Approximation results for a bicriteria job scheduling problem on a single machine without preemption

We study the problem of minimizing the average weighted completion time on a single machine under the additional constraint that the sum of completion times does not exceed a given bound B(1|∑Cj⩽B|∑wjCj) for which we propose a (2,1)-approximation algorithm. We also address the problem 1|∑cjCj⩽B|∑wjCj for which we present a (2,2)-approximation algorithm. After showing that the problem of minimizing two different sums of weighted completion times is intractable, we present an algorithm which computes a (2(1+ɛ),1) (respectively (2(1+ɛ),2))-approximate Pareto curve for the problem 1‖(∑Cj,∑wjCj) (respectively 1‖(∑cjCj,∑wjCj)).