| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Perform one Simplex iteration |
| Difficulty | Moderate -0.5 This is a routine application of the Simplex algorithm requiring two iterations with standard pivot selection rules. While it involves multiple steps and careful arithmetic, it follows a completely mechanical procedure taught directly in D2 with no problem-solving or insight required—students simply apply the algorithm as learned. The method is explicitly scaffolded (pivot column selection rule given, row operations to be stated). This is easier than average A-level maths questions which typically require some problem-solving, but not trivial since it requires accurate execution of a multi-step procedure. |
| Spec | 7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| Basic Variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 4 | 3 | \(\frac { 5 } { 2 }\) | 1 | 0 | 0 | 50 |
| \(s\) | 1 | 2 | 1 | 0 | 1 | 0 | 30 |
| \(t\) | 0 | 5 | 1 | 0 | 0 | 1 | 80 |
| \(P\) | - 25 | - 40 | - 35 | 0 | 0 | 0 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(M1 A1\) | — |
| \(B1\) | — | Row operations CAO; allow if given in terms of old row 2 |
| \(M1 A1\) | — | (ft) Correct row operations used at least once, column \(x\), \(z\), \(s\) or value correct |
| \(A1\) | — | CAO on numbers (ignore row operations and b.v.) |
| \(M1\) | — | Correct pivot located and b.v. changed. If choosing negative pivot 2B0 3M0 |
| \(B1\) | — | Row operations CAO |
| \(M1 A1\) | — | (ft) Correct row operations used at least once, column \(x\), \(r\), \(s\) or value correct |
| \(A1\) | — | CAO on numbers (ignore row operations and b.v.) |
| \(B1\) | (9) | — |
| \(B1\) | — | — |
| \(M1 A1\) | — | — |
| (b) | \(B1\) | (1) |
| (c) | \(B2, 1, 0\) | (2) |
| 12 marks |
**(a)** | $M1 A1$ | — | Correct pivot located, attempt to divide row. If choosing negative pivot no marks |
| $B1$ | — | Row operations CAO; allow if given in terms of old row 2 |
| $M1 A1$ | — | (ft) Correct row operations used at least once, column $x$, $z$, $s$ or value correct |
| $A1$ | — | CAO on numbers (ignore row operations and b.v.) |
| $M1$ | — | Correct pivot located and b.v. changed. If choosing negative pivot 2B0 3M0 |
| $B1$ | — | Row operations CAO |
| $M1 A1$ | — | (ft) Correct row operations used at least once, column $x$, $r$, $s$ or value correct |
| $A1$ | — | CAO on numbers (ignore row operations and b.v.) |
| $B1$ | (9) | — |
| $B1$ | — | — |
| $M1 A1$ | — | — |
**(b)** | $B1$ | (1) | — |
**(c)** | $B2, 1, 0$ | (2) | **Explanation.** Must have gained at least 2 M marks in (a), must refer to increasing $x$, $r$ and $s$ (condone no ref to $y = z = t = 0$), must have correct signs in equation in (b). Do not accept 'negatives in profit row' o.e. alone |
| | 12 marks | |
---4. The tableau below is the initial tableau for a three-variable linear programming problem in $x , y$ and $z$. The objective is to maximise the profit, $P$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
Basic Variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 4 & 3 & $\frac { 5 } { 2 }$ & 1 & 0 & 0 & 50 \\
\hline
$s$ & 1 & 2 & 1 & 0 & 1 & 0 & 30 \\
\hline
$t$ & 0 & 5 & 1 & 0 & 0 & 1 & 80 \\
\hline
$P$ & - 25 & - 40 & - 35 & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Taking the most negative number in the profit row to indicate the pivot column at each stage, perform two complete iterations of the simplex algorithm to obtain tableau T. Make your method clear by stating the row operations you use.
\item Write down the profit equation given by T .
\item Use your answer to (b) to determine whether T is optimal, justifying your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2014 Q4 [12]}}