| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Formulate LP from context |
| Difficulty | Moderate -0.5 Part (a) is straightforward translation of a word problem into LP format with clear constraints from a table. Parts (b) and (c) involve mechanical application of the Simplex algorithm following a prescribed pivot rule and basic interpretation of slack variables. This is a standard D2 question requiring careful arithmetic but no novel insight or problem-solving beyond textbook procedures. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06c Working with constraints: algebra and ad hoc methods7.06d Graphical solution: feasible region, two variables7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| \cline { 2 - 5 } \multicolumn{1}{c|}{} | Processing | Blending | Packing | Profit ( \(\pounds 100\) ) |
| Morning blend | 3 | 1 | 2 | 4 |
| Afternoon blend | 2 | 3 | 4 | 5 |
| Evening blend | 4 | 2 | 3 | 3 |
| \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | 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 |
9. 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{center}
\begin{tabular}{ | l | c | c | c | c | }
\cline { 2 - 5 }
\multicolumn{1}{c|}{} & Processing & Blending & Packing & Profit ( $\pounds 100$ ) \\
\hline
Morning blend & 3 & 1 & 2 & 4 \\
\hline
Afternoon blend & 2 & 3 & 4 & 5 \\
\hline
Evening blend & 4 & 2 & 3 & 3 \\
\hline
\end{tabular}
\end{center}
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)
An initial Simplex tableau for the above situation is
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
\begin{tabular}{ c }
Basic \\
variable \\
\end{tabular} & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 3 & 2 & 4 & 1 & 0 & 0 & 35 \\
\hline
$s$ & 1 & 3 & 2 & 0 & 1 & 0 & 20 \\
\hline
$t$ & 2 & 4 & 3 & 0 & 0 & 1 & 24 \\
\hline
$P$ & - 4 & - 5 & - 3 & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\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.
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.
\item Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2002 Q9 [17]}}