Scheduling fully parallel jobs

We consider the following scheduling problem. We have m identical machines, where each machine can accomplish one unit of work at each time unit. We have a set of n fully parallel jobs, where each job j has s_j units of workload, and each unit workload can be executed on any machine at any time unit. A job is considered complete when its entire workload has been executed. The objective is to find a schedule that minimizes the total weighted completion time ∑ w_jC_j, where w_j is the weight of job j and C_j is the completion time of job j. We provide theoretical results for this problem. First, we give a PTAS of this problem with fixed m. We then consider the special case where w_j= s_j for each job j, and we show that it is polynomial solvable with fixed m. Finally, we study the approximation ratio of a greedy algorithm, the Largest-Ratio-First algorithm. For the special case, we show that the approximation ratio depends on the instance size, i.e. n and m, while for the general case where jobs have arbitrary weights, we prove that the upper bound of the approximation ratio is 1+m-1m+2.