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 $$s_j$$ 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 $$\sum w_j C_j$$ ∑ w j C j , where $$w_j$$ w j is the weight of job j and $$C_j$$ 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 $$w_j = s_j$$ 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 $$1 + \frac{m-1}{m+2}$$ 1 + m - 1 m + 2 .