6 Sandie makes tanning lotions which she sells to beauty salons. She makes three different lotions using the same basic ingredients but in different proportions. These lotions are called amber, bronze and copper.
To make one litre of tanning lotion she needs one litre of fluid. This can either be water or water mixed with hempseed oil. One litre of amber lotion uses one litre of water, one litre of bronze lotion uses 0.8 litres of water and one litre of copper lotion uses 0.5 litres of water. Any remainder is made up of hempseed oil. Sandie has 40 litres of water and 7 litres of hempseed oil available.
- By defining appropriate variables \(a , b\) and \(c\), show that the constraint on the amount of water available can be written as \(10 a + 8 b + 5 c \leqslant 400\).
- Find a similar constraint on the amount of hempseed oil available.
The tanning lotions also use two colourants which give two further availability constraints. Sandie wants to maximise her profit, \(\pounds P\). The problem can be represented as a linear programming problem with the initial Simplex tableau below. In this tableau \(s , t , u\) and \(v\) are slack variables.
| \(P\) | \(a\) | \(b\) | \(c\) | \(s\) | \(t\) | \(u\) | \(v\) | RHS |
| 1 | -8 | -7 | -4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 8 | 5 | 1 | 0 | 0 | 0 | 400 |
| 0 | 0 | 2 | 5 | 0 | 1 | 0 | 0 | 70 |
| 0 | 2 | 4 | 1 | 0 | 0 | 1 | 0 | 176 |
| 0 | 5 | 1 | 3 | 0 | 0 | 0 | 1 | 80 |
- Use the initial Simplex tableau to write down two inequalities to represent the availability constraints for the colourants.
- Write down the profit that Sandie makes on each litre of amber lotion that she sells.
- Carry out one iteration of the Simplex algorithm, choosing a pivot from the \(a\) column. Show the operations used to calculate each row.
After a second iteration of the Simplex algorithm the tableau is as given below.
| \(P\) | \(a\) | \(b\) | \(c\) | \(s\) | \(t\) | \(u\) | \(v\) | RHS |
| 1 | 0 | 0 | 14.3 | 0 | 2.7 | 0 | 1.6 | 317 |
| 0 | 0 | 0 | -16 | 1 | -3 | 0 | -2 | 30 |
| 0 | 0 | 1 | 2.5 | 0 | 0.5 | 0 | 0 | 35 |
| 0 | 0 | 0 | -9.2 | 0 | -1.8 | 1 | -0.4 | 18 |
| 0 | 1 | 0 | 0.1 | 0 | -0.1 | 0 | 0.2 | 9 |
- Explain how you know that the optimal solution has been achieved.
- How much of each lotion should Sandie make and what is her maximum profit? Why might the profit be less than this?
- If none of the other availabilities change, what is the least amount of water that Sandie needs to make the amounts of lotion found in part (vii)?