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={J 1,J 2,…,J n } to be processed on a single machine. Each job J i is associated with a due date d i . The processing time of any job depends on the position at which it is arranged in a particular schedule. If job J i ∈N is scheduled in the jth position (1⩽j⩽n), then its processing time is p ij . Owing to the learning effect, the processing time of a job is non-increasing with respect to the positions, that is, p i1⩾p i2⩾⋯⩾p in for any J i ∈N. The completion time of job J i is denoted by C i . The job is late if C i >d i . 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|∑U j .

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 non-negative integer B and a set of 3m non-negative integers A={x 1,x 2,…,x 3m } with B/4<x i <B/2 for each x i and ∑ i=1 3m x i =mB, is there a partition A 1,A 2,…,A m of set A such that for each subset

Based on an instance of 3-PARTITION, the author created an instance of 4m jobs (3m ordinary and m enforcer). For each element x i ∈A, the author created job J i , 1⩽i⩽3m, such that:

where v=2mB, ω=v m and D is a due date that is the same for all ordinary jobs. Since the definition of the ordinary jobs is enough to show that the given proof is incorrect, then we will omit the definition of enforcer jobs.

Recall now a condition from a definition of a pseudopolynomial reduction. Let denote their sets of all possible instances, Max(I) denotes the maximum value for an instance I and N(I) is the size of I. Let f: denote the reduction from π 2 to π 1. One of the requirements for f to be pseudopolynomial is such that there must exist a polynomial Q of two variables that holds:

It means that the values of any instance I of the problem π 2 cannot increase in an exponential manner if π 2 is reduced to π 1.

Let π 2 denote 3-PARTITION and π 1 is the considered scheduling problem. It is obvious that for the given reduction and I∈D π2, we have Max(I)=B, N(I)=3m, Max(f(I))>ω=(2mB)m and N(f(I))=4m (ie 3m ordinary and m enforcer jobs). Thus, there does not exist such Q for which (1) holds, thereby the reduction cannot be pseudopolynomial. Therefore, the proof of the strong NP-hardness is incorrect. Observe that in this way it is proved that the scheduling problem is at least NP-hard, but it is already established by Theorem 1 (Lin, 2007).

Nevertheless, the strong NP-hardness of 1|LE|∑U j , follows straightforward from the strong NP-hardness of the problem 1|LE|L max proved by Cheng and Wang (2000).