| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2013 |
| Session | January |
| 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 requiring formulation of constraints, graphical solution, and sensitivity analysis. While it involves multiple constraints and a three-part structure, the techniques are routine for D1: plotting inequalities, finding corner points, and evaluating an objective function. The integer solution requirement and price change scenario add minor complexity but follow textbook methods without requiring novel insight. |
| 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 |
|---|---|---|
| Let \(x\) be the number of hats which Jean knits | B1 | must say "number of" |
| Let \(y\) be the number of scarves which Jean knits | B1 | or vice-versa of course |
| \(1.5x + 3y \leq 75\), i.e. \(x + 2y \leq 50\) | B1 | simplification not required |
| \(4x + 2.5y \leq 100\), i.e. \(8x + 5y \leq 200\) | B1 | both |
| \(x \leq 20\) and \(y \leq 20\) | B1 | |
| Graph with lines drawn correctly | B1 | lines (cao) |
| Correct feasible region shaded (hexagon in bottom left corner) | B1 | shading ... follow any set of two horizontal, two vertical and two negatively inclined lines which give a hexagon in the bottom left corner |
| Vertices labelled: \((10,20)\) value \(270\); \((13.64, 18.18)\) value \(277.27\); \((20,8)\) value \(220\) | B1 B1 B1 | |
| [10] |
| Answer | Marks | Guidance |
|---|---|---|
| Objective \(= 7x + 10y\) | B1 | objective |
| Best non-integer point | M1 | considering profits at their three points as indicated |
| Solution: \((12, 19)\) 274, \((13, 18)\) 271 or \((14, 17)\) 268 | A1 | cao |
| So 12 hats and 19 scarves | B1 | cao |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| 10 hats and 20 scarves | B1 | cao |
| £34 | B1 | FT ... *their answer* \(- 240\) |
| [2] |
# Question 6:
## Part (i):
| Let $x$ be the number of hats which Jean knits | B1 | must say "number of" |
| Let $y$ be the number of scarves which Jean knits | B1 | or vice-versa of course |
| $1.5x + 3y \leq 75$, i.e. $x + 2y \leq 50$ | B1 | simplification not required |
| $4x + 2.5y \leq 100$, i.e. $8x + 5y \leq 200$ | B1 | both |
| $x \leq 20$ and $y \leq 20$ | B1 | |
| Graph with lines drawn correctly | B1 | lines (cao) |
| Correct feasible region shaded (hexagon in bottom left corner) | B1 | shading ... follow any set of two horizontal, two vertical and two negatively inclined lines which give a hexagon in the bottom left corner |
| Vertices labelled: $(10,20)$ value $270$; $(13.64, 18.18)$ value $277.27$; $(20,8)$ value $220$ | B1 B1 B1 | |
| | **[10]** | |
## Part (ii):
| Objective $= 7x + 10y$ | B1 | objective |
| Best non-integer point | M1 | considering profits at their three points as indicated |
| Solution: $(12, 19)$ **274**, $(13, 18)$ **271** or $(14, 17)$ **268** | A1 | cao |
| So 12 hats and 19 scarves | B1 | cao |
| | **[4]** | |
## Part (iii):
| 10 hats and 20 scarves | B1 | cao |
| £34 | B1 | FT ... *their answer* $- 240$ |
| | **[2]** | |6 Jean knits items for charity. Each month the charity provides her with 75 balls of wool.\\
She knits hats and scarves. Hats require 1.5 balls of wool each and scarves require 3 balls each. Jean has 100 hours available each month for knitting. Hats require 4 hours each to make, and scarves require 2.5 hours each.
The charity sells the hats for $\pounds 7$ each and the scarves for $\pounds 10$ each, and wants to gain as much income as possible.
Jean prefers to knit hats but the charity wants no more than 20 per month. She refuses to knit more than 20 scarves each month.\\
(i) Define appropriate variables, construct inequality constraints, and draw a graph representing the feasible region for this decision problem.\\
(ii) Give the objective function and find the integer solution which will give Jean's maximum monthly income.\\
(iii) If the charity drops the price of hats in a sale to $\pounds 4$ each, what would be an optimal number of hats and scarves for Jean to knit? Assuming that all hats and scarves are sold, by how much would the monthly income drop?
\hfill \mbox{\textit{OCR MEI D1 2013 Q6 [16]}}