
MARTA would provide free (to the rider) 2 way
transportation between the track and essentially
any point served by MARTA, all weekend. Roget
figures, since folks like to drink and raise hell
at the races, this might draw a lot of people who
would rather not have a DUI on their license. On
a dry day, he estimates that it would boost
attendance by 10. On a wet day, when people
risk getting their cars stuck in the infield mud,
it's probably worth a 40 boost in attendance.
That would really help cut the risk from
rain. IndyCars have never really pulled big
crowds in the South this is NASCAR country.
NASCAR has offered him the possibility of a Busch
Grand National taxicab race for a total cost to
him of 500,000. Roget is tempted. It might be
a way of educating some NASCAR fans about
IndyCars, and he thinks that a BGN support race
might boost attendance 20. It is worth
considering. So Roget's alternatives are to put
the race on with or without MARTA and with or
without a BGN support race. See spreadsheet for
calculations but he gets the following payoff
table
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Roget Pinkys Payoff Table
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LaPlace-Bayes
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Maximax
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Maximin
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Net Payoff Table
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Regret Table
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Decision Making Under Risk

• Expected Monetary Value (EMV)
• Si The ith state of nature
• Aj The jth alternative action
• P(Si) The probability that Si will occur
• Vij The payoff if Aj and Si occurs
• EMVj The long-term average payoff
• EMVj ? P(Si) S Vi
• Variance ?P(Si) S (EMVj - Vij)2

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Expected Value Under Initial Information

• EVUII is the value of the decision you would make
  with the initial information available. It is
  the payoff (EMV) associated with the decision
  which generates the best or maximum EMV.
• EVUII Max(EMVj)

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Expected Value Under Perfect Information

• EVUPI measures what the payoff or outcome would
  be if you could know which State of Nature would
  in fact occur.
• EVUPI ?P(Si) S Max(Vij)

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Expected Value of Perfect Information

• EVPI measures how much better you could do on
  this decision if you could know which State of
  Nature would occur. In other words, it measures
  how much better off you are with Perfect
  Information than you were under Initial
  Information, and therefore represents the value
  of the additional information.
• EVPI EVUPI - EVUII

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Net Payoff w/EMV Variance
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Net Payoff Table - EVPI
20
Expected Opportunity Loss

• EOL is an alternative to EMV and produces the
  same results
• Si The ith state of nature
• Ai The jth alternative action
• P(Si) The probability that Si will occur
• Vij The payoff if Aj and Si occurs
• OLij OL if DM chooses Aj and Si occurs
• EOLj The long-term average opportunity loss
• OLij Max(Vij) - Vi
• EOLj ?P(Si) S OLij

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Opportunity Loss
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Decision Trees
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Decision Trees

• A method of visually structuring the problem
• Effective for sequential decision problems

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Decision Trees

• Components of a tree
• Two types of branches
• Decision nodes
• Chance nodes
• Terminal points
• Solving the tree involves pruning all but the
  best decisions
• Completed tree forms a decision rule

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Decision Node

• Decision nodes are represented by Squares
• Each branch refers to an Alternative Action

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Decision Node

• The expected monetary value (EMV) for the branch
  is
• The payoff if it is a terminal node, or
• The EMV of the following node
• The EMV of a decision node is the alternative
  with the maximum EMV

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Chance Node

• Chance nodes are represented by Circles
• Each branch refers to a State of Nature

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Chance Node

• The expected monetary value (EMV) for the branch
  is
• The payoff if it is a terminal node, or
• The EMV of the following node
• The EMV of a chance node is the sum of the
  probability weighted EMVs of the branches
• EMV ? P(Si) Vi

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Terminal Node

• Terminal nodes are optionally represented by
  Triangles
• The node refers to a payoff
• The value for the node is the payoff

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Solving the Tree

• Start at terminal node at the end and work
  backward
• Using the EMV calculation for decision nodes,
  prune branches (alternative actions) that are not
  the maximum EMV
• When completed, the remaining branches will form
  the sequential decision rules for the problem

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LaLa Lovely

• LaLa Lovely is a romantic actress. Mega Studios
  wants to sign her for a movie to be filmed next
  spring. The Turnip Network wants her to star in
  a mini-series to be shot during the same period.
  Turnip has offered her a fixed fee of 900,000,
  but Mega wants to give her a percentage of the
  Gross. Unfortunately, as usual, the Gross is not
  certain. Depending upon the success of the
  film(small, medium, or great), he may earn
  respectively 200,000, 1 million, or 3 million.
  Based upon Megas past productions, she assesses
  the probabilities of a small, medium, or great
  production to be respectively .3, .6, .1
• She may choose either the offer from Turnip or
  Mega but not both. Who should she sign with?

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LaLa Lovely Decision Tree
Small Gross
200000
.3
Medium Gross
Mega Studios
1000000
.6
Great Gross
EMV 960000
3000000
.1
EMV 960000
Turnip Network
900000
EMV 900000
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Bayes Theorem
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The Theorem

• Bayes' Theorem is used to revise the probability
  of a particular event happening based on the fact
  that some other event had already happened.

35
Review of Basic Probabilities
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Gender Discrimination Case?
2 X 2 Cross-Tabs Table of Gender Vs. Promotion
Gender and Promotion Status Related????
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Lecture Flow Bottom to Top
Relative Frequency and Cross-Tabs
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2 X 2 Cross-Tabs Table
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Unconditional Probabilities from Cross-Tabs Table

• Written As P(Event A)
• Frequency of Event A/Total Sample Size
• P(Male) 120/200 .60
• P(Promoted) ______
• P(Not Promoted) ______

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Conditional Probabilities from Cross-Tabs Table

• Written As P(A Given or if B)
• P(Promoted Male)
• P(Female Not Promoted)
• Frequency of Event A/Sample Space B
• For P(Prom Male), Denominator is Not 200, But
  Number of Males (120).

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Compute P(Promoted Given Male)
P(Promoted Male)
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Compute P(Promoted Given Female)
P(Promoted Female)
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Comparing Three Probabilities From Previous Slides

• Unconditional Probability
• P(Prom) ______
• Cond. Probability
• P(Prom Male) ______
• Cond. Probability
• P(Prom Female) ______

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Does Preponderance of Evidence Favor
Discrimination?

• Conclusions from Previous Slide?
• Intervening Variables
• What Other Variables Could Affect Promotion Other
  Than Gender?
• What if n 200 is Only Sample Taken From the
  Firm?

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How Should the Table Have Looked if Not
Statistically Related?
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How Should the Table Have Looked if Not
Statistically Related?
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How Should the Table Have Looked if Not
Statistically Related?
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Other Types of ProbabilitiesJoint Probabilities
P(Prom and Male) P(Not Prom and Female)
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Other Types of Probabilities Union Probabilities
P(Prom or Male) P(Not Prom or Female)
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Probability Information

• Prior Probabilities
• Initial beliefs or knowledge about an event
  (frequently subjective probabilities)
• Likelihoods
• Conditional probabilities that summarize the
  known performance characteristics of events
  (frequently objective, based on relative
  frequencies)

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Probabilities Involved

• P(Event)
• Prior probability of this particular situation
• P(Prediction Event)
• Predictive power of the information source
• P(Prediction ? Event)
• Joint probabilities where both Prediction Event
  occur
• P(Prediction)
• Marginal probability that this prediction is made
• P(Event Prediction)
• Posterior probability of Event given Prediction

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Circumstances for using Bayes Theorem

• You have the opportunity, usually at a price, to
  get additional information before you commit to a
  choice.
• You have likelihood information that describes
  how well you should expect that source of
  information to perform.
• You wish to revise your prior probabilities.

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The Choice Phase
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An Apartment Complex Contains 40 Monthly
Furnished Rental Units. The Lease Is Typically
for a Month and Is Renewable in One Month
Increments. Our Firm Is Considering Purchasing
the Complex and is Considering a Five Year Time
Horizon. It Wants to Know What Is the Potential
Profit From the Investment. It Anticipates
Renting the Units at 950 per Month. They
Anticipate Spending About 30,000 per Month for
Expenses.Lets First Focus on Profitability.
Ultimately We Will Compute Expected Return on
Investment to Determine If This Project Meets Our
Firms Minimum Target Value.
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Apartment Decision Purchase Model
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Types of Variation for Uncontrollable Variables

• Assignable-Cause Variation -- Use Regression
  Modeling or Data Analysis Methods.
• Did Use Regression Analysis to Estimate Annual
  Demand for EOQ Model.
• Did Use Mean and Standard Deviation for Stock
  Returns and Risk in Portfolio Model.
• Common-Cause Variation (Uncertainty)
• Use Monte Carlo Simulation (Crystal Ball)

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Simulation Modeling

• Monté Carlo Simulation is used to model the
  random behavior of components.
• Some systems with random components are too
  complex to solve for a closed-form solution.
• Steady state solution may not provide the
  information desired.
• Monte Carlo simulation is a fast and inexpensive
  way to obtain empirical results.

59
Building a Simulation Model

• Required Elements
• The basic logic of the system
• The known (or estimated) distributions of the
  random variables

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Probability Definitions -1 of 2

• Random Variable
• A consistent procedure for assigning numbers to
  random events
• Random Process
• The underlying system that gives rise to random
  events
• Probability Distribution
• The combination of a particular random variable
  and a particular random process

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Probability Definitions - 2 of 2

• Probability density function (pdf)
• A mathematical description of the relative
  likelihood of occurance of each random value
• Distribution function (DF)
• The cumulative form of the pdf
• Variate
• A single observation of the random variable for a
  pdf

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Distributions

• A distribution defines the behavior of a variable
  by defining its limits, central tendency and
  nature
• Mean
• Standard Deviation
• Upper and Lower Limits
• Continuous or Discrete

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Uniform Distribution

• All values between minimum and maximum occur with
  equal likelihood
• Conditions
• Minimum Value is Fixed
• Maximum Value is Fixed
• All values occur with equal likelihood
• Examples - Value of Property, Cost

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Normal Distribution

• Define uncertain variables
• Conditions
• Some value of the uncertain variable is most
  likely (mean)
• Uncertain variable is symmetric about the mean
• Uncertain variable is more likely to be in
  vicinity of the mean than far away
• Examples - inflation rates, future prices

65
Triangular Distribution

• Used when we know where the minimum, maximum and
  most likely values occur
• Conditions
• Minimum number of items is fixed
• Maximum number of items is fixed
• The most likely value is between the min and max,
  forming a triangle
• Examples - Number of goods sold per week, or
  quarter, etc.

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Binomial Distribution

• Used to define the behavior of a variable that
  takes on one of two values
• Conditions
• For each trial, only two outcomes are possible
• The trials are independent
• Chances of an event occurring remain the same
  from one trial to the other
• Examples - defective items in manufacturing, coin
  tosses, etc.

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Generating Simulation Data

• When we do not have ample data to conduct an
  analysis, we run an iterated simulation by
  generating input values
• Each value for an input variable is based on an
  assumption about its distribution
• For example,
• Profit can be uniformly distributed
• Defects in manufacturing can be binomially
  distributed

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What If Uncontrollable Variables Best Case /
Worst Case

• If Best Case for UNCONTROLLABLE Variables
  Generates an Exceptional Good Target Cell Value,
  Is It Worth Attempting to Gain Control over
  Previously Uncontrollable Variables?
• If Worst Case is Very Bad, Is It Worth Developing
  a Contingency Plan?

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Apartment Decision Purchase Model
Uncontrollable
Controllable
Output
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Uncontrollable Variable Cells in Crystal Ball

• Uncontrollable Variables AKA Assumption Cells.
• Use Probability Distributions to Represent
  Uncontrollable Variables.
• Number of Units Rented per Month in Cell B3
• Expected Expenses in Cell B5

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Output Cells in Model

• Output Cells Called Forecast Cells in Crystal
  Ball.
• Profit/Loss Five Years , Cell B8
• Click on Cell and Then Define Forecast.

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Apartment Decision Purchase Model
Click on B3 and then Define Assumption or
Distribution to Model It.
Click on B5 and then Define Assumption or
Distribution to Model It.
Click on B8 and then Define Forecast.
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No
Yes
Display Stats
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Which Distribution to Use?
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Parameters for Distributions

• Uniform
• Minimum and Maximum Values
• Normal
• Most Likely Value and Standard Deviation
• Standard Deviation Max - Min/6
• Triangular
• Most Likely, Minimum, and Maximum Values
• Weibull Distribution
• Location, Scale, Shape Parameters.
• Skewed Right and Left Distribution and
  Exponential Distributions

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Why Monte Carlo Simulation?

• Helps Examine the Simultaneous Impacts of the
  Possible Variation in Uncontrollable Variables on
  Output Variable(s).
• Variation is Additive!!!!
• Determines the Worst and Best Possible Values for
  Output Variable.
• Assigns Probabilities to Output Variable(s)
  Ranges.

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Conclusion

• Monté Carlo simulation can be used when data is
  hard to come by.
• Monté Carlo simulation can also be used to test
  the range of inputs to get a reliable outcome.

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Major Problems in Making Managerial Decisions

• Dont Understand Decision Environment.
• Consider Too Few Alternatives.
• Problem Solving Meetings Ineffectively Run.
• Alternatives Clones of One Another.
• Decision Making Not a Formal Analysis.