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 The NETFLOW Procedure

## Output Data Sets

The procedure determines the flow that should pass through each arc as well as the value assigned to each nonarc variable. The goal is that the minimum flow bounds, capacities, lower and upper value bounds, and side constraints are not violated. This goal is reached when total cost incurred by such a flow pattern and value assignment is feasible and optimal. The solution found must also conserve flow at each node.

The ARCOUT= data set contains a solution obtained when performing optimization that does not consider any constraints. The NODEOUT= data set contains nodal dual variable information for this type of solution. You can choose to have PROC NETFLOW create the ARCOUT= data set and the NODEOUT= data set and save the optimum of the network or the nodal dual variable values before any optimization that considers the side constraints is performed.

If there are side constraints, the CONOUT= data set can be produced and contains a solution obtained after performing optimization that considers constraints. The DUALOUT= data set contains dual variable information for nodes and side constraints from the solution obtained after optimization that considers the constraints. The CONOUT= data set and DUALOUT= data set can be used to save the constrained optimal solution.

### ARCOUT= and CONOUT= Data Sets

The ARCOUT= and CONOUT= data sets contain the same variables. These variables and their possible values in an observation are

_FROM_
a tail node of an arc. This is a missing value if an observation has information about a nonarc variable
_TO_
a head node of an arc. This is a missing value if an observation has information about a nonarc variable
_COST_
the cost of an arc or the objective function coefficient of a nonarc variable
_CAPAC_
the capacity of an arc or upper value bound of a nonarc variable
_LO_
the lower flow bound of an arc or lower value bound of a nonarc variable
_NAME_
a name of an arc or nonarc variable
_SUPPLY_
the supply of the tail node of the arc in the observation. This is a missing value if an observation has information about a nonarc variable
_DEMAND_
the demand of the head node of the arc in the observation. This is a missing value if an observation has information about a nonarc variable
_FLOW_
the flow through the arc or value of the nonarc variable
_FCOST_
flow cost, the product of _COST_ and _FLOW_
_RCOST_
the reduced cost of the arc or nonarc variable
_ANUMB_
the number of the arc (positive) or nonarc variable (nonpositive). Used for warm starting PROC NETFLOW
_TNUMB_
the number of the tail node in the network basis spanning tree. Used for warm starting PROC NETFLOW
_STATUS_
the status of the arc or nonarc variable

The variables present in the ARCDATA= data set are present in an ARCOUT= data set or a CONOUT= data set. For example, if there is a variable called tail in the ARCDATA= data set and you specified the SAS variable list
```from tail;
```
then tail is a variable in the ARCOUT= and CONOUT= data sets instead of _FROM_. Any ID list variables also appear in the ARCOUT= and CONOUT= data sets.

### NODEOUT= and DUALOUT= Data Sets

There are two types of observations in the NODEOUT= and DUALOUT= data sets. One type of observation contains information about a node. These are called type N observations. There is one such observation of this type for each node. The _NODE_ variable has a name of a node, and the _CON_ variable values in these observations are missing values.

The other type of observation contains information about constraints. These are called the type C observations. There is one such observation for each constraint. The _CON_ variable has a name of a constraint, and the _NODE_ variable values in these observations are missing values.

Many of the variables in the NODEOUT= and DUALOUT= data sets contain information used to warm start PROC NETFLOW. The variables _NODE_, _SD_, _DUAL_, _VALUE_, _RHS_, _TYPE_, and _CON_ contain information that might be of interest to you.

The NODEOUT= and DUALOUT= data sets look similar, as the same variables are in both. These variables and their values in an observation of each type are

_NODE_
Type N: the node name.
Type C: a missing value.
_SD_
Type N: the supply (positive) or demand (negative) of the node.
Type C: a missing value.
_DUAL_
Type N: the dual variable value of the node in _NODE_.
Type C: the dual variable value of the constraint named in _CON_.
_NNUMB_
Type N: the number of the node named in _NODE_.
Type C: the number of the constraint named in _CON_.
_PRED_
Type N: the predecessor in the network basis spanning tree of the node named in _NODE_.
Type C: the number of the node toward which the arc with number in _ARCID_ is directed, or the constraint number associated with the slack, surplus, or artificial variable basic in this row.
_TRAV_
Type N: the traversal thread label of the node named in _NODE_.
Type C: a missing value.
_SCESS_
Type N: the number of successors (including itself) in the network basis spanning tree of the node named in _NODE_.
Type C: a missing value.
_ARCID_
Type N: if _ARCID_ is nonnegative, _ARCID_ is the number of the network basis spanning tree arc directed from the node with number _PRED_ to the node named in _NODE_. If _ARCID_ is negative, minus _ARCID_ is the number of the network basis spanning tree arc directed from the node named in _NODE_ to the node with number _PRED_.
Type C: if _ARCID_ is positive, _ARCID_ is the number of the arc basic in a constraint row. If nonpositive, minus _ARCID_ is the number of the nonarc variable basic in a constraint row.
_FLOW_
Type N: the flow minus the lower flow bound of the arc _ARCID_.
Type C: the flow minus lower flow bound of the arc _ARCID_ or value lower bound of the nonarc variable value minus _ARCID_.
_FBQ_
Type N: if _FBQ_ is positive, then _FBQ_ is the subscript in arc length arrays of the first arc directed toward the node named in _NODE_. PROC NETFLOW's arc length arrays are sorted so that data of arcs directed toward the same head node are together. If _FBQ_ is negative, no arcs are directed toward the node named in _NODE_. Arcs directed toward node i have subscripts in the arc length arrays between observations FBQ(i) abd (FBQ(i+1))- 1, inclusive.
Type C: missing value.
_VALUE_
Type N: missing value.
Type C: the lhs value (the sum of the products of coefficient and flows or values) of the constraint named in _CON_.
_RHS_
Type N: missing value.
Type C: the rhs value of the constraint named in _CON_.
_TYPE_
Type N: missing value.
Type C: the type of the constraint named in _CON_.
_CON_
Type N: missing value.
Type C: the name of the constraint.

If specified in variable lists, the variables in the input data sets are used instead of some of the previous variables. These variables are specified in the NODE, SUPDEM, RHS, TYPE, and ROW (if there is only one variable in the ROW list) lists and are used instead of _NODE_, _SD_, _RHS_, _TYPE_, and _CON_, respectively.

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