The Simplex algorithmEdexcel A-Level Further Maths: Flashcards

Exam questions on The Simplex algorithm

1. A firm makes $x$ units of product X and $y$ units of product Y. It wishes to maximise the profit $P=2x+3y$ subject to $x+y\le10$ and $x+2y\le16$, with $x\ge0$ and $y\ge0$. Slack variables $r$ and $s$ are added to the first and second constraints and the Simplex algorithm is used.
   State which variable enters the basis and which leaves, and write down the new pivot row after the first iteration.2 marks
2. A maximising problem in $x$ and $y$, with slack variables $r$ and $s$, has been partly solved by the Simplex algorithm. The current tableau has columns in the order $x$, $y$, $r$, $s$, then the value: r row: $\frac12$, $0$, $1$, $-\frac12$ | $2$ y row: $\frac12$, $1$, $0$, $\frac12$ | $8$ P row: $-\frac12$, $0$, $0$, $\frac32$ | $24$
   Perform the next iteration and state the optimal values of $x$, $y$ and $P$.2 marks
3. A firm makes $x$, $y$ and $z$ units of three products each day. Three resources give the constraints $2x+3y+z\le5$, $4x+y+2z\le11$ and $3x+4y+2z\le8$, with $x,y,z\ge0$. The profit is $P=5x+4y+3z$ and the firm wishes to maximise it. Slack variables $r$, $s$ and $t$ are added to the three constraints in order.
   Write down the initial Simplex tableau.3 marks