1. Prepared by Lee Revere and John Large
Chapter 3
Decision Analysis
Prepared by Lee Revere and John Large
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-1

2. Learning Objectives Students will be able to:
List the steps of the decision-making process.
Describe the types of decision-making environments.
Make decisions under uncertainty.
Use probability values to make decisions under risk.
Develop accurate and useful decision trees.
Revise probabilities using Bayesian analysis.
Use computers to solve basic decision-making problems.
Understand the importance and use of utility theory in decision theory.
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3-2

3. Chapter Outline 3.1 Introduction 3.2 The Six Steps in Decision Theory
3.3 Types of Decision-Making Environments
3.4 Decision Making under Uncertainty
3.5 Decision Making under Risk
3.6 Decision Trees
3.7 How Probability Values Are Estimated by Bayesian Analysis
3.8 Utility Theory
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4. Introduction
Decision theory is an analytical and systematic way to tackle problems.
A good decision is based on logic.
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5. The Six Steps in Decision Theory
Clearly define the problem at hand.
List the possible alternatives.
Identify the possible outcomes.
List the payoff or profit of each combination of alternatives and outcomes.
Select one of the mathematical decision theory models.
Apply the model and make your decision.
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3-5

6. John Thompson’s Backyard Storage Sheds
Define problem
To manufacture or market backyard storage sheds
List alternatives
Construct a large new plant
A small plant
No plant at all
Identify outcomes
The market could be favorable or unfavorable for storage sheds
List payoffs
List the payoff for each state of nature/decision alternative combination
Select a model
Decision tables and/or trees can be used to solve the problem
Apply model and make decision
Solutions can be obtained and a sensitivity analysis used to make a decision
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3-6

7. Decision Table for Thompson Lumber
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-7

8. Types of Decision-Making Environments
Type 1: Decision making under certainty.
Decision maker knows with certainty the consequences of every alternative or decision choice.
Type 2: Decision making under risk.
The decision maker does know the probabilities of the various outcomes.
Decision making under uncertainty.
The decision maker does not know the probabilities of the various outcomes.
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3-8

9. Decision Making under Uncertainty
Maximax
Maximin
Equally likely (Laplace)
Criterion of realism
Minimax
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10. Decision Table for Thompson Lumber
Maximax: Optimistic Approach
Find the alternative that maximizes the maximum outcome for every alternative.
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-10

11. Thompson Lumber: Maximax Solution
Alternative
State of Nature
Maximax
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-11

12. Decision Table for Thompson Lumber
Maximin: Pessimistic Approach
Choose the alternative with maximum minimum output.
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-12

13. Thompson Lumber: Maximin Solution
Alternative
State of Nature
Maximin
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-13

14. Thompson Lumber: Hurwicz
Criterion of Realism (Hurwicz)
Decision maker uses a weighted average based on optimism of the future.
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-14

15. Thompson Lumber: Hurwicz Solution
CR = α*(row max)+(1- α)*(row min)
Alternative
State of Nature
Criterion of Realism or Weighted Average (α = 0.8) ($)
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
124,000
Construct a small plant
100,000
-20,000
76,000
Do nothing
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3-15

16. Decision Making under Uncertainty
Equally likely (Laplace)
Assume all states of nature to be equally likely, choose maximum Average.
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-16

17. Decision Making under Uncertainty
Alternative
State of Nature
Avg.
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
10,000
Construct a small plant
100,000
-20,000
40,000
Do nothing
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3-17

18. Thompson Lumber; Minimax Regret
Choose the alternative that minimizes the maximum opportunity loss .
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
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3-18

19. Thompson Lumber: Opportunity Loss Table
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000 – 200,000 = 0
0- (-180,000) = 180,000
Construct a small plant
200, ,000 = 100,000
0- (-20,000) = 20,000
Do nothing
200,000 – 0 = 0
0 – 0 = 0
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20. Thompson Lumber: Minimax Regret Solution
Alternative
State of Nature
Maximum Opportunity Loss
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
180,000
Construct a small plant
100,000
20,000
Do nothing
200,000
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21. In-Class Example 1
Let’s practice what we’ve learned. Use the decision table below to compute
(1) Mazimax (2) Maximin (3) Minimax regret
Alternative
State of Nature
Good
Market
($)
Average
Poor
Construct a large plant
75,000
25,000
-40,000
Construct a small plant
100,000
35,000
-60,000
Do nothing
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3-21

22. In-Class Example 1: Maximax
Alternative
State of Nature
Maximax
Good
Market
($)
Average
Poor
Construct a large plant
75,000
25,000
-40,000
Construct a small plant
100,000
35,000
-60,000
Do nothing
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3-22

23. In-Class Example 1: Maximin
Alternative
State of Nature
Maximin
Good
Market
($)
Average
Poor
Construct a large plant
75,000
25,000
-40,000
Construct a small plant
100,000
35,000
-60,000
Do nothing
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3-23

24. In-Class Example 1: Minimax Regret Opportunity Loss Table
Alternative
State of Nature
Maximum Opp. Loss
Good
Market
($)
Average
Poor
Construct a large plant
25,000
75,000
40,000
Construct a small plant
60,000
Do nothing
100,000
35,000
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25. Decision Making under Risk
Expected Monetary Value:
In other words:
EMVAlternative n = Payoff 1 * PAlt. 1 + Payoff 2 * PAlt. 2 + … + Payoff n * PAlt. n
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26. Unfavorable Market ($)
Thompson Lumber: EMV
Alternative
State of Nature
EMV
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
200,000*0.5 +
(-180,000)*0.5 = 10,000
Construct a small plant
100,000
-20,000
100,000*0.5 +
(-20,000)*0.5 = 40,000
Do nothing
0* *0.5 = 0
Probabilities
0.50
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27. Thompson Lumber: EV|PI and EMV Solution
Alternative
State of Nature
EMV
Favorable Market ($)
Unfavorable Market
($)
Construct a large plant
200,000
-180,000
10,000
Construct a small plant
100,000
-20,000
40,000
Do nothing
EV׀PI
200,000*0.5 = 100,000
0*0.5 = 0
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28. Expected Value of Perfect Information (EVPI)
EVPI places an upper bound on what one would pay for additional information.
EVPI is the expected value with perfect information minus the maximum EMV.
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29. Expected Value with Perfect Information (EV|PI)
In other words
EV׀PI = Best Outcome of Alt 1 * PAlt. 1 + Best Outcome of Alt 2 * PAlt. 2 +… + Best Outcome of Alt n * PAlt. n
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30. Expected Value of Perfect Information
EVPI = EV|PI - maximum EMV
Expected value with perfect information
Expected value with no additional information
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31. Thompson Lumber: EVPI Solution
EVPI = expected value with perfect information - max(EMV)
= $200,000* * $40,000
= $60,000
From previous slide
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32. In-Class Example 2
Let’s practice what we’ve learned. Using the table below compute EMV, EV׀PI, and EVPI.
Alternative
State of Nature
Good
Market
($)
Average
Poor
Construct a large plant
75,000
25,000
-40,000
Construct a small plant
100,000
35,000
-60,000
Do nothing
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3-32

33. In-Class Example 2: EMV and EV׀PI Solution
Alternative
State of Nature
EMV
Good
Market
($)
Average
Poor
Construct a large plant
75,000
25,000
-40,000
21,250
Construct a small plant
100,000
35,000
-60,000
27,500
Do nothing
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3-33

34. In-Class Example 2: EVPI Solution
EVPI = expected value with perfect information - max(EMV)
= $100,000* ,000* *0.25
= $ 42, ,500
= $ 15,000
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35. Expected Opportunity Loss
EOL is the cost of not picking the best solution. EOL = Expected Regret
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36. Thompson Lumber: EOL The Opportunity Loss Table
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000 – 200,000
0- (-180,000)
Construct a small plant
200, ,000
0 – (-20,000)
Do nothing
200,
0-0
Probabilities
0.50
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37. Thompson Lumber: EOL Table
Alternative
State of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000
-180,000
Construct a small plant
100,000
-20,000
Do nothing
Probabilities
0.50
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3-37

38. Thompson Lumber: EOL Solution
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3-38

39. Thompson Lumber: Sensitivity Analysis
EMV(Large Plant):
= $200,000P - (1-P)$180,000
EMV(Small Plant):
= $100,000P - $20,000(1-P)
EMV(Do Nothing):
= $0P + 0(1-P)
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3-39

40. Thompson Lumber: Sensitivity Analysis (continued)
250000
200000
Point 1
Point 2
150000
Small Plant
100000
50000
EMV Values
-50000
0.2
0.4
0.6
0.8 1
Large Plant EMV
Values of P
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3-40

41. Marginal Analysis P = probability that demand > a given supply.
1-P = probability that demand < supply.
MP = marginal profit.
ML = marginal loss.
Optimal decision rule is:
P*MP  (1-P)*ML or
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42. Marginal Analysis - Discrete Distributions
Steps using Discrete Distributions:
Determine the value for P.
Construct a probability table and add a cumulative probability column.
Keep ordering inventory as long as the probability of selling at least one additional unit is greater than P.
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43. Café du Donut: Marginal Analysis
Café du Donut sells a dozen donuts for $6. It costs $4 to make each dozen. The following table shows the discrete distribution for Café du Donut sales.
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44. Café du Donut: Marginal Analysis Solution
Marginal profit = selling price cost
= $6 - $4 = $2
Marginal loss = cost
Therefore:
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45. Café du Donut: Marginal Analysis Solution
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46. In-Class Example 3
Let’s practice what we’ve learned. You sell cases of goods for $15/case, the raw materials cost you $4/case, and you pay $1/case commission.
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47. In-Class Example 3: Solution
MP = $15-$4-$1 = $10 per case ML = $4
P>= $4 / $10+$4 = .286
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48. Marginal Analysis Normal Distribution
 = average or mean sales
 = standard deviation of sales
MP = marginal profit
ML = Marginal loss
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49. Marginal Analysis - Discrete Distributions
Steps using Normal Distributions:
Determine the value for P.
Locate P on the normal distribution. For a given area under the curve, we find Z from the standard Normal table.
Using we can now solve for: MP ML P + = s m - = * X Z X*
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50. Marginal Analysis: Normal Curve Review
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51. Marginal Analysis - Normal Curve Review
area = .30
Use table to find Z
area = .70 MP ML .3 + =
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3-51

52. Joe’s Newsstand Example
Joe sells newspapers for $1.00 each. Papers cost him $.40 each. His average daily demand is 50 papers with a standard deviation of 10 papers. Assuming sales follow a normal distribution, how many papers should Joe stock?
ML = $0.40
MP = $0.60
 = Average demand = 50 papers per day
 = Standard deviation of demand = 10
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53. Joe’s Newsstand Example (continued)
.40 ML
Step 1: = = = P . 40 ML + MP
.40 +
.60 .
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54. Joe’s Newsstand Example (continued)
Step 2: Look on the Normal table for
P = 0.6 (i.e., )  Z = 0.25,
and
or: 10 50 25 - = * X
X* = 10 * = 52.5 or 53 newspapers
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55. Joe’s Newsstand Example B
Joe also offers his clients the “Times” for $1.00. This paper is flown in from out of state, which greatly increases its costs. Joe pays $.80 for the “Times.” The “Times” has average daily sales of 100 papers with a standard deviation of 10. Assuming sales follow a normal distribution, how many “Times” papers should Joe stock?
ML = $0.80
MP = $0.20
 = Average demand = 100 papers per day
 = Standard deviation of demand = 10
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56. Joe’s Newsstand Example B (continued)80 .8 .2 . MP ML P = +
Step 1: .
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57. Joe’s Newsstand Example B (continued)
Step 2:
Z = 0.80
= for an area of 0.80
And
or: X= or 92 newspapers 10
100
.84 - = * X
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58. Decision Making with Uncertainty: Using the Decision Trees
Decision trees enable one to look at decisions:
With many alternatives and states of nature,
which must be made in sequence.
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59. Five Steps to Decision Tree Analysis
Define the problem.
Structure or draw the decision tree.
Assign probabilities to the states of nature.
Estimate payoffs for each possible combination of alternatives and states of nature.
Solve the problem by computing expected monetary values (EMVs) for each state of nature node.
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3-59

60. Structure of Decision Trees
A graphical representation where:
A decision node from which one of several alternatives may be chosen.
A state-of-nature node out of which one state of nature will occur.
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3-60

61. Thompson’s Decision Tree
Step 1: Define the problem
Lets re-look at John Thompson’s decision regarding storage sheds. This simple problem can be depicted using a decision tree.
Step 2: Draw the tree 1 2
A Decision Node
A State of Nature Node
Favorable Market
Unfavorable Market
Construct Large Plant
Construct Small Plant
Do Nothing
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62. Thompson’s Decision Tree
Step 3: Assign probabilities to the states of nature.
Step 4: Estimate payoffs.
A State of Nature Node
$200,000
Favorable (0.5)
Market 1
Construct Large Plant
-$180,000
Unfavorable (0.5)
Market
A Decision Node
$100,000
Favorable (0.5)
Market
Construct Small Plant 2
-$20,000
Unfavorable (0.5)
Market
Do Nothing
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63. Thompson’s Decision Tree
Step 5: Compute EMVs and make decision.
A State of Nature Node
$200,000
Favorable (0.5)
Market 1
EMV
=$10,000
Construct Large Plant
Unfavorable (0.5)
Market
-$180,000
A Decision Node
Favorable (0.5)
Market
Construct Small Plant
$100,000 2
EMV
=$40,000
Unfavorable (0.5)
Market
-$20,000
Do Nothing
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64. Thompson’s Decision: A More Complex Problem
John Thompson has the opportunity of obtaining a market survey that will give additional information on the probable state of nature. Results of the market survey will likely indicate there is a percent change of a favorable market. Historical data show market surveys accurately predict favorable markets 78 % of the time. Thus P(Fav. Mkt / Fav. Survey Results) = .78
Likewise, if the market survey predicts an unfavorable market, there is a 13 % chance of its occurring. P(Unfav. Mkt / Unfav. Survey Results) = .13
Now that we have redefined the problem (Step 1), let’s use this additional data and redraw Thompson’s decision tree (Step 2).
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65. Thompson’s Decision Tree
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66. Thompson’s Decision Tree
Step 3: Assign the new probabilities to the states of nature.
Step 4: Estimate the payoffs.
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67. Thompson’s Decision Tree
Step 5: Compute the EMVs and make decision.
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68. John Thompson Dilemma
John Thompson is not sure how much value to place on market survey. He wants to determine the monetary worth of the survey. John Thompson is also interested in how sensitive his decision is to changes in the market survey results. What should he do?
Expected Value of Sample Information
Sensitivity Analysis
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69. Expected Value of Sample Information
Expected value of best decision with sample information, assuming no cost to gather it
Expected value of best decision without sample information
EVSI =
EVSI for Thompson Lumber = $59,200 - $40,000
= $19,200
Thompson could pay up to $19,200 and come out ahead.
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70. Calculations for Thompson Lumber Sensitivity Analysis
2,400
$104,000
($2,400)
($106,400) 1)
EMV(node + = - p ) ( 1
Equating the EMV(node 1) to the EMV of not conducting the survey, we have
$104,000 p +
$2,400 =
$40,000
$104,000 p =
$37,600 or
$37,600 p = =
0.36
$104,000
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71. In-Class Problem 3 Let’s practice what we’ve learned
Leo can purchase a historic home for $200,000 or land in a growing area for $50,000. There is a 60% chance the economy will grow and a 40% change it will not. If it grows, the historic home will appreciate in value by 15% yielding a $30,00 profit. If it does not grow, the profit is only $10,000. If Leo purchases the land he will hold it for 1 year to assess the economic growth. If the economy grew during the first year, there is an 80% chance it will continue to grow. If it did not grow during the first year, there is a 30% chance it will grow in the next 4 years. After a year, if the economy grew, Leo will decide either to build and sell a house or simply sell the land. It will cost Leo $75,000 to build a house that will sell for a profit of $55,000 if the economy grows, or $15,000 if it does not grow. Leo can sell the land for a profit of $15,000. If, after a year, the economy does not grow, Leo will either develop the land, which will cost $75,000, or sell the land for a profit of $5,000. If he develops the land and the economy begins to grow, he will make $45,000. If he develops the land and the economy does not grow, he will make $5,000.
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72. In-Class Problem 3: Solution
Economy grows (.6) 1 2 3 4 5 6 7
No growth (.4)
Purchase historic home
Economy grows (.8)
Build house
No growth (.2)
Sell land
Economy grows (.6)
Purchase land
Economy grows (.3)
Develop land
No growth (.4)
No growth (.7)
Sell land
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73. In-Class Problem 3: Solution1 2 3 4 5 6 7
Purchase historic home
Purchase land
$35,000
$22,000
Economy grows (.6)
$30,000
No growth (.4)
$10,000
$47,000
Build house
Economy grows (.8)
$55,000
$15,000
No growth (.2)
Sell land
$17,000
Develop land
$5,000
Economy grows (.3)
No growth (.7)
$45,000
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74. Estimating Probability Values with Bayesian
Management experience or intuition
History
Existing data
Need to be able to revise probabilities based upon new data
Posterior
probabilities
Prior
New data
Baye’s Theorem
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75. Bayesian Analysis
The probabilities of a favorable / unfavorable state of nature can be obtained by analyzing the Market Survey Reliability in Predicting Actual States of Nature.
Market Survey Reliability in Predicting
Actual States of Nature
Actual States of Nature
Result of Survey
Favorable
Unfavorable
Market (FM)
Market (UM)
Positive (predicts P
(survey positive|FM) P
(survey positive|UM)
= 0.70 =
0.20
favorable market
for product)
Negative (predicts P
(survey P
(survey negative|UM)
negative|FM) = 0.30
= 0.80
unfavorable
market for
product)
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76. Bayesian Analysis (continued): Favorable Survey
Probability Revisions Given a Favorable Survey
Conditional
Probability
Posterior Probability
State of
Nature
P(Survey positive|State of Nature
Prior Probability
Joint Probability FM
0.70
* 0.50
0.35
0.45
= 0.78 UM
0.20
0.10
= 0.22
1.00
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-76

77. Bayesian Analysis (continued): Unfavorable Survey
Probability Revisions Given an Unfavorable Survey
Conditional
Probability
Posterior Probability
State of
Nature
P(Survey
negative|State
of Nature)
Prior
Joint FM
0.30
* 0.50
0.15
0.55
= 0.27 UM
0.80
0.40
= 0.73
1.00
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-77

78. Decision Making Using Utility Theory
Utility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1.
A standard gamble is used to determine utility values.
When you are indifferent, the utility values are equal.
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-78

79. Standard Gamble for Utility Assessment
Best outcome
Utility = 1
Worst outcome
Utility = 0
Other outcome
Utility = ??
(p)
(1-p)
Alternative 1
Alternative 2
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-79

80. Simple Example: Utility Theory
Let’s say you were offered $2,000,000 right now on a chance to win $5,000,000. The $5,000,000 is won only if you flip a coin and get tails. If you get heads you lose and get $0. What should you do?
$5,000,000 $0
$2,000,000
Accept Offer
Reject Offer
Heads
(0.5)
Tails
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-80

81. Real Estate Example: Utility Theory
Jane Dickson is considering a 3-year real estate investment. There is an 80 % chance the real estate market will soar and a 20 % chance it will bust. In a good market the real estate investment will pay $10,000, in an unfavorable market it is $0. Of course, she could leave her money in the bank and earn a $5,000 return. What should she do?
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-81

82. Real Estate Example: Solution
$10,000
U($10,000) = 1.0
U(0)=0
$5,000
U($5,000)=p
=0.80
p= 0.80
(1-p)= 0.20
Invest in
Real Estate
Invest in Bank
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-82

83. Utility Curve for Jane Dickson
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-83

84. Preferences for Risk Risk Avoider Risk Indifference Utility Seeker
Monetary Outcome
Risk
Avoider
Seeker
Risk Indifference
Utility
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-84

85. Decision Facing Mark Simkin
Tack lands
point up (0.45)
point down (0.55)
$10,000
-$10,000
Alternative 1
Mark plays
the game
Alternative 2
Mark does not play the game
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-85

86. Utility Curve for Mark Simkin
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-86

87. Thompson Decision Tree Problem Using QM for Windows
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-87

88. Thompson Decision Tree Problem Using Excel
To accompany Quantitative Analysis for Management, 9e
by Render/Stair/Hanna
3-88