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
8.0
Solution E : The expected value of the market research
15
19
information
9.0
Solution F : The efficiency of the information
20
10.0 Conclusion
21
11.0 References
23
3
CHAPTER
II
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.
4
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.
5
CHAPTER
III
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?
6
CHAPTER
IV
Solution A : Decision Based On A Decision Tree Diagram
Expected Value =  ( Profit
1.
EV ( d1 ) =
EV ( d2 ) =
Profit’s probability )
-20(0.35) + 40(0.35) + 100(0.30)
=
2.

37
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
7
CHAPTER
V
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
8
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.
9
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.
10
CHAPTER
VI
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( Fs1) = P( F | s1 )  P( s1 ) = 0.10  0.35 = 0.035
P( Fs2) = P( F | s2 )  P( s2 ) = 0.40  0.35 = 0.140
P( Fs3) = 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 ( Fsj )
= P ( Fs1) + P ( Fs2 ) + P ( Fs3 )
= 0.035 + 0.140 + 0.180
= 0.355
11
1.5 The Posterior Probabilities Associated With Favorable Report:
P(s1 | F ) 
P ( F  s 1 ) 0.035

 0.099
P(F)
0.355
P(s 2 | F ) 
P ( F  s 2 ) 0.140

 0.394
P(F)
0.355
P(s 3 | F ) 
P ( F  s 3 ) 0.180

 0.507
P(F)
0.355
1.6 The Obtained Information based on a Favorable Report:
12
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( Us1) = P( U | s1 )  P( s1 ) = 0.90  0.35 = 0.315
P( Us2) = P( U | s2 )  P( s2 ) = 0.60  0.35 = 0.210
P( Us3) = 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 ( Usj )
= P ( Us1 ) + P ( Us2 ) + P ( Us3 )
= 0.315 + 0.210 + 0.120
= 0.645
13
2.5
2.6
The Posterior Probabilities Associated With Favorable Report:
P(s1 | U ) 
P ( U  s1 ) 0.315

 0.488
P(U)
0.645
P(s 2 | U ) 
P ( U  s 2 ) 0.210

 0.326
P(U)
0.645
P(s 3 | U ) 
P ( U  s 3 ) 0.120

 0.186
P(U)
0.645
The Obtained Information based on a Unfavorable Report:
14
3.0 Decision Tree Based On The Test Market Study
15
CHAPTER
VII
Solution D : Optimal Decision Strategy
Decision Tree Representing The Revised Or Posterior Probabilities
16
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 d 1 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 d 2 that give EV (Node 9) = 32.57 instead of EV (Node 8) = 21.88
17
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:
EV (Node 2)
= (0.355)(EVNode 3) + (0.645)(EV Node 4 )
= 0.355(64.48) + 0.645(32.57)
= 43.89805
8.
At node 5, the decision alternative d 2 , to purchase the component, with an
expected value of 40.25 is the optimal decision..Hence, EV (Node 5) = 40.25.
9.
Finally, the decision can be made at decision node 1 by selecting the best
expected values from nodes 2 and 5. This action leads to the decision alternative
to conduct the market research study, which provides an overall expected value,
EVNode 1 = 43.89805
10. Therefore, the optimal decision strategy for this problem is to go for the market
research study before making the final decision .
11. The decision alternative d 1 (manufacture the component part at its plant) is
carried out if the study reports a favorable condition.
12. If the study reports an unfavorable condition, the company is best to carry out
decision alternative d2 (purchase the component part from a supplier).
13. The expected profit is $43890.5
18
14. The overview of the expected value of each node
– 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
19
CHAPTER
VIII
Solution E : The Expected Value Of The Market Research Information
1.
In decision theory, the expected value of sample information (EVSI) is the
price that one would be willing to pay in order to gain access to a sample from
the distribution about which the prediction has to be made.
2.
Expected Value of Sample Information ( Market Research Information ), EVSI is
the difference between the Expected Value best decision with market research
information and Expected Value of best decision without market research
information.
EVSI = | EV (market study ) –
EV ( without market study ) |
= | EV( node2 ) –
EV ( node5 ) |
= | 43.89805 –
40.25 |
= 3.65
3.
Therefore, conducting the market research study will adds $3650 to the Gorman
Manufacturing Company expected value. In other words, the company is
suggested pay only up to $3650 to conduct the market study for acquiring the
sample information.
20
CHAPTER
IX
Solution F : The Efficiency Of The Information
1.
An efficiency measure is to express the value of the market study information.
2.
Low efficiency ratings for sample information might lead the decision maker to
look for other types of information.
3.
Whereas, high Efficiency ratings indicate that the sample is almost as good as
perfect information.
4.
Efficiency is the ratio between Expected Value of Sample Information, EVSI and
Expected Value of Perfect Information, EVPI times with 100 percent.
5.
From the calculation, we conclude that the information from the market study is
40.55% as efficiency as perfect information.
21
CHAPTER
X
CONCLUSION
Initially, the company does not run market study to obtain the component part.
From the decision tree diagram, the expected value of profit should be higher if the
company decides to purchase the component parts. The expected value of profit for
purchasing is $40,250 which is higher than the expected value of profit for
manufacturing $37,000.Hence the Gorman Manufacturing Company should decide to
purchase the component parts.
Calculating the Expected Value of Perfect Information which shows $9000, the
company is willing to pay not more than $9000 to obtain sample information from
market study. In this case, the EVPI is $9000 which means that if the company do or
buy the market study info, it will earn extra $9000. 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.
By using Bayer’s Theorem, the probability that the market study report will be
favorable is 0.355. The probability that the market research report will be unfavorable
is 0.645. the company is taking risk since the probability of favorable is lower than
unfavorable.
22
The optimal decision strategy is to run a market study. If the report is favorable,
the company should manufacture the component parts. Otherwise, unfavorable report
will lead the company to purchase the component part for better profit.
.
The company is suggested pay only up to $3650 to conduct the market study for
acquiring the sample information. The information from the market study is 40.55%
as efficiency as perfect information.
23
CHAPTER
XI
REFERENCES
1.
Douglas C. Montgomery, Geroge C.Runger. Applied Statistics and Probability
for Enginners, 4 th edition. John Wiley & Sons ( Asia ) Pte Ltd, 2007
2.
Rachel Brown. Expected Value Method in Decision Making.
http://www.freequality.org/sites/www.../Expected%20Value%20Method.ppt
November, 2005
3.
Mike Middleton. Value of Information in Decision Trees.
http://www.treeplan.com/chapters/19_decan_20071029_1042.pdf
October,2007
4.
Kenneth C. Levine. Expected Value of Sample Information.
http://www2.gsu.edu/~wwwkcl/MGS3100/MGS3100_Slides8c.ppt
August, 2005.