Abstract
Kise et al. (Oper. Res. 26:121–126, 1978) give an O(n2) time algorithm to find an optimal schedule for the single-machine number of late jobs problem with agreeable job release dates and due dates. Li et al. (Oper. Res. 58:508–509, 2010a) point out that their proof of optimality for their algorithm is incorrect by giving a counter-example. In this paper, using the concept of “tower-of-sets” from Lawler (Math. Comput. Model. 20:91–106, 1994), we construct the tower-of-sets of the early job set generated by the algorithm. Then we give a correct proof of optimality for the algorithm and show a new result that the early job set by the algorithm obtains not only the maximum number of jobs but also the smallest total processing time among all optimal schedules. The result can be applied to study the problems of the hard real-time systems.
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