Batch Scheduling and Common Due Date Assignment Problem: an NP-hard Case

An important special case of the problem studied in Cheng and Kovalyov (1996) arises when there are equal setup times and equal job processing times. Computational complexity of this case was indicated to be open, however. I prove its NP-hardness. Cheng and Kovalyov [l] introduced the following due date assignment and scheduling problem. There are n groups of jobs to be scheduled for processing on a single machine. All jobs are available at time zero. Each group j consists of qj > I identical jobs with a same processing requirement pi 3 0 and a weight w, 3 0. Each job is completed immediately when its processing is finished. Each group may be partitioned into batches containing contiguously scheduled jobs. A setup time si 3 0 is required before a batch of group j is processed if it is processed first on the machine or immediately after a batch of another group. Thus, setup times are sequence independent. The machine can handle at most one job at a time and cannot process any job whilst a setup is being performed. A schedule specifies the size of each batch, i.e. the number of jobs it contains, and the processing order for the batches. A due date value u'>O common to all jobs has to be determined. For any schedule and any due date value d, a job i with completion time C'; is euul~, if C, dd and it is tavdls if Ci > d. The number of tardy jobs of group j is denoted by L$. The objective is to determine a value for the common due date d and a schedule so as to minimize the sum of the due date assignment penalty and the weighted number of tardy jobs: r(d)+ EYE, w,U,. The due date assignment penalty function r(d) is defined as follows: given a threshold value *