| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Slack variable interpretation |
| Difficulty | Moderate -0.8 This is a routine D1 Simplex question requiring standard procedures: writing constraints, interpreting slack variables, reading tableau values, and performing pivot operations. Parts (a)-(d) are textbook exercises with no novel insight required. Only part (f) requires minor problem-solving to find an integer solution, but this is a common D1 task. Overall easier than average A-level maths. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.07a 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 | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(r\) | 3 | 5 | 6 | 1 | 0 | 50 |
| \(s\) | 1 | 2 | 4 | 0 | 1 | 24 |
| \(P\) | - 1 | - 3 | - 4 | 0 | 0 | 0 |
| Answer | Marks |
|---|---|
| \(x + 2y + 4z \leq 24\) | B1 |
| Answer | Marks |
|---|---|
| i. \(x + 2y + 4z + s = 24\) | \(\text{B1}\sqrt{}\) |
| ii. \(s\ (\geq 0)\) is the slack time on the machine in hours | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 Euro | B1 | (4 total for a–c) |
| Answer | Marks | Guidance |
|---|---|---|
| b.v. | \(x\) | \(y\) |
| \(r\) | \(\frac{3}{2}\) | \(\mathbf{2}\) |
| \(z\) | \(\frac{1}{4}\) | \(\frac{1}{2}\) |
| \(p\) | 0 | \(-1\) |
| \(R_1 - 6R_2\), \(R_2 \div 4\), \(R_3 + 4R_2\) | M1 | \(\text{A1}\sqrt{}\), A1 |
| Answer | Marks | Guidance |
|---|---|---|
| b.v. | \(x\) | \(y\) |
| \(y\) | \(\frac{3}{4}\) | 1 |
| \(z\) | \(-\frac{1}{8}\) | 0 |
| \(p\) | \(\frac{3}{4}\) | 0 |
| \(R_1 \div 2\), \(R_2 - \frac{1}{2}R_1\), \(R_3 + R_1\) | M1 | \(\text{A1}\sqrt{}\), A1 |
| Profit \(= 31\) Euros; \(y=7\) (medium), \(z=2.5\) (large), \(x=r=s=0\) | M1, \(\text{A1}\sqrt{}\), \(\text{A1}\sqrt{}\) | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Cannot make \(\frac{1}{2}\) a lamp | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. \((0,10,0)\) or \((0,6,3)\) or \((1,7,2)\); checks in both inequalities | B1, B1 | (2) |
# Question 8:
## Part (a)
$x + 2y + 4z \leq 24$ | B1 |
## Part (b)
i. $x + 2y + 4z + s = 24$ | $\text{B1}\sqrt{}$ |
ii. $s\ (\geq 0)$ is the slack time on the machine in hours | B1 |
## Part (c)
1 Euro | B1 | (4 total for a–c)
## Part (d)
First tableau:
| b.v. | $x$ | $y$ | $z$ | $r$ | $s$ | value |
|------|-----|-----|-----|-----|-----|-------|
| $r$ | $\frac{3}{2}$ | $\mathbf{2}$ | 0 | 1 | $-\frac{3}{2}$ | 14 |
| $z$ | $\frac{1}{4}$ | $\frac{1}{2}$ | 1 | 0 | $\frac{1}{4}$ | 6 |
| $p$ | 0 | $-1$ | 0 | 0 | 1 | 24 |
$R_1 - 6R_2$, $R_2 \div 4$, $R_3 + 4R_2$ | M1 | $\text{A1}\sqrt{}$, A1 | (3)
Second tableau:
| b.v. | $x$ | $y$ | $z$ | $r$ | $s$ | value |
|------|-----|-----|-----|-----|-----|-------|
| $y$ | $\frac{3}{4}$ | 1 | 0 | $\frac{1}{2}$ | $-\frac{3}{4}$ | 7 |
| $z$ | $-\frac{1}{8}$ | 0 | 1 | $-\frac{1}{4}$ | $\frac{5}{8}$ | $\frac{5}{2}$ |
| $p$ | $\frac{3}{4}$ | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{4}$ | 31 |
$R_1 \div 2$, $R_2 - \frac{1}{2}R_1$, $R_3 + R_1$ | M1 | $\text{A1}\sqrt{}$, A1 | (3)
Profit $= 31$ Euros; $y=7$ (medium), $z=2.5$ (large), $x=r=s=0$ | M1, $\text{A1}\sqrt{}$, $\text{A1}\sqrt{}$ | (3)
## Part (e)
Cannot make $\frac{1}{2}$ a lamp | B1 | (1)
## Part (f)
e.g. $(0,10,0)$ or $(0,6,3)$ or $(1,7,2)$; checks in **both** inequalities | B1, B1 | (2)
**Total: 16 marks**8. A company makes three sizes of lamps, small, medium and large. The company is trying to determine how many of each size to make in a day, in order to maximise its profit. As part of the process the lamps need to be sanded, painted, dried and polished. A single machine carries out these tasks and is available 24 hours per day. A small lamp requires one hour on this machine, a medium lamp 2 hours and a large lamp 4 hours.
Let $x =$ number of small lamps made per day,
$$\begin{aligned}
& y = \text { number of medium lamps made per day, } \\
& z = \text { number of large lamps made per day, }
\end{aligned}$$
where $x \geq 0 , y \geq 0$ and $z \geq 0$.
\begin{enumerate}[label=(\alph*)]
\item Write the information about this machine as a constraint.
\item \begin{enumerate}[label=(\roman*)]
\item Re-write your constraint from part (a) using a slack variable $s$.
\item Explain what $s$ means in practical terms.
Another constraint and the objective function give the following Simplex tableau. The profit $P$ is stated in euros.
\begin{center}
\begin{tabular}{ | c | r | r | r | r | r | c | }
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & Value \\
\hline
$r$ & 3 & 5 & 6 & 1 & 0 & 50 \\
\hline
$s$ & 1 & 2 & 4 & 0 & 1 & 24 \\
\hline
$P$ & - 1 & - 3 & - 4 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\end{enumerate}\item Write down the profit on each small lamp.
\item Use the Simplex algorithm to solve this linear programming problem.
\item Explain why the solution to part (d) is not practical.
\item Find a practical solution which gives a profit of 30 euros. Verify that it is feasible.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2003 Q8 [16]}}