| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| 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 Simplex algorithm question requiring two iterations with clear pivot selection rules. While it involves multiple steps and careful arithmetic, it follows a completely mechanical procedure taught directly in D2 with no problem-solving or insight required—making it slightly easier than average for A-level maths overall. |
| 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 |
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 16 | 2 | 4 | 1 | 0 | 0 | 350 |
| \(s\) | 18 | - 2 | 6 | 0 | 1 | 0 | 480 |
| \(t\) | 5 | 0 | 5 | 0 | 0 | 1 | 360 |
| \(P\) | - 18 | - 7 | - 20 | 0 | 0 | 0 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(16x + 2y + 4z + r = 350\) | M1 | |
| \(18x - 2y + 6z + s = 480\) | A1 | |
| \(5x + 5z + t = 360\) | A2, 1, 0(4) | |
| \(P - 18x - 7y - 20z = 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| First pivot: \(R_3 \div 5\), \(R_1 - 4R_3\), \(R_2 - 6R_3\), \(R_4 + 20R_3\) | M1 A1, M1 A1, B1 | |
| Second iteration: \(R_1 \div 2\), \(R_2 + 2R_1\), \(R_4 + 7R_1\) | M1 A1, M1 A1(9) | |
| Final tableau: \(y=31\), \(s=110\), \(z=72\), \(P=217\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Not optimal since a negative value in the \(p\) row | B1(1) |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $16x + 2y + 4z + r = 350$ | M1 | |
| $18x - 2y + 6z + s = 480$ | A1 | |
| $5x + 5z + t = 360$ | A2, 1, 0(4) | |
| $P - 18x - 7y - 20z = 0$ | | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| First pivot: $R_3 \div 5$, $R_1 - 4R_3$, $R_2 - 6R_3$, $R_4 + 20R_3$ | M1 A1, M1 A1, B1 | |
| Second iteration: $R_1 \div 2$, $R_2 + 2R_1$, $R_4 + 7R_1$ | M1 A1, M1 A1(9) | |
| Final tableau: $y=31$, $s=110$, $z=72$, $P=217$ | | |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Not optimal since a negative value in the $p$ row | B1(1) | |
---6. The tableau below is the initial tableau for a maximising linear programming problem.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 16 & 2 & 4 & 1 & 0 & 0 & 350 \\
\hline
$s$ & 18 & - 2 & 6 & 0 & 1 & 0 & 480 \\
\hline
$t$ & 5 & 0 & 5 & 0 & 0 & 1 & 360 \\
\hline
$P$ & - 18 & - 7 & - 20 & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Write down the four equations represented in the initial tableau.
\item Taking the most negative number in the profit row to indicate the pivot column at each stage, perform two complete iterations of the Simplex algorithm. State the row operations that you use.
\item State whether or not your last tableau is optimal. Give a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 Q6 [14]}}