7.
\begin{figure}[h]
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\caption{Figure 6}
\includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-10_1018_1557_342_214}
\end{figure}
Figure 6 shows a capacitated, directed network of pipes flowing from two oil fields \(\mathrm { F } _ { 1 }\) and \(\mathrm { F } _ { 2 }\) to three refineries \(\mathrm { R } _ { 1 } , \mathrm { R } _ { 2 }\) and \(\mathrm { R } _ { 3 }\). The number on each arc represents the capacity of the pipe and the numbers in the circles represent a possible flow of 65.
- Find the value of \(x\) and the value of \(y\).
- On Diagram 1 in the answer book, add a supersource and a supersink, and arcs showing their minimum capacities.
- Taking the given flow of 65 as the initial flow pattern, use the labelling procedure on Diagram 2 to find the maximum flow. State clearly your flow augmenting routes.
- Show the maximum flow on Diagram 3 and write down its value.
- Verify that this is the maximum flow by finding a cut equal to the flow.