Research on single-machine scheduling with past-sequence-dependent setup times and effects of deterioration and learning

:Some result in a recent paper(Wang et al.,Int J Adv Manuf Technol,41:1221–1226,2009) are incorrect.In this note, we show by a counterexample that the published results are incorrect.


Introduction
The recent paper"single-machine scheduling with past-sequence-dependent setup times and effects of deterioration and learning" [1] addresses the single machine scheduling problems with past-sequence-dependent setup times and effects of deterioration and learning.They showed that the makespan minimization problem, the total completion time minimization problem,the number of tardy jobs minimization problem and the total weighted completion time minimization problem remain polynomially solvable, respectively.As we observe, some results for the method in Wang et al. [1] are incorrect.In this note we will give a counterexamples to show the incorrectness of some results in Wang et al. [1].
We shall follow the notations and terminologies given in Wang et al. [1]. There are given a single machine and n independent and non-preemptive jobs that are available for processing at some The machine can handle one job at a time and preemption is not allowed. Let j  be the deterioration rate of job j J in a sequence. In addition, let   A k p be the actual processing time of a job if it is scheduled in the kth position in a sequence. As in Wang and Cheng [2], we assume that the actual processing time of job j J if it is started at time t and scheduled in position r is given by: where a ≤ 0 is a constant learning effect. Also, as in Koulamas and Kyparisis [3] and Kuo and Yang [4], we assume that the p-s-d setup time of job   r J if it is scheduled in position r is given by: .For convenience, we denote by psd s the p-s-d setup given by Eq. 2 (see Koulamas and Kyparisis [3]).Let j C be the completion time of job j J . For a given , represent the number of tardy jobs of a given permutation, where j d is the due date of job j J .Using the conventional notation (Graham et al. [5]),the number of tardy jobs scheduling problem is denoted as  .

A counterexample
Let N denote the set of jobs already scheduled, e N be the set of jobs already considered for scheduling but having been discarded because they will not meet their due dates in the optimal schedule, and f N denote the set of jobs not yet considered for scheduling. The problem  j U 1 is known to be solved by Moore's algorithm [6] as follows:

Moore's algorithm
Step 1 Order the jobs in non-decreasing order of j d (the earliest due date (EDD) rule).
Step 2 If no jobs in the sequence are late, stop. The schedule is optimal.
Step 3 Find the first late job in the schedule. Denote this job by  J .
Step , an optimal schedule can be obtained by Moore's algorithm.
The following example shows that Theorem1' and Theorem2' are incorrect.

Counterexample1.Let
.Consider a problem containing    Table 2). In Step 4, job 1 J is removed from N and placed in e N .In the next pass at Steps 3 and 4, job 2 J is removed from N and placed in e N (See Table 3). In the next pass at Steps 3 and 4, job 3 J is removed from N and placed in e N (See Table 4).