269 EMSE - Elements of Problem Solving and Decision Making
EXTRA PROBLEM 6:
SOLVING DECISION TREES
Read the following decision problem and answer the questions below.
A manufacturer produces items that have a probability p of being defective.
These items are formed into batches of 150. Past experience indicates that some
are of good quality (i.e. p=0.05) and others are of bad quality (i.e. p=0.25).
Furthermore, 80% of the batches produced are of good quality and 20% of the
batches are of bad quality. These items are then used in an assembly, and
ultimately their quality is determined before the final assembly leaves the plant.
The manufacturer can either screen each item in a batch and replace defective
items at a total average cost of $10 per item or use the items directly without
screening. If the latter action is chosen, the cost of rework is ultimately $100 per
defective item. For these data, the costs per batch can be calculated as follows:
Screen
Do not Screen
p = 0.05
$1500
$750
p = 0.25
$1500
$3750
Because screening requires scheduling of inspectors and equipment, the
decision to screen or not screen must be made 2 days before the potential
screening takes place. However, the manufactures may take one item taken from
a batch and sent it to a laboratory, and the test results (defective or nondefective) can be reported before the screen/no-screen decision must be made.
After the laboratory test, the tested item is returned to its batch. The cost of this
initial inspection is $125. Also note that the probability that a random sample item
is defective is
0.8 * 0.05 + 0.2 * 0.25 = 0.09,
and the probability that an item in a lot is of good quality given a randomly
sampled item is defective is 0.444 and the probability that an item in a lot is of
good quality given a randomly sampled item is not defective is 0.835. wants The
manufactures to minimize his cost.
A. Derive the cost figures in the table above. Clearly show your calculations
Instructor: Dr. J. R. van Dorp
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EMSE 269 - Elements of Problem Solving and Decision Making
B. What Law of Probability was used in deriving. Provide an Explanation
Pr(Sampled Item is Defective) = 0.09
C. Show that:
Pr(Good Quality|Sampled Item is Defective) = 0.444
Pr(Good Quality|Sampled Item is Not Defective) = 0.835
Instructor: Dr. J. R. van Dorp
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EMSE 269 - Elements of Problem Solving and Decision Making
D. Calculate and show your calculations:
Pr(Bad Quality|Sampled Item is Defective) = ?
Pr(Bad Quality|Sampled Item is Not Defective) = ?
Instructor: Dr. J. R. van Dorp
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EMSE 269 - Elements of Problem Solving and Decision Making
E. Model the decision problem in a decision tree and fill in ALL the details. Solve
the tree using EMV and clearly show your calculations in the tree.
Instructor: Dr. J. R. van Dorp
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EMSE 269 - Elements of Problem Solving and Decision Making
F. How many cumulative risk profiles can be drawn for the decision tree under E?
Provide an Explanation.
Instructor: Dr. J. R. van Dorp
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EMSE 269 - Elements of Problem Solving and Decision Making
G. Solve the tree under E using EMV and clearly show your calculations in the
tree under E
Instructor: Dr. J. R. van Dorp
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EMSE 269 - Elements of Problem Solving and Decision Making
H. Describe in words the optimal decision strategy.
I. Draw the Cumulative Risk Profile for each alternative of the immediate
decision, taking optimal decisions from thereon. What can you conclude with
respect to dominance considerations? (Hint: You should be drawing 2
Cumulative Risk Profiles).
Instructor: Dr. J. R. van Dorp
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EMSE 269 - Elements of Problem Solving and Decision Making
Instructor: Dr. J. R. van Dorp
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