Approximation Algorithm for Scheduling Parallel Machines with Machine Eligibility Restrictions and special jobs

This paper addresses the scheduling problem of parallel machines with machine eligibility restrictions and special jobs with the objective of minimizing the makespan. Each job can only be assigned to a specific subset of the machines. And the processing times of jobs are restricted to one of two values, 1 andε. A semi-matching model G=[J∪M,E,W] is presented to formulate this scheduling problem. We propose an approximation algorithm, which is composed of two steps, that is, initial solution construction and initial solution improvement. The initial solution construction algorithm is developed to build a feasible solution by performing a simple greedy heuristic method. The initial solution is used as a starting point by the improvement algorithm. The main idea of the improvement algorithm is to construct alternating tree, then to find the optimal alternating path for each vertex in M iteratively. In order to improve efficiency, the length of each path in alternating tree is limited to 4 at most.