A note on "minimizing the maximum deviation of job completion time about a common due-date"

A recent paper by Cheng analyzed the problem of simultaneous determination of a common due-date and a sequence of n jobs to minimize the maximum deviation of job completion time around the common due-date. It is assumed that all jobs are available at time zero and their processing times, t_i's, are known in advance. The analysis in Cheng used an LP formulation and duality results to prove the theorems. This note provides an alternate simpler proof for the same results. We also give some results for the case when splitting and preemption are allowed. A recent paper by Cheng [1] gives an optimal solution to the problem of setting a common due-date and sequence of n jobs on a single machine. All jobs are available at time zero and their processing times, t_i's, are known in advance. The objective is to determine the optimal common due-date value k^*, and the sequence ( σ ) to minimize the maximum deviation of job completion time about the common due-date. We here give a much simpler proof of the results derived in [1]. The notations used here are the same as in [1]. Let C_[1] denote the completion time (decision variable) and t_[i] denote the given processing time of a job in the i^th position of sequence σ.