| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2007 |
| Session | January |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Perform one Simplex iteration |
| Difficulty | Moderate -0.8 This is a routine, mechanical application of the simplex algorithm requiring standard pivot operations. Students follow a prescribed procedure (identify pivot column from most negative, calculate ratios, perform row operations) with no problem-solving or insight needed. The question is straightforward for D1 students who have practiced the algorithm, making it easier than average A-level maths questions overall. |
| Spec | 7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(r\) | 2 | 0 | 4 | 1 | 0 | 80 |
| \(s\) | 1 | 4 | 2 | 0 | 1 | 160 |
| \(P\) | \(-2\) | \(-8\) | \(-20\) | 0 | 0 | 0 |
| Answer | Marks |
|---|---|
| Table with b.v., x, 9, z, r, s, Value, Rhs expl columns showing: Z=\(\frac{1}{2}\), 0, 1, \(\frac{1}{4}\), 0, 2.0, (\(R_1 + 4\)); S=0, 4, 0, \(-\frac{7}{2}\), 1, 12.0, (\(R_2 - 2R_1\)); P=12, -8, 0, 5, 0, 4.00, (\(R_3 + 20R_1\)) | M1, A1, A1 (5) |
| Answer | Marks |
|---|---|
| \(P + 8x - 8y + 5r = 400\) | BW (1) |
| Answer | Marks |
|---|---|
| Not optimal since there is a negative number in the pivot row | BW (1), [7] |
## 4(a)
| Table with b.v., x, 9, z, r, s, Value, Rhs expl columns showing: Z=$\frac{1}{2}$, 0, 1, $\frac{1}{4}$, 0, 2.0, ($R_1 + 4$); S=0, 4, 0, $-\frac{7}{2}$, 1, 12.0, ($R_2 - 2R_1$); P=12, -8, 0, 5, 0, 4.00, ($R_3 + 20R_1$) | M1, A1, A1 (5) |
## 4(b)
| $P + 8x - 8y + 5r = 400$ | BW (1) |
## 4(c)
| Not optimal since there is a negative number in the pivot row | BW (1), [7] |A three-variable linear programming problem in $x$, $y$ and $z$ is to be solved. The objective is to maximise the profit $P$. The following initial tableau was obtained.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & Value \\
\hline
$r$ & 2 & 0 & 4 & 1 & 0 & 80 \\
$s$ & 1 & 4 & 2 & 0 & 1 & 160 \\
$P$ & $-2$ & $-8$ & $-20$ & 0 & 0 & 0 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm, to obtain tableau $T$. State the row operations that you use. (5)
\item Write down the profit equation shown in tableau $T$. (1)
\item State whether tableau $T$ is optimal. Give a reason for your answer. (1)
\end{enumerate}
(Total 7 marks)
\hfill \mbox{\textit{Edexcel D1 2007 Q4}}