Scheduling problem to minimize the maximum lateness for parallel processors

The problem of minimizing the maximum lateness while scheduling tasks to parallel identical processors is a classical combinatorial optimization problem. It has many applications, and it is NP-hard. This problem relates to the scheduling problem and it is denoted by $P|r_j |L_{\mathrm{max}}$. The multiprocessor scheduling problem is defined as follows: tasks have to be executed on several parallel identical processors. We must find where and when each task will be executed, such that the maximum lateness is minimum. The duration, release time and due date of each task are known. Preemption on processors is not allowed. A lot of research in scheduling has concentrated on the construction of the nondelay schedule. A nondelay schedule is a feasible schedule in which no processor is kept idle at a time when it could begin processing a task. An inserted idle time schedule (IIT) as a feasible schedule in which the processor is kept idle at a time when it could begin processing a task. The goal of this paper is to propose an IIT schedule for the $P|r_j |L_{\rm max}$ problem. We propose an approximate IIT algorithm named ELS/IIT (earliest latest start/ inserted idle time) and depth first branch and bound algorithm, which produces a feasible IIT (inserted idle time) schedule for a fixed maximum lateness $L$. The algorithm may be used in a binary search mode to find the smallest maximum lateness. The branch and bound algorithm is based on the earliest latest start/ inserted idle time branching strategy. Two new dominance criterions between decision nodes are used. A new method for evaluating unfeasible partial solutions was designed. To illustrate the effectiveness of this approach we tested algorithms on instances, which were randomly generated with the number of jobs from 50 to 300. Refs 14. Tables 5.