### Reasons for Infeasibility

Before optimization commences, PROC NETFLOW tests to ensure that the problem is not infeasible by ensuring that, with respect to supplies, demands, and arc flow bounds, flow conservation can be obeyed at each node.

• Let IN be the sum of lower flow bounds of arcs directed toward a node plus the node’s supply. Let OUT be the sum of capacities of arcs directed from that node plus the node’s demand. If IN exceeds OUT, not enough flow can leave the node.

• Let OUT be the sum of lower flow bounds of arcs directed from a node plus the node’s demand. Let IN be the total capacity of arcs directed toward the node plus the node’s supply. If OUT exceeds IN, not enough flow can arrive at the node.

Reasons why a network problem can be infeasible are similar to those previously mentioned but apply to a set of nodes rather than for an individual node. Consider the network illustrated in Figure 6.13.

Figure 6.13: An Infeasible Network

```
NODE_1----------------->NODE_2
/          capac=55           \
/              lo=50            \
/                                 \
/                                   \
/                                     \
NODE_3                                      NODE_4
supply=100 \                                     / demand=120
\                                   /
\                                 /
\           capac=62            /
\             lo=60           /
NODE_5----------------->NODE_6
```

The demand of NODE_4 is 120. That can never be satisfied because the maximal flow through arcs (NODE_1, NODE_2) and (NODE_5, NODE_6) is 117. More specifically, the implicit supply of NODE_2 and NODE_6 is only 117, which is insufficient to satisfy the demand of other nodes (real or implicit) in the network.

Furthermore, the lower flow bounds of arcs (NODE_1, NODE_2) and (NODE_5, NODE_6) are greater than the flow that can reach the tail nodes of these arcs, that, by coincidence, is the total supply of the network. The implicit demand of nodes NODE_1 and NODE_5 is 110, which is greater than the amount of flow that can reach these nodes.

When PROC NETFLOW detects that the problem is infeasible, it indicates why the solution, obtained after optimization stopped, is infeasible. It can report that the solution cannot obey flow conservation constraints and which nodes these conservation constraints are associated with. If applicable, the side constraints that the solution violates are also output.

If stage 1 optimization obtains a feasible solution to the network, stage 2 optimization can determine that the problem is infeasible and note that some flow conservation constraint is broken while all side constraints are satisfied. The infeasibility messages issued by PROC NETFLOW pertain to why the current solution is infeasible, not quite the same as the reasons why the problem is infeasible. However, the messages highlight areas in the problem where the infeasibility can be tracked down. If the problem is infeasible, make PROC NETFLOW do a stage 1 unconstrained optimization by removing the CONDATA= data set specification in the PROC NETFLOW statement. If a feasible network solution is found, then the side constraints are the source of the infeasibility in the problem.