One way to write this constraint is to require that the total flow leaving Austin be equal to the total flow leaving Gainesville. Using the arc numbers of Fig. 2, this constraint is:

This is called a side constraint. The majority of the problem is described using a network model, but this constraint cannot be handled within the network construct. The addition of side constraints makes the special purpose network flow programming algorithms inapplicable, but linear programming can still solve the model. We simply add the side constraint to the linear programming model of the network and solve it with a general purpose linear programming algorithm. When using Excel we must use the Excel Solver with the side constraint option chosen for the network model. The network model is shown below.

To create the side constraint we add the expression shown in the figure in cell K22. We add the requirement that the contents of this cell be 0 into the Excel model. The Solver dialog is shown below with the side constraint listed first. The other constraints are automatically provided by the Math Programming add-in. Clicking the Solve button provides the solution shown on the worksheet.

The optimum for the restricted problem is shown in Fig. 7. The production at Austin is reduced, eliminating the flows on the links from Austin to Atlanta, and Atlanta to Chicago, and not meeting the optional demand at Chicago. The change reduces the profit by $100. Forcing equality between the two plants is clearly not advisable.

Figure 7. The effect of an equality constraint between Austin and Gainesville. z = -1500