3 Three products, A, B and C are to be made.
Three supplements are included in each product. Product A has 10 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z .
Product B has 5 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 3 g per kg of supplement Z .
Product C has 12 g per kg of supplement \(\mathrm { X } , 7 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z .
There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .
Product A will sell at \(\pounds 7\) per kg and costs \(\pounds 3\) per kg to produce. Product B will sell at \(\pounds 5\) per kg and costs \(\pounds 2\) per kg to produce. Product C will sell at \(\pounds 4\) per kg and costs \(\pounds 3\) per kg to produce.
The profit is to be maximised.
- Explain how the initial feasible tableau shown in Fig. 3 models this problem.
\begin{table}[h]
| P | a | b | c | s 1 | s 2 | s 3 | RHS |
| 1 | - 4 | - 3 | - 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 5 | 12 | 1 | 0 | 0 | 12000 |
| 0 | 5 | 5 | 7 | 0 | 1 | 0 | 12000 |
| 0 | 5 | 3 | 5 | 0 | 0 | 1 | 9000 |
\captionsetup{labelformat=empty}
\caption{Fig. 3}
- Use the simplex algorithm to solve this problem, and interpret the solution.
- In the solution, one of the basic variables appears at a value of 0 . Explain what this means.
There is a contractual requirement to provide at least 500 kg of product A .
- Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method.
Briefly describe how the method works. You are not required to perform the iterations.