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EMGT 501
Mid-Term Exam
Due Day: Oct. 20 (Noon)
Note:
(a) Do not send me after copying your computer
results. Answer formulations and these solutions.
(b) Put your mailing address so that I will be able to
return your exam result via US postal service.
(c) Answer on a PPS that has many slides.
1. Slim-Down Manufacturing makes a line of nutritionally
complete, weight-reduction beverages. One of their products
is a strawberry shake which is designed to be a complete meal.
The strawberry shake consists of several ingredients. Some
information about each of these ingredients is given below.
Calories Total
Vitamin
from Fat Calories Content Thickeners Cost
Ingredient
(per tbsp) (per tbsp) (mg/tbsp) (mg/tbsp) (¢/tbsp)
Strawberry flavoring
1
50
20
3
10
Cream
75
100
0
8
8
Vitamin supplement
0
0
50
1
25
Artificial sweetener
0
120
0
2
15
Thickening agent
30
80
2
25
6
The nutritional requirements are as follows. The beverage must
total between 380 and 420 calories (inclusive). No more than 20
percent of the total calories should come from fat. There must be
at least 50 milligrams (mg) of vitamin content. For taste reasons,
there must be at least 2 tablespoons (tbsp) of strawberry
flavoring for each tablespoon of artificial sweetener. Finally, to
maintain proper thickness, there must be exactly 15 mg of
thickeners in the beverage.
Management would like to select the quantity of each
ingredient for the beverage which would minimize cost while
meeting the above requirements.
(a) Formulate a linear programming model for this problem.
(b) Solve this model by your computer.
2. Consider the following problem.
Maximize
Z  4x1  3x2  x3  2x4 ,
subject to
4 x1  2 x2  x3  x4   5
3x1  x2  2 x3  x4   4
x1  0, x2  0, x3  0, x4  0.
Letx5 and x6 denote the slack variables for the respective
constraints. After you apply the simplex method, a portion of
the final simplex tableau is as follows:
Basic
Variable
Z
x2
x4
Coefficient of:
Eq.
(0)
(1)
(2)
Z
1
0
0
x1 x2 x3 x4 x5 x6
1
1
-1
1
-1
2
(a) Solve the problem
(b) What is B-1 ? How about B-1b and CBB-1b ?
Right
Side
3. The Lockhead Aircraft Co. is ready to begin a project to develop
a new fighter airplane for the U.S. Air Force. The company’s
contract with the Department of Defense calls for project
completion within 100 weeks, with penalties imposed for late
delivery.
The project involves 10 activities (labeled A, B, …, J), where
their precedence relationships are shown in the following project
network.
A
C
J
START
FINISH
F
H
B
E
I
D
G
Using the PERT three-estimate approach, the usual three
estimates of the duration of each activity have been obtained as
given below.
Activity
A
B
C
D
E
F
G
H
I
J
Optimistic
Estimate
28 weeks
22 weeks
26 weeks
14 weeks
32 weeks
40 weeks
12 weeks
16 weeks
26 weeks
12 weeks
Most Likely
Estimate
32 weeks
28 weeks
36 weeks
16 weeks
32 weeks
52 weeks
16 weeks
20 weeks
34 weeks
16 weeks
Pessimistic
Estimate
36 weeks
32 weeks
46 weeks
18 weeks
32 weeks
74 weeks
24 weeks
26 weeks
42 weeks
30 weeks
(a) Find the estimate of the mean and variance of the duration of
each activity.
(b) Find the mean critical path.
(c) Find the approximate probability that the project will finish
within 100 weeks.
(d) Is the approximate probability obtained in part (c) likely to
be higher or lower than the true value?
4. Consider the game having the following payoff table.
Strategy
Player 1
1
2
3
4
1
1
2
0
-4
2
-3
3
4
0
Player 2
3
2
0
-1
-2
4
-2
3
-3
2
5
1
-2
2
1
(a) Formulate the problem to find an optimal mixed strategy
according to the minimax criterion as a linear programming
problem.
(b) Show its dual formulation.
(c) Use the simplex method to find these optimal mixed strategies
of the primal and dual models.
5. The Hit-and-Miss Manufacturing Company produces items that
have a probability p of being defective. These items are produced
in lots of 150. Past experience indicates that p for an entire lot is
either 0.05 or 0.25. Furthermore, in 80 percent of the lots
produced, p equals 0.05 (so p equals 0.25 in 20 percent of the lots).
These items are then used in an assembly, and ultimately their
quality is determined before the final assembly leaves the plant.
Initially the company can either screen each item in a lot at a cost
of $10 per item and replace defective items or use the items
directly without screening. If the latter action is chosen, the cost of
network is ultimately $100 per defective item. Because screening
requires scheduling of inspectors and equipment, the decision to
screen or not screen must be made 2 days before the screening is to
take place. However, one item can be taken from the lot and sent to
a laboratory for inspection, and its quality (defective or
nondefective) can be reported before the screen/no screen decision
must be made. The cost of this initial inspection is $125.
(a) Develop a decision analysis formulation of this problem by
identifying the alternative actions, the states of nature, and the
payoff table if the single item is not inspected in advance.
(b) Assuming the single item is not inspected in advance, use
Bayes’ decision rule to determine which decision alternative
should be chosen.
(c) Find EVPI. Does this answer indicate that consideration should
be given to inspecting the single item in advance?
(d) Assume now that the single item is inspected in advance. Find
the posterior probabilities of the respective states of nature for
each of the two possible outcomes of this inspection.
(e) Find EVE. Is inspecting the single item worthwhile?
(f) Determine the optimal policy.
EMGT 501
HW #5
Answer
15.3-11
(a)
State of Nature
Alternative
Success
Unsuccess
Depelop new product
1,500,000 -1,800,000
Don't develop new product
0
0
Prior probabilities
0.667
0.333
(b)
State of Nature
Expected
Alternative
Success
Unsuccess Payoff
Depelop new product
1,500,000 -1,800,000 400,000 Maximum
Don't develop new product
0
0
0
Prior probabilities
0.667
0.333
Choose to develop new product (expected payoff
is $400,000).
(c)
State of Nature
Alternative
Success
Unsuccess
Depelop new product
1,500,000 -1,800,000
Don't develop new product
0
0
Prior probabilities
0.667
0.333
Maximum Payoff
1,500,000
0
Expected Payoff with Perfect Information = 1,000,000
EVPI = EP (with perfect info) - EP (without more info)
= 1,000,000 - 400,000 = $600,000
This indicates that consideration should be given to
conducting the market survey.
(d)
Data:
State of
Nature
Success
Unsuccess
P(Finding | State)
Prior
Finding
Probability Success Unsuccess
0.6666667
0.8
0.2
0.3333333
0.3
0.7
Posterior
Probabilities
Finding
Success
Unsuccess
P(State | Finding)
State of Nature
P(Finding) Success
Unsuccess
0.6666667 0.842105 0.157894737
0.3333333 0.363636 0.636363636
(e)
E[ p (a1 , | pred.succ.)]
 0.8421(1,500,000)  0,1579(1,800,000)
 $979,000
E[ p (a2 , | pred.succ.)]  0
E[ p (a1 , | pred.no succ.)]
 0.3636(1,500,000)  0.6364(1,800,000)
 $  600,000
E[ p (a2 , | pred.no succ.)]  0
Optimala1 if pred.success, a2 if not.
p(pred.success)
 p(pred.success | 1)p(1)  p(pred.success | 2 )p(2 )
 3  0.313   0.633
 0.8 2
Expect edpayoffgiven info
 0.633(979,000)  0.367(0)  $620,000
So EVE  620,000 400,000  $220,000
Cost of survey  $300,000  $214,000
So opt imalst rat egyis :
Do not conduct survey,marketproduct .