Two-machine and three-machine problems

Abstract

In this chapter we discuss problems involving two or three machines. We shall be mainly interested in a further exploration of the borderline between ‘easy’ and ‘hard’ problems as discussed in section 3.6. The crucial result here is Johnson’s efficient algorithm for the n|2|F|C_max problem. In section 5.1 we try to extend Johnson’s approach to solve other n\2\γ,Γ\c_max or n\3\γ,Γ\c_max problems; with a few exceptions, many of these turn out to be NP-complete. A similar situation exists with respect to the n|2|F|∑c _i problem discussed in section 5.2. In both sections, we shall occasionally suggest implicit enumeration approaches to solve NP-complete problems. Finally, in section 5.3 we show that Johnson’s approach can be used to solve a variety of two-machine problems involving time lags, whereby jobs do not have to be finished on M₁ before starting on M₂.