James Madison, president of Madison Manufacturing, Inc., is considering whether
ID: 3265091 • Letter: J
Question
James Madison, president of Madison Manufacturing, Inc., is considering whether to build more manufacturing plants in Madison, Wisconsin. He is considering three sizes of plant: Small, Medium, or Large. At the same time, an uncertain economy makes ascertaining the demand for the new plants difficult. His management team has prepared the following profit (in Millions) payoff table.
Decision Alternatives
States of Nature
Good Economy
Fair Economy
Poor Economy
Small Plant, d1
14,000
15,000
10,000
Medium Plant, d2
12,000
18,000
8,000
Large Plant, d3
21,000
16,000
6,000
James Madison is considering a 6-month Market Research Survey to learn more about the potential market acceptance of his plant project and anticipates that the market research study will provide either a Positive or Negative report.) Conditional Probabilities are as follows:
Market Research Report
State of Nature
Positive
Negative
Good Economy, S1
90%
10%
Fair Economy, S2
50%
50%
Poor Economy, S3
20%
80%
1. Calculate the joint and posterior probabilities for each state of nature.
Decision Alternatives
States of Nature
Good Economy
Fair Economy
Poor Economy
Small Plant, d1
14,000
15,000
10,000
Medium Plant, d2
12,000
18,000
8,000
Large Plant, d3
21,000
16,000
6,000
Explanation / Answer
Answer to A
Small
Medium
Large
Fair Economy
15000
18000
16000
Poor Economy
10000
8000
6000
Management Estimate
14000
12000
21000
Under conservative approach we look at the alternative which provides maximum profit in worst scenario also. Therefore, in poor economy, small size plant would provide the maximum profit.
Answer to B
Expected Opportunity Loss Table
Regret Table
Small
Medium
Large
Max Regret
Fair Economy
3000
0
2000
3000
Poor Economy
0
2000
4000
4000
Management Estimate (Moderate)
7000
9000
0
9000
Probablistic Value Table
Probablity
Small
Medium
Large
Fair Economy
40%
15000
18000
16000
Poor Economy
20%
10000
8000
6000
Management Estimate (Moderate)
40%
14000
12000
21000
13600
13600
19200
Using probablistic values given, Alternative to have large plant is benificial.
1. Calculate the joint and posterior probabilities for each state of nature.
As we know that posterior probabilities can be find through using Bayes Theorem which is as follows:
P(A/B) = P( ) / P(B)
We know that there is direct relation between P(A/B) and P(B/A) which is
P(A/B) = P( ) / P(B)
P(B/A) = P( ) / P(A)
which gives,
P(A/B) P(B) = P(B/A) P(A)
We can now write prior and posterior probability concerning Favourable report will same for each state of nature:
State of nature (sj)
Prior Probability P(sj)
conditional Probability(P(Favourable/sj))
Weak(s1)
0.33
0.60
Moderate(s2)
0.33
0.50
Strong(s3)
0.33
0.80
We can find the probabilities based on our criteria:
P(s1 favourable ) = P(s1) * P(favourable/s1) = 0.33*0.60 = 0.198
P(s2 favourable ) = P(s2)*P(favourable/s2) = 0.33*0.50 = 0.165
P(s3 favourable ) = P(s3)*P(favourable/s3) = 0.33*0.80 = 0.264
Probability of favourable report
P(favourable) = P(s1 favourable ) + P(s2 favourable ) + P(s3 favourable )
=0.198 + 0.165 + 0.264 = 0.627
Now,
P(s1/favourable) = P(s1 favourable ) / P(favourable) = 0.198/0.627 = 0.316
P(s2/favourable) = P(s2 favourable ) / P(favourable) = 0.165/0.627 = 0.263
P(s3/favourable) = P(s3 favourable ) / P(favourable) = 0.264/0.627 = 0.421
All these probabilities can be summarized in a table:
States of Nature
Prior (P(sj)
Conditional(P(fav/sj))
Joint (P(fav sj))
Posterior(P(sj/fav)
Weak
0.33
0.60
0.198
0.316
Moderate
0.33
0.50
0.165
0.263
Strong
0.33
0.80
0.264
0.421
P(favourable) = 0.627
Similarly we will do it for unfavourable report also:
We can now write prior and posterior probability concerning Unfavourable report will same for each state of nature:
State of nature (sj)
Prior Probability P(sj)
conditional Probability(P(Favourable/sj))
Weak(s1)
0.33
0.40
Moderate(s2)
0.33
0.50
Strong(s3)
0.33
0.20
We can find the probabilities based on our criteria:
P(s1 favourable ) = P(s1) * P(unfavourable/s1) = 0.33*0.40 = 0.132
P(s2 favourable ) = P(s2)*P(ufavourable/s2) = 0.33*0.50 = 0.165
P(s3 favourable ) = P(s3)*P(unfavourable/s3) = 0.33*0.20 = 0.066
Probability of unfavourable report
P(unfavourable) = P(s1 favourable) + P(s2 favourable) + P(s3 favourable)
=0.132 + 0.165 + 0.066 = 0.363
Now,
P(s1/unfavourable) = P(s1 favourable)/ P(unfavourable) = 0.132/0.363 = 0.364
P(s2/unfavourable) = P(s2 favourable) / P(unfavourable) = 0.165/0.363 = 0.454
P(s3/unfavourable) = P(s3 favourable) / P(unfavourable) = 0.066/0.363 = 0.182
All these probabilities can be summarized in a table:
States of Nature
Prior (P(sj)
Conditional(P(unfav/sj))
Joint (P(unfavsj))
Posterior(P(sj/unfav)
Weak
0.33
0.40
0.132
0.364
Moderate
0.33
0.50
0.165
0.454
Strong
0.33
0.20
0.066
0.182
P(unfavourable) = 0.363
Small
Medium
Large
Fair Economy
15000
18000
16000
Poor Economy
10000
8000
6000
Management Estimate
14000
12000
21000
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