| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| 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 identification of pivot column (most negative in P row), pivot row (minimum ratio test), and standard row operations. It's a standard D2 textbook exercise with no problem-solving or insight required, just following a learned procedure, making it easier than average A-level maths questions. |
| Spec | 7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations |
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 2 | -4 | 1 | 1 | 0 | 0 | 15 |
| \(s\) | 4 | 2 | -8 | 0 | 1 | 0 | 20 |
| \(t\) | 1 | -1 | 4 | 0 | 0 | 1 | 8 |
| \(P\) | -3 | 2 | 7 | 0 | 0 | 0 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Pivot row correct including change of b.v., divide row by 4: \(x \mid 1 \mid \frac{1}{2} \mid -2 \mid 0 \mid \frac{1}{4} \mid 0 \mid 5\) | M1, A1 | Correct pivot located (4 in column \(x\)), attempt to divide row; pivot row correct including change of b.v. |
| \(R_1 - 2R_2\): \(r \mid 0 \mid -5 \mid 5 \mid 1 \mid -\frac{1}{2} \mid 0 \mid 5\) | M1 | All values in one non-pivot row correct or one non-zero and one column correct following through from pivot |
| \(R_3 - R_2\): \(t \mid 0 \mid -\frac{3}{2} \mid 6 \mid 0 \mid -\frac{1}{4} \mid 1 \mid 3\) | A1ft | Two of the non-zero and one columns correct following through |
| \(R_4 + 3R_2\): \(P \mid 0 \mid \frac{7}{2} \mid 1 \mid 0 \mid \frac{3}{4} \mid 0 \mid 15\) | A1 | CAO – all values and row operations correctly stated; ignore b.v. column for this mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P + \frac{7}{2}y + z + \frac{3}{4}s = 15\) | B1ft | Follow their profit equation from (a), dependent on scoring both M marks in (a) |
| \(r = 5,\ s = 0,\ t = 3\) | B1 | CAO (no follow through) for slack variables |
# Question 1:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Pivot row correct **including change of b.v.**, divide row by 4: $x \mid 1 \mid \frac{1}{2} \mid -2 \mid 0 \mid \frac{1}{4} \mid 0 \mid 5$ | M1, A1 | Correct pivot located (4 in column $x$), attempt to divide row; pivot row correct including change of b.v. |
| $R_1 - 2R_2$: $r \mid 0 \mid -5 \mid 5 \mid 1 \mid -\frac{1}{2} \mid 0 \mid 5$ | M1 | All values in one non-pivot row correct **or** one non-zero and one column correct following through from pivot |
| $R_3 - R_2$: $t \mid 0 \mid -\frac{3}{2} \mid 6 \mid 0 \mid -\frac{1}{4} \mid 1 \mid 3$ | A1ft | Two of the non-zero and one columns correct following through |
| $R_4 + 3R_2$: $P \mid 0 \mid \frac{7}{2} \mid 1 \mid 0 \mid \frac{3}{4} \mid 0 \mid 15$ | A1 | CAO – all values and row operations correctly stated; **ignore b.v. column for this mark** |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P + \frac{7}{2}y + z + \frac{3}{4}s = 15$ | B1ft | Follow their profit equation from (a), dependent on scoring **both** M marks in (a) |
| $r = 5,\ s = 0,\ t = 3$ | B1 | CAO (no follow through) for slack variables |
---\begin{enumerate}
\item The tableau below is the initial tableau for a linear programming problem in $x , y$ and $z$. The objective is to maximise the profit, $P$.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 2 & -4 & 1 & 1 & 0 & 0 & 15 \\
\hline
$s$ & 4 & 2 & -8 & 0 & 1 & 0 & 20 \\
\hline
$t$ & 1 & -1 & 4 & 0 & 0 & 1 & 8 \\
\hline
$P$ & -3 & 2 & 7 & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
(a) Perform one iteration of the Simplex algorithm to obtain a new tableau, $T$. State the row operations you use.\\
(5)\\
(b) Write down the profit equation given by $T$ and state the current values of the slack variables.\\
\hfill \mbox{\textit{Edexcel D2 2015 Q1 [7]}}