Holding Arcs, Movement Arcs, and Throughput Arcs

In a time-expanded network, every node has the form $(v, t)$, where $v$ is a location and $t$ is a time period. A holding arc (also called a storage or waiting arc) is an arc of the form

$$(v,\, t) \;\to\; (v,\, t+1)$$

that represents keeping one unit of flow at location $v$ from period $t$ to period $t+1$.

The two key parameters of a holding arc are:

• Capacity: the storage capacity available at $v$ per period.
• Cost: the per-unit holding cost at $v$ for one period.

For example, if a warehouse $W$ has storage capacity $30$ and holding cost \2$per unit per period, then over time periods$t = 0, 1, 2$the holding arcs at$W$ are

$$(W,0)\to(W,1), \quad (W,1)\to(W,2),$$

each with capacity $30$ and unit cost \2$.