| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2003 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Complete Simplex solution |
| Difficulty | Challenging +1.2 This is a standard Simplex algorithm question requiring mechanical application of the method with 2-3 iterations. Part (a) tests basic setup recognition, parts (b)-(c) require routine pivot operations following the algorithm. While it's a Further Maths topic (D2), the execution is algorithmic with no novel problem-solving required, making it moderately above average difficulty primarily due to the computational care needed. |
| 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 |
| \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value | ||
| \(r\) | 2 | 3 | 4 | 1 | 0 | 8 | ||
| \(s\) | 3 | 3 | 1 | 0 | 1 | 10 | ||
| \(P\) | - 8 | - 9 | - 5 | 0 | 0 | 0 |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(P\) |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(P\) |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(P\) |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(P\) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(2x + 3y + 4z \leq 8\); \(3x + 3y + z \leq 10\); \(P = 8x + 9y + 5z\) | B1 B1 B1 |
| (b) | First tableau with basic variables \(r, s\) and Value 8, 10, 0; showing \(y\) enters (most negative in P row). Second tableau with basic variables \(y, s\) and pivot on \(y\) column: \(R_1 \div 3\), \(R_2 - 3R_1\), \(R_3 + 9R_1\). Third tableau (optimal) with basic variables \(y, x\) and Value \(\frac{4}{3}, 2, 28\) | M1 A1 M1 A1 |
| (c) | \(P = 28\); \(x = 2, y = \frac{4}{3}\); \(z = 0, r = 0, s = 0\) | M1 A1 A1 |
(a) | $2x + 3y + 4z \leq 8$; $3x + 3y + z \leq 10$; $P = 8x + 9y + 5z$ | B1 B1 B1 | 3 |
(b) | First tableau with basic variables $r, s$ and Value 8, 10, 0; showing $y$ enters (most negative in P row). Second tableau with basic variables $y, s$ and pivot on $y$ column: $R_1 \div 3$, $R_2 - 3R_1$, $R_3 + 9R_1$. Third tableau (optimal) with basic variables $y, x$ and Value $\frac{4}{3}, 2, 28$ | M1 A1 M1 A1 | 8 |
(c) | $P = 28$; $x = 2, y = \frac{4}{3}$; $z = 0, r = 0, s = 0$ | M1 A1 A1 | 3 |8. The tableau below is the initial tableau for a maximising linear programming problem.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
\begin{tabular}{ c }
Basic \\
variable \\
\end{tabular} & $x$ & $y$ & $z$ & $r$ & $s$ & Value \\
\hline
$r$ & 2 & 3 & 4 & 1 & 0 & 8 \\
$s$ & 3 & 3 & 1 & 0 & 1 & 10 \\
\hline
$P$ & - 8 & - 9 & - 5 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item For this problem $x \geq 0 , y \geq 0 , z \geq 0$. Write down the other two inequalities and the objective function.
\item Solve this linear programming problem.
You may not need to use all of these tableaux.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & Value \\
\hline
& & & & & & \\
\hline
$P$ & & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & Value \\
\hline
& & & & & & \\
\hline
$P$ & & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & Value \\
\hline
& & & & & & \\
\hline
$P$ & & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & Value \\
\hline
& & & & & & \\
\hline
$P$ & & & & & & \\
\hline
\end{tabular}
\end{center}
\item State the final value of $P$, the objective function, and of each of the variables.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2003 Q8 [14]}}