Two-stage Simplex and big-M methodsEdexcel A-Level Further Maths: Mind map

Exam questions on Two-stage Simplex and big-M methods

1. A maximising problem has the constraint $2x+y\ge8$, to which a surplus variable $s$ and an artificial variable $t$ are added. The objective is $P=3x+2y$ and the big-M method is used, where $M$ is a very large positive number.
   Explain the purpose of the term $-Mt$, and what it means if $t>0$ in the final optimal tableau.2 marks
2. A maximising problem has the constraints $x+y\le6$ and $x+y\ge9$, with $x,y\ge0$. It is solved using the two-stage Simplex method, with a slack variable $s_1$ in the first constraint and a surplus variable $s_2$ and an artificial variable $t$ in the second.
   Explain, without using the Simplex algorithm, why the stage 1 result is correct.2 marks
3. A company makes $x$ tonnes of product X and $y$ tonnes of product Y. It wishes to maximise the profit $P=3x+4y$ subject to $x+y\le12$ and $x+2y\ge6$, with $x\ge0$ and $y\ge0$. The big-M method is used, with slack variable $s_1$, surplus variable $s_2$ and artificial variable $t$. Tableau columns are in the order $x$, $y$, $s_1$, $s_2$, $t$, then the value.
   Write the constraints as equations and write down the objective function for the big-M method.3 marks