PERT scheduling with convex cost functions

This paper deals with the problem of ,nding a minimum cost schedule for a set of dependent activities when a convex cost function is attached to the starting time of each activity. A ,rst optimality necessary and su/cient condition bearing on the head and tail blocks of a schedule is ,rst established. A second such condition that uses the spanning active equality trees of a schedule leads to design a generic algorithm for the general case. When the cost function is the usual earliness–tardiness linear function with assymetric and independent penalty coe/cients, the problem is shown to be solved in O(nmax{n; m}). Finally, the special cases when the precedence graph is an intree or a family of chains are then also shown to be solved by e/cient polynomial algorithms. c © 2002 Elsevier Science B.V. All rights reserved.