Mathematics IA
Worked Examples
ALGEBRA: OPTIMISATION AND
CONVEX SETS
Produced by the Maths Learning Centre,
The University of Adelaide.
May 1, 2013
The questions on this page have worked solutions and links to videos on
the following pages. Click on the link with each question to go straight to
the relevant page.
Questions
1. See Page 3 for worked solutions.
By drawing sketches, decide if the following sets in R2 are convex or not.
Also decide whether each set is bounded or unbounded.
(a)
(c)
(e)
(g)
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
|
|
|
|
y ≤ x2 }
x + y ≤ 1}
x2 + y 2 ≤ 1}
x2 + y 2 = 0}
(b)
(d)
(f)
(h)
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
|
|
|
|
y ≥ x2 }
x + y = 1}
x2 + y 2 ≥ 1}
x2 + y 2 < 0}
2. See Page 6 for worked solutions.
Sketch the following convex sets and write down any vertices they have.
(a) {(x, y) ∈ R2 | x + y ≤ 3, y + 2x ≥ 2, y ≥ 0}
(b) {(x, y) ∈ R2 | x + y ≥ 3, y + 2x ≥ 2, y ≥ 0}
(c) {(x, y) ∈ R2 | x + y ≥ 3, y + 2x ≥ 2, y ≤ 0}
(d) {(x, y) ∈ R2 | x + y ≤ 3, 2 ≤ y + 2x ≤ 4, y ≥ 0}
(e) {(x, y) ∈ R2 | 2 ≤ y + 2x ≤ 4}
3. See Page 9 for worked solutions.
Let v = (3, 2). Show that the set {u ∈ R2 | u · v ≤ 9} is a convex set.
4. See Page 10 for worked solutions.
Solve the following problem graphically: Maximise 3x1 + 5x2 subject to
x1 + 2x2 ≤ 16, 2x1 + x2 ≤ 12, x1 + 2x2 ≥ 2, x1 ≥ 0, x2 ≥ 0.
1
5. See Page 11 for worked solutions.
Consider the convex region C in R4 defined by x1 + x2 + x3 ≤ 6,
x2 + x3 + x4 ≥ 2 and x1 , x2 , x3 , x4 ≥ 0. Use slack variables to find the
vertices of C.
6. See Page 13 for worked solutions.
A small company manufactures and sells two models of lamps FUNKY
and SNAZZY, the profit per lamp being $15 and $10 respectively. They
have two workers who make the lamps. Brian assembles the lamps and
takes 20 minues to assemble each FUNKY lamp, and 30 minutes to assemble each SNAZZY lamp. Bruce paints the lamps and takes 20 minutes
to paint each FUNKY lamp and 10 minutes to paint each SNAZZY lamp.
Brian and Bruce can work at most 100 hours and 80 hours per month
respectively. How many of each lamp should be made each month, assuming that every lamp that is made can be sold?
7. See Page 16 for worked solutions.
Maximise F = 4x − y + 5z subject to the constraints x + y + z = 7,
3x + 4y + 6z ≤ 36, 6x − y − 15z ≤ 0 and x, y, z ≥ 0.
8. See Page 18 for worked solutions.
Formulate the following problem as a linear program:
A company producing canned mixed fruit has a stock of 10 000 kg of pears,
12 000 kg of peaches and 8000 kg of cherries. The company produces
three fruit mixtures, which it sells in 1-kg cans. The first mixture is half
pears and half peaches and sells for $1.20. The second mixture has equal
amounts of the three fruits and sells for $1.70. The third mixture is threequarters peaches and one-quarter cherries and sells for $2.10. How many
cans of each mixture should be produced to maximise the return on the
current stock?
2
1. Click here to go to question list.
By drawing sketches, decide if the following sets in R2 are convex or not.
Also decide whether each set is bounded or unbounded.
(a)
(c)
(e)
(g)
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
|
|
|
|
y ≤ x2 }
x + y ≤ 1}
x2 + y 2 ≤ 1}
x2 + y 2 = 0}
(b)
(d)
(f)
(h)
Link to video on YouTube
3
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
{(x, y) ∈ R2
|
|
|
|
y ≥ x2 }
x + y = 1}
x2 + y 2 ≥ 1}
x2 + y 2 < 0}
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5
2. Click here to go to question list.
Sketch the following convex sets and write down any vertices they have.
(a) {(x, y) ∈ R2 | x + y ≤ 3, y + 2x ≥ 2, y ≥ 0}
(b) {(x, y) ∈ R2 | x + y ≥ 3, y + 2x ≥ 2, y ≥ 0}
(c) {(x, y) ∈ R2 | x + y ≥ 3, y + 2x ≥ 2, y ≤ 0}
(d) {(x, y) ∈ R2 | x + y ≤ 3, 2 ≤ y + 2x ≤ 4, y ≥ 0}
(e) {(x, y) ∈ R2 | 2 ≤ y + 2x ≤ 4}
Link to video on YouTube
6
7
8
3. Click here to go to question list.
Let v = (3, 2). Show that the set {u ∈ R2 | u · v ≤ 9} is a convex set.
Link to video on YouTube
9
4. Click here to go to question list.
Solve the following problem graphically: Maximise 3x1 + 5x2 subject to
x1 + 2x2 ≤ 16, 2x1 + x2 ≤ 12, x1 + 2x2 ≥ 2, x1 ≥ 0, x2 ≥ 0.
Link to video on YouTube
10
5. Click here to go to question list.
Consider the convex region C in R4 defined by x1 + x2 + x3 ≤ 6,
x2 + x3 + x4 ≥ 2 and x1 , x2 , x3 , x4 ≥ 0. Use slack variables to find the
vertices of C.
Link to video on YouTube
11
12
6. Click here to go to question list.
A small company manufactures and sells two models of lamps FUNKY
and SNAZZY, the profit per lamp being $15 and $10 respectively. They
have two workers who make the lamps. Brian assembles the lamps and
takes 20 minues to assemble each FUNKY lamp, and 30 minutes to assemble each SNAZZY lamp. Bruce paints the lamps and takes 20 minutes
to paint each FUNKY lamp and 10 minutes to paint each SNAZZY lamp.
Brian and Bruce can work at most 100 hours and 80 hours per month
respectively. How many of each lamp should be made each month, assuming that every lamp that is made can be sold?
Link to video on YouTube
NOTE: There is an error in the solution here! I said that 80 hours times
60 minutes was 4000 minutes, when in fact it’s 4800 minutes! Oops!
Consequently the answer given is wrong. Sorry! Hopefully you can still
learn how the process works until I fix the actual answer.
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15
7. Click here to go to question list.
Maximise F = 4x − y + 5z subject to the constraints x + y + z = 7,
3x + 4y + 6z ≤ 36, 6x − y − 15z ≤ 0 and x, y, z ≥ 0.
Link to video on YouTube
16
17
8. Click here to go to question list.
Formulate the following problem as a linear program:
A company producing canned mixed fruit has a stock of 10 000 kg of pears,
12 000 kg of peaches and 8000 kg of cherries. The company produces
three fruit mixtures, which it sells in 1-kg cans. The first mixture is half
pears and half peaches and sells for $1.20. The second mixture has equal
amounts of the three fruits and sells for $1.70. The third mixture is threequarters peaches and one-quarter cherries and sells for $2.10. How many
cans of each mixture should be produced to maximise the return on the
current stock?
Link to video on YouTube
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