Complexity results for single-machine scheduling with positional learning effects

In the paper Lin (2007), it is claimed that a single machine scheduling problem of minimizing the number of late jobs with a positional learning effect is strongly NP-hard. To prove it, the author provided a reduction from 3-PARTITION to the decision version of this problem. However, we will show that the proof is incorrect, since the reduction is not a pseudopolynomial one. Nevertheless, we will provide a correct proof. Throughout the paper, we will keep the notation and terminology used by Lin (2007). There is given a set of jobs N={J1, J2, . . . , Jn} to be processed on a single machine. Each job Ji is associated with a due date di . The processing time of any job depends on the position at which it is arranged in a particular schedule. If job Ji ∈ N is scheduled in the j th position (1 j n), then its processing time is pi j . Owing to the learning effect, the processing time of a job is non-increasing with respect to the positions, that is, pi1 pi2 · · · pin for any Ji ∈ N . The completion time of job Ji is denoted by Ci . The job is late if Ci > di . The problem is to determine a schedule that has the minimum number of late jobs. The problem according to the three-field notation scheme is denoted by 1|LE|∑Uj . Let us recall the significant fragments of the strong NP-hardness proof. First, the definition of 3-PARTITION will be given. 3-PARTITION (Garey and Jonson, 1979): Given nonnegative integer B and a set of 3m non-negative integers A = {x1, x2, . . . , x3m} with B/4< xi < B/2 for each xi and ∑3m i=1 xi = mB, is there a partition A1, A2, . . . , Am of set A such that for each subset Ak , ∑ xi∈Ak = B? Based on an instance of 3-PARTITION, the author created an instance of 4m jobs (3m ordinary and m enforcer). For each element xi ∈ A, the author created job Ji , 1 i 3m, such that: