| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.8 This is a standard textbook linear programming problem with straightforward constraint formulation (sweets, stickers, market research), routine graphical representation, and vertex testing. The extensions in parts (d) and (e) require only basic interpretation of shadow prices and understanding of constraints. No novel problem-solving insight needed—pure application of D1 algorithms. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Maximise \((P =) 15x + 15y\) subject to \(3x + 10y \leq 3600\), \(3x + 6y \leq 1200\), \(x \geq 2y\), \(x, y \geq 0\) | B1, B1 B3, 2, I, O | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph showing feasible region with vertices identified including profit line and constraint lines labeled | B6, 5, 4, 3, 2, 1P | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Profit line or vertex testing, \((300, 150)\), profit \(\pm 67.50\) | m1 A'/ÂT' (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Production of striker should be increased since the need more the intersection point further from the origin. | B2, I/O (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. The constraint line would be for outside the feasible region as they would not affect it. | B2, y/e (1) |
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Maximise $(P =) 15x + 15y$ subject to $3x + 10y \leq 3600$, $3x + 6y \leq 1200$, $x \geq 2y$, $x, y \geq 0$ | B1, B1 B3, 2, I, O | (5) |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph showing feasible region with vertices identified including profit line and constraint lines labeled | B6, 5, 4, 3, 2, 1P | (1) |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Profit line or vertex testing, $(300, 150)$, profit $\pm 67.50$ | m1 A'/ÂT' (1) | |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Production of striker should be increased since the need more the intersection point further from the origin. | B2, I/O (2) | |
## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. The constraint line would be for outside the feasible region as they would not affect it. | B2, y/e (1) | |
[18]A company produces two types of party bag, Infant and Junior. Both types of bag contain a balloon, a toy and a whistle. In addition the Infant bag contains 3 sweets and 3 stickers and the Junior bag contains 10 sweets and 2 stickers.
The sweets and stickers are produced in the company's factory. The factory can produce up to 3000 sweets per hour and 1200 stickers per hour. The company buys a large supply of balloons, toys and whistles.
Market research indicates that at least twice as many Infant bags as Junior bags should be produced.
Both types of party bag are sold at a profit of 15p per bag. All the bags are sold. The company wishes to maximise its profit.
Let $x$ be the number of Infant bags produced and $y$ be the number of Junior bags produced per hour.
\begin{enumerate}[label=(\alph*)]
\item Formulate the above situation as a linear programming problem. [5]
\item Represent your inequalities graphically, indicating clearly the feasible region. [6]
\item Find the number of Infant bags and Junior bags that should be produced each hour and the maximum hourly profit. Make your method clear. [3]
\end{enumerate}
In order to increase the profit further, the company decides to buy additional equipment. It can buy equipment to increase the production of either sweets or stickers, but not both.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Using your graph, explain which equipment should be bought, giving your reasoning. [2]
\end{enumerate}
The manager of the company does not understand why the balloons, toys and whistles have not been considered in the above calculations.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Explain briefly why they do not need to be considered. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2006 Q6 [18]}}