| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Formulate LP from context |
| Difficulty | Moderate -0.3 This is a standard linear programming problem using the Simplex algorithm with straightforward formulation and execution. Part (a) requires routine constraint writing, part (b) involves mechanical application of the Simplex method with clear pivot selection rules, and part (c) asks for basic interpretation of slack variables. While it has multiple parts worth 17 marks total, each step follows textbook procedures without requiring novel insight or complex problem-solving, making it slightly easier than average for A-level. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| Processing | Blending | Packing | Profit (£100) | |
| Morning blend | 3 | 1 | 3 | 4 |
| Afternoon blend | 2 | 3 | 4 | 5 |
| Evening blend | 4 | 2 | 3 | 3 |
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 3 | 2 | 4 | 1 | 0 | 0 | 35 |
| \(s\) | 1 | 3 | 2 | 0 | 1 | 0 | 20 |
| \(t\) | 2 | 4 | 3 | 0 | 0 | 1 | 24 |
| \(P\) | \(-4\) | \(-5\) | \(-3\) | 0 | 0 | 0 | 0 |
T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
\begin{tabular}{c|cccc}
& Processing & Blending & Packing & Profit (£100) \\
\hline
Morning blend & 3 & 1 & 3 & 4 \\
Afternoon blend & 2 & 3 & 4 & 5 \\
Evening blend & 4 & 2 & 3 & 3
\end{tabular}
The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit.
Let $x$, $y$ and $z$ be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
\begin{enumerate}[label=(\alph*)]
\item Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. [4]
\end{enumerate}
An initial Simplex tableau for the above situation is
\begin{tabular}{c|ccccccc}
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 3 & 2 & 4 & 1 & 0 & 0 & 35 \\
$s$ & 1 & 3 & 2 & 0 & 1 & 0 & 20 \\
$t$ & 2 & 4 & 3 & 0 & 0 & 1 & 24 \\
$P$ & $-4$ & $-5$ & $-3$ & 0 & 0 & 0 & 0
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{1}
\item Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage. [11]
\end{enumerate}
T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{2}
\item Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 Q9 [17]}}