Efficient approximations of univariate nonlinearities for linear planning models

Univariate nonlinearities occur either if the variables or parameters of a planning model are nonlinear functions of time, or if linear functions of time are multiplied by each other. Moreover, cost functions or revenue functions consist in most cases of univariate nonlinear terms. These nonlinearities are approximated by several types of piecewise-linear functions either particularly or simultaneously. In the case of linearizing nonlinearities in the decision variables it should be mentioned that the results of the approximation can only be used for linear models, if the convexity of the set of feasible solutions is guaranteed. Since the approximating procedure is based on variable nodes, the fit may be improved by optimizing the positions of these nodes. Therefore, this approach yields a far better fit than an approximation on the basis of equidistant and fixed nodes. As the number of variables and restrictions in linearized models increases according to the number of approximating intervals, the number of nodes determines substantially the size of a LP-model and therefore the expenses of a planning approach. Using the procedure described here, the number of approximating intervals will be minimal at a given goodness of fit and the model's size may be effectively reduced. On the other hand, for a certain number of intervals the most efficient approximation is found.