Minimizing the Weighted Number of Late Jobs with Batch Setup Times and Delivery Costs on a Single Machine

Abstract

We study a single machine scheduling problem with batch setup time and batch delivery cost. In this problem, n jobs have to be scheduled on a single machine and delivered to a customer. Each job has a due date, a processing time and a weight. To save delivery cost, several jobs can be delivered together as a batch including the late jobs. The completion (delivery) time of each job in the same batch coincides with the batch completion (delivery) time. A batch setup time has to be added before processing the first job in each batch. The objective is to find a batching schedule which minimizes the sum of the weighted number of late jobs and the delivery cost. Since the problem of minimizing the weighted number of late jobs on a single machine is already -hard [Karp, 1972], the above problem is also hard. We propose a new dynamic programming algorithm (DP), which runs in pseudopolynomial time. The DP runs in O(n5) time for the special cases of equal processing times or equal weights. By combining the techniques of binary range search and static interval partitioning, we convert the DP into a fully polynomial time approximation scheme (FPTAS) for the general case. The time complexity of this FPTAS is O(n4/ + n4logn). Minimizing the total weighted number of late jobs on a single machine, denoted by [Graham et. al, 1979], is a classic scheduling problem that has been well studied in the last forty years. Moore [1968] proposed an algorithm for solving the unweighted problem on n jobs in O(nlogn) time. The weighted problem was in the original list of hard problems of Karp [1972]. Sahni [1976] presented a dynamic program and a fully polynomial time approximation scheme (FPTAS) for the maximization version of the weighted problem in which we want to maximize the total weight of on-time jobs. Gens and Levner [1979] developed an FPTAS solving the minimization version of the weighted problem in O(n3/ ) time. Later on, they developed another FPTAS that improved the time complexity to O(n2logn + n2/ ) [Gens and Levner, 1981]. In the batching version of the problem, denoted by , jobs are processed in batches which require setup time s, and every job's completion time is the completion time of the last job in its batch. Hochbaum and Landy [1994] proposed a dynamic programming algorithm for this problem, which runs in pseudopolynomial time. Brucker and Kovalyov