Transcript document

The Choice Phase
Decision
Science
Foundations
Intelligence
Phase
Qualitative
Method
Quantitative
Methods
Univariate
Data
Analysis
Design
Phase
Choice Phase
Understanding
the Relations
Decision
Analysis,
Decision Trees
Bayesian Analysis
Modeling the
Problem
Risk Analysis
& Simulation
Bivariate
Data
Analysis
Avoiding
GroupThink
1
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.
2
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
3
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.
4
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.
5
Roget Pinky’s 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
6
won't bring in any revenue.
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
7
calculations but he gets the following payoff table:
Roget Pinky’s Payoff Table
Cold
Wet
Cold
Dry
Warm
Wet
Warm
Dry
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
MARTA
and BUSCH
($270,000)
($640,500)
$549,000
$2,362,500
8
LaPlace-Bayes
Cold
Wet
Cold
Dry
Warm
Wet
Warm
Dry
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
MARTA
and BUSCH
($270,000)
($640,500)
$549,000
$2,362,500
$500,250
9
Maximax
Cold
Wet
Cold
Dry
Warm
Wet
Warm
Dry
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
MARTA
and BUSCH
($270,000)
($640,500)
$549,000
$2,362,500
$2,362,500
10
Maximin
Cold
Wet
Cold
Dry
Warm
Wet
Warm
Dry
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)
MARTA
and BUSCH
($270,000)
($640,500)
$549,000
$2,362,500
($640,500)
11
Net Payoff Table
C o ld
W et
C o ld
D ry
W a rm
W et
W a rm
D ry
N o B usch,
N o M AR T A
($ 3 7 5 ,0 0 0 )
($ 2 1 2 ,5 0 0 )
$ 1 1 2 ,5 0 0
$ 2 ,0 6 2 ,5 0 0
B usch,
no M AR T A
($ 5 5 0 ,0 0 0 )
($ 3 5 5 ,0 0 0 )
$ 3 5 ,0 0 0
$ 2 ,3 7 5 ,0 0 0
M AR T A,
no B usch
($ 2 2 5 ,0 0 0 )
($ 5 3 3 ,7 5 0 )
$ 4 5 7 ,5 0 0
$ 1 ,9 6 8 ,7 5 0
M AR T A
and B U SC H
($ 2 7 0 ,0 0 0 )
($ 6 4 0 ,5 0 0 )
$ 5 4 9 ,0 0 0
$ 2 ,3 6 2 ,5 0 0
Id e a l
($ 2 2 5 ,0 0 0 )
($ 2 1 2 ,5 0 0 )
$ 5 4 9 ,0 0 0
$ 2 ,3 7 5 ,0 0 0
12
Regret Table
Cold
Wet
Cold
Dry
Warm
Wet
Warm
Dry
Max
Regret
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
$0
$321,250
$91,500
$406,250
$406,250
$45,000
$426,000
$0
$12,500
$428,000
MARTA,
no Busch
MARTA
and BUSCH
13
Decision Making Under Risk
Expected Monetary Value (EMV)
–
–
–
–
–
Si
Aj
P(Si)
Vij
EMVj
The ith state of nature
The jth alternative action
The probability that Si will occur
The payoff if Aj and Si occurs
The long-term average payoff
• EMVj =  P(Si) S Vi
• Variance = P(Si) S (EMVj - Vij)2
14
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)
15
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)
16
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
17
Net Payoff w/EMV & Variance
Cold
Wet
Cold
Dry
Warm
Wet
Warm
Dry
EMV
Variance/
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
MARTA
and BUSCH
($270,000)
($640,500)
$549,000
$2,362,500
$923,400
$1,290,211
0.1
0.15
0.4
0.35
Probability
18
Net Payoff Table - EVPI
Cold
W et
Cold
Dry
W arm
W et
W arm
Dry
EM V
No Busch,
No M ART A
($375,000)
($212,500)
$112,500
$2,062,500
$697,500
Busch,
no M ART A
($550,000)
($355,000)
$35,000
$2,375,000
$737,000
M ART A,
no Busch
($225,000)
($533,750)
$457,500
$1,968,750
$769,500
M ART A
and BUSCH
($270,000)
($640,500)
$549,000
$2,362,500
$923,400
Ideal
($225,000)
($212,500)
$549,000
$2,375,000
$996,475
0.1
0.15
0.4
Probability
0.35
19
Expected Opportunity Loss
EOL is an alternative to EMV and produces
the same results
–
–
–
–
–
–
Si
Ai
P(Si)
Vij
OLij
EOLj
The ith state of nature
The jth alternative action
The probability that Si will occur
The payoff if Aj and Si occurs
OL if DM chooses Aj and Si occurs
The long-term average opportunity loss
• OLij = Max(Vij) - Vi
• EOLj = P(Si) S OLij
20
Opportunity Loss
Cold
Wet
Cold
Dry
Warm
Wet
Warm
Dry
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
MARTA
and BUSCH
($270,000)
($640,500)
$549,000
$2,362,500
$73,075
0.1
0.15
0.4
Probability
0.35
21
Decision Trees
22
Decision Trees
A method of visually structuring the
problem
Effective for sequential decision problems
23
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
24
Decision Node
Decision nodes are represented by Squares
Each branch refers to an Alternative Action
25
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
26
Chance Node
Chance nodes are represented by Circles
Each branch refers to a State of Nature
27
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
28
Terminal Node
Terminal nodes are optionally represented
by Triangles
The node refers to a payoff
The value for the node is the payoff
29
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
30
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?
31
LaLa Lovely Decision Tree
Mega Studios
EMV 960000
EMV 960000
Turnip Network
Small Gross
.3
200000
Medium Gross
.6 1000000
Great Gross
.1 3000000
900000
EMV 900000
32
Bayes’ Theorem
33
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.

P( B  A) P( A | B)  P( B)
P( B | A) 

P( A)
P( A)
34
Review of Basic
Probabilities
35
Gender Discrimination Case?
2 X 2 Cross-Tabs Table of
Gender Vs. Promotion
Male
Female
Total
Promoted
40
10
50
Not Promoted
80
70
150
Total
120
80
200
Gender and Promotion Status Related????
36
Lecture Flow: Bottom to Top
Statistical Independence
Unconditional
Probabilities
Conditional
Probability
Relative Frequency and Cross-Tabs
37
2 X 2 Cross-Tabs Table
Male
Female
Total
Promoted
40
10
50
Not Promoted
80
70
150
Total
120
80
200
38
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) = ______
39
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).
40
Compute P(Promoted Given
Male)
Promoted
Not Promoted
Total
Male
40
80
120
Female
Total
10
50
70
150
80
200
P(Promoted | Male) =
41
Compute P(Promoted Given
Female)
Male
Promoted
40
Not Promoted
80
Total
120
Female
10
70
80
Total
50
150
200
P(Promoted | Female) =
42
Comparing Three Probabilities
From Previous Slides
Unconditional Probability
– P(Prom) =
______
Cond. Probability
– P(Prom | Male) =
______
Cond. Probability
– P(Prom | Female) =
______
43
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?
44
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
45
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
0.6
0.4
1
Total
46
How Should the Table Have
Looked if Not Statistically
Related?
Male
Female
Total
Promoted
30
20
50
Not Promoted
90
60
150
120
80
200
Total
47
Other Types of Probabilities:
Joint Probabilities
Male
Female
Total
Promoted
40
10
50
Not Promoted
80
70
150
120
80
200
Total
P(Prom and Male) =
P(Not Prom and Female) =
48
Other Types of Probabilities:
Union Probabilities
Male
Female
Total
Promoted
40
10
50
Not Promoted
80
70
150
120
80
200
Total
P(Prom or Male) =
P(Not Prom or Female) =
49
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)
50
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
51
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.
52
The Choice Phase
Decision
Science
Foundations
Intelligence
Phase
Qualitative
Method
Quantitative
Methods
Univariate
Data
Analysis
Design
Phase
Choice Phase
Understanding
the Relations
Decision
Analysis,
Decision Trees
Bayesian Analysis
Modeling the
Problem
Risk Analysis
&
Monté Carlo
Simulation
Bivariate
Data
Analysis
Avoiding
GroupThink
53
Diagnosis
Build Model
Validate Model
What-If: Evaluate
Alternatives
Enhance or Enrich
Model
54
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.
55
Apartment Decision
Purchase Model
Purchase Apartment Complex
Units Rented per Month
Rental per Unit
Expected Expenses
35
$950
$30,500
Profit or Loss per Month
Profit or Loss - Five Years
$2,750
$165,000
56
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)
57
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.
58
Building a Simulation Model
Required Elements
– The basic logic of the system
– The known (or estimated) distributions of the
random variables
59
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
60
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
61
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
62
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
63
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
64
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.
65
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.
66
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
67
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?
68
Apartment Decision
Purchase Model
Purchase Apartment Complex
Units Rented per Month
Rental per Unit
Expected Expenses
35
$950
$30,500
Profit or Loss per Month
Profit or Loss - Five Years
$2,750
$165,000
Uncontrollable
Controllable
Output
69
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
70
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.
71
Apartment Decision
Purchase Model
Click on B3 and then
Define Assumption or
Purchase Apartment Complex
Distribution to Model It.
Units Rented per Month
Click on B5 and then
Rental per Unit
Define Assumption or Expected Expenses
35
$950
$30,500
Distribution to Model It.
Profit or Loss per Month
Profit or Loss - Five Years
$2,750
$165,000
Click on B8 and then
Define Forecast.
72
73
74
75
Start
Generate
Values for
Assumption Cells
A/R and Demand X
Recalculate
Spreadsheet
Store Results
for Forecast Cell
Net Cash Flow
Yes
Display Stats
Runs Complete?
No
76
77
Which Distribution to Use?
All Values Equally
Likely?
Yes
Uniform
Distribution
No
Distribution
Symmetric?
Yes
Most Data
Values
Near Mean?
Yes
Normal
Distribution
No
Weibull
or
Triangular
No
Equilateral
Triangular
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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
Don’t Understand Decision
Environment.
Consider Too Few Alternatives.
Problem Solving Meetings Ineffectively
Run.
Alternatives Clones of One Another.
Decision Making Not a Formal
Analysis.
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