| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Network Flows |
| Type | Transportation problem: north-west corner |
| Difficulty | Moderate -0.3 This is a standard textbook transportation problem using well-defined algorithms (north-west corner rule, improvement indices, stepping-stone method). While it requires multiple steps and careful arithmetic across four parts, each part follows a mechanical procedure taught directly in D2 with no novel problem-solving or insight required. The algorithms are algorithmic rather than conceptual, making this slightly easier than average A-level maths. |
| Spec | 7.04f Network problems: choosing appropriate algorithm |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Business | |||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | \(B _ { 1 }\) | \(B _ { 2 }\) | \(B _ { 3 }\) | |
| \multirow{3}{*}{Factory} | \(F _ { 1 }\) | 10 | 4 | 11 |
| \cline { 2 - 5 } | \(F _ { 2 }\) | 12 | 5 | 8 |
| \cline { 2 - 5 } | \(F _ { 3 }\) | 9 | 6 | 7 |
7. A steel manufacturer has 3 factories $F _ { 1 } , F _ { 2 }$ and $F _ { 3 }$ which can produce 35,25 and 15 kilotonnes of steel per year, respectively. Three businesses $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ have annual requirements of 20,25 and 30 kilotonnes respectively. The table below shows the cost $C _ { i j }$ in appropriate units, of transporting one kilotonne of steel from factory $F _ { i }$ to business $B _ { j }$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{Business} \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & $B _ { 1 }$ & $B _ { 2 }$ & $B _ { 3 }$ \\
\hline
\multirow{3}{*}{Factory} & $F _ { 1 }$ & 10 & 4 & 11 \\
\cline { 2 - 5 }
& $F _ { 2 }$ & 12 & 5 & 8 \\
\cline { 2 - 5 }
& $F _ { 3 }$ & 9 & 6 & 7 \\
\hline
\end{tabular}
\end{center}
The manufacturer wishes to transport the steel to the businesses at minimum total cost.
\begin{enumerate}[label=(\alph*)]
\item Write down the transportation pattern obtained by using the North-West corner rule.
\item Calculate all of the improvement indices $I _ { i j }$, and hence show that this pattern is not optimal.
\item Use the stepping-stone method to obtain an improved solution.
\item Show that the transportation pattern obtained in part (c) is optimal and find its cost.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2002 Q7 [10]}}