Example 1: **Furniture store opening special**

As part of their opening special, a furniture store has promised to give away at least 40 prizes with a total value of at least $4000. They intend to give away kettles and toasters. They decide there will be at least 10 units of each prize. A kettle costs the company $120 and a toaster costs $100.

Determine how many of each prize will represent the cheapest option for the company. Calculate how much this combination of kettles and toasters will cost.

Use a suitable strategy to organise the information and solve the problem.

Solution:

In this situation there are two variables that we need to consider: let the number of kettles produced be *K* and let the number of toasters produced be *T*.

Write down a summary of the information given in the problem so that we consider all the different components in the situation.

minimum number for K =10 minimum number for T =10 cost of K =$120 cost of T =$100 minimum number of prizes =40 minimum total value of prizes =$4000 |

Use the summary to draw up a table of the number of kettles and toasters that are needed. We will consider multiples of 5 to simplify the calculations:

We need to have at least 40 kettles and toasters together.

With this limitation, we are able to eliminate some of the combinations in the table where *K+T*≥40:

These combinations have been excluded as possible answers.

We can express the minimum cost as: *C*=120(*K*)+100(*T*).

By substituting the different combinations for *K* and *T*, we can find the value that gives the minimum cost.

We are looking for a pair of values that give the minimum cost. We can easily check the non—multiples of 5 for kettles and toasters to confirm that 10 kettles and 30 toasters meet the minimum cost.

For example if 11 kettles and 29 toasters are given away, the cost will be $4220. And if 12 kettles and 28 toasters are given away, the cost will be $4340.

The minimum cost of $4000 can be obtained if 10 kettles and 30 toasters are given away. This will result in a cost of $4200.

This is the satisfying graph.

Example 2 (**Worked Example**): **Prizes**!

As part of their opening specials, a furniture store has promised to give away at least 40 prizes with a total value of at least $2,000. The prizes are kettles and toasters.

1. If the company decides that there will be at least 10 of each prize, write down two more inequalities from these constraints.

2. If the cost of manufacturing a kettle is $60 and a toaster is $50, write down an objective function *C* which can be used to determine the cost to the company of both kettles and toasters.

3. Sketch the graph of the feasibility region that can be used to determine all the possible combinations of kettles and toasters that honour the promises of the company.

4. How many of each prize will represent the cheapest option for the company?

5. How much will this combination of kettles and toasters cost?

Solution:

Step 1: **Identify the decision variables**

Let the number of kettles be *x* and the number of toasters be *y* and write down two constraints apart from ask *x*≥0 and *y*≥0 that must be adhered to.

Step 2: **Write constraint equations**

Since there will be at least 10 of each prize we can write:

*x*≥10

and

*y*≥10

Also the store has promised to give away at least 40 prizes in total. Therefore:

*x*+

*y*≥40

Step 3:

**Write the objective function**

The cost of manufacturing a kettle is $60 and a toaster is $50. Therefore the cost the total cost

*C*is:

*C*=60

*x*+50

*y*

Step 4:

**Sketch the graph of the feasible region**

Step 5:

**Determine vertices of feasible region**

From the graph, the coordinates of vertex A are (3,1) and the coordinates of vertex B are (1,3).

Step 6:

**Calculate cost at each vertex**

At vertex A, the cost is:

*C*=60

*x*+50

*y*

=60(30)+50(10)

=1800+500

=2300

At vertex B, the cost is:

*C*=60

*x*+50

*y*

=60(10)+50(30)

=600+1500

=2100

Step 7:

**Write the final answer**

The cheapest combination of prizes is 10 kettles and 30 toasters, costing the company $2,100.

💎 If a Feasible Region is Unbounded — LP keywords: cost, vitamin, diet 👀