A technique is developed in order to find best possible bounds under given conditions for the expected critical time in network planning. This is equivalent to solving extremum problems in certain function spaces. The purpose of the paper is to show how such an inifinite dimensional variational problem can be reduced to a maximum or minimum problem for a function of a finite number of variables. The case with independent activities is discussed in some detail and it is indicated how the procedure should be modified to cover the case of dependent activities.