1 the choice phase. 2 decision analysis: the three components na set of alternative actions: –we...
TRANSCRIPT
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Q u a lita t iveM eth od
U n ivaria teD ata
A n a lys is
Q u an tita tiveM eth od s
In te llig en ceP h ase
U n d ers tan d in gth e R e la tion s
M od e lin g th eP rob lem
B ivaria teD ata
A n a lys is
D es ig nP h ase
D ec is ionA n a lys is ,
D ec is ion TreesB ayes ian A n a lys is
R isk A n a lys is& S im u la tion
A vo id in gG rou p Th in k
C h o ice P h ase
D ec is ionS c ien ce
F ou n d ation s
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 don’t 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 Knowledge:Decision 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 Pinky’s ProblemRoget 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 Pinky’s Payoff Table
ColdWet
ColdDry
WarmWet
WarmDry
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500
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LaPlace-Bayes
ColdWet
ColdDry
WarmWet
WarmDry
Mean
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500 $396,875
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000 $376,250
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750 $416,875
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500 $500,250
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Maximax
ColdWet
ColdDry
WarmWet
WarmDry
Maximum
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500 $2,062,500
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000 $2,375,000
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750 $1,968,750
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500 $2,362,500
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Maximin
ColdWet
ColdDry
WarmWet
WarmDry
Minimum
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500 ($375,000)
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000 ($550,000)
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750 ($533,750)
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500 ($640,500)
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Net Payoff TableColdWet
ColdDry
WarmWet
WarmDry
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500
Ideal ($225,000) ($212,500) $549,000 $2,375,000
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Regret Table
ColdWet
ColdDry
WarmWet
WarmDry
MaxRegret
No Busch,No MARTA
$150,000 $0 $436,500 $312,500 $436,500
Busch,no MARTA
$325,000 $142,500 $514,000 $0 $514,000
MARTA,no Busch
$0 $321,250 $91,500 $406,250 $406,250
MARTAand BUSCH
$45,000 $426,000 $0 $12,500 $428,000
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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) Vi
• Variance = P(Si) (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) 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 & VarianceColdWet
ColdDry
WarmWet
WarmDry
EMVVariance/1,000,000
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500 $697,500 $1,028,259
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000 $737,000 $1,480,694
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750 $769,500 $895,980
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500 $923,400 $1,290,211
Probability 0.1 0.15 0.4 0.35
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Net Payoff Table - EVPIColdWet
ColdDry
WarmWet
WarmDry
EMV
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500 $697,500
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000 $737,000
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750 $769,500
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500 $923,400
Ideal ($225,000) ($212,500) $549,000 $2,375,000 $996,475
Probability 0.1 0.15 0.4 0.35
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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) OLij
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Opportunity LossColdWet
ColdDry
WarmWet
WarmDry
EOL
No Busch,No MARTA
($375,000) ($212,500) $112,500 $2,062,500 $296,975
Busch,no MARTA
($550,000) ($355,000) $35,000 $2,375,000 $259,475
MARTA,no Busch
($225,000) ($533,750) $457,500 $1,968,750 $226,975
MARTAand BUSCH
($270,000) ($640,500) $549,000 $2,362,500 $73,075
Probability 0.1 0.15 0.4 0.35
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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
1) 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 Mega’s past productions, she assesses the probabilities of a small, medium, or great production to be respectively .3, .6, & .1
2) 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
1000000 .6 Great Gross
3000000 .1
Mega Studios
900000 Turnip Network
EMV 960000
EMV 900000
EMV 960000
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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.
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AP
BPBAP
AP
ABPABP
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Review of Basic Probabilities
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Gender Discrimination Case?Gender Discrimination Case?2 X 2 Cross-Tabs Table of Gender Vs. Promotion
Gender and Promotion Status Related????Gender and Promotion Status Related????
Male Female Total
Promoted 40 10 50
Not Promoted 80 70 150
Total 120 80 200
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Lecture Flow: Bottom to Top
U nconditionalP robabilities
C onditionalP robability
S tatistica l Independence
Relative Frequency and Cross-Tabs
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2 X 2 Cross-Tabs Table
Male Female Total
Promoted 40 10 50
Not Promoted 80 70 150
Total 120 80 200
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Written As P(Event A)P(Event A)
Frequency of Event A/Total Sample Size
P(Male) = 120/200 =.60 P(Promoted) = ______ P(Not Promoted) = ______
Unconditional Probabilities from Cross-Tabs Table
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Written As: P(A Given or if B)P(A Given or if B) P(Promoted | Male)P(Promoted | Male) P(Female | Not Promoted)P(Female | Not Promoted)
Frequency of Event A/SampleSample Space BSpace B
For P(Prom | Male), Denominator is Not 200, For P(Prom | Male), Denominator is Not 200, But Number of Males (120).But Number of Males (120).
Conditional Probabilities from Cross-Tabs Table
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Compute P(Promoted Given Male)
P(Promoted | Male) =
Male Female Total
Promoted 40 10 50
Not Promoted 80 70 150
Total 120 80 200
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Compute P(Promoted Given Female)
P(Promoted | Female) =
Male Female Total
Promoted 40 10 50
Not Promoted 80 70 150
Total 120 80 200
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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 VariablesWhat 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?Male Female Total
Promoted 0.25
Not Promoted 0.75
Total 0.6 0.4 1
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How Should the Table Have Looked if Not Statistically
Related?Male Female Total
Promoted .25*.6 .25*.4 0.25
Not Promoted .75*.6 .75*.4 0.75
Total 0.6 0.4 1
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How Should the Table Have Looked if Not Statistically
Related?Male Female Total
Promoted 30 20 50
Not Promoted 90 60 150
Total 120 80 200
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Other Types of Probabilities:
Joint Probabilities
P(Prom andand Male) =
P(Not Prom andand Female) =
Male Female Total
Promoted 40 10 50
Not Promoted 80 70 150
Total 120 80 200
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Other Types of Probabilities:
Union Probabilities
P(Prom or Male) =
P(Not Prom or Female) =
Male Female Total
Promoted 40 10 50
Not Promoted 80 70 150
Total 120 80 200
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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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Q u alita tiveM eth od
U n ivaria teD ata
A n a lys is
Q u an tita tiveM eth od s
In te llig en ceP h ase
U n d ers tan d in gth e R e la tion s
M od e lin g th eP rob lem
B ivaria teD ata
A n a lys is
D es ig nP h ase
D ec is ionA n a lys is ,
D ec is ion TreesB ayes ian A n a lys is
R isk A n a lys is&
M on té C arloS im u la tion
A vo id in gG rou p Th in k
C h o ice P h ase
D ec is ionS c ien ce
F ou n d ation s
The Choice Phase
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Enhance or EnrichM odel
W hat-If: Eva luateA lte rna tives
Valida te M ode l
Bu ild M ode l
D iagnosis
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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.Let’s First Focus on Profitability. Ultimately We Will Compute Expected Return on Investment to Determine If This Project Meets Our Firm’s Minimum Target Value.
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Apartment Decision Purchase Model
Purchase Apartment Complex
Units Rented per Month 35Rental per Unit $950
Expected Expenses $30,500
Profit or Loss per Month $2,750Profit or Loss - Five Years $165,000
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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) Common-Cause Variation (Uncertainty) – Use Monte Carlo Simulation (Crystal Ball)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.
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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
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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
Purchase Apartment Complex
Units Rented per Month 35Rental per Unit $950
Expected Expenses $30,500
Profit or Loss per Month $2,750Profit or Loss - Five Years $165,000
Controllable
Uncontrollable
Output
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Uncontrollable Variable Cells in Crystal Ball
Uncontrollable Variables AKA Assumption Assumption Cells.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
OutputOutput Cells Called ForecastForecast 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
Purchase Apartment Complex
Units Rented per Month 35Rental per Unit $950
Expected Expenses $30,500
Profit or Loss per Month $2,750Profit or Loss - Five Years $165,000
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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76 R u n s C om p le te?
S to re R esu ltsfo r F orecas t C e llN e t C a sh F lo w
R eca lcu la teS p read sh ee t
G en era teV a lu es fo r
A ssu m p tion C e lls A /R a n d D e m a n d X
S tart
NoYesDisplay Stats
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Which Distribution to Use?
Y esU n ifo rm
D is trib u tion
Y esN orm a l
D is trib u tion
N oE q u ila te ra lTrian g u la r
Y esM os t D a ta
V a lu esN ear M ean ?
N oW eib u ll
o rTrian g u la r
N oD is trib u tionS ym m etric?
A ll V a lu es E q u a llyL ike ly?
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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 SimultaneousSimultaneous Impacts of the Possible Variation in Uncontrollable Uncontrollable VariablesVariables on Output VariableOutput Variable(s).(s).– Variation is Additive!!!!
Determines the Worst and Best Possible Values for OutputOutput Variable.
Assigns Probabilities to OutputOutput 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
Don’t Understand Decision Don’t Understand Decision Environment.Environment.
Consider Too Few AlternativesConsider Too Few Alternatives. Problem Solving Meetings Ineffectively
Run. Alternatives Clones of One Another. Decision Making Not a Formal Decision Making Not a Formal
Analysis.Analysis.