| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.8 This is a standard D1 linear programming question requiring routine formulation skills and mechanical application of the Simplex algorithm. Parts (i) and (ii) involve straightforward translation of word problem to mathematical constraints, while part (iii) requires textbook Simplex setup and one iteration. Reading the final tableau is also routine. No novel insight or problem-solving required—purely procedural application of a well-practiced algorithm. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations |
| P | \(x\) | \(y\) | \(z\) | \(s\) | \(t\) | \(u\) | \(v\) | RHS |
| 1 | 0 | 9 | 0 | 20 | 0 | 80 | 0 | 2000 |
| 0 | 1 | 0.5 | 0 | 0.25 | 0 | -0.25 | 0 | 12.5 |
| 0 | 0 | -0.5 | 0 | -0.25 | 1 | 0.25 | 0 | 12.5 |
| 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 10 |
| 0 | 0 | 0.5 | 0 | -0.25 | 0 | -0.75 | 1 | 17.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P = 80x + 31y + 100z\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4x+2y+z \leq 60\): planting hours constraint | B1 | With explanation |
| \(x + y + z \leq 40\) | B1 | Land constraint |
| \(x \leq 25\) | B1 | Wheat constraint |
| \(z \leq 10\) | B1 | Soya beans constraint (3 marks for any 3 correct) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct initial tableau set up with slack variables \(s, t, u, v\) | M1A1 | |
| Most negative in objective row is \(z\) column (\(-100\)), pivot on \(z\) column | M1 | Correct pivot column identified |
| Correct ratios calculated, correct pivot element identified | M1 | |
| Pivot row correctly divided | A1 | |
| Other rows correctly updated | A1A1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 12.5\), \(y = 0\) (read from tableau; \(y\) not basic) | B1 | |
| \(z = 10\) | B1 | |
| Unplanted = \(40 - 12.5 - 0 - 10 = 17.5\) acres | B1 |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P = 80x + 31y + 100z$ | B1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4x+2y+z \leq 60$: planting hours constraint | B1 | With explanation |
| $x + y + z \leq 40$ | B1 | Land constraint |
| $x \leq 25$ | B1 | Wheat constraint |
| $z \leq 10$ | B1 | Soya beans constraint (3 marks for any 3 correct) |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct initial tableau set up with slack variables $s, t, u, v$ | M1A1 | |
| Most negative in objective row is $z$ column ($-100$), pivot on $z$ column | M1 | Correct pivot column identified |
| Correct ratios calculated, correct pivot element identified | M1 | |
| Pivot row correctly divided | A1 | |
| Other rows correctly updated | A1A1A1 | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 12.5$, $y = 0$ (read from tableau; $y$ not basic) | B1 | |
| $z = 10$ | B1 | |
| Unplanted = $40 - 12.5 - 0 - 10 = 17.5$ acres | B1 | |4 A farmer has 40 acres of land that can be used for growing wheat, potatoes and soya beans. The farmer can expect a profit of $\pounds 80$ for each acre of wheat, $\pounds 31$ for each acre of potatoes and $\pounds 100$ for each acre of soya beans. Land that is left unplanted incurs no cost and generates no profit. The farmer wants to choose how much land to use for growing each crop to maximise the profit.
It takes 4 hours to plant each acre of wheat, 2 hours to plant each acre of potatoes and 1 hour to plant each acre of soya beans. There are 60 hours available in total for planting.
At most 25 acres can be used for wheat and at most 10 acres can be used for soya beans.\\
Let $x$ denote the number of acres used for wheat, $y$ denote the number of acres used for potatoes and $z$ denote the number of acres used for soya beans.\\
\begin{enumerate}[label=(\roman*)]
\item Express the profit, $\pounds P$, as a function of $x , y$ and $z$.
\item Explain why the constraint $4 x + 2 y + z \leqslant 60$ is needed. Write down three more constraints on the values of $x , y$ and $z$, other than that they must be non-negative.
\item Set up an initial Simplex tableau to represent the farmer's problem. Perform one iteration of the Simplex algorithm, choosing a pivot from the column with the most negative value in the objective row. Show how each row that has changed was calculated.
Julie uses the Simplex algorithm to solve the farmer's problem. Her final tableau is given below. The order of the rows and the use of the slack variables in Julie's tableau may be different from yours.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
P & $x$ & $y$ & $z$ & $s$ & $t$ & $u$ & $v$ & RHS \\
\hline
1 & 0 & 9 & 0 & 20 & 0 & 80 & 0 & 2000 \\
\hline
0 & 1 & 0.5 & 0 & 0.25 & 0 & -0.25 & 0 & 12.5 \\
\hline
0 & 0 & -0.5 & 0 & -0.25 & 1 & 0.25 & 0 & 12.5 \\
\hline
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 10 \\
\hline
0 & 0 & 0.5 & 0 & -0.25 & 0 & -0.75 & 1 & 17.5 \\
\hline
\end{tabular}
\end{center}
\item Write down the values of $x , y$ and $z$ from Julie's final tableau. Hence advise the farmer on how many acres to use for each crop and how much land should be left unplanted.
\end{enumerate}
\hfill \mbox{\textit{OCR D1 2015 Q4 [15]}}