PERT scheduling with convex cost functions

This paper deals with the problem of finding 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 first optimality necessary and sufficient condition bearing on the head and tail blocks of a schedule is first 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 coefficients, 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 efficient polynomial algorithms. (C) 2002 Elsevier Science B.V. All rights reserved.