Flow Conservation in Time-Expanded Networks

In a time-expanded network, every node represents a (location, time) pair, and arcs carry flow between these pairs. Flow conservation is the rule that, at every node which is neither a source nor a sink, the total flow entering the node equals the total flow leaving it:

$$\sum_{a \,\in\, \delta^-(v)} x_a \;=\; \sum_{a \,\in\, \delta^+(v)} x_a,$$

where $\delta^-(v)$ denotes the set of arcs ending at $v$ and $\delta^+(v)$ denotes the set of arcs starting at $v.$

For example, suppose node $(B,2)$ has exactly one incoming movement arc carrying $8$ units, one outgoing movement arc carrying $x$ units, and one outgoing holding arc carrying $3$ units. Conservation requires

$$8 \;=\; x + 3,$$

so $x = 5.$