### Decision Theory Part-1

```Graduate Program in Business Information Systems
Decision Analysis
Part 1
Aslı Sencer
Analytical Decision Making
 Can Help Managers to:
 Gain deeper insight into the nature of
 Find better ways to assess values in such
relationships; and
 See a way of reducing, or at least
understanding, uncertainty that
2
Steps to Analytical DM
Define problem and influencing factors
Establish decision criteria
Select decision-making tool (model)
Identify and evaluate alternatives using
decision-making tool (model)
 Select best alternative
 Implement decision
 Evaluate the outcome




3
Models
 Are less expensive and disruptive than experimenting
with the real world system
 Allow operations managers to ask “What if” types of
questions
 Are built for management problems and encourage
management input
 Force a consistent and systematic approach to the
analysis of problems
 Require managers to be specific about constraints and
goals relating to a problem
 Help reduce the time needed in decision making
4
Limitations of the Models
 They may be expensive and timeconsuming to develop and test
 Often misused and misunderstood (and
feared) because of their mathematical and
logical complexity
 Tend to downplay the role and value of
nonquantifiable information
 Often have assumptions that oversimplify
the variables of the real world
5
The Decision-Making Process
Quantitative Analysis
Problem
Logic
Historical Data
Marketing Research
Scientific Analysis
Modeling
Decision
Qualitative Analysis
Emotions
Intuition
Personal Experience
and Motivation
Rumors
6
Displaying a Decision Problem
 Decision trees
 Decision tables
Outcomes
States of Nature
Alternatives
Decision Problem
7
Types of Decision Models
 Decision making under uncertainty
 Decision making under risk
 Decision making under certainty
8
Fundamentals of Decision Theory
Terms:
 Alternative: course of action or choice
 State of nature: an occurrence over
which the decision maker has no control
Symbols used in a decision tree:
 A decision node from which one of several
alternatives may be selected
 A state of nature node out of which one
state of nature will occur
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Decision Table
States of Nature
Alternatives
State 1
State 2
Alternative 1
Outcome 1
Outcome 2
Alternative 2
Outcome 3
Outcome 4
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Getz Products Decision Tree
Favorable market
A state of nature node
1
Unfavorable market
Favorable market
A decision node
Construct
small plant 2
Unfavorable market
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Decision Making under Uncertainty
 Maximax - Choose the alternative that
maximizes the maximum outcome for every
alternative (Optimistic criterion)
 Maximin - Choose the alternative that
maximizes the minimum outcome for every
alternative (Pessimistic criterion)
 Equally likely - chose the alternative with
the highest average outcome.
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Example:
States of Nature
Alternatives Favorable Unfavorable Maximum Minimum
Construct
large plant
Construct
small plant
Do nothing
Market
\$200,000
\$100,000
\$0
Row
Market
in Row
in Row Average
-\$180,000 \$200,000 -\$180,000 \$10,000
-\$20,000 \$100,000
\$0
Maximax
-\$20,000 \$40,000
\$0
Maximin
\$0
\$0
Equally
likely
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Decision criteria

The maximax choice is to construct a large plant.
This is the maximum of the maximum number within
each row or alternative.

The maximin choice is to do nothing. This is the
maximum of the minimum number within each row or
alternative.

The equally likely choice is to construct a small plant.
This is the maximum of the average outcomes of
each alternative. This approach assumes that all
outcomes for any alternative are equally likely.
14
Decision Making under Risk
 Probabilistic decision situation
 States of nature have probabilities of
occurrence
 Maximum Likelihood Criterion
 Maximize Expected Monitary Value
(Bayes Decision Rule)
15
Maximum Likelihood Criteria

Maximum Likelihood: Identify most likely event,
ignore others, and pick act with greatest payoff.
 Personal decisions are often made that way.


Collectively, other events may be more likely.
Ignores lots of information.
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Bayes Decision Rule
 It is not a perfect criterion because it can
lead to the less preferred choice.
 Consider the Far-Fetched Lottery decision:
EVENTS
Tail
Probability
.5
.5
ACTS
Gamble
Don’t Gamble
+\$10,000
-5,000
\$0
0
Would you gamble?
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The Far-Fetched Lottery
Decision
ACTS
Gamble
Don’t Gamble
EVENTS
Probability
Payoff × Prob.
Payoff × Prob
.5
+\$5,000
\$0
Tail
.5
-2,500
0
\$2,500
\$0
Expected Payoff:
Most people prefer not to gamble!


That violates the Bayes decision rule.
But the rule often indicates preferred choices even
though it is not perfect.
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Expected Monetary Value
N: Number of states of nature
k: Number of alternative decisions
Xij: Value of Payoff for alternative i in state
of nature j, i=1,2,...,k and j=1,2,...,N.
Pj: Probability of state of nature j
EMV ( Ai )   j 1 X ij P j
N
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Example:
States of Nature
Alternatives
Construct
large plant
Construct
small plant
Do nothing
Favorable
Unfavorable
Market
Market P(0.5)
P(0.5)
\$200,000
-\$180,000
\$100,000
-\$20,000
\$0
\$0
Expected
value
\$10,000
\$40,000 Best choice
\$0
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Decision Making under Certainty
 What if Getz knows the state of the
nature with certainty?
 Then there is no risk for the state of
the nature!
 A marketing research company
requests \$65000 for this information
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Questions:
 Should Getz hire the firm to make
this study?
 How much does this information
worth?
 What is the value of perfect
information?
22
Expected Value With
Perfect Information (EVPI)
EVPI = Expected Payoff - Maximum expected payoff
under Certainty
with no information
Let N: Number of states of nature and k: Number of actions,
N
Expected Payoff under Ceratinty=
 (Max {X
i
ij}).Pj
j1
Maximum expected payoff with no information=Max {EMVi; i=1,..,k}
EVPI
places an upper bound on what one would pay for additional information
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Example: Expected Value of
Perfect Information
S ta te o f N a tu r e
A lte r n a tiv e
Favorable Unfavorable
Market (\$) Market (\$)
EMV
Construct a
large plant
Construct a
small plant
200,000
-\$180,000
\$10,000
\$100,000
-\$20,000
\$40,000
Do nothing
\$0
\$0
\$0
P r o b a b ilitie s
0.50
0.50
24
Expected Value of Perfect
Information
Expected Value Under Certainty
=(\$200,000*0.50 + 0*0.50)= \$100,000
Max(EMV)= Max{10,000, 40,000, 0}=\$40,000
EVPI = Expected Value Under Certainty - Max(EMV)
= \$100,000 - \$40,000
= \$60,000
So Getz should not be willing to pay more than \$60,000
25
Ex: Toy Manufacturer
 How to choose among 4 types of
tippi-toes?
 Demand for tippi-toes is uncertain:
Light demand: 25,000 units (10%)
Moderate demand: 100,000 units
(70%)
Heavy demand: 150,000 units
(20%)
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Payoff Table
ACT (choice)
Event
(State of
nature)
Gears and Spring
Action
Probability levers
Weights and
pulleys
Light
0.10
\$25,000
-\$10,000
-\$125,000
Moderate
0.70
400,000
440,000
400,000
Heavy
0.20
650,000
740,000
750,000
27
Maximum Expected Payoff Criteria
ACT (choice)
Gears and
levers
Expected
Payoff
Spring
Action
\$412,500 \$455,500
Weights and
pulleys
\$417,000
Maximum expected payoff occurs at Spring Action!
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Decision Trees
 Graphical display of decision process, i.e., alternatives,
states of nature, probabilities, payoffs.
 Decision tables are convenient for problems
with one set of alternatives and states of nature.
 With several sets of alternatives and states of nature
(sequential decisions), decision trees are used!
 EMV criterion is the most commonly used criterion in
decision tree analysis.
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Softwares for Decision Tree
Analysis
 DPL
 Tree Plan
 Supertree
Analysis with less effort.
Full color presentations for managers
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Steps of 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 for each state-of-nature
node




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Decision Tree
State 1
1
State 2
State 1
2
Decision
Node
State 2
Outcome 1
Outcome 2
Outcome 3
Outcome 4
State of Nature Node
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Ex1:Getz Products Decision Tree
EMV for node 1 = \$10,000
1
Payoffs
Favorable market (0.5)
Unfavorable market (0.5) -\$180,000
Favorable market (0.5)
Construct
small plant 2
\$200,000
Unfavorable market (0.5)
\$100,000
-20,000
EMV for node 2 = \$40,000
0
33
A More Complex Decision Tree
Let’s say Getz Products has two
sequential decisions to make:
 Conduct a survey for \$10000?
 Build a large or small plant or not
build?
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Ex1:Getz Products Decision Tree
\$49,200
\$106,400
2nd decision point
1st decision
point
2
\$63,600
3
-\$87,400
1
\$2,400
4
\$2,400
5
\$10,000
\$40,000
\$49,200
\$106,400
6
\$40,000
7
Fav. Mkt (0.78)
\$190,000
Unfav. Mkt (0.22)
-\$190,000
Fav. Mkt (0.78)
Unfav. Mkt (0.22)
Fav. Mkt (0.27)
Unfav. Mkt (0.73)
Fav. Mkt (0.27)
Unfav. Mkt (0.73)
Fav. Mkt (0.5)
Unfav. Mkt (0.5)
Fav. Mkt (0.5)
Unfav. Mkt (0.5)
\$90,000
-\$30,000
-\$10,000
\$190,000
-\$190,000
\$90,000
-\$30,000
-\$10,000
\$200,000
-\$180,000
\$100,000
-\$20,000
\$0
35
Resulting Decision
 EMV of conducting the survey=\$49,200
 EMV of not conducting the survey=\$40,000
So Getz should conduct the survey!
If the survey results are favourable, build
large plant.
If the survey results are infavourable, build
small plant.
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Ex2: Ponderosa Record Company
 Decide whether or not to market the
recordings of a rock group.
 Alternative1: test market 5000 units
and if favorable, market 45000 units
nationally
 Alternative2: Market 50000 units
nationally
 Outcome is a complete success (all
are sold) or failure
37
Ex2: Ponderosa-costs, prices




Fixed payment to group: \$5000
Production cost: \$5000 and \$0.75/cd
Handling, distribution: \$0.25/cd
Price of a cd: \$2/cd
Cost of producing 5,000 cd’s
=5,000+5,000+(0.25+0.75)5,000=\$15,000
Cost of producing 45,000 cd’s
=0+5,000+(0.25+0.75)45,000=\$50,000
Cost of producing 50,000 cd’s
=5,000+5,000+(0.25+0.75)50,000=\$60,000
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Ex2: Ponderosa-Event Probabilities
 Without testing P(success)=P(failure)=0.5
 With testing
P(success|test result is favorable)=0.8
P(failure|test result is favorable)=0.2
P(success|test result is unfavorable)=0.2
P(failure|test result is unfavorable)=0.8
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Decision Tree for Ponderosa Record
Company
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Backward Approach
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Optimal Decision Policy
 Precision Tree provides excell add-ins.
 Optimal decision is:
Test market
 If the market is favorable, market nationally
 Else, abort
 Risk Profile
Possible outcomes for the opt. soln.
\$35,000 with probability 0.4
-\$55,000 with probability 0.1
-\$15,000 with probability 0.5
42
Risk Profile
for Ponderosa Record Co.
R isk P r ofile For P onder osa R ecor d C ompany
0.6
P rob a bility
0.5
0.4
0.3
0.2
0.1
0
- 70000
- 60000
- 50000
- 40000
- 30000
- 20000
- 10000
0
10000
20000
30000
40000
50000
Ex p e cte d V a lu e , \$
43
Sensitivity Analysis
The optimal solution depends on many
factors. Is the optimal policy robust?
Question:
-How does \$1000 payoff change with respect
to a change in
 success probability (0.8 currently)?
 earnings of success (\$90,000 currently)?
 test marketing cost (\$15,000 currently)?
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Application Areas of Decision
Theory
Investments in
research and development
plant and equipment
new buildings and structures
Production and Inventory control
Aggregate Planning
Maintenance
Scheduling, etc.
45
References

Lapin L.L., Whisler W.D., Quantitative Decision Making, 7e,
2002.

Heizer J., Render, B., Operations Management, 7e, 2004.

Render, B., Stair R. M., Quantitative Analysis for
Management, 8e, 2003.

Anderson, D.R., Sweeney D.J, Williams T.A., Statistics for