Download Ch 4A

Anderson
Sweeney
Williams
QUANTITATIVE
METHODS FOR
BUSINESS 8e
Slides Prepared by JOHN LOUCKS
© 2001 South-Western College Publishing/Thomson Learning
Slide 1
Chapter 4
Decision Analysis, Part A
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

Problem Formulation
Decision Making without Probabilities
Decision Making with Probabilities
Slide 2
Problem Formulation
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A decision problem is characterized by decision
alternatives, states of nature, and resulting payoffs.
The decision alternatives are the different possible
strategies the decision maker can employ.
The states of nature refer to future events, not under
the control of the decision maker, which may occur.
States of nature should be defined so that they are
mutually exclusive and collectively exhaustive.
Slide 3
Influence Diagram
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An influence diagram is a graphical device showing the
relationships among the decisions, the chance events,
and the consequences.
Squares or rectangles depict decision nodes.
Circles or ovals depict chance nodes.
Diamonds depict consequence nodes.
Lines or arcs connecting the nodes show the direction of
influence.
Slide 4
Payoff Tables
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The consequence resulting from a specific combination
of a decision alternative and a state of nature is a
payoff.
A table showing payoffs for all combinations of
decision alternatives and states of nature is a payoff
table.
Payoffs can be expressed in terms of profit, cost, time,
distance or any other appropriate measure.
Slide 5
Decision Trees
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A decision tree is a chronological representation of
the decision problem.
Each decision tree has two types of nodes; round
nodes correspond to the states of nature while square
nodes correspond to the decision alternatives.
The branches leaving each round node represent the
different states of nature while the branches leaving
each square node represent the different decision
alternatives.
At the end of each limb of a tree are the payoffs
attained from the series of branches making up that
limb.
Slide 6
Decision Making without Probabilities

Three commonly used criteria for decision making
when probability information regarding the
likelihood of the states of nature is unavailable are:
• the optimistic approach
• the conservative approach
• the minimax regret approach.
Slide 7
Optimistic Approach
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The optimistic approach would be used by an
optimistic decision maker.
The decision with the largest possible payoff is
chosen.
If the payoff table was in terms of costs, the decision
with the lowest cost would be chosen.
Slide 8
Conservative Approach
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The conservative approach would be used by a
conservative decision maker.
For each decision the minimum payoff is listed and
then the decision corresponding to the maximum of
these minimum payoffs is selected. (Hence, the
minimum possible payoff is maximized.)
If the payoff was in terms of costs, the maximum
costs would be determined for each decision and
then the decision corresponding to the minimum of
these maximum costs is selected. (Hence, the
maximum possible cost is minimized.)
Slide 9
Minimax Regret Approach
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The minimax regret approach requires the
construction of a regret table or an opportunity loss
table.
This is done by calculating for each state of nature
the difference between each payoff and the largest
payoff for that state of nature.
Then, using this regret table, the maximum regret for
each possible decision is listed.
The decision chosen is the one corresponding to the
minimum of the maximum regrets.
Slide 10
Example
Consider the following problem with three decision
alternatives and three states of nature with the
following payoff table representing profits:
States of Nature
s1
s2
s3
d1
Decisions d2
d3
4
0
1
4
3
5
-2
-1
-3
Slide 11
Example

Optimistic Approach
An optimistic decision maker would use the
optimistic approach. All we really need to do is to
choose the decision that has the largest single value in
the payoff table. This largest value is 5, and hence the
optimal decision is d3.
Maximum
Decision
Payoff
d1
4
d2
3
choose d3
d3
5
maximum
Slide 12
Example

1
2
3
4
5
6
7
8
9
Formula Spreadsheet for Optimistic Approach
A
B
PAYOFF TABLE
Decision
Alternative
d1
d2
d3
C
State
s1
4
0
1
D
of Nature
s2
s3
4
-2
3
-1
5
-3
Best Payoff
E
F
Maximum
Recommended
Payoff
Decision
=MAX(B5:D5) =IF(E5=$E$9,A5,"")
=MAX(B6:D6) =IF(E6=$E$9,A6,"")
=MAX(B7:D7) =IF(E7=$E$9,A7,"")
=MAX(E5:E7)
Slide 13
Example

1
2
3
4
5
6
7
8
9
Solution Spreadsheet for Optimistic Approach
A
B
PAYOFF TABLE
Decision
Alternative
d1
d2
d3
C
State
s1
4
0
1
D
of Nature
s2
s3
4
-2
3
-1
5
-3
Best Payoff
E
F
Maximum Recommended
Payoff
Decision
4
3
5
d3
5
Slide 14
Example

Conservative Approach
A conservative decision maker would use the
conservative approach. List the minimum payoff for
each decision. Choose the decision with the maximum
of these minimum payoffs.
Minimum
Decision
Payoff
d1
-2
choose d2
d2
-1
maximum
d3
-3
Slide 15
Example

Formula Spreadsheet for Conservative Approach
A
B
C
D
1 PAYOFF TABLE
2
3 Decision
State of Nature
4 Alternative
s1
s2
s3
5
d1
4
4
-2
6
d2
0
3
-1
7
d3
1
5
-3
8
9
Best Payoff
E
F
Minimum
Recommended
Payoff
Decision
=MIN(B5:D5) =IF(E5=$E$9,A5,"")
=MIN(B6:D6) =IF(E6=$E$9,A6,"")
=MIN(B7:D7) =IF(E7=$E$9,A7,"")
=MAX(E5:E7)
Slide 16
Example

Solution Spreadsheet for Conservative Approach
A
B
C
D
1 PAYOFF TABLE
2
3
Decision
State of Nature
4 Alternative
s1
s2
s3
5
d1
4
4
-2
6
d2
0
3
-1
7
d3
1
5
-3
8
9
Best Payoff
E
F
Minimum
Payoff
-2
-1
-3
Recommended
Decision
d2
-1
Slide 17
Example

Minimax Regret Approach
For the minimax regret approach, first compute a
regret table by subtracting each payoff in a column
from the largest payoff in that column. In this
example, in the first column subtract 4, 0, and 1 from
4; in the second column, subtract 4, 3, and 5 from 5;
etc. The resulting regret table is:
s1 s2 s3
d1
d2
d3
0
4
3
1
2
0
1
0
2
Slide 18
Example

Minimax Regret Approach (continued)
For each decision list the maximum regret. Choose
the decision with the minimum of these values.
choose d1
Decision
d1
d2
d3
Maximum Regret
1
minimum
4
3
Slide 19
Example
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Formula Spreadsheet for Minimax Regret Approach
A
B
PAYOFF TABLE
Decision
Altern.
s1
d1
4
d2
0
d3
1
C
D
1
2
State of Nature
3
s2
s3
4
4
-2
5
3
-1
6
5
-3
7
8 OPPORTUNITY LOSS TABLE
9 Decision
State of Nature
10 Altern.
s1
s2
s3
11
d1
=MAX($B$4:$B$6)-B4 =MAX($C$4:$C$6)-C4 =MAX($D$4:$D$6)-D4
12
d2
=MAX($B$4:$B$6)-B5 =MAX($C$4:$C$6)-C5 =MAX($D$4:$D$6)-D5
13
d3
=MAX($B$4:$B$6)-B6 =MAX($C$4:$C$6)-C6 =MAX($D$4:$D$6)-D6
14
Minimax Regret Value
E
F
Maximum
Recommended
Regret
Decision
=MAX(B11:D11) =IF(E11=$E$14,A11,"")
=MAX(B12:D12) =IF(E12=$E$14,A12,"")
=MAX(B13:D13) =IF(E13=$E$14,A13,"")
=MIN(E11:E13)
Slide 20
Example
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Solution Spreadsheet for Minimax Regret Approach
1
2
3
4
5
6
7
8
9
10
11
12
13
14
PAYOFF TABLE
Decision
State
Alternative
s1
d1
4
d2
0
d3
1
of Nature
s2
s3
4
-2
3
-1
5
-3
OPPORTUNITY LOSS TABLE
Decision
State of Nature
Alternative
s1
s2
s3
d1
0
1
1
d2
4
2
0
d3
3
0
2
Minimax Regret Value
Maximum Recommended
Regret
Decision
1
d1
4
3
1
Slide 21
Decision Making with Probabilities

Expected Value Approach
• If probabilistic information regarding he states of
nature is available, one may use the expected
value (EV) approach.
• Here the expected return for each decision is
calculated by summing the products of the payoff
under each state of nature and the probability of
the respective state of nature occurring.
• The decision yielding the best expected return is
chosen.
Slide 22
Expected Value of a Decision Alternative
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The expected value of a decision alternative is the sum
of weighted payoffs for the decision alternative.
The expected value (EV) of decision alternative di is
defined as:
N
EV( d i )   P( s j )Vij
j 1
where:
N = the number of states of nature
P(sj ) = the probability of state of nature sj
Vij = the payoff corresponding to decision
alternative di and state of nature sj
Slide 23
Example: Burger Prince
Burger Prince Restaurant is contemplating
opening a new restaurant on Main Street. It has three
different models, each with a different seating
capacity. Burger Prince estimates that the average
number of customers per hour will be 80, 100, or 120.
The payoff table for the three models is as follows:
Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120
Model A
Model B
Model C
$10,000
$ 8,000
$ 6,000
$15,000
$18,000
$16,000
$14,000
$12,000
$21,000
Slide 24
Example: Burger Prince

Expected Value Approach
Calculate the expected value for each decision. The
decision tree on the next slide can assist in this
calculation. Here d1, d2, d3 represent the decision
alternatives of models A, B, C, and s1, s2, s3 represent the
states of nature of 80, 100, and 120.
Slide 25
Example: Burger Prince

Decision Tree
Payoffs
2
d1
d3
.4
.2
10,000
15,000
.4
14,000
d2
1
s1
s2
s3
3
s1
s2
s3
.4
.2
8,000
18,000
.4
12,000
4
s1
s2
s3
.4
6,000
.2
16,000
.4
21,000
Slide 26
Example: Burger Prince
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Expected Value For Each Decision
d1
EMV = .4(10,000) + .2(15,000) + .4(14,000)
= $12,600
2
Model A
1
Model B
Model C
d2
EMV = .4(8,000) + .2(18,000) + .4(12,000)
= $11,600
3
d3
EMV = .4(6,000) + .2(16,000) + .4(21,000)
= $14,000
4
Choose the model with largest EV, Model C.
Slide 27
Example: Burger Prince
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Formula Spreadsheet for Expected Value Approach
A
B
C
D
E
F
1 PAYOFF TABLE
2
3 Decision
State of Nature
Expected
Recommended
4 Alternative s1 = 80 s2 = 100 s3 = 120
Value
Decision
5 Model A 10,000 15,000 14,000 =$B$8*B5+$C$8*C5+$D$8*D5 =IF(E5=$E$9,A5,"")
6 Model B
8,000 18,000 12,000 =$B$8*B6+$C$8*C6+$D$8*D6 =IF(E6=$E$9,A6,"")
7 Model C
6,000 16,000 21,000 =$B$8*B7+$C$8*C7+$D$8*D7 =IF(E7=$E$9,A7,"")
8 Probability 0.4
0.2
0.4
9
Maximum Expected Value
=MAX(E5:E7)
Slide 28
Example: Burger Prince
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Solution Spreadsheet for Expected Value Approach
A
B
C
D
E
F
1 PAYOFF TABLE
2
3 Decision
State of Nature
Expected Recommended
4 Alternative s1 = 80 s2 = 100 s3 = 120 Value
Decision
5
Model A
10,000 15,000 14,000
12600
6
Model B
8,000
18,000 12,000
11600
7
Model C
6,000
16,000 21,000
14000
Model C
8 Probability
0.4
0.2
0.4
9
Maximum Expected Value
14000
Slide 29
The End of Chapter 4, Part A
Slide 30