An optimal method for the preemptive job shop scheduling problem

In this study, a powerful solution methodology is developed for minimizing makespan in the preemptive Job Shop Scheduling Problem (pJSSP). Some new properties of the problem are stated and proved via theorems on the basis of which a new dominant set is introduced for the problem. These properties give rise to a dramatic decrease in the search space and provide the potential for exact methods to be successfully used in the solution of this notoriously NP-hard problem. The exact method presented here is a branch and bound algorithm developed on the basis of a new disjunctive graph. Its efficiency is enhanced by the effective use of such techniques as dominance rules or lower bounds. The capability of the approach is investigated by using it to solve the well-known benchmark problems and comparing the results obtained with those from the best methods in common use. The results indicate that the proposed method is capable of optimally solving 24 open benchmark problems including the famous 10×10 problems. Additionally, it is the first optimal method ever developed to find optimal solutions to some large-scale problems of the size 30×10 and 50×10.