# Linear Programming Problems and Solutions

### Solutions

1A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of 20 m^{3} and a non-refrigerated capacity of 40 m^{3} while Type B has the same overall volume with equal sections for refrigerated and non-refrigerated stock. A grocer needs to hire trucks for the transport of 3,000 m^{3} of refrigerated stock and 4 000 m^{3} of non-refrigerated stock. The cost per kilometer of a Type A is $30, and $40 for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

2A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is $800 and $600 for the small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.

3A store wants to liquidate 200 of its shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a pair of pants which will sell for $30. Offer B is a package of three shirts and a pair of pants, which will sell for $50. The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many packages of each do they have to sell to maximize the money generated from the promotion?

## 1

A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of 20 m^{3} and a non-refrigerated capacity of 40 m^{3} while Type B has the same overall volume with equal sections for refrigerated and non-refrigerated stock. A grocer needs to hire trucks for the transport of 3,000 m^{3} of refrigerated stock and 4,000 m^{3} of non-refrigerated stock. The cost per kilometer of a Type A is $30, and $40 for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

1**Choose the unknowns.**

**x = Type A trucks**

**y = Type B trucks**

2**Write the objective function**.

**f(x,y) = 30x + 40y**

3Write the constraints as a system of inequalities.

A | B | Total | |
---|---|---|---|

Refrigerated | 20 | 30 | 3 000 |

Non-refrigerated | 40 | 30 | 4 000 |

**20x + 30y ≥ 3 000**

**40x + 30y ≥ 4 000**

**x ≥ 0**

**y ≥ 0**

4 **Find the set of feasible solutions that graphically represent the constraints.**

5 **Calculate the coordinates of the vertices** from the compound of feasible solutions.

6 **Calculate the value of the objective function at each of the vertices** to determine which of them has the maximum or minimum values.

f(0, 400/3) = 30** · **0 + 40** · ** 400/3 = 5,333.332

f(150, 0) = 30** · **150 + 40 **· ** 0 = 4,500

As x and y must be natural numbers round the value of y.

f(50, 67) = 30** · **50 + 40** ·**67 = 4,180

By default, we see what takes the value x to y = 66 in the equation 20x + 30y = 3,000. x = 51 which it is within the feasible solutions.

f(51, 66) = 30** · **51 + 40** · **66 = 4,170

The minimum cost is $4,170. To achieve this 51 trucks of Type A and 66 trucks of Type B are needed.

## 2

A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is $800 and $600 for the small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.

1**Choose the unknowns.**

x = small buses

y = big buses

2**Write the objective function**.

**f(x, y) = 600x + 800y**

3Write the constraints as a system of inequalities.

**40x + 50y ≥ 400**

**x + y ≤ 9**

**x ≥ 0**

**y ≥ 0**

4 **Find the set of feasible solutions that graphically represent the constraints.**

5 **Calculate the coordinates of the vertices** from the compound of feasible solutions.

6 **Calculate the value of the objective function at each of the vertices** to determine which of them has the maximum or minimum values.

f(0, 8) = 600 ** · ** 0 + 800 ** · ** 8 = $6,400

f(0, 9) = 600 ** · ** 0 + 800 ** ·** 9 = $7,200

f(5, 4) = 6 00 ** · ** 5 + 800 ** ·** 4 = $6,200 € **Minimum**

The minimum cost is $6,200. This is acheived with **4 large** and** 5 small buses.**

## 3

A store wants to liquidate 200 of its shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a pair of pants which will sell for $30. Offer B is a package of three shirts and a pair of pants, which will sell for $50. The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many packages of each do they have to sell to maximize the money generated from the promotion?

1**Choose the unknowns.**

x = number of packages of Offer A

y = number of packages of Offer B

2**Write the objective function**.

**f(x, y) = 30x + 50y**

3Write the constraints as a system of inequalities.

A | B | Minimal | |
---|---|---|---|

Shirts | 1 | 3 | 200 |

Pants | 1 | 1 | 100 |

**x + 3y ≤ 200**

**x + y ≤ 100**

**x ≥ 20**

**y ≥ 10**

4 **Find the set of feasible solutions that graphically represent the constraints.**

5 **Calculate the coordinates of the vertices** from the compound of feasible solutions.

6 **Calculate the value of the objective function at each of the vertices** to determine which of them has the maximum or minimum values.

f(x, y) = 30** · **20 + 50** · ** 10 = $1,100

f(x, y) = 30** · **90 + 50** · ** 10 = $3,200

f(x, y) = 30** · **20 + 50** · ** 60 = $3,600

f(x, y) = 30** · **50 + 50** · ** 50 = $4,000 **Maximum**

**50 packages** of each offer generates a maximum amount of **$4,000 in sales.**