A note on "Single-machine scheduling with general learning functions"

The recent paper ‘‘Single-machine scheduling with general learning functions’’ (Wang [1]) addresses the single-machine scheduling problems with a sum-of-processing-times-based learning effect. The author showed that that the shortest processing time (SPT) rule is optimal for the sum of completion times square minimization problem. He also showed by examples that the optimal schedule for the classical version of the problem is not optimal in the presence of a sum-ofprocessing-times-based learning effect for the following three objective functions: the weighted sum of completion times, the maximum lateness and the number of tardy jobs. But for some special cases, he proved that the weighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule and Moore’s algorithm can construct an optimal schedule for the problem to minimize these objective functions, respectively. In this note we will give a counter-example to show the incorrectness of some results in Wang [1]. We shall follow the notations and terminologies given in Wang [1]. There are given a single machine and n independent and non-preemptive jobs that are immediately available for processing. The machine can handle one job at a time and preemption is not allowed. Let pj be the normal processing time of job j and p[k] the normal processing time of a job if it is scheduled in the kth position in a sequence. Associated with each job j (j = 1, 2, . . . , n) is a due-date dj. Let pj,r be the processing time of job j if it is scheduled in position r in a sequence. Then