| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Effect of parameter changes |
| Difficulty | Standard +0.8 This question requires understanding of the Simplex algorithm with parameter analysis, multiple iterations, and interpretation of optimal tableaux. Part (b)(ii) demands algebraic reasoning about when optimality is reached based on parameter k, which goes beyond routine application. While systematic, it requires deeper conceptual understanding than standard Simplex questions. |
| 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 objective7.07d Simplex terminology: basic feasible solution, basic/non-basic variable |
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { z }\) | \(s\) | \(\boldsymbol { t }\) | value |
| 1 | -4 | -1 | \(- k\) | 0 | 0 | 0 |
| 0 | 1 | 2 | 3 | 1 | 0 | 7 |
| 0 | 2 | 1 | 4 | 0 | 1 | 10 |
| Answer | Marks | Guidance |
|---|---|---|
| I appreciate your request, but the content you've provided appears to be just a row of numbers (4 | 3 | 4 |
I appreciate your request, but the content you've provided appears to be just a row of numbers (4 | 3 | 4 | 0 | 2) rather than a mark scheme with marking annotations, guidance notes, or mathematical symbols that need cleaning.
Could you please provide the actual mark scheme content you'd like me to clean up? It should include elements like:
- Marking points (M1, A1, B1, etc.)
- Mathematical notation or unicode symbols to convert
- Guidance notes or criteria
Once you share the full mark scheme, I'll be happy to format it according to your specifications.4 A linear programming problem involving variables $x , y$ and $z$ is to be solved. The objective function to be maximised is $P = 4 x + y + k z$, where $k$ is a constant. The initial Simplex tableau is given below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
$\boldsymbol { P }$ & $\boldsymbol { x }$ & $\boldsymbol { y }$ & $\boldsymbol { z }$ & $s$ & $\boldsymbol { t }$ & value \\
\hline
1 & -4 & -1 & $- k$ & 0 & 0 & 0 \\
\hline
0 & 1 & 2 & 3 & 1 & 0 & 7 \\
\hline
0 & 2 & 1 & 4 & 0 & 1 & 10 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item In addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, write down two inequalities involving $x , y$ and $z$ for this problem.
\item \begin{enumerate}[label=(\roman*)]
\item The first pivot is chosen from the $\boldsymbol { x }$-column. Identify the pivot and perform one iteration of the Simplex method.
\item Given that the optimal value of $P$ has not been reached after this first iteration, find the possible values of $k$.
\end{enumerate}\item Given that $k = 10$ :
\begin{enumerate}[label=(\roman*)]
\item perform one further iteration of the Simplex method;
\item interpret the final tableau.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2009 Q4 [14]}}