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ESE304  Introduction to Optimization (Homework #1)
Fall Semester, 2010
M. Carchidi
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Problem #1 (20 points)  Maximization of Total Revenue
Farmer Jones must determine how many acres of corn and wheat to plant
this year. An acre of wheat yields 25 bushels of wheat and requires 10 hours
of labor per week. An acre of corn yields 10 bushels of corn and requires 4
hours of labor per week. All wheat can be sold at $4 a bushel, and all corn
can be sold for $3 a bushel. Seven (7) acres of land for the year, and 40
hours per week of labor are available. Government regulations require that
at least 30 bushels of corn must be produced during the current year. Using
x
1
as the number of acres of corn planted and
x
2
as the number of acres of
wheat planted, formulate an LP whose solution will tell Farmer Jones how
to maximize total revenue from wheat and corn, and solve your problem
using graphical methods.
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Problem #2 (20 points)  Convex Sets
Let
x
1
,
x
2
,
x
3
,
...
,
x
m
be a set of
m
elements in the set
R
n
and let
A
=
(
m
X
k
=1
c
k
x
k
¯
¯
¯
¯
¯
c
1
,c
2
,c
3
,...,c
m
≥
0
and
m
X
k
=1
c
k
=1
)
.
Prove that
A
is a convex set.
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Problem #3 (20 points)  A Diet Problem
Peg and Al Fundy have a limited food budget, so Peg is trying to feed the
family as cheaply as possible. However, she still wants to make sure her
family members meet their daily nutritional requirements. Peg can buy two
foods. Food 1 sells for $7 per pound, and each pound contains 3 units of
vitamin A and 1 unit of vitamin C. Food 2 sells for $1 per pound, and each
pound contains 1 unit of each vitamin. Each day, the family members needs
at least 12 units of vitamin A and 6 units of vitamin C.
a.)
(10 points)
Verify that Peg should purchase 12 units of Food 2 each day,
and thus over satisfy the vitamin C requirements by 6 units.
b.)
(10 points)
Al has put his foot down and demanded that Peg ful
f
ll the fam
ily’s daily nutritional requirements exactly by obtaining precisely 12 units
of vitamin A and 6 units of vitamin C. The optimal solution to the new
problem will involve ingesting less vitamin C, but it will be more expensive.
Why?
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Problem #4 (20 points)  Minimization Of Labor Costs
During each 4hour period, the Smalltown police force requires (at least)
the following number of onduty police o
ﬃ
ces.
Time
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This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.
 Winter '11
 MichaelA.Carchidi

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