Tribhuvan University Institute of Science and Technology Irreducibility on Open Shop Scheduling Problems Thesis Submitted to Central Department of Mathematics Kirtipur, Kathmandu, Nepal In partial fulfillment of the requirements for the Master's Degree in Mathematics by Raju Prasad Bhusal Date: March 2010 ABSTRACT We consider the classical open shop scheduling problem, where each job must be processed on each machine at least once. Our task is to determine the feasible combination of all job orders and machine orders minimizing the given objective function. Shop problems are modeled with the pair of mathematical models. As most of the problems belong to class NP-hard, our focus is to study the irreducibility theory and deal with the complexity of shop problem. There is no polynomial time algorithm for irreducibility test in general case but the problem is solved for two machines. We study some necessary and sufficient conditions for irreducibility and conclude that sourcesink irreducibility test is efficient among them. Moreover, irreducibility with implication classes and decomposition approach are studied. We study the generalized concept of irreducibility as dominance relation.