This problem is ready made for a network flow model, and
we use it to describe the several components of this model
type. The network model is graphical in that it is presented
as a collection of the nodes and arcs drawn in the figure.
The nodes represent the cities of this problem, and we name
them with the shortened names of the supply and demand cities.
Arcs are the directed line segments of the figure. The nodes
at its ends identify an arc. One arc in the figure originates
at node Phoenix and terminates at node Chicago.

Flow is associated with the network, entering and leaving
at the nodes and passing through the arcs. Flows entering
or leaving a node from external sources are called *external
flows* and are shown adjacent to the nodes in the square
brackets. A positive external flow is a supply, flow that
enters the network, and a negative external flow is a demand,
flow that leaves the network. Flow is conserved at each node,
implying that the total flow entering a node, either from
arcs or external supplies, must equal the total flow leaving
the node, either to arcs or to external demands. The arc flows
are the decision variables for the network flow programming
model.

Flow is limited in an arc by the lower and upper bounds on
flow. In this example we specify 0 as the lower bound for
all arcs and 200 as the upper bound. Sometimes the term *capacity*
refers to the upper bound on flow. Within the restrictions
imposed by conservation of flow for each node and the bounds
on flow for each arc, there are usually many feasible flows
(a *flow* is an assignment of an arc flow to each arc).
The problem is to find a feasible flow, if one exists, and
an optimal flow from the set of feasible flows.

The criterion for optimality is cost. Associated with each
arc is a cost per unit of flow (the number in the parenthesis).
If a flow *x* passes through the arc with unit cost *c*,
a cost *cx* is incurred. The total cost for the network
is the sum of the arc costs, and the goal of optimization
is to find the feasible flow that minimizes this measure.

We call the model of this section a *pure network flow
model* because the flow entering an arc at its origin node
is equal to the flow leaving the arc at its terminal node.
This is contrasted later to the *generalized network flow
model* that does not have this limitation. The pure model
has the important feature of integral optimum solutions. Whenever
all node external flows and all arc upper and lower bounds
are integer, the flow solution to the pure model is also integer.
As we will see, this has important ramifications.

The model defined in Fig. 1 can be solved to find the optimum
flows. The solution is shown on the next page.