Transcript Chapter 12 - Decision Analysis Chapter Topics

Decision Analysis
What is it?
(to p2)
(to p5)
What is the objective?
More example
Tutorial:
(to p50)
8th ed:: 5, 18, 26, 37
9th ed: 3, 12, 17, 24
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What is it?
It concerns with making a decision or an
action based on a series of possible
outcomes
The possible outcomes are normally
presented in a table format called
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“ Payoff Table”
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Payoff Table
A payoff table consists of
Columns: state of nature (ie events)
Rows:
choice of decision
It corresponding entries are “payoff values”
Example
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Payoff Table
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Objective
Our objective here is to determine which
decision should be chosen based on the
payoff table
How to achieve it?
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Simple two-dimensional cases
Posterior analysis with Bayesian rules
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How to achieve it?
• The final decision could be based on one of the
following criteria:
1) maximax,
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2) maximin,
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3) minimax regret,
4) Hurwicz,
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5) equal likelihood
• Each one is better one?
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– Ex 4 and 7
• QM programs
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The Maximax Criterion
Known as “optimistic” person, he/she will pick the decision that has
the maximum of maximum payoffs .
Example:
How to derive this decision?
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Procedural Steps:
Step1: For each row (or decision), select its max values
Step 2: Choose the decision that has the max value of
Step1
Example:
Max
100,000
50,000
100,000
30,000
max
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The Maximin Criterion
Known as the “pessimistic” person, she/he will select the decision that
has the maximum of the minimum from the payoff table
Example
Select this action
How to obtain it?
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Procedural steps:
Step 1: For each row (or decision), obtain its min values
Step 2: Choose the decision that has the max value from step 1
Min
30,000
-40,000
10,000
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The Minimax: Regret Criterion
Regret is the difference between the payoff from the best
decision and all other decision payoffs.
The decision maker attempts to avoid regret by selecting
the decision alternative that minimizes the maximum
regret.
Solution
How to obtain it?
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Procedural steps:
Step 1: Find out the max value for each column
Step 2: Subtract value of step 1 to all entry of that column
Step 3: Find out the max value for each row
Step 4: Choose the decision that has smallest value of step 3
Example
Step 1:
Max
100,000
30,000
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Step 2: Subtract value of step 1 to all entry of that column
(100,000 - 50,000)
Note: this is a regret table.
(30,000 - 30,000)
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Step 3: Find out the max value for each row
Step 4: Choose the decision that has smallest value of step 3
Max
50,000Min
70,000
70,000
Select this decision
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Decision Making without Probabilities
The Hurwicz Criterion
- The Hurwicz criterion is a compromise between the maximax and maximin criterion.
- A coefficient of optimism, , is a measure of the decision maker’s optimism.
- The Hurwicz criterion multiplies the best payoff by  and the worst payoff by 1- .,
for each decision, and the best result is selected.
Example:
Decision
Values
Apartment building
$50,000(.4) + 30,000(.6) = 38,000
Office building
$100,000(.4) - 40,000(.6) = 16,000
Warehouse
$30,000(.4) + 10,000(.6) = 18,000
Select this,max
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Decision Making without Probabilities
The Equal Likelihood Criterion
- The equal likelihood ( or Laplace) criterion multiplies the decision payoff for each
state of nature by an equal weight, thus assuming that the states of nature are equally
likely to occur.
Example:
Decision
Values
Apartment building
$50,000(.5) + 30,000(.5) = 40,000
Office building
$100,000(.5) - 40,000(.5) = 30,000
Warehouse
Select this, Max
$30,000(.5) + 10,000(.5) = 20,000
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Decision Making without Probabilities
Summary of Criteria Results
A dominant decision is one that has a better payoff than another decision under each
state of nature.
The appropriate criterion is dependent on the “risk” personality and philosophy of the
decision maker.
Criterion
Decision (Purchase)
Maximax
Office building
Maximin
Apartment building
Minimax regret
Apartment building
Hurwicz
Apartment building
Equal liklihood
Apartment building
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Decision Making without Probabilities
Solutions with QM for Windows (1 of 2)
Exhibit 12.1
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Decision Making without Probabilities
Solutions with QM for Windows (2 of 2)
Exhibit 12.2
Exhibit 12.3
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Posterior analysis with Bayesian rules
Here, we need to understand the following concepts:
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1) Expected values
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2) Expected Opportunity loss
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3) Expected values of Perfect Information
(Ex. 18 and 26)
4) Decision Tree
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a) basic tree
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b) sequential tree
5) Posterior analysis with Bayesian rules
(Ex. 33, 35 and 37)
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Expected Value
Expected value is computed by multiplying each decision outcome under
each state of nature by the probability of its occurrence.
Table 12.7 Payoff table with Probabilities for States of Nature
EV(Apartment) = $50,000(.6) + 30,000(.4) = 42,000
Select this, Max
EV(Office) = $100,000(.6) - 40,000(.4) = 44,000
EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000
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Decision Making with Probabilities
Expected Opportunity Loss
The expected opportunity loss is the expected value of the regret for
each decision (Minimax)
Table 12.8 Regret (Opportunity Loss) Table with Probabilities for States of Nature
EOL(Apartment) = $50,000(.6) + 0(.4) = 30,000
EOL(Office) = $0(.6) + 70,000(.4) = 28,000
Select this, Min
EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000
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Decision Making with Probabilities
Solution of Expected Value Problems with QM for Windows
Exhibit 12.4
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Decision Making with Probabilities
Solution of Expected Value Problems with Excel and Excel QM
(1 of 2)
Exhibit 12.5
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Decision Making with Probabilities
Solution of Expected Value Problems with Excel and Excel QM
(2 of 2)
Exhibit 12.6
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Expected Value of Perfect Information
The expected value of perfect information (EVPI) is the maximum
amount a decision maker would pay for additional information.
EVPI = (expected value with perfect information) –(expected value
without perfect information)
EVPI = the expected opportunity loss (EOL) for the best decision.
How to compute them?
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Decision Making with Probabilities
EVPI Example
Table 12.9 Payoff Table with Decisions, Given Perfect Information
Best outcome for each column
Best outcome
Worst outcome
Decision with perfect information: $100,000(.60) + 30,000(.40) = $72,000
Decision without perfect information: EV(office) = $100,000(.60) - 40,000(.40) = $44,000
EVPI = $72,000 - 44,000 = $28,000
EOL(office) = $0(.60) + 70,000(.4) = $28,000
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(back p47)
Decision Making with Probabilities
EVPI with QM for Windows
Exhibit 12.7
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Decision Trees (1 of 2)
A decision tree is a diagram consisting of decision nodes (represented as
squares), probability nodes (circles), and decision alternatives
(branches).
Table 12.10
Payoff Table for Real
Estate Investment Example
Figure 12.1
Decision tree for real
estate investment
example
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Decision Trees (2 of 2)
The expected value is computed at each probability node:
EV(node 2) = .60($50,000) + .40(30,000) = $42,000
EV(node 3) = .60($100,000) + .40(-40,000) = $44,000
EV(node 4) = .60($30,000) + .40(10,000) = $22,000
And
Choose branch with max value:
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Use to denote not chosen path(s)
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Decision Making with Probabilities
Decision Trees with QM for Windows
Exhibit 12.8
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan
(1 of 4)
Exhibit 12.9
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan
(2 of 4)
Exhibit 12.10
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan
(3 of 4)
Exhibit 12.11
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan
(4 of 4)
Exhibit 12.12
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Sequential Decision Trees
(1 of 2)
A sequential decision tree uses to illustrate a series of decisions.
.
How to make use of them?
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Sequential Decision Trees
(2 of 2)
Compute the expected values for each stage
Then choose action that produces the highest value.
Extra information
0.6*2M+0.4*0.225M
Max (1.29M-0.8M, 1.36M-0.2M)
=1.16M
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Sequential Decision Tree Analysis with QM for Windows
Exhibit 12.13
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Sequential Decision Tree Analysis with Excel and TreePlan
Exhibit 12.14
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Bayesian Analysis
(1 of 3)
Bayesian analysis deals with posterior information, and it can be
computed based on their conditional probabilities
Example: Consider the following example. Using expected value
criterion, best decision was to purchase office building with expected
value of $444,000, and EVPI of $28,000.
Now, what
posterior information
can be obtained here?
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Bayesian Analysis
(2 of 3)
Let consider the following information:
g = good economic conditions
p = poor economic conditions
P = positive economic report
New
N = negative economic report
From Economic analyst,we can obtain the following “conditional probability”, the probability
that an event will occur given that another event has already occurred.
P(Pg) = .80
Total = 0.8+0.2 = 1.0
P(Ng) = .20
P(Pp) = .10
P(Np) = .90
From about information, we can further compute the posterior information such as:
P(gP)
How it works?
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Bayesian Analysis
(3 of 3)
(A posterior probability is the altered marginal probability of an event based on additional
information)
Example for its application:
Let P(g) = .60; P(p) = .40
Then, the posterior probabilities is:
P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923
Systematic way to compute it?
(to p43)
Other posterior (revised) probabilities are:
P(gN) = .250
P(pP) = .077
P(pN) = .750
Where are these values located/appeared in our tree diagram?
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Form this table to compute them
0.52
Table 12.12
Computation of Posterior Probabilities
P(P) = P(P/g)P(g) +P(P/p) P(p) =0.8* 0.6 +0.1* 0.4 = 0.52
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Decision Trees with Posterior Probabilities
(1 of 2)
- Decision tree below differs from earlier versions in that :
1. Two new branches at beginning of tree represent report outcomes;
2. Newly introduced
Figure 12.5
Decision tree with posterior
probabilities
Old one
Making Decision
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Decision Trees with Posterior Probabilities
(2 of 2)
- EV (apartment building) = $50,000(.923) + 30,000(.077) = $48,460
- EV (strategy) = $89,220(.52) + 35,000(.48) = $63,194
The derivative is obtained
from slide44
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What good are these values?
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• We can make use these information to determine
how much should we pay for, say, sampling a survey
How? Example
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The Expected Value of Sample Information
The expected value of sample information (EVSI)
= expected value with information – expect value without information.:
For example problem, EVSI = $63,194 - 44,000 = $19,194
From the decision tree
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From without
perfect information
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How efficiency is this value?
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Efficiency
• The efficiency of sample information is the
ratio of the expected value of sample
information to the expected value of perfect
information:
• efficiency = EVSI /EVPI = $19,194/ 28,000
= 0.68
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Decision Analysis with Additional Information Utility
Table 12.13 Payoff Table for Auto Insurance Example
Expected Cost (insurance) = .992($500) + .008(500) = $500
Expected Cost (no insurance) = .992($0) + .008(10,000) = $80
- Decision should be do not purchase insurance, but people almost always do purchase insurance.
- Utility is a measure of personal satisfaction derived from money.
- Utiles are units of subjective measures of utility.
- Risk averters forgo a high expected value to avoid a low-probability disaster.
- Risk takers take a chance for a bonanza on a very low-probability event in lieu of a sure thing.
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Example Problem Solution
(1 of 7)
Consider:
States of Nature
Decision
Expand
Maintain Status Quo
Sell now
Good Foreign
Competitive
Conditions
$800,000
1,300,000
320,000
Poor Foreign
Competitive
Conditions
$500,000
-150,000
320,000
a. Determine the best decision without probabilities using the 5 criteria of the chapter.
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b. Determine best decision with probabilities assuming .70 probability of good conditions, .30 of poor
conditions. Use expected value and expected opportunity loss criteria.
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c. Compute expected value of perfect information.
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d. Develop a decision tree with expected value at the nodes.
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e. Given following, P(Pg) = .70, P(Ng) = .30, P(Pp) = 20, P(Np) = .80, determine posterior
probabilities using Bayesians rule.
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f. Perform a decision tree analysis using the posterior probability obtained in part e.
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Example Problem Solution
(2 of 7)
Step 1 (part a): Determine Decisions Without Probabilities
Maximax Decision: Maintain status quo
Decisions
Maximum Payoffs
States of Nature
Expand
$800,000
Decision
Status quo
1,300,000 (maximum) Expand
Sell
320,000
Maintain Status Quo
Sell now
Good Foreign
Competitive
Conditions
$800,000
1,300,000
320,000
Poor Foreign
Competitive
Conditions
$500,000
-150,000
320,000
Maximin Decision: Expand
Decisions
Minimum Payoffs
Expand
$500,000 (maximum)
Status quo
-150,000
Sell
320,000
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Example Problem Solution
(3 of 7)
Minimax Regret Decision: Expand
Decisions
Expand
States of Nature
Maximum Regrets
$500,000 (minimum)
Status quo
650,000
Sell
980,000
Decision
Expand
Maintain Status Quo
Sell now
Good Foreign
Competitive
Conditions
$800,000
1,300,000
320,000
Poor Foreign
Competitive
Conditions
$500,000
-150,000
320,000
Hurwicz ( = .3) Decision: Expand
Expand
Status quo
Sell
$800,000(.3) + 500,000(.7) = $590,000
$1,300,000(.3) - 150,000(.7) = $285,000
$320,000(.3) + 320,000(.7) = $320,000
(to p53)
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States of Nature
Decision
Example Problem Solution
(4 of 7)
Expand
Maintain Status Quo
Sell now
Good Foreign
Competitive
Conditions
$800,000
1,300,000
320,000
Poor Foreign
Competitive
Conditions
$500,000
-150,000
320,000
Equal Liklihood Decision: Expand
Expand
Status quo
Sell
$800,000(.5) + 500,000(.5) = $650,000
$1,300,000(.5) - 150,000(.5) = $575,000
(to p50)
$320,000(.5) + 320,000(.5) = $320,000
Step 2 (part b): Determine Decisions with EV and EOL
Expected value decision: Maintain status quo
Expand
Status quo
Sell
$800,000(.7) + 500,000(.3) = $710,000
$1,300,000(.7) - 150,000(.3) = $865,000
$320,000(.7) + 320,000(.3) = $320,000
(to p54)
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States of Nature
Decision
Example Problem Solution
(5 of 7)
Expand
Maintain Status Quo
Sell now
Good Foreign
Competitive
Conditions
$800,000
1,300,000
320,000
Poor Foreign
Competitive
Conditions
$500,000
-150,000
320,000
Expected opportunity loss decision: Maintain status quo
Expand
Status quo
Sell
$500,000(.7) + 0(.3) = $350,000
0(.7) + 650,000(.3) = $195,000
(to p50)
$980,000(.7) + 180,000(.3) = $740,000
Step 3 (part c): Compute EVPI
EV given perfect information = 1,300,000(.7) + 500,000(.3) = $1,060,000
EV without perfect information = $1,300,000(.7) - 150,000(.3) = $865,000
EVPI = $1.060,000 - 865,000 = $195,000
(to p50)
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Example Problem Solution
(6 of 7)
Step 4 (part d): Develop a Decision Tree
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Example Problem Solution
(7 of 7)
Step 5 (part e): Determine Posterior Probabilities
P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.70)(.70)/[(.70)(.70) + (.20)(.30)] = .891
P(p P) = .109
P(gN) = P(Ng)P(g)/[P(Ng)P(g) + P(Np)P(p)] = (.30)(.70)/[(.30)(.70) + (.80)(.30)] = .467
P(pN) = .533
Step 6 (part f): Perform Decision tree Analysis with Posterior Probabilities
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