| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Complete Simplex solution |
| Difficulty | Standard +0.8 This is a complete Simplex algorithm problem requiring multiple iterations and interpretation. While the mechanical steps are routine for Further Maths students, it's above average difficulty for A-level due to: (1) being a 3-variable problem requiring careful arithmetic across multiple tableaux, (2) the guided structure still requiring understanding of pivot selection rules, and (3) being from D2 which is optional Further Maths content. The execution is procedural but error-prone with many calculations. |
| Spec | 7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 3 rows correct (must include slack variables) | M1 | |
| All correct | A1 | |
| 20/1, 24/3, 30/2 ALL seen; '3' in z-col identified | E1, B1 | Correct value may be highlighted in table |
| Correct values from tableau shown | M1 | SCA - Row reduction, 1 row correct (other than (shaded) pivot row) |
| Any 3 rows correct | A1 | |
| All correct | A1 | |
| Row reduction, 1 row correct (other than (shaded) pivot row) | B1F, M1 | |
| All correct | A1 | |
| In part (c), FT ONLY IF all non-negative in profit row. All answers must be exact. (isw) | ||
| Max \(P = 36\) | B1F | Max/optimal oe stated in part (c) or end of part (b) |
| \(x = 6, y = 0, z = 6\) | B1F | FT their values, must be non-negative |
| \(r = 8, s = 0, t = 0\) | B1F | must be non-negative |
| Answer | Marks | Guidance |
|--------|-------|----------|
| 3 rows correct (must include slack variables) | M1 | |
| All correct | A1 | |
| 20/1, 24/3, 30/2 ALL seen; '3' in z-col identified | E1, B1 | Correct value may be highlighted in table |
| | | |
| Correct values from tableau shown | M1 | SCA - Row reduction, 1 row correct (other than (shaded) pivot row) |
| Any 3 rows correct | A1 | |
| All correct | A1 | |
| Row reduction, 1 row correct (other than (shaded) pivot row) | B1F, M1 | |
| All correct | A1 | |
| In part (c), FT ONLY IF all non-negative in profit row. All answers must be exact. (isw) | | |
| Max $P = 36$ | B1F | Max/optimal oe stated in part (c) or end of part (b) |
| $x = 6, y = 0, z = 6$ | B1F | FT their values, must be non-negative |
| $r = 8, s = 0, t = 0$ | B1F | must be non-negative |3\\
Maximise $\quad P = 2 x - 3 y + 4 z$\\
subject to $\quad x + 2 y + z \leqslant 20$\\
$x - y + 3 z \leqslant 24$\\
$3 x - 2 y + 2 z \leqslant 30$\\
and $\quad x \geqslant 0 , y \geqslant 0 , z \geqslant 0$.
\begin{enumerate}[label=(\alph*)]
\item Display the linear programming problem in a Simplex tableau.
\item \begin{enumerate}[label=(\roman*)]
\item The first pivot to be chosen is from the $z$-column. Identify the pivot and explain why this particular value is chosen.
\item Perform one iteration of the Simplex method.
\item Perform one further iteration.
\end{enumerate}\item Interpret your final tableau and state the values of your slack variables.\\[0pt]
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2016 Q3 [13]}}