6.
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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.
- 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.
- State the maximum flow along
- \(W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R\),
- \(W W _ { 3 } \quad C \quad R _ { 2 } \quad R\).
- 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 flow-augmenting route you use, together with its flow.
- From your final flow pattern, determine the number of van loads passing through \(B\) each day.
The company has the opportunity to increase the number of vans loads from one of the warehouses \(W _ { 1 } , W _ { 2 } , W _ { 3 }\), to \(A , B\) or \(C\).
- Determine how the company should use this opportunity so that it achieves a maximal flow.