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Convex Costs and Concave Revenues

Consider an arc that carries flow between two nodes. We learn that the first 10 units of flow have a unit cost of $5, the next 5 units of flow have a cost of $8, and any additional amount has a cost of $10.

With unit cost is increasing with flow, the arc cost is a convex function of flow. Fig. 25 shows a linear model that has an arc for each level of unit cost. In cases where the cost function is continuous, the model uses a piecewise linear approximation with the number of arcs determining the accuracy of the approximation. Concave cost functions cannot be addressed with linear models.

Nonlinear concave revenue functions can be handled in a similar way. The revenues are represented with negative arc costs.

Figure 25. Representation of nonlinear costs

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