A note on minimizing makespan on a single batch processing machine with nonidentical job sizes

In a relatively recent paper (G. Zhang, X. Cai, C.Y. Lee, C.K. Wong, Minimizing makespan on a single batch processing machine with nonidentical job sizes, Naval Research Logistics 48 (2001) 226–240), authors considered minimizing makespan on a single batching machine having unit capacity. For the restricted version of the problem in which the processing times of the jobs with sizes greater than 1/2 are not less than those of jobs with sizes not greater than 1/2, they proposed an O(nlogn) algorithm with absolute worst-case ratio 3/2. We propose an algorithm with absolute worst-case ratio 3/2 and asymptotic worst-case ratio (m+1)/m (m≥2 and integer) for a more general version in which the processing times of the jobs of sizes greater than 1/m are not less than the remaining (the case of m=2 has been considered by Zhang et al.). This general assumption is particularly held for those problem instances in which the job sizes and job processing times are agreeable. We obtain an O(nlogn) algorithm with asymptotic worst-case ratio 4/3 for these problems leading to a more dependable algorithm than that of Zhang et al.