11. A company wishes to transport its products from 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) to a single retail outlet \(R\). The capacities of the possible routes, in van loads per day, are shown in Fig. 5.
\begin{figure}[h]
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\caption{Figure 5}
\includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-05_470_766_447_1695}
\end{figure}
- On Diagram 1 in the answer booklet add a supersource \(S\) to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added.
- State the maximum flow along \(S F _ { 1 } A B R\) and \(S F _ { 3 } C R\).
- Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles.
Taking your answer to part (b)(ii) as the initial flow pattern,
- use the labelling procedure to find a maximum flow from \(S\) to \(R\). Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow.
- Prove that your final flow is maximal.