Single machine controllable scheduling with bounded makespan

In a controllable scheduling environment, the processing time of a job can be shortened by allocating extra resource at a cost, or the job can be declined for processing by paying a penalty. We investigate the single machine controllable scheduling to minimize the sum of the total resource consumption cost, the total job rejection cost, and the makespan of the accepted jobs, where the makespan is upper bounded and the job processing time is a decreasing linear function in the amount of allocated resource. We first show that the studied problem is polynomial solvable if the makespan is unbounded, but otherwise is NP-hard, and characterize important structural properties for the optimal solution; we then take advantage of the structural properties to design several algorithms for the problem, including a pseudo-polynomial time dynamic programming exact algorithm, an O ( n 2 )-time n -approximation algorithm where n is the number of jobs, and building on top of the dynamic programming exact algorithm, the n -approximation algorithm and the bound improvement procedure, two fully polynomial time approximation schemes.