Batch delivery scheduling with limited waiting time constraint on a single machine

In this paper, we study a batch delivery scheduling problem on a single machine in which the waiting time of a finished job on the machine is no more than a limited value. The waiting time of a job is defined as the difference between its completion time and the delivery date of the batch to which it belongs. The objective is to minimize the sum of the batch delivery cost and the makespan. We analyze some properties for the optimal schedule and prove the strong NP-hardness of the problem. When there is no long job, we propose a heuristic algorithm with worst-case 3. When there are some long jobs, we propose a heuristic with worst-case ratio 4.