Supplement to Part II: Decision Analysis
The more it seems difficult to determine by reason what is uncertain by chance, the more
the Algebra which determines this result appears admirable.
— Christianus Huygens (1629-1695).
Definitions:
Decision Analysis is a tool for studying situations in which there are several decision
alternatives and a set of uncertain future events.
Decision Alternatives are elements of a set of possible choices, represented by
d1, d2, d3, … di
States of Nature are elements of a set of N random future events, represented by
s1 , s2 , s3 , … sN
A Payoff Table lists outcomes associated with some combination of decision alternative
and state of nature. The payoff for decision alternative i under state of nature j is
symbolized by vij.
Decision analysis is a natural extension of our previous work with conditional
probability and systems of probabilities and payoffs; the only new element here is the
opportunity for a decision maker to make choices at certain discrete points in time.
Example: Stereo Industries, Ltd., must decide to build either a large or small plant to
produce a new mini-disk player. A large plant will cost $2.8 million to build and put
into operation, while a small plant will cost only $1.4 million. In this example, the
decision alternatives are the large and small plants.
The states of nature are different levels of customer demand. The company’s best
estimate of a discrete distribution of sales over the relevant planning horizon of 10 years
is given in this table:
High Demand
Probability = 0.5
Moderate Demand
Probability = 0.3
Low Demand
Probability = 0.2
Cost-volume-profit analysis by Stereo Industries suggests the following conditional
outcomes under various combinations of plant size and market demand:
1. A large plant with high demand would yield $1 million annually in profits.
2. A large plant with moderate demand would yield $0.6 million annually in
profits.
3. A large plant with low demand would lose $0.2 million annually, because of
production inefficiencies.
4. A small plant with high demand would yield only $0.25 million annually in
profits, considering the cost of lost sales because of inability to supply customers.
5. A small plant with moderate demand would yield $0.45 million annually in
profits because the cost of lost sales would be somewhat lower.
6. A small plant with low demand would yield $0.55 million annually, because the
plant size and the market size would be matched fairly optimally.
Therefore, a 10-year payoff table for this problem (not taking into account the cost of
building a plant) would look something like this:
s1 = High Demand
s2 = Moderate Demand
s3 = Low Demand
d1 = Large Plant d2 = Small Plant
$10,000,000
$2,500,000
$6,000,000
$4,500,000
-$2,000,000
$5,500,000
If we include the costs of construction ($2.8 million for a large plant or $1.4 million for a
small plant), the payoff table looks like this:
s1 = High Demand
s2 = Moderate Demand
s3 = Low Demand
Managerial Statistics
d1 = Large Plant d2 = Small Plant
$7,200,000
$1,100,000
$3,200,000
$3,100,000
-$4,800,000
$4,100,000
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Prof. Juran
Decision Trees
A decision tree is a graphical tool useful for representing a decision analysis problem.
The “tree” grows from left to right, with different branches representing different
possible outcomes. By convention, the nodes (places where the tree splits) are identified
by their shape as to whether they are probability nodes (circles, indicating random
outcomes) or decision nodes (squares, indicating a choice to be made by some decision
maker).
Here is a decision tree representing these possible outcomes. The square at the left of
the diagram indicates a decision node (whether to build a large or a small plant) and the
two circles in the center of the diagram indicate random outcomes (whether demand is
high, moderate, or low).
High Demand
Large Plant TRUE
0
Stereo Industries
0.50
Expected Value
$0
Moderate Demand
0.30
Low Demand
0.20
High Demand
0.50
Moderate Demand
0.30
Expected Value
$0
Small Plant FALSE
0
Expected Value
$0
Low Demand
0.20
Note that this example contains some fairly restrictive assumptions:

Demand is a discrete variable with only three possible values, and the
probabilities associated with the three values are known in advance.

There are only two possible sizes for the plant, and not building the plant is not
considered to be a possible decision.

Once built, the size of the plant is fixed.

The future is considered to consist of a single ten-year period.

The time value of money is not considered.
The decision tree could have been altered to make each of these assumptions more
realistic.
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Prof. Juran
Decision Making with Probabilities
Expected Value of an Alternative
The decision tree can be used as the basis for calculating an expected value for each of
the decision alternatives. In this case, the expected value of building the large plant is
the probability-weighted sum of all of the payoffs, under all of the states of nature,
given that the large plant is built:
EV d 1 
N
 
  P s j v 1 j  (cost of building the large plant)
j 1
 0.5$7 ,200,000   0.3$3,200,000   0.2 $4 ,800,000 
 $3,600,000  $960,000  $960,000
 $3,600 ,000
We can say that the expected value of the large plant decision is $3,600,000.
For the small plant:
EV d 2 
N
 
  P s j v 2 j  (cost of building the small plant)
j 1
 0.5$1,100,000   0.3$3,100,000   0.2$4 ,100,000 
 $550,000  $930,000  $820,000
 $2 ,300,000
In this example, the large plant alternative has the highest expected value.
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Prof. Juran
Software is available to construct decision tree models in a spreadsheet. The decision
trees in these notes were drawn with PrecisionTree, an Excel add-in from Palisade
Corporation. Here is a picture of the Stereo Industries problem solved using
PrecisionTree.
A
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
B
C
High Demand
D
0.50
$7,200,000
Large Plant TRUE
0
Expected Value
$3,600,000
Moderate Demand
0.30
$3,200,000
Low Demand
0.20
-$4,800,000
Stereo Industries
Expected Value
$3,600,000
High Demand
0.50
$1,100,000
Moderate Demand
0.30
$3,100,000
Small Plant FALSE
0
Expected Value
$2,300,000
Low Demand
0.20
$4,100,000
Once the decision tree model is created, it can be interpreted by reading backwards
from right to left.

In column D of the spreadsheet, we see the payoffs for each possible combination
of a state of nature and a decision alternative, along with the probability
associated with the state of nature.

In column C we see the expected value of each decision alternative.

In column B the decision alternative with the highest expected value is identified
with the word “TRUE”, and the expected value of the entire problem is given as
$3,600,000.
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Prof. Juran
Decision Analysis with Perfect Information
A more advanced concept in decision analysis involves considering how the optimal
choice is affected by knowing in advance what the future state of nature will be.
In the Stereo Industries example there is clearly a “best” choice of plant size for each of
the possible customer demand states. Here, the optimal decision for each state of nature
is identified with a non-shaded cell:
Demand Probabilities Net Payout Large Plant Net Payout Small Plant Optimal Decision
High
0.5
$7,200,000
$1,100,000
Large Plant
Moderate
0.3
$3,200,000
$3,100,000
Large Plant
Low
0.2
$(4,800,000)
$4,100,000
Small Plant
We calculate the expected value with perfect information by summing up the
probability-weighted best payoffs for each state of nature. For this example:
EVwPI
 0.50 * $7 ,200,000  0.30 * $3,200,000  0.20 * $4,100,000
 $3,600,000  $960,000  $820,000
 $5,380 ,000
This result can be interpreted as follows: If we know ahead of time that the true state of
nature will be revealed before we make the decision, then the expected value of the
problem is $5,380,000 instead of $3,600,000. Therefore, perfect information (if it were
available) would be worth up to $5,380,000 - $3,600,000 = $1,780,000 to Stereo
Industries. This is referred to as expected value of perfect information.
This number may not appear to have much practical significance, but it does provide
some basis for considering whether to collect additional information before making the
decision. If the expected value of perfect information is small, then there is little to be
gained from additional research, no matter what the results of the new information
might be. However, if the expected value of perfect information is large, then there is an
opportunity to create value by conducting a research project before making the
decision.
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Prof. Juran
Here is another way to think about perfect information. Imagine that the order of events
could be reversed in the Stereo Industries problem, so that we could see the results of
the probability node before having to make a decision about which plant to build (see
decision tree below).
A
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
B
C
D
TRUE
Large Plant
$
50.0%
High Demand
7,200,000
7200000
Expected Value
0
$7,200,000
FALSE
Small Plant
$
Perfect Information
0.5
1,100,000
0
1100000
Expected Value
$5,380,000
TRUE
Large Plant
$
30.0%
Moderate Demand
3,200,000
Expected Value
0
$3,200,000
FALSE
Small Plant
$
3,100,000
FALSE
Large Plant
$
20.0%
Low Demand
0.3
3200000
(4,800,000)
0
3100000
0
-4800000
Expected Value
0
$4,100,000
Small Plant
$
TRUE
4,100,000
0.2
4100000
Note that (a) the correct choice of plants is going to be obvious if we know the demand
level ahead of time, and (b) we can quantify the expected benefit from having this
information even before it is revealed to us.
Decision Analysis with Sample Information
Perfect information may be impossible to obtain, but we can often get sample
information about the future states of nature, for example by performing a market
research project. New information from the results of such a project might make us
more confident in choosing one of the decision alternatives. The following analysis is
aimed at placing a monetary value on this improvement in confidence.
Example: Suppose that Stereo Industries’ market research department could conduct a
survey to predict the level of future demand, and that the survey could have two
possible outcomes, either favorable or unfavorable (indicated, respectively, by I1 and I2).
Suppose further that historical data can be used to estimate the following conditional
probabilities for the results of the research project, given the possible states of nature:
Research Results
I1 = Favorable
I2 = Unfavorable
High Demand
PI 1 s 1   0.96
PI 2 s 1   0.04
Moderate Demand
PI 1 s 2   0.40
PI 2 s 2   0.60
Low Demand
PI 1 s 3   0.00
PI 2 s 3   1.00
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Prof. Juran
Developing a Decision Strategy
Using Bayes’ Law, we can use these conditional probabilities to calculate posterior
probabilities (probabilities for each state of nature given each possible outcome of the
market research project):
For a Favorable Report:
Probabilities
Joint
States
Prior
sj
P sj
P I1 s j
High
0.5
0.96
0.48
0.80
Moderate
Low
0.3
0.2
0.40
0.00
0.12
0.00
0.20
0.00
 
Conditional


  

P s j  I1  P s j P I1 s j

Posterior


P s j I1 
Total P I 1   0.60

P s j  I1
P I 1 

For an Unfavorable Report:
Probabilities
Joint
States
Prior
sj
P sj
P I2 sj
High
0.5
0.04
0.02
0.05
Moderate
Low
0.3
0.2
0.60
1.00
0.18
0.20
0.45
0.5
 
Conditional


  

P sj  I2  P sj P I2 sj
Total PI 2   0.40

Posterior


P sj I2 

P sj  I2
P I 2 

Now we can calculate an expected value for each decision alternative for each possible
outcome of the market research study, and we can calculate an overall expected value.
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Prof. Juran
An Optimal Decision Strategy
For this problem, the best size of plant to build depends on the results of the research
project, as shown in the expanded decision tree, showing the two possible outcomes
from the market research study.
High Demand
0.80
$7,200,000
TRUE
Large Plant
0
Expected Value
$6,400,000
Moderate Demand
0.20
$3,200,000
Low Demand
0.00
-$4,800,000
Favorable Report
60.00%
Expected Value
$6,400,000
0
High Demand
0.80
$1,100,000
Moderate Demand
0.20
$3,100,000
Small Plant
FALSE
0
Expected Value
$1,500,000
Low Demand
0.00
$4,100,000
Stereo Industries
Market Research Project
$5,240,000
High Demand
0.05
$7,200,000
Large Plant
FALSE
0
Expected Value
-$600,000
Moderate Demand
0.45
$3,200,000
Low Demand
0.50
-$4,800,000
Unfavorable Report
40.0%
Expected Value
$3,500,000
0
High Demand
0.05
$1,100,000
Moderate Demand
0.45
$3,100,000
Small Plant
TRUE
0
Expected Value
$3,500,000
Low Demand
0.50
$4,100,000
The optimal strategy appears to be (1) wait and see what the report says, (2) if the report
is favorable, build the large plant, and (3) if the report is unfavorable, build the small
plant. Here is a summary of the various decision alternatives and their expected values:
Favorable Report
Unfavorable Report
The overall expected value is:
EV I 1 PI 1   EV I 2 PI 2 
Decision Alternative
Expected Value
Large Plant
$6,400,000
Small Plant
$1,500,000
Large Plant
-$600,000
Small Plant
$3,500,000
← Optimal
← Optimal
 $6 ,400 ,000 * 0.60   $3,500 ,000 * 0.40   $5,240 ,000
Note that the diagram above is really just two copies of the original one from page 5,
with revised probabilities. You could solve the problem just as well by entering the
revised probabilities into the original tree diagram and tabulating the results as shown
above.
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Prof. Juran
Expected Value of Sample Information
The expected value of sample information is calculated using this formula:
EVSI = EVwSI - EVwoSI
where
EVSI = expected value of sample information
EVwSI = expected value with sample information about the states of nature
EVwoSI = expected value without sample information about the states of nature
In our example, the expected value of sample information is:
EVSI = EVwSI - EVwoSI
 $5,240,000  $3,600,000
 $1,640 ,000
This is one way to address the question, “How much should Stereo Industries be
prepared to pay for the research study?” Clearly, it is not worth any more than $1.64
million.
Efficiency of Sample Information
The efficiency of sample information is calculated using this formula:

E
EVSI
EVPI
For this example,
E

EVSI
EVPI

$1,640 ,000
$1,780 ,000
 0.9213
In other words, the market research project gives us information with about 92% of the
utility of having perfect information.
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Prof. Juran