| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Standard +0.3 This is a standard D1 linear programming formulation question with straightforward constraints derived from a word problem. While it requires careful reading and systematic setup of inequalities, the mathematical techniques are routine (forming constraints, graphing, finding optimal vertex). The multi-part structure and real-world context add length but not conceptual difficulty—this is a typical textbook exercise testing standard LP methodology rather than requiring novel insight. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations |
| Month | September | October | November |
| Manufacturing costs per set during each month (£) | 500 | 800 | 600 |
| Demand for sets at the end of each month | 800 | 1000 | 700 |
| Cost of storing sets during each month ( £ ) | - | 100 | 150 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) | \(x + y + z = 800 + 1000 + 700\) | M1 A1 |
| (a) (ii) | \(z = 2500 - x - y\); costs = \(500x + 800y + 600z + 100(x - 800) + 150(x + y - 1800)\); sub in for \(z\) giving: costs = \(150x + 350y + 1150000\) | M1 A1, M1 A1 |
| (b) | \(x + y \geq 1800\) and \(x + y \leq 2500\) | A2 |
| (c) | Graph showing constraints: \(x = 800\), \(x + y = 2500\), \(x + y = 1800\), and \(y = 0\), with feasible region shaded and vertices labeled \(A, B, C, D\) | B4 |
| (d) | considering vertices \(A, B, C\) and \(D\); minimum cost at \(A\): \(y = 0\) meets \(x + y = 1800\); \(\therefore\) should produce 1800 in Sep, 0 in Oct and 700 in Nov; total cost = £1 420 000 | M1 A1, A1 |
**(a) (i)** | $x + y + z = 800 + 1000 + 700$ | M1 A1 | |
**(a) (ii)** | $z = 2500 - x - y$; costs = $500x + 800y + 600z + 100(x - 800) + 150(x + y - 1800)$; sub in for $z$ giving: costs = $150x + 350y + 1150000$ | M1 A1, M1 A1 | |
**(b)** | $x + y \geq 1800$ and $x + y \leq 2500$ | A2 | |
**(c)** | Graph showing constraints: $x = 800$, $x + y = 2500$, $x + y = 1800$, and $y = 0$, with feasible region shaded and vertices labeled $A, B, C, D$ | B4 | |
**(d)** | considering vertices $A, B, C$ and $D$; minimum cost at $A$: $y = 0$ meets $x + y = 1800$; $\therefore$ should produce 1800 in Sep, 0 in Oct and 700 in Nov; total cost = £1 420 000 | M1 A1, A1 | (15) |6. A company makes lighting sets to be sold to stores for use during the Christmas period. As the product is only required at this time of year, all manufacturing takes place during September, October and November.
The sets are delivered to stores at the end of each of these months. Any sets that have been made but do not need to be delivered at the end of each of September and October are put into storage which the company must pay for.
Let $x , y$ and $z$ be the number of sets manufactured in September, October and November respectively.
The demand for lighting sets and the relevant costs are shown in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Month & September & October & November \\
\hline
Manufacturing costs per set during each month (£) & 500 & 800 & 600 \\
\hline
Demand for sets at the end of each month & 800 & 1000 & 700 \\
\hline
Cost of storing sets during each month ( £ ) & - & 100 & 150 \\
\hline
\end{tabular}
\end{center}
The company must be able to meet the demand at the end of each month and there must be no unsold articles at the end of November.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $z$ in terms of $x$ and $y$.
\item Hence, find an expression for the total costs incurred in terms of $x$ and $y$.
The company wishes to minimise its total costs by modelling this situation as a linear programming problem.
\end{enumerate}\item Find as inequalities the constraints that apply in addition to $x \geq 800$ and $y \geq 0$.\\
(2 marks)
\item On graph paper, illustrate these inequalities and label clearly the feasible region.\\
(4 marks)
\item Use your graph to solve the problem. You must state how many sets should be produced in each month and the total costs incurred by the company.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 Q6 [15]}}