Bi-criteria scheduling on a single parallel-batch machine

This paper considers bi-criteria scheduling on a single parallel-batch machine to minimizing two regular scheduling criteria that are non-decreasing in the job completion times. For minimizing the total weighted completion time and the makespan simultaneously, we prove that the problem is NP-hard in the ordinary sense and possesses a fully polynomial-time approximation scheme. For minimizing the weighted total number of tardy jobs and the makespan simultaneously, we provide a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme. Furthermore, we identify that all Pareto-optimal solutions for ∑j=1nwjCj,Cmax and ∑j=1nwjUj,Cmax can be found in pseudo-polynomial time, respectively.