Abstract: This paper deals with the total weighted completion time single machine scheduling with constraints of series-parallel digraph and processing time dependent on starting time.We apply Lawler’s algorithm which is used to solve the total weighted completion time single machine with constraints of series-parallel digraph to our problem by proving the following theorem.Under the condition that job processing time being a linear function of the starting time,jobs in the ρ-maximal initial set I of module M should be processed prior to other jobs in module M,and the schedule is the optimal if the jobs in set I aren’t be preempted by the jobs in N/I.

Key words: scheduling, single machine, linear processing time, series-parallel digraph, the total weighted completion time