Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints

Abstract

Suppose n jobs are to be sequenced for processing by a single machine, with the object of minimizing total weighted completion time. It is shown that the problem is NP-complete if there are arbitrary precedence constraints. However, if precedence constraints are “series parallel”, the problem can be solved in O( n log n ) time. This result generalizes previous results for the more special case of rooted trees. It is also shown how a decomposition procedure suggested by Sidney can be implemented in polynomial-bounded time. Equivalence of the sequencing problem with the optimal linear ordering problem for directed graphs is discussed.