| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.8 This is a standard textbook linear programming question requiring routine formulation of constraints, graphical solution by plotting lines and finding intersection points, and sensitivity analysis. All techniques are algorithmic with no novel insight required—easier than average A-level material. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Let \(f\) be the number of litres of Flowerbase produced. Let \(g\) be the number of litres of Growmuch produced. Max \(9f + 20g\) s.t. \(0.75f + 0.5g \le 12000\) \(f + 2g \le 25000\) | B1 M1 A1 M1 A1 A1 | |
| (ii) [Graph showing constraints and feasible region with shading] Optimal point \((11500, 6750)\) Max profit \(= £2500\) by producing \(12500\) litres of Growmuch | B1 B1 B1 M1 A1 | labels + scales B1 B1 lines shading |
| (iii) No effect | B1 | |
| (iv) No effect. The profit on Flowerbase will be reduced by more than that suffered by Growmuch, since it uses more fibre. The objective gradient will thus increase from \(–9/20\), making it even less attractive to produce any Flowerbase. | M1 A1 | |
| (v) \(£3000\) | B1 |
| (i) Let $f$ be the number of litres of Flowerbase produced. Let $g$ be the number of litres of Growmuch produced. Max $9f + 20g$ s.t. $0.75f + 0.5g \le 12000$ $f + 2g \le 25000$ | B1 M1 A1 M1 A1 A1 | |
| (ii) [Graph showing constraints and feasible region with shading] Optimal point $(11500, 6750)$ Max profit $= £2500$ by producing $12500$ litres of Growmuch | B1 B1 B1 M1 A1 | labels + scales B1 B1 lines shading |
| (iii) No effect | B1 | |
| (iv) No effect. The profit on Flowerbase will be reduced by more than that suffered by Growmuch, since it uses more fibre. The objective gradient will thus increase from $–9/20$, making it even less attractive to produce any Flowerbase. | M1 A1 | |
| (v) $£3000$ | B1 | |6 A company manufactures two types of potting compost, Flowerbase and Growmuch. The weekly amounts produced of each are constrained by the supplies of fibre and of nutrient mix. Each litre of Flowerbase requires 0.75 litres of fibre and 1 kg of nutrient mix. Each litre of Growmuch requires 0.5 litres of fibre and 2 kg of nutrient mix. There are 12000 litres of fibre supplied each week, and 25000 kg of nutrient mix.
The profit on Flowerbase is 9 p per litre. The profit on Growmuch is 20 p per litre.
\begin{enumerate}[label=(\roman*)]
\item Formulate an LP to maximise the weekly profit subject to the constraints on fibre and nutrient mix.
\item Solve your LP using a graphical approach.
\item Consider each of the following separate circumstances.\\
(A) There is a reduction in the weekly supply of fibre from 12000 litres to 10000 litres. What effect does this have on profit?\\
(B) The price of fibre is increased. Will this affect the optimal production plan? Justify your answer.\\[0pt]
(C) The supply of nutrient mix is increased to 30000 kg per week. What is the new profit? [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI D1 2005 Q6 [16]}}