| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Constraint derivation verification |
| Difficulty | Easy -1.2 This is a straightforward constraint formulation question requiring direct translation of verbal statements into algebraic inequalities. All parts involve routine recall of linear programming terminology and standard techniques (identifying constraints, objective function, slack variables) with no problem-solving or novel insight required. The multi-part structure is typical but each part is mechanical. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06c Working with constraints: algebra and ad hoc methods |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(d + f + g = 120\) | B1 |
| (ii) | Area of grass is not more than 4 times (area of) decking | B1 |
| (iii) | \(d < f\) | B1 |
| (iv) | \(g > 40\) | B1 |
| \(\min d = 10\) \(\min f = 20\) | B1 | |
| \(5g + 10d + 20f\) or any positive multiple of this | B1, 3 | |
| (v) | \(5g + 10d + 20f\) or \(g + 2d + 4f\) | B1 |
| (vi) | Minimise \(g + 2d + 4f\) Subject to \(d + f + g = 120\) \(g - 4d - s = 0\) \(d - f + t = 0\) and \(d \geq 40\), \(d \geq 10\), \(f \geq 20\), \(s \geq 0\), \(t \geq 0\) | M1, B1, A1, 3 |
(i) | $d + f + g = 120$ | B1 | For this equality. Condone an inequality
(ii) | Area of grass is not more than 4 times (area of) decking | B1 | Identify the constraint in words (not just "grass is less than or equal to 4 times decking" though)
(iii) | $d < f$ | B1 | Do not accept $d < f$
(iv) | $g > 40$ | B1 | Do not accept $g > 40$
| $\min d = 10$ $\min f = 20$ | B1 |
| $5g + 10d + 20f$ or any positive multiple of this | B1, 3 |
(v) | $5g + 10d + 20f$ or $g + 2d + 4f$ | B1 |
(vi) | Minimise $g + 2d + 4f$ Subject to $d + f + g = 120$ $g - 4d - s = 0$ $d - f + t = 0$ and $d \geq 40$, $d \geq 10$, $f \geq 20$, $s \geq 0$, $t \geq 0$ | M1, B1, A1, 3 | For a reasonable attempt at setting up the minimisation problem using their expressions For dealing with this slack variable correctly (variables on LHS and constant on RHS) For a completely correct formulation (accept d and $f \geq 0$, or their min values for d, f)
---2 A landscape gardener is designing a garden. Part of the garden will be decking, part will be flowers and the rest will be grass. Let d be the area of decking, f be the area of flowers and g be the area of grass, all measured in $\mathrm { m } ^ { 2 }$.
The total area of the garden is $120 \mathrm {~m} ^ { 2 }$ of which at least $40 \mathrm {~m} ^ { 2 }$ must be grass. The area of decking must not be greater than the area of flowers. Also, the area of grass must not be more than four times the area of decking.
Each square metre of grass will cost $\pounds 5$, each square metre of decking will cost $\pounds 10$ and each square metre of flowers will cost $\pounds 20$. These costs include labour. The landscape gardener has been instructed to come up with the design that will cost the least.\\
(i) Write down a constraint in d , f and g from the total area of the garden.\\
(ii) Explain why the constraint $\mathrm { g } \leqslant 4 \mathrm {~d}$ is required.\\
(iii) Write down a constraint from the requirement that the area of decking must not be greater than the area of flowers.\\
(iv) Write down a constraint from the requirement that at least $40 \mathrm {~m} ^ { 2 }$ of the garden must be grass and write down the minimum feasible values for each of $d$ and $f$.\\
(v) Write down the objective function to be minimised.\\
(vi) Write down the resulting LP problem, using slack variables to express the constraints from parts (ii) and (iii) as equations.\\
(You are not required to solve the resulting LP problem.)
\hfill \mbox{\textit{OCR D1 2007 Q2 [10]}}