| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Complete Simplex solution |
| Difficulty | Standard +0.3 This is a standard textbook Simplex algorithm question requiring mechanical application of the method: setting up the initial tableau, performing two iterations with the pivot column specified for the first iteration, and interpreting the final tableau. While it involves multiple steps and careful arithmetic, it requires no problem-solving insight or novel thinking—just systematic application of a learned algorithm. Slightly easier than average since the first pivot is given and it's a routine 3-variable, 3-constraint problem. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct initial tableau with slack variables \(s_1, s_2, s_3\): \(P - 4x - 3y - z = 0\); rows: \(2x+y+z+s_1=25\); \(x+2y+z+s_2=40\); \(x+y+2z+s_3=30\) | B1 B1 | B1 for objective row, B1 for constraint rows |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Pivot column is \(x\); ratios: \(25/2, 40/1, 30/1\); smallest is \(25/2\), pivot on row 1 | M1 | M1 for correct pivot selection |
| Correct row operations to produce new tableau with \(x\) as basic variable | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify new pivot column (most negative in \(P\)-row); perform correct ratio test; correct row operations | M1 M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Read off values of slack variables from final tableau; interpret as unused capacity/slack in each constraint | B1 B1 B1 | One mark per correct slack variable value with interpretation |
## Question 6:
**(a)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct initial tableau with slack variables $s_1, s_2, s_3$: $P - 4x - 3y - z = 0$; rows: $2x+y+z+s_1=25$; $x+2y+z+s_2=40$; $x+y+2z+s_3=30$ | B1 B1 | B1 for objective row, B1 for constraint rows |
**(b)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Pivot column is $x$; ratios: $25/2, 40/1, 30/1$; smallest is $25/2$, pivot on row 1 | M1 | M1 for correct pivot selection |
| Correct row operations to produce new tableau with $x$ as basic variable | M1 A1 | |
**(c)(i)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify new pivot column (most negative in $P$-row); perform correct ratio test; correct row operations | M1 M1 A1 | |
**(c)(ii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Read off values of slack variables from final tableau; interpret as unused capacity/slack in each constraint | B1 B1 B1 | One mark per correct slack variable value with interpretation |6
\begin{enumerate}[label=(\alph*)]
\item Display the following linear programming problem in a Simplex tableau.\\
Maximise $\quad P = 4 x + 3 y + z$\\
subject to
$$\begin{aligned}
& 2 x + y + z \leqslant 25 \\
& x + 2 y + z \leqslant 40 \\
& x + y + 2 z \leqslant 30
\end{aligned}$$
and $x \geqslant 0 , \quad y \geqslant 0 , \quad z \geqslant 0$.
\item The first pivot to be chosen is from the $x$-column.
Perform one iteration of the Simplex method.
\item \begin{enumerate}[label=(\roman*)]
\item Perform one further iteration.
\item Interpret your final tableau and state the values of your slack variables.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2013 Q6 [11]}}