Asymptotic optimality in probability of a heuristic schedule for open shops with job overlaps

Abstract

The assumption that different operations of a given job cannot be processed simultaneously is accepted in most classical multi-operation scheduling models. There are, however, real-life applications where parallel processing (within a given job) is possible. In this paper we study open shops with job overlaps, i.e., scheduling systems in which each job requires processing operations by M different machines, and different operations of a single job may be processed in parallel. The problem of computing an optimal schedule, that is, ordering the jobs’ operations for each machine with the objective of minimizing the sum of job completion times, is NP-hard. A simple heuristic, based on ordering the jobs by the average processing time of the M operations required for each job, and a lower bound on the optimal cost are introduced. The lower bound is used to prove asymptotic optimality (in probability) of the heuristic when the processing times are i.i.d. from any distribution with a finite variance.

Introduction

Multi-operation scheduling models incorporate jobs which require execution on more than one machine. Each of these jobs consists of a set of operations which need to be processed on different machines in order to accomplish the entire job. Several multi-operation models have been widely studied: the open-shop, the flow-shop and the job-shop, see, e.g., Lawler et al. [6]. These models differ in their underlying assumption with respect to the machine ordering of the jobs: in an open-shop this order is immaterial, in a flow-shop the order is identical for all jobs, and in the job-shop the machine ordering may be job-dependent. A common assumption of all these models is that different operations of a given job cannot be processed simultaneously on different machines. This assumption relates to production applications in which a typical product develops gradually along the production line as each machine reflects a different stage of the process. There are, however, many real-life scheduling settings where simultaneous processing of different operations of the same job are acceptable and sometimes preferable. These relate, e.g., to cases in which the finished product is actually defined as a group of items produced independently along parallel lines. In this paper we consider the so-called open shop with job overlaps (see [7]), i.e., open shop settings where such parallel processing (within a given job) is permitted. This is a “relaxed” version of the open-shop problem, a setting with an option to process different operations of a given job simultaneously on different dedicated machines. The objective is to minimize the sum of the jobs’ completion times. This problem was recently shown to be unary NP-hard even for two machines [7]. Therefore, we focus in this paper on a polynomial heuristic requiring a computational effort of no more than $\text{max}\text{{}\text{O}\text{(MN),}\text{O}\text{(N}\text{log}\text{N)}}$, where M is the number of the machines and N is the number of jobs. In Section 2we formulate the problem and present the heuristic and a lower bound on the optimal solution which requires the same computational effort. Asymptotic optimality of the heuristic under i.i.d. processing times is proved in Section 3by showing that the ratio of the sum of the jobs’ completion times under the proposed heuristic schedule and the lower bound converges to 1 in probability. A short numerical study of the heuristic is given in Section 4.