Mathematics IAWorked Examples

ALGEBRA: OPTIMISATION ANDCONVEX 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 onthe following pages. Click on the link with each question to go straight tothe 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) {(x, y) ∈ R2 | y ≤ x2} (b) {(x, y) ∈ R2 | y ≥ x2}(c) {(x, y) ∈ R2 | x + y ≤ 1} (d) {(x, y) ∈ R2 | x + y = 1}(e) {(x, y) ∈ R2 | x2 + y2 ≤ 1} (f) {(x, y) ∈ R2 | x2 + y2 ≥ 1}(g) {(x, y) ∈ R2 | x2 + y2 = 0} (h) {(x, y) ∈ R2 | x2 + y2 < 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 tox1 + 2x2 ≤ 16, 2x1 + x2 ≤ 12, x1 + 2x2 ≥ 2, x1 ≥ 0, x2 ≥ 0.

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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 thevertices of C.

6. See Page 13 for worked solutions.A small company manufactures and sells two models of lamps FUNKYand SNAZZY, the profit per lamp being $15 and $10 respectively. Theyhave two workers who make the lamps. Brian assembles the lamps andtakes 20 minues to assemble each FUNKY lamp, and 30 minutes to as-semble each SNAZZY lamp. Bruce paints the lamps and takes 20 minutesto 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 monthrespectively. How many of each lamp should be made each month, as-suming 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 producesthree fruit mixtures, which it sells in 1-kg cans. The first mixture is halfpears and half peaches and sells for $1.20. The second mixture has equalamounts of the three fruits and sells for $1.70. The third mixture is three-quarters peaches and one-quarter cherries and sells for $2.10. How manycans of each mixture should be produced to maximise the return on thecurrent stock?

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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) {(x, y) ∈ R2 | y ≤ x2} (b) {(x, y) ∈ R2 | y ≥ x2}(c) {(x, y) ∈ R2 | x + y ≤ 1} (d) {(x, y) ∈ R2 | x + y = 1}(e) {(x, y) ∈ R2 | x2 + y2 ≤ 1} (f) {(x, y) ∈ R2 | x2 + y2 ≥ 1}(g) {(x, y) ∈ R2 | x2 + y2 = 0} (h) {(x, y) ∈ R2 | x2 + y2 < 0}Link to video on YouTube

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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

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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

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4. Click here to go to question list.Solve the following problem graphically: Maximise 3x1 + 5x2 subject tox1 + 2x2 ≤ 16, 2x1 + x2 ≤ 12, x1 + 2x2 ≥ 2, x1 ≥ 0, x2 ≥ 0.

Link to video on YouTube

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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 thevertices of C.

Link to video on YouTube

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6. Click here to go to question list.A small company manufactures and sells two models of lamps FUNKYand SNAZZY, the profit per lamp being $15 and $10 respectively. Theyhave two workers who make the lamps. Brian assembles the lamps andtakes 20 minues to assemble each FUNKY lamp, and 30 minutes to as-semble each SNAZZY lamp. Bruce paints the lamps and takes 20 minutesto 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 monthrespectively. How many of each lamp should be made each month, as-suming 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 times60 minutes was 4000 minutes, when in fact it’s 4800 minutes! Oops!Consequently the answer given is wrong. Sorry! Hopefully you can stilllearn how the process works until I fix the actual answer.

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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

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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 producesthree fruit mixtures, which it sells in 1-kg cans. The first mixture is halfpears and half peaches and sells for $1.20. The second mixture has equalamounts of the three fruits and sells for $1.70. The third mixture is three-quarters peaches and one-quarter cherries and sells for $2.10. How manycans of each mixture should be produced to maximise the return on thecurrent stock?

Link to video on YouTube

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