Scheduling of a two-stage differentiation flowshop to minimize weighted sum of machine completion times

This paper considers the problem of scheduling a two-stage flowshop that consists of a common critical machine in stage one and two independent dedicated machines in stage two. All the jobs require processing first on the common critical machine. Each job after completing its critical operation in stage one will proceed to the dedicated machine of its type for further processing in stage two. The objective is to minimize the weighted sum of stage-two machine completion times. We show that the problem is strongly NP-hard, and develop an O(n3)O(n3) polynomial time algorithm to solve the special case where the sequences of both types of jobs are given. We also design an approximation algorithm with a tight performance ratio of 43 for the general case.