In this paper, we consider the online parallel-machine customer order scheduling with delivery time and penalties. Given a set of customer orders \(\mathcal {O}\) and a set of dedicated machines \(\mathcal {M}\) , each customer order consists of m different types of products, and each dedicated machine can only process one type of products. Each customer order can either be accepted and processed by the dedicated machines or rejected and paid a penalty. If a customer order is processed, an additional delivery time is required after processing, so the delivery completion time of a customer order is the sum of its processing completion time and delivery time. In the online setting, customer orders arrive one by one. When a customer order arrives, we must immediately and irrevocably determine whether to process or reject this customer order without any information about the following customer orders. This problem is to schedule an accepted set A and a rejected set R to minimize the sum of the maximum delivery completion time and the penalty cost for rejected customer orders. We present an online algorithm with a competitive ratio of \(m + 2\) , where m is the number of machines in \(\mathcal {M}\) .
