# Network

**DOI:**https://doi.org/10.1007/1-4020-0611-X_662

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A network is a pair of sets (*N*, *A*), where *N* is a set of nodes (points, vertices) and *A* is a set of arcs (edges, lines, links). If *i* and *j* are nodes, then the arc joining them is denoted by the ordered pair (*i*, *j*). An arc may have a cost *c*_{ij} that denotes the cost per unit flow across that arc, and an upper bound flow capacity denoted by *u*_{ij}. For some applications, a node may be a supply node in which goods enter the network, a demand node in which goods leave the network, or a transshipment node through which goods are shipped without a gain or a loss. In most network applications, it is assumed that the flow of goods that enter a node is equal to the flow that leaves the node. This is the *conservation of flow assumption.*How-ever, in some applications, the amount of goods that enter a node can be more than the amount that leaves the node (e.g., due to the expansion of a liquid) or can be less than the amount that leaves a node (e.g., due to a leak or pilferage). These latter...