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Side Constraints

We return to the model without losses for this illustration. The logistics manager is receiving complaints from the Gainesville plant. While Austin is producing and shipping all its capacity, the Gainesville plant is only shipping 2/3 of its capacity. Since both plants have a capacity of 300, the manager wants to see the effect of a constraint that requires both to ship the same proportion of capacity.

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. In this case, the optimum in Fig. 7 reduces production at Austin, 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

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