12.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
  \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-11_618_1211_253_253}
\end{center}
\end{figure}

A company has 3 warehouses $W _ { 1 } , W _ { 2 }$, and $W _ { 3 }$. It needs to transport the goods stored there to 2 retail outlets $R _ { 1 }$ and $R _ { 2 }$. The capacities of the possible routes, in van loads per day, are shown in Fig 2. Warehouses $W _ { 1 } , W _ { 2 }$ and $W _ { 3 }$ have 14, 12 and 14 van loads respectively available per day and retail outlets $R _ { 1 }$ and $R _ { 2 }$ can accept 6 and 25 van loads respectively per day.
\begin{enumerate}[label=(\alph*)]
\item On Diagram 1 on the answer sheet add a supersource $W$, a supersink $R$ and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added.
\item State the maximum flow along
\begin{enumerate}[label=(\roman*)]
\item $W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R$,
\item $W W _ { 3 } \quad C \quad R _ { 2 } \quad R$.
\end{enumerate}\item Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from $W$ to $R$. Show your working on Diagram 2. List each flowaugmenting route you use, together with its flow.
\item From your final flow pattern, determine the number of van loads passing through $B$ each day.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D2 2002 Q12 [11]}}