| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Complete Simplex solution |
| Difficulty | Standard +0.3 This is a standard, methodical Simplex algorithm question requiring routine application of the pivot procedure. While it involves multiple steps, each follows a well-defined algorithmic process taught directly in D2. The question guides students through each stage (identifying slack variables, performing iterations, interpreting results) with no novel problem-solving or insight required—just careful arithmetic and procedure following. |
| 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 variable7.07e Graphical interpretation: iterations as edges of convex polygon7.07f Algebraic interpretation: explain simplex calculations |
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { Z }\) | \(s\) | \(t\) | \(\boldsymbol { u }\) | value |
| 1 | -2 | -4 | -3 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 1 | 1 | 0 | 0 | 14 |
| 0 | -1 | 1 | 2 | 0 | 1 | 0 | 6 |
| 0 | 4 | 4 | 3 | 0 | 0 | 1 | 29 |
4 A linear programming problem involving variables $x , y$ and $z$ is to be solved. The objective function to be maximised is $P = 2 x + 4 y + 3 z$. The initial Simplex tableau is given below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$\boldsymbol { P }$ & $\boldsymbol { x }$ & $\boldsymbol { y }$ & $\boldsymbol { Z }$ & $s$ & $t$ & $\boldsymbol { u }$ & value \\
\hline
1 & -2 & -4 & -3 & 0 & 0 & 0 & 0 \\
\hline
0 & 2 & 2 & 1 & 1 & 0 & 0 & 14 \\
\hline
0 & -1 & 1 & 2 & 0 & 1 & 0 & 6 \\
\hline
0 & 4 & 4 & 3 & 0 & 0 & 1 & 29 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item What name is given to the variables $s , t$ and $u$ ?
\item Write down an equation involving $x , y , z$ and $s$ for this problem.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item By choosing the first pivot from the $\boldsymbol { y }$-column, perform one iteration of the Simplex method.
\item Explain how you know that the optimal value has not been reached.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Perform one further iteration.
\item Interpret the final tableau, stating the values of $P , x , y$ and $z$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2010 Q4 [14]}}