Scheduling two-machine preemptive open shops to minimize total completion time

This paper deals with the problem of scheduling two-machine preemptive open shops to minimize total completion time. The unweighted version of this problem is known to be NP-hard in the ordinary sense, while the weighted version of this problem is known to be NP-hard in the strong sense. Based on the analysis of problem characteristics, several fundamental properties are presented. A dynamic programming (DP) algorithm is developed to optimally solve both the un-weighted and weighted versions of the problem. Also, an efficient heuristic is proposed for solving large-sized problems. Computational results show that the proposed DP algorithm can handle problems with up to 30 jobs within a reasonable amount of time, and that the proposed heuristic has an average percentage deviation of less than 0.5% from the optimal solution value for problems with up to 30 jobs.