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Understanding Network Flows: The Labelling Procedure for Capacity and Flow Management

This guide explores the labelling procedure for network flows, focusing on how to visually represent spare capacity and flow direction using forward and backward facing arrows. It illustrates how to indicate excess capacity and back capacity, along with the saturation points of various arcs within the network. The procedure also delves into flow augmenting paths and how to assess the initial flow, culminating in a diagram that represents final flows on each arc. By understanding these concepts, one can effectively manage and optimize network flows.

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Understanding Network Flows: The Labelling Procedure for Capacity and Flow Management

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  1. NETWORK FLOWS THE LABELLING PROCEDURE Use the forward facing arrow to indicate the spare capacity in that arc - the amount by which the flow can be increased - the excess capacity. Use the backward facing arrow to indicate the flow along the arc, i.e. the amount by which the flow could be decreased - the back capacity.

  2. 0 SB is now saturated AC is now saturated BC is now saturated CT is now saturated 1 3 2 4 3 1 1 3 2 0 4 1 0 1 1 2 2 6 3 3 5 4 2 4 1 2 3 4 5 Flow out of S and into T is now 9 Flow out of S and into T is now 8 The Initial Flow is 7. 7 out of S and 7 into T 2 6 4 Flow out of S and into T is now 10 1 BD is now saturated 0 0 We now look for Flow Augmenting Paths There are no other flows through the network as every route is now blocked by a saturated arc. Initial Flow: SACT Flow 3 SBDT Flow 4 Next path is SBCDT Flow 1 Next path is SABCDT Flow 1 One such path is SBCT Flow 1

  3. The final flow along each arc should then be represented on a fresh diagram: The Maximum Flow is 10

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