1 Contents CHAPTERS PAGES 1.0 Agreement 2 2.0 Introduction 3 3.0 Question review 5 4.0 Solution A : Decision based on a decision tree diagram 6 5.0 Solution B : Using EVPI to determine whether attempt to 7 obtain a better estimate of demand is required 6.0 Solution C : The probability that the market research report 10 will be favorable 7.0 Solution D : Optimal decision strategy 15 8.0 Solution E : The expected value of the market research 19 information 9.0 Solution F : The efficiency of the information 20 10.0 Conclusion 21 11.0 References 23
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1
Contents
CHAPTERS PAGES
1.0 Agreement 2
2.0 Introduction 3
3.0 Question review 5
4.0 Solution A : Decision based on a decision tree diagram 6
5.0 Solution B : Using EVPI to determine whether attempt to 7
obtain a better estimate of demand is required
6.0 Solution C : The probability that the market research report 10
will be favorable
7.0 Solution D : Optimal decision strategy 15
8.0 Solution E : The expected value of the market research 19
information
9.0 Solution F : The efficiency of the information 20
10.0 Conclusion 21
11.0 References 23
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INTRODUCTION
Probability is the chance of something happening. For example, this chance
could be getting a heads when we toss a coin. Here something is "getting a heads".
Probability is expressed as a fractional value between "0" and "1". A "0" probability
means something can never happen whereas a "1" probability indicates something
always happens.
Decision making is the study of identifying and choosing alternatives based on
the values and preferences of the decision maker. Making a decision implies that there
are alternative choices to be considered, and in such a case we want not only to
identify as many of these alternatives as possible but to choose the one that has the
highest probability of success or effectiveness and best fits with our goals, desires,
lifestyle, values, and so on.
In mathematics, we have learned about statistic. We were given a problem that
needs to solve based on statistic. By using appropriate methods, we have to select the
best decision to solve the problem which is given as our program based learning
project. The main objective for this project is to revise the study of decision analysis.
The problem given need us to help Gorman Manufacturing Company makes a
choice either to manufacture a component part or purchase the component part from a
supplier. We are required to decide what the best decision for the company in gaining
largest profit. Data on the payoff table and the state-of-nature denotes s.are given.
CHAPTER II
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The problem were given can be categories as difficult problem. So, it requires
more knowledge about business. The study on decision making with probabilities and
decision analysis with sample information should be well understood in order to solve
the question. To make this problem solution easier, we need to form a decision
strategy refer from analysis.
In solving this problem, we need more knowledge about engineering as it was so
important. The theorem given such as Bayes theorem and conditional probability can
be used to calculate the probabilities. Therefore, those methods are very useful to get
the best solution for this problem. So, we should study more about decision analysis
and expert in this method.
In our report, we will combine these two features which are probability and
decision making to purpose the best decision based on the consequences. The analysis
consists of Expected Value, Expected Value of Perfect Information, Expected Value of
Sample Information and Efficiency of Sample Information. Besides, decision tree
diagram is being computed as an overview in decision making.
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PBL QUESTION 1 KQ 2014
The Gorman Manufacturing Company is deciding whether to manufacture a
component part at its Milan, Michigan plant or purchase the component part from a
supplier. The resulting profit is dependent upon the demand of the product.
The following payoff table shows the projected profit ( in thousands of dollars)
Question:
A) Use a decision tree to recommend a decision.
B) Use EVPI to determine whether Gorman should attempt to obtain a better
estimate of demand.
C) A test market study of the potential demand for the product is expected to report
either a favorable (F) or unfavorable (U) condition. The relevant conditional
probability is as follow. What is the probability that the market research report
will be favorable?
D) What is Gorman’s optimal decision strategy?
E) What is the expected value of the market research information?
F) What is the efficiency of the information?
CHAPTER III
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Solution A : Decision Based On A Decision Tree Diagram
Expected Value = ( Profit Profit’s probability )
1. EV ( d1 ) = -20(0.35) + 40(0.35) + 100(0.30)
= 37
2. EV ( d2 ) = 10(0.35) + 45(0.35) + 70(0.30)
= 40.25
Since EV ( Node 2 ) < EV ( Node 3 ), The Gorman Manufacturing Company should
choose to purchase the component since it gives the highest profit, $ 40,250
CHAPTER IV
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Solution B : Using EVPI To Determine Whether Attempt To Obtain A Better
Estimate Of Demand Is Required
EVPI is denoted as Expected Value of Perfect Info
EVwPI is denoted as Expected Value with Perfect Info
EVw/oPI is denoted as Expected Value without Perfect Info
1. EVwPI is a better estimation of profit .It sums the profit by selecting the highest
profit from manufacturing and purchasing decision for each state of nature.
i.e. If the demand is low, the best decision alternative is to purchase component
If the demand is medium, the best decision alternative is to purchase component
If the demand is high, the best decision alternative is to manufacture component
CHAPTER V
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2. EVw/oPI is the highest value of the EV of the decision alternative. It is a rough
estimation of profit which takes the highest profit from the decision alternatives
without going through the highest value of profit for each state of nature.
3. EVPI is difference between the payoff under certainty and the payoff under risk
It means the extra profit if perfect info is taken to calculate the expected value of
profit.
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4. It is would be worth $9000 for the Gorman Manufacturing Company to run a
market study before selecting a decision alternative. In other words, the EVPI is
$9000 which means that if the company do or buy the market study info, it will
earn extra $9000.
5. Hence the price for buying or doing the market study info should not exceed
$9000. If the price of market study info is equal or more than $9000, it is useless
because the company spend more than it earns. In this case, the Gorman
Manufacturing Company should obtain a better estimate of demand.
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Solution C : The Probability That The Market Research Report Will Be
Favorable
1.0 The Branch Probabilities Based On Favorable Report:
1.1 Using The Conditional Probability Formula To Find The Necessary Information:
1.2 The Given Relevant Conditional Probabilities:
P( F | s1 ) = 0.10
P( F | s2 ) = 0.40
P( F | s3 ) = 0.60
1.3 The Calculation of Joint Probabilities:
P( Fs1) = P( F | s1 ) P( s1 ) = 0.10 0.35 = 0.035
P( Fs2) = P( F | s2 ) P( s2 ) = 0.40 0.35 = 0.140
P( Fs3) = P( F | s3 ) P( s3 ) = 0.60 0.30 = 0.180
1.4 The Probabilities That The Market Study Report Will Be Favorable:
P( F ) = P ( Fsj )
= P ( Fs1) + P ( Fs2 ) + P ( Fs3 )
= 0.035 + 0.140 + 0.180
= 0.355
CHAPTER VI
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1.5 The Posterior Probabilities Associated With Favorable Report:
1.6 The Obtained Information based on a Favorable Report:
099.0355.0035.0
) F ( P)s F ( P
) F | P(s 11
394.0355.0140.0
) F ( P)s F ( P ) F | P(s 2
2
507.0355.0180.0
) F ( P)s F ( P
) F | P(s 33
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2.0 The Branch Probabilities Based On Favorable Report:
2.1 Using The Conditional Probability Formula To Find The Necessary Information.
2.2 The Given Relevant Conditional Probabilities:
P( U | s1 ) = 0.90
P( U | s2 ) = 0.60
P( U | s3 ) = 0.40
2.3 The Calculation of Joint Probabilities:
P( Us1) = P( U | s1 ) P( s1 ) = 0.90 0.35 = 0.315
P( Us2) = P( U | s2 ) P( s2 ) = 0.60 0.35 = 0.210
P( Us3) = P( U | s3 ) P( s3 ) = 0.40 0.30 = 0.120
2.4 The Probabilities That The Market Study Report Will Be Unfavorable:
P( U ) = P ( Usj )
= P ( Us1 ) + P ( Us2 ) + P ( Us3 )
= 0.315 + 0.210 + 0.120
= 0.645
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488.0645.0315.0
) U( P)s U( P ) U| P(s 1
1
326.0645.0210.0
) U( P)s U( P
) U| P(s 22
186.0645.0120.0
) U( P)s U( P
) U| P(s 33
2.5 The Posterior Probabilities Associated With Favorable Report:
2.6 The Obtained Information based on a Unfavorable Report:
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3.0 Decision Tree Based On The Test Market Study
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Solution D : Optimal Decision Strategy
Decision Tree Representing The Revised Or Posterior Probabilities
CHAPTER VII
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1. Starting the backward pass calculations by computing the expected values (EV)
at nodes 6 to 9 provides the following results:
EV (Node 6) = 0.099(-20) + 0.394(40) + 0.507(100)
= 64.48
EV (Node 7) = 0.099(10) + 0.394(45) + 0.507(70)
= 54.21
EV (Node 8) = 0.488(-20) + 0.326(40) + 0.186(100)
= 21.88
EV (Node 9) = 0.488(10) + 0.326(45) + 0.186(70)
= 32.57
2. Now we move to decision nodes 3 and 4. For each of these nodes, we select the
decision alternative branch that leads to the best expected value.
3. At node 3 we have the choice of the decision alternative d1 with EV (Node 6) =
64.48 , the decision alternative d2 with EV (Node7) = 54.21
4. Thus we select the decision alternative d1 with the large value and so the
expected value at node 3 becomes : EV (Node 3) = 64.48
5. For node 4, we select the best expected value from nodes 8 and 9. The best
decision alternative is d2 that give EV (Node 9) = 32.57 instead of EV (Node 8) = 21.88
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6. So the expected value at node 4 becomes : EV (Node 4) = 32.57
7. The expected value at node 2 can now be computed as follows: