On the Solvability of Open Shop Scheduling Problems

The problem of scheduling n jobs each of which must be processed by m machines with shop dependent processing order is examined. If the order in which a given job is processed on the machine is not fixed, the system is called an open shop. This situation might occur in testing components of an automobile. The computational difficulty of solving most open shop problem is known with the majority being NP- hard. In shop scheduling problem, a combination of machine order and job order is represented by a shop graph or by the corresponding rank matrix pair (MO, JO). A task is to determine an optimal feasible sequence. Various models of mathematics have been presented to solve different shop scheduling problems. Here, the disjunctive graph model, block matrices model without preemption and with preemption and linear programming models are studied. Some problems have also been represented by Gantt charts. Polynomial time algorithms have been presented with complexity analysis. Some NP- hard problems are also discussed in both cases with preemption and without preemption.