| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matchings and Allocation |
| Type | Transportation problem: balanced problem |
| Difficulty | Moderate -0.8 This is a standard algorithmic transportation problem requiring mechanical application of taught methods (north-west corner, stepping-stone) with no conceptual insight or problem-solving. The question explicitly guides students through each step, making it easier than average A-level maths questions which typically require some independent thinking. |
| Spec | 7.07a Simplex tableau: initial setup in standard format |
| \(\mathbf { A }\) | \(\mathbf { B }\) | \(\mathbf { C }\) | Supply | |
| \(\mathbf { X }\) | 17 | 8 | 7 | 22 |
| \(\mathbf { Y }\) | 16 | 12 | 15 | 17 |
| \(\mathbf { Z }\) | 6 | 10 | 9 | 15 |
| Demand | 16 | 15 | 23 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | The supply is equal to the demand | B1 (1) |
| (b) | \(\begin{array}{c | ccc} & A & B & C \\ \hline X & 16 & 6 & \\ Y & & 9 & 8 \\ Z & & & 15 \end{array}\) |
| (c) | \(\begin{array}{c | ccc} & A & B & C \\ \hline X & 16- \theta & 6+ \theta & \\ Y & & 9- \theta & 8+ \theta \\ Z & \theta & & 15- \theta \end{array}\) with Value of \(\theta = 9\), exiting cell is YB |
| (d) | \(\begin{array}{c | ccc} & 17 & 8 & 20 \\ & A & B & C \\ \hline 0 & X & 7 & 15 \\ -5 & Y & & & 17 \\ -11 & Z & 9 & & 6 \end{array}\) |
| \(XC = 7 - 0 - 20 = -13\); \(YA = 16 + 5 - 17 = 4\); \(YB = 12 + 5 - 8 = 9\); \(ZB = 10 + 11 - 8 = 13\) | A1 (3) | |
| \(\begin{array}{c | ccc} & A & B & C \\ \hline X & 7- \theta & 15 & \theta \\ Y & & & 17 \\ Z & 9+ \theta & & 6- \theta \end{array}\) with Value of \(\theta = 6\), entering cell XC, exiting cell ZC | M1 A1 |
| \(\begin{array}{c | ccc} & A & B & C \\ \hline X & 1 & 15 & 6 \\ Y & & & 17 \\ Z & 15 & & \end{array}\) | A1 (3) |
| Cost (£) 524 | B1 (1) |
(a) | The supply is equal to the demand | B1 (1) | |
(b) | $\begin{array}{c|ccc} & A & B & C \\ \hline X & 16 & 6 & \\ Y & & 9 & 8 \\ Z & & & 15 \end{array}$ | B1 (1) | |
(c) | $\begin{array}{c|ccc} & A & B & C \\ \hline X & 16- \theta & 6+ \theta & \\ Y & & 9- \theta & 8+ \theta \\ Z & \theta & & 15- \theta \end{array}$ with Value of $\theta = 9$, exiting cell is YB | M1 A1 A1 (3) | |
(d) | $\begin{array}{c|ccc} & 17 & 8 & 20 \\ & A & B & C \\ \hline 0 & X & 7 & 15 \\ -5 & Y & & & 17 \\ -11 & Z & 9 & & 6 \end{array}$ | M1 A1 | |
| $XC = 7 - 0 - 20 = -13$; $YA = 16 + 5 - 17 = 4$; $YB = 12 + 5 - 8 = 9$; $ZB = 10 + 11 - 8 = 13$ | A1 (3) | |
| $\begin{array}{c|ccc} & A & B & C \\ \hline X & 7- \theta & 15 & \theta \\ Y & & & 17 \\ Z & 9+ \theta & & 6- \theta \end{array}$ with Value of $\theta = 6$, entering cell XC, exiting cell ZC | M1 A1 | |
| $\begin{array}{c|ccc} & A & B & C \\ \hline X & 1 & 15 & 6 \\ Y & & & 17 \\ Z & 15 & & \end{array}$ | A1 (3) | |
| Cost (£) 524 | B1 (1) | |
**Total [12]**
---6. The table below shows the cost, in pounds, of transporting one unit of stock from each of three supply points, $\mathrm { X } , \mathrm { Y }$ and Z to three demand points, $\mathrm { A } , \mathrm { B }$ and C . It also shows the stock held at each supply point and the stock required at each demand point.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
& $\mathbf { A }$ & $\mathbf { B }$ & $\mathbf { C }$ & Supply \\
\hline
$\mathbf { X }$ & 17 & 8 & 7 & 22 \\
\hline
$\mathbf { Y }$ & 16 & 12 & 15 & 17 \\
\hline
$\mathbf { Z }$ & 6 & 10 & 9 & 15 \\
\hline
Demand & 16 & 15 & 23 & \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item This is a balanced problem. Explain what this means.
\item Use the north west corner method to obtain a possible solution.
\item Taking ZA as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear and state your exiting cell.
\item Perform one more iteration of the stepping-stone method to find a further improved solution. You must make your shadow costs, improvement indices, entering cell, exiting cell and route clear.
\item State the cost of the solution you found in part (d).
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2009 Q6 [12]}}