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Topic 1
Decision Analysis (Chapter 3)
Source: Render et al., 2012. Quantitative Analysis Management, 11 editions, Pearson.
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.
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
Introduction
Decision theory is an analytical and systematic way to tackle problems.
A good decision is based on logic.
The Six Steps in Decision Theory
1. Clearly define the problem at hand.
2. List the possible alternatives.
3. Identify the possible outcomes.
4. List the payoff or profit of each combination of alternatives and outcomes.
5. Select one of the mathematical decision theory models.
6. Apply the model and make your decision.
Example: Thompson Lumber Company Case
John Thompson is the founder and president of Thompson Lumber Company, a profitable firm located in Portland, Oregon.
The problem that John Thompson identifies is whether to expand his product line by manufacturing and marketing a new product, backyard sheds.
Define problem To manufacture or market backyard storage
sheds
List alternatives 1. Construct a large new plant
2. A small plant
3. 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
Example: Thompson Lumber Company Case
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 0 0
Example: Thompson Lumber Company Case Decision Table
Types of Decision-Making Environments
Decision making under certainty.
Decision maker knows with certainty the consequences of every alternative or decision choice.
Decision making under uncertainty.
The decision maker does not know the probabilities of the various outcomes.
Decision making under risk.
The decision maker knows the probabilities of the various outcomes.
Decision Making under certainty:
Decision makers know within certainty the consequence of every alternative or decision choice.
E.g.: You have $1000 to invest for a 1-year period. Alternatives are to open a savings or a fixed deposit account paying 3% or 5% interest per year respectively. If both investments are secure and guaranteed, there is a certainty that a fixed deposit will pay a higher return.
Decision Making under Uncertainty: Criteria
Optimistic (Maximax)
Pessimistic (Maximin)
Criterion of Realism (Hurwicz)
Equally likely (Laplace Criterion)
Minimax Regret (Opportunity Loss or Regret)
Alternative
State of Nature
Favorable
Market ($)
Unfavorable
Market ($)
Construct a large plant 200,000 -180,000
Construct a small plant 100,000 -20,000
Do nothing 0 0
Decision Making under Uncertainty
Example: Thompson Lumber Company Case.
Alternative
State of Nature
Maximum in
Row Favorable
Market ($)
Unfavorable
Market ($)
Construct a large
plant 200,000 -180,000 200,000
Construct a small
plant 100,000 -20,000 100,000
Do nothing 0 0 0
Decision Making under Uncertainty: Optimistic (Maximax) Criterion
Example: Thompson Lumber Company Case
Optimistic (Maximax)
Alternative
State of Nature
Minimum in
Row Favorable
Market ($)
Unfavorable
Market ($)
Construct a large
plant 200,000 -180,000 -180,000
Construct a small
plant 100,000 -20,000 -20,000
Do nothing 0 0 0
Decision Making under Uncertainty: Pessimistic (Maximin) Criterion
Example: Thompson Lumber Company Case
Pessimistic (Maximin)
Weighted Average = (best in row) + (1- ) (worst in row)
where: is the coefficient of realism (or the degree of optimism of the decision maker), 0 1.
Alternative
State of Nature Weighted
Average in
Row
( = 0.8)
Favorable
Market ($)
Unfavorable
Market ($)
Construct a large
plant 200,000 -180,000 124,000
Construct a small
plant 100,000 -20,000 76,000
Do nothing 0 0 0
Decision Making under Uncertainty: Criterion of Realism (Hurwicz) Criterion
Example: Thompson Lumber Company Case (given = 0.8)
Criterion of Realism (Hurwicz)
Alternative
State of Nature
Average in
Row Favorable
Market ($)
Unfavorable
Market ($)
Construct a large
plant 200,000 -180,000 10,000
Construct a small
plant 100,000 -20,000 40,000
Do nothing 0 0 0
Decision Making under Uncertainty: Equally likely (Laplace) Criterion
Example: Thompson Lumber Company Case
Assume all states of nature to be equally likely, choose
maximum Average.
Equally likely
(Laplace)
Alternative
State of Nature
Favorable Market ($) Unfavorable Market ($)
Construct a
large plant 200,000 200,000 = 0 0 (180,000) = 180,000
Construct a
small plant 200,000 100,000 = 100,000 0 (20,000) = 20,000
Do nothing 200,000 0 = 200,000 0 0 = 0
Step 1: Create an opportunity loss table by subtracting each payoff in the column from the best payoff in the same column.
Opportunity Loss Optimal payoff Actual payoff
Decision Making under Uncertainty: Minimax Regret (Opportunity Loss) Criterion
Example: Thompson Lumber Company Case
Alternative
State of Nature Maximum in
Row Favorable Market
($)
Unfavorable
Market ($)
Construct a large
plant 0 180,000 180,000
Construct a
small plant 100,000 20,000 100,000
Do nothing 200,000 0 200,000
Step 2: Choose the minimum alternative out of all the maximum opportunity losses.
Decision Making under Uncertainty: Minimax Regret (Opportunity Loss) Criterion
Example: Thompson Lumber Company Case
Minimax Regret
(Opportunity Loss)
Decision Making under Risk Expected Monetary Value (EMV)
The probabilities of the states of nature P(S) are known.
EMV(Alternative) = Payoffs1*P(S1) + Payoffs2*P(S2) ++ Payoffsn*P(Sn)
Alternative
State of Nature Expected Monetary Value
(EMV) Favorable
Market ($)
Unfavorable
Market ($)
Construct a
large plant 200,000 180,000
(200,000)(0.5) +
(180,000)(0.5) = 10,000
Construct a
small plant 100,000 20,000
(100,000)(0.5) +
(20,000)(0.5) = 40,000
Do nothing 0 0 (0)(0.5) + (0)(0.5) = 0
Probabilities 0.50 0.50
Example: Thompson Lumber Company Case
Alternative
State of Nature
Expected Monetary Value
(EMV) Favorable
Market, S1 ($)
Unfavorable
Market, S2 ($)
Construct a large plant 200,000 -180,000 10,000
Construct a small plant 100,000 -20,000 40,000
Do nothing 0 0 0
Probabilities 0.50 0.50
With Perfect
Information
Best payoff
in S1
200,000
Best payoff in
S2
0
EVwPI = (200,000)(0.5)
+ (0)(0.5) = 100,000
Decision Making under Risk Expected Value with Perfect Information (EVwPI)
Example: Thompson Lumber Company Case
EVwPI = best Payoffs1*P(S1) + best Payoffs2*P(S2) ++ best Payoffsn*P(Sn)
EVPI places an upper bound on what one would pay for additional information.
EVPI = EVwPI Best EMV
where:
EVwPI is Expected Value with Perfect Information.
Best EMV is expected value without perfect information.
Decision Making under Risk Expected Value of Perfect Information (EVPI)
Decision Making under Risk Expected Value of Perfect Information (EVPI)
Hence, the EVPI = EVwPI best EMV
= 100,000 40,000 = $60,000
Expected Opportunity Loss (EOL)
EOL is the cost of not picking the best solution.
EOL = Expected Regret
Thompson Lumber: Payoff Table
Alternative
State of Nature
Favorable Market
($)
Unfavorable
Market ($)
Construct a large
plant 200,000 -180,000
Construct a small
plant 100,000 -20,000
Do nothing 0 0
Probabilities 0.50 0.50
Thompson Lumber: EOL The Opportunity Loss Table
Alternative
State of Nature
Favorable Market ($) Unfavorable Market
($)
Construct a large plant 200,000 200,000 0- (-180,000)
Construct a small
plant 200,000 - 100,000 0 (-20,000)
Do nothing 200,000 - 0 0-0
Probabilities 0.50 0.50
Thompson Lumber: Opportunity Loss Table
Alternative
State of Nature
Favorable Market
($)
Unfavorable
Market ($)
Construct a large
plant 0 180,000
Construct a small
plant 100,000 20,000
Do nothing 200,000 0
Probabilities 0.50 0.50
Thompson Lumber: EOL Solution
Alternative EOL
Large Plant (0.50)*$0 +
(0.50)*($180,000)
$90,000
Small Plant (0.50)*($100,000)
+ (0.50)(*$20,000)
$60,000
Do Nothing (0.50)*($200,000)
+ (0.50)*($0)
$100,000
Note:
1. The minimum EOL and maximum EMV will suggest the same decision.
2. The value of EVPI will always equal to the minimum EOL value.
Thompson Lumber: Sensitivity Analysis
Let P = probability of favorable market EMV(Large Plant): = $200,000P + (-$180,000)(1-P) = 380,000P - 180,000 EMV(Small Plant): = $100,000P + (-$20,000)(1-P) = 120,000P - 20,000 EMV(Do Nothing): = $0P + 0(1-P) = 0
Thompson Lumber: Sensitivity Analysis (continued)
Decision Making with Uncertainty: Using the Decision Trees
Decision trees are most beneficial when a
sequence of decisions must be made.
All information included in a payoff table is
also included in a decision tree.
Five Steps to Decision Tree Analysis
1. Define the problem.
2. Structure or draw the decision tree.
3. Assign probabilities to the states of nature.
4. Estimate payoffs for each possible combination of alternatives and states of nature.
5. Solve the problem by computing expected monetary values (EMVs) at each state of nature node.
Structure of Decision Trees Trees start from left to right and represent decisions and
outcomes in sequential order.
A decision node (indicated by a square ) from which one of several alternatives may be chosen.
A state-of-nature node (indicated by a circle ) out of which one state of nature will occur.
Lines or branches connect the decisions nodes and the states of nature.
Thompsons Decision Tree
1
2
A Decision
Node
A State of
Nature Node Favorable Market
Unfavorable Market
Favorable Market
Unfavorable Market
Construct Small Plant
Step 1: Define the problem
Lets re-look at John Thompsons decision regarding storage sheds.
This simple problem can be depicted using a decision tree.
Step 2: Draw the tree
Thompsons Decision Tree
Step 3: Assign probabilities to the states of nature.
Step 4: Estimate payoffs.
1
2
A Decision
Node
A State of
Nature Node Favorable (0.5) Market
Unfavorable (0.5) Market
Favorable (0.5) Market
Unfavorable (0.5) Market
Construct Small Plant
$200,000
-$180,000
$100,000
-$20,000
0
Alternatives
Outcomes Payoffs
Thompsons Decision Tree
1
2
A Decision
Node
A State of
Nature Node Favorable (0.5) Market
Unfavorable (0.5) Market
Favorable (0.5) Market
Unfavorable (0.5) Market
Construct Small Plant
$200,000
-$180,000
$100,000
-$20,000
0
EMV
=$40,000
EMV
=$10,000
Step 5: Compute EMVs and make decision.
Thompsons 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.
P(Fav. Mkt | Results Favourable) = 0.78
Likewise, if the market survey predicts an unfavorable market, there is a 73 % chance of its occurring.
P(Unfav. Mkt | Results Negative) = 0.73
Assumed the cost of conduct market survey is $10,000, and is deducted from each payoff under Conduct Market Survey.
Thompsons Decision Tree
Now that we have redefined the problem (Step 1), lets use this additional data and redraw Thompsons decision tree (Step 2).
Thompsons Decision Tree Step 3: Assign the new probabilities to the states of nature.
Step 4: Estimate the payoffs.
Step 5: Compute the EMVs and make decision.
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
Expected Value of Sample Information
o The survey cost $10,000.
o The expected value of $49,200 (when the survey is used) is based on payoffs after the $10,000 cost was subtracted.
o The expected value with sample information (EV with SI) is the expected value of using the survey assuming no cost to gather it. Thus, in this example:
EV with SI = $49,200 + $10,000 (cost) = $59,200.
o Without the sample information, the best expected value is $40,000. Thus, the expected value would increase by $19,200 if the survey was available free.
Expected Value of Sample Information (EVSI)
Expected value with
sample information
+ study cost
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 for a market study.
Since it costs only $10,000, the survey is indeed worthwhile.
Sensitivity Analysis
How sensitive are the decisions to changes in the
probabilities?
e.g. If the probability of a favorable survey were
less than the current value (0.45), would the survey
still be selected? How low would this have to be to
cause a change in the decision?
Let p = probability of favorable market
Calculations for Thompson Lumber Sensitivity Analysis
2,400 $104,000
($2,400) ($106,400) 1) EMV(node
+ =
- + =
p
) p ( p 1
Equating the EMV with the survey to the EMV without the
survey, we have
0.36 $104,000
$37,600 or
$37,600 $104,000
$40,000 $2,400 $104,000
= =
=
= +
p
p
p
Hence, our decision will stay the same as long as the
probability of favorable survey results, p, is greater than 0.36.
Estimating Probability Values with Bayes Theorem
Management experience or intuition History Existing data Need to be able to revise probabilities based
upon new data
Posterior
probabilities Prior
probabilities Bayes Theorem
Information about accuracy
of sample information.
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.
STATE OF NATURE
RESULT OF SURVEY
FAVORABLE MARKET (FM)
UNFAVORABLE MARKET (UM)
Positive (predicts favorable market for product)
P (survey positive | FM)
= 0.70
P (survey positive | UM)
= 0.20
Negative (predicts unfavorable market for product)
P (survey negative | FM)
= 0.30
P (survey negative | UM)
= 0.80
Bayesian Analysis (continued): Favorable Survey
POSTERIOR PROBABILITY
STATE OF NATURE
CONDITIONAL PROBABILITY
P(SURVEY POSITIVE | STATE
OF NATURE) PRIOR
PROBABILITY JOINT
PROBABILITY
P(STATE OF NATURE | SURVEY
POSITIVE)
FM 0.70 X 0.50 = 0.35 0.35/0.45 = 0.78
UM 0.20 X 0.50 = 0.10 0.10/0.45 = 0.22
P(survey results positive) = 0.45 1.00
Bayesian Analysis (continued): Unfavorable Survey
POSTERIOR PROBABILITY
STATE OF NATURE
CONDITIONAL PROBABILITY
P(SURVEY NEGATIVE | STATE
OF NATURE) PRIOR
PROBABILITY JOINT
PROBABILITY
P(STATE OF NATURE | SURVEY
NEGATIVE)
FM 0.30 X 0.50 = 0.15 0.15/0.55 = 0.27
UM 0.80 X 0.50 = 0.40 0.40/0.55 = 0.73
P(survey results positive) = 0.55 1.00
Decision Making Using Utility Theory
There are many occasions in which people make decisions that would appear to be inconsistent with the EMV criterion.
E.g.: 1. When people buy insurance, the amount of the premium is greater than the expected payout. 2. A person involved in a law suit may choose to settle out of court rather than go trial even if the expected value of going to trial is greater hat the proposed settlement.
This is because monetary value is not always a true indicator of the overall value of the result of a decision.
The overall worth of a particular decision is called utility.
Rational people make decisions that maximize the expected utility.
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, your utility values are equal.
Expected utility of alternative 2 = Expected utility of alternative 1
Utility of other outcome = (p)(utility of best outcome, which is
1) + (1p)(utility of the worst
outcome, which is 0)
Utility of other outcome = (p)(1) + (1p)(0) = p
Where p is the probability of obtaining the best outcome, and (1p) is the
probability of obtaining the worst
outcome.
Standard Gamble for Utility Assessment
Best outcome
Utility = 1
Worst outcome
Utility = 0
Other outcome
Utility = ??
p
1p
Real Estate Example: Utility Theory Jane Dickson wants to construct a utility curve revealing her
preference for money between $0 and $10,000.
A utility curve plots the utility value versus the monetary value.
An investment in a bank will result in $5,000.
An investment in real estate will result in $0 or $10,000.
Unless there is an 80% chance of getting $10,000 from the real estate deal, Jane would prefer to have her money in the bank.
So if p = 0.80, Jane is indifferent between the bank or the real estate investment.
Real Estate Example: Solution
$10,000
U($10,000) = 1.0
0
U(0) = 0
$5,000
U($5,000) = p = 0.80
p= 0.80
(1 p)= 0.20
Hence, Janes Utility for $5,000 = U($5,000) = p
= p*U($10,000) + (1 p)*U($0)
= (0.8)(1) + (0.2)(0) = 0.8
Real Estate Example: Solution
Utility for $7,000 = 0.90
Utility for $3,000 = 0.50
We can assess other utility values in the same way.
For Jane, let say these are:
Using the three utilities for different dollar amounts, she can construct a utility curve.
Note: In setting the value of probability p, one should be aware that utility assessment is completely subjective. It is a value set by the decision maker that cannot be measured on an objective scale.
Real Estate Example: Utility Curve
U ($7,000) = 0.90
U ($5,000) = 0.80
U ($3,000) = 0.50
U ($0) = 0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
| | | | | | | | | | |
$0 $1,000 $3,000 $5,000 $7,000 $10,000
Monetary Value
Uti
lity
U ($10,000) = 1.0