| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.3 This is a standard linear programming question covering all routine steps: formulating inequalities, graphing the feasible region, and finding optimal solutions using the objective line method. While multi-part with several constraints, it requires only straightforward application of textbook techniques with no novel problem-solving or geometric insight beyond the standard D1 curriculum. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06c Working with constraints: algebra and ad hoc methods7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Let \(x\) be the number of type X motors produced. Let \(y\) be the number of type Y motors produced. | M1, A1 | Adequate definition of "number of". Strict inequalities equally OK. |
| \(10x + 12y \leq 200\) | B1 | |
| \(x \geq 5\) and \(y \geq 5\) | B1 | |
| \(0.5x + 0.3y \leq 7\) | B1 | |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Inclined line | B1 | |
| Inclined line | B1 | |
| \(x=5\) and \(y=5\) | B1 | |
| Shading, follow line errors if shape correct | B1 | Guidance accuracy throughout is \(\pm 0.25\) on \(x\) and \(\pm 0.25\) on \(y\). Look at \((8,10)\) first. Inaccurate sketch with axis intercepts given is OK. |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Profit \(= 100X + 70Y\) | B1 | |
| \((5,12.5)\) or \((5,12)\): 1375 or 1340; \((8,10)\): 1500; \((11,5)\): 1450 | M1 | Optimisation — either profit line or evaluating and comparing at their 3 appropriate points (OK if on graph) |
| £1500 profit | A1 | 1500 seen cao. SC: B1 for 1500 without the preceding M mark |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solution in range \(\left(10 \pm \frac{1}{4},\ 6\frac{2}{3} \pm \frac{1}{4}\right) = \left(9.75\text{–}10.25,\ 6.416\text{–}6.916\right)\) | B1 | cao — looking for \(\left(10, 6\frac{2}{3}\right)\) |
| Identification of one of \((9,7)\), \((10,6)\) and \((11,5)\) | B1 | cao |
| Evaluation at all three of \((9,7)\): 1390; \((10,6)\): 1420; \((11,5)\): 1450 | M1 | |
| So 11 of X and 5 of Y | A1 | cao |
| [4] |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $x$ be the number of type X motors produced. Let $y$ be the number of type Y motors produced. | M1, A1 | Adequate definition of "number of". Strict inequalities equally OK. |
| $10x + 12y \leq 200$ | B1 | |
| $x \geq 5$ and $y \geq 5$ | B1 | |
| $0.5x + 0.3y \leq 7$ | B1 | |
| **[5]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Inclined line | B1 | |
| Inclined line | B1 | |
| $x=5$ and $y=5$ | B1 | |
| Shading, follow line errors if shape correct | B1 | Guidance accuracy throughout is $\pm 0.25$ on $x$ and $\pm 0.25$ on $y$. Look at $(8,10)$ first. Inaccurate sketch with axis intercepts given is OK. |
| **[4]** | | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Profit $= 100X + 70Y$ | B1 | |
| $(5,12.5)$ or $(5,12)$: 1375 or 1340; $(8,10)$: 1500; $(11,5)$: 1450 | M1 | Optimisation — either profit line or evaluating and comparing at their 3 appropriate points (OK if on graph) |
| £1500 profit | A1 | 1500 seen cao. SC: B1 for 1500 without the preceding M mark |
| **[3]** | | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solution in range $\left(10 \pm \frac{1}{4},\ 6\frac{2}{3} \pm \frac{1}{4}\right) = \left(9.75\text{–}10.25,\ 6.416\text{–}6.916\right)$ | B1 | cao — looking for $\left(10, 6\frac{2}{3}\right)$ |
| Identification of one of $(9,7)$, $(10,6)$ and $(11,5)$ | B1 | cao |
| Evaluation at all three of $(9,7)$: 1390; $(10,6)$: 1420; $(11,5)$: 1450 | M1 | |
| So 11 of X and 5 of Y | A1 | cao |
| **[4]** | | |
---4 In a factory, two types of motor are made. Each motor of type X takes 10 man hours to make and each motor of type Y takes 12 man hours to make.
In each week there are 200 man hours available.
To satisfy customer demand, at least 5 of each type of motor must be made each week.\\
Once a motor has been started it must be completed; no unfinished motors may be left in the factory at the end of each week.
When completed, the motors are put into a container for shipping. The volume of the container is $7 \mathrm {~m} ^ { 3 }$. A type X motor occupies a volume of $0.5 \mathrm {~m} ^ { 3 }$ and a type Y motor occupies a volume of $0.3 \mathrm {~m} ^ { 3 }$.\\
(i) Define appropriate variables and from the above information derive four inequalities which must be satisfied by those variables.\\
(ii) Represent your inequalities on a graph and shade the infeasible region.
The profit on each type X is $\pounds 100$ and on each type Y is $\pounds 70$.\\
(iii) The weekly profit is to be maximised. Write down the objective function and find the maximum profit.\\
(iv) Because of absenteeism, the manager decides to organise the work in the factory on the assumption that there will be only 180 man hours available each week. Find the number of motors of each type that should now be made in order to maximise the profit.
\hfill \mbox{\textit{OCR MEI D1 2012 Q4 [16]}}