*Mathematical
Model of the Maximum Flow Problem*

The decision variables are defined *x _{ij}*,
the number of units of flow on arc

*x _{6,7} + x_{6,8} = x_{3,6}
+ x_{5,6}*

After transposing all variables to the left hand side of the equation, the result is

*x _{6,7} + x_{6,8} - x_{3,6}
- x_{5,6} = 0*

The maximum flow problem has no specified net stock
position for any node. However, the goal is to maximize the flow from Node
1 to Node 8. Let the net stock position at Node 1 be +*Z*, and let
the net stock position at Node 8 be -*Z*. The amount *Z* is a
decision variable, and you define it as the maximum flow amount. In the
maximum flow problem, the decision variables are *Z* and the flows
for each arc.

The constraint for Node 1 is

*x _{1,2} + x_{1,3} = Z*

After transposing a decision variable *Z* to
the left hand side of the equation, the result is

*x _{1,2} + x_{1,3} – Z = 0*

The entire linear programming model is