Total Supply and Total Demand

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Total Supply and Total Demand: How Generalized Networks Differ from Pure Networks

For a pure network, it is easy to check for excess supply or excess demand. If the sum of positive `supdem`

values exceeds (is less than) the absolute value of the sum of negative `supdem`

values, then the network has excess supply (demand).

However, in a generalized network you need to specify whether the network should have excess supply or excess demand. To do
that you can specify the option EXCESS=SUPPLY or EXCESS=DEMAND, respectively.

Although the total supply and total demand of a generalized network can be determined, you may not know beforehand if excess
flow must be added to, removed from, or left unused by the network. For example, consider a simple network, one consisting
of two nodes, A and B, connected by a single arc, A —> B. Suppose the supply of node A is 10 and the demand of node B is 30.
If this is a pure network, then the network solved must be either `_EXCESS_`

—> A —> B if the THRUNET option is not specified and the flow through the arc between A and B is 30 units, or A —> B <— `_EXCESS_`

if the THRUNET option is specified and the flow through the arc from A to B is 10 units. `_EXCESS_`

is the name of an extra node that is set up by the procedure behind the scenes, and in both cases it would have a supply
capacity of 20 units, which is the flow through the excess arc. However, if the network is generalized, and the arc from A
to B has a multiplier of 3.0, then the flow through the arc from A to B would be 10 units, and the network would be feasible
without any excess node and arcs. Indeed, no excess node and arcs would be created, even though total supply and total demand
are unequal. Therefore, once the NETFLOW procedure detects that the network has arc multipliers that are not 1.0, it might
not set up the excess node and the excess arcs.

In Example 6.11 we illustrate the use of the EXCESS= option to solve generalized networks that have total supply equal to total demand, but
have arcs with varying multipliers.

In the section Handling Missing Supply and Demand Simultaneously, we discuss the case where a network has both nodes with missing S supply values and nodes with missing D demand values.
In the next two subsections we analyze scenarios where a network has nodes with either missing S supply values or missing
D demand values, but not both.

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