||In this research, we consider the problem of scheduling n preemptive jobs with distinct release time on m identical parallel machines under machine availability and eligibility constraints to minimize total completion time. To find the optimal solution, we propose a branch and bound algorithm. First, using the idea of time epochs which was proposed from Liao and Sheen (2008) to separate machine available intervals. Then adopt the branching scheme from Mellouli et al. (2009) to branch out a node by assigning an additional job on each available interval specified by two adjacent time epochs. Second, according to our bounding scheme, we develop an iterated heuristic upper bound based on bipartite matching algorithm to find an initial feasible upper bound. By relaxing the restriction of machine eligibility constraint, we used extended shortest remaining processing time (SRPT) which proposed from Yalaoui and Chu (2006) to solve the reduced problem optimally, and set the optimal solution of it to be our lower bound. Computational analysis shows that the efficiency of our branch and bound algorithm can eliminate unnecessary nodes up to 99%. Due to the result of the average percentage gap of our upper bound, it can show that this upper bound is not only a polynomial time bound but also a tight bound.|
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