A genetic algorithm for scheduling open shops with conflict graphs to minimize the makespan

The open shop problem with conflict graph consists of scheduling jobs on an open shop system subject to conflict constraints given by a simple undirected graph G, called the conflict graph. In this graph, each vertex represents a job , jobs that are adjacent in G are in conflict, i.e. they cannot be processed at the same time on different machines. The problem of finding a feasible schedule that minimizes the maximum completion time is known to be NP-hard even on two machines. In this paper, we present mixed integer linear programming models, lower bounds, , genetic algorithms for this problem. Extensive computational experiments are conducted on a large set of instances derived from well-known benchmarks of the basic open shop problem. The results show the effectiveness of the genetic algorithm that solves to optimality at least 93.490% of the instances and the average deviation from the lower bounds is within 0.475%. Furthermore, the algorithm improves the upper bounds obtained by the mathematical formulations and outperforms the existing heuristics.