Project scheduling with processing time compression cost and lateness penalties

We consider a project scheduling problem where the precedence constraints graph is an out-tree. The processing times can be compressed by using additional resources, which causes cost and simultaneously reduces the processing times of jobs. The objective is to minimize the sum of total lateness penalties and total compression costs where the cost function for compressing the processing times is linear in the amount of compression. The problem can be decomposed into two types of subproblems. We show that both subproblems can be solved in polynomial time. Using these results, we show that the main problem is solvable in polynomial time if the number of chains in the out-tree is fixed.