Jacek Błażewicz

Scheduling in Job Shops

In this chapter we are going to consider scheduling tasks on dedicated processors or machines. We assume that tasks belong to a set of jobs, each of which is characterized by its own machine sequence. We will assume that any two consecutive tasks of the same job are to be processed on different machines. The type of factory layout is the job shop. It provides the most flexible form of manufacturing, however, frequently accepting unsatisfactory machine utilization and a large amount of work-in-process. Hence, makespan minimization is one of the objectives in order to schedule job shops effectively, see e.g. [Pin95].

Definition, Analysis and Classification of Scheduling Problems

Throughout this book we are concerned with scheduling computer and manufacturing processes. Despite the fact that we deal with two different areas of applications, the same model could be applied. This is because the above processes consist of complex activities to be scheduled, which can be modeled by means of tasks (or jobs), relations among them, processors, sometimes additional resources (and their operational functions), and parameters describing all these items in greater detail. The purpose of the modeling is to find optimal or sub-optimal schedules in the sense of a given criterion, by applying best suited algorithms. These schedules are then used for the original setting to carry out the various activities. In this chapter we introduce basic notions used for such a modeling of computer and manufacturing processes.

Scheduling on Parallel Processors

This chapter is devoted to the analysis of scheduling problems in parallel processor environment. As before the three main criteria to be analyzed are schedule length, mean flow time and lateness. Then, some more developed models of multiprocessor systems are described, including semi-identical processors, imprecise computations and lot size scheduling. Corresponding results are presented in the four following sections.

Communication Delays and Multiprocessor Tasks

One of the assumptions imposed in Chapter 3 was that each task is processed on at most one processor at a time. However, in recent years, with the rapid development of manufacturing as well as microprocessor and especially multi-microprocessor systems, the above assumption has ceased to be justified in some important applications. There are, for example, self-testing multi-microprocessor systems in which one processor is used to test others, or diagnostic systems in which testing signals stimulate the tested elements and their corresponding outputs are simultaneously analyzed [Avi78, DD81]. When formulating scheduling problems in such systems, one must take into account the fact that some tasks have to be processed on more than one processor at a time. On the other hand, communication issues must be also taken into account in systems where tasks (e. g. program modules) are assigned to different processors and exchange information between each other.

Scheduling in Flow and Open Shops

Consider scheduling tasks on dedicated processors or machines. We assume that tasks belong to a set of n jobs, each of which is characterized by the same machine sequence. For convenience, let us assume that any two consecutive tasks of the same job are to be processed on different machines. The type of factory layout in the general case — handled in Chapter 8 — is the job shop; the particular case where each job is processed on a set of machines in the same order is the flow shop. A part of this chapter will also consider the situation where no predefined machine or processor sequences are existing. This results in the case of scheduling a set of jobs on a set of machines in any order — the open shop. The most commonly used performance measure will be makespan minimization.

Formulation of Scheduling Problems

In general, scheduling problems considered in this book are characterized by three sets: set T = {T 1, T 2,…,T n} of n tasks, set P = {P 1, P 2,…,P m} of m processors (machines) and set R = {R 1, R 2,…,R s} of s types of additional resources R. Scheduling, generally speaking, means to assign processors from P and (possibly) resources from R to tasks from T in order to complete all tasks under the imposed constraints. There are two general constraints in classical scheduling theory. Each task is to be processed by at most one processor at a time (plus possibly specified amounts of additional resources) and each processor is capable of processing at most one task at a time. In Sections 5.4 and 7.2 we will show some new applications in which the first constraint will be relaxed.

Knowledge-Based Scheduling

Within all activities of production management, production scheduling is a major part of production planning and control. By production management we mean all activities which are necessary to carry out production. The two main activities in this field are production planning and production control. Production scheduling is a common activity of these two areas because scheduling is needed not only on the planning level — as mainly treated in the preceding chapters — but also on the control level. From the different aspects of production scheduling problems we can further distinguish predictive production scheduling or offline planning (OFP) and reactive production scheduling or online-control (ONC). Predictive production scheduling serves to provide guidance in achieving global coherence in the process of local decision making. Reactive production scheduling is concerned with revising predictive schedules when unexpected events force changes. OFP generates the requirements for ONC and ONC creates feedback to OW. The relationship between these functions are shown in Figure 9.0.1.

Single Processor Scheduling

Single machine scheduling (SMS) problems seem to have received substantial attention because of several reasons. These type of problems are important both because of their own intrinsic value, as well as their role as building blocks for more generalized and complex problems. In a multi-processor environment single processor schedules may be used in bottlenecks, or to organize task assignment to an expensive processor; sometimes an entire production line may be treated as a single processor for scheduling purposes. Also, compared to multiple processor scheduling, SMS problems are mathematically more tractable. Hence more problem classes can be solved in polynomial time, and a larger variety of model parameters, such as various types of cost functions, or an introduction of change-over cost, can be analyzed. Single processor problems are thus of rather fundamental character and allow for some insight and development of ideas when treating more general scheduling problems.

Static Shop Scheduling

In this chapter we will consider scheduling tasks on dedicated processors (machines). As we said in Section 3.1 we assume that tasks form n subsets (or jobs), belonging to set J, and two adjacent tasks of a job are to be performed on different machines. Unfortunately, most scheduling problems of this kind are NP-hard, which is especially true for optimality criteria other than C max. In the first two sections we will concentrate first on polynomial time algorithms, where special cases of flow shop and open shop scheduling problems will be considered. Then the job shop scheduling problem will be discussed and two approaches, a heuristic based on simulated annealing and an exact based on branch and bound will be presented.