The Scheduling Problem

In contrast, interface timing constraints are concerned with the relative timing of a small set of individual signaling operators. Interface protocols specify minimum and maximum values for these operations but do not specify the timing of unrelated operators. For this reason, interface timing constraints are best expressed on individual pairs of operators rather than intervals containing several operators. 

CSTEP supports interface timing constraints which specify minimum and maximum times between operations. Constraint declarations in CSTEP describe a constraint name, a time interval between operators, and an inequality with a constant that is expressed in either nanoseconds or clock periods. For example, the following maximum-time constraint specifies that operator xl must execute at least 80ns before operator x2  

Similarly, the following minimum-time constraint specifies that operator x3 must execute at least one clock period before after operator x4  

Constraints on equality and inequality are accepted by CSTEP and treated as combinations of minimum and maximum timing constraints. 

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Optimal schedules are given in Table 7.4-20. Clearly, for cleaning times of five hours, single-product campaigns are the best solution of the scheduling problem, while for zero cleaning times (theoretical case) mixed-product campaigns would be best. 

The scheduling problem that is considered in this chapter can be stated as follows. Given ... 

During normal operation of the copper plant, there are a number of regular maintenance jobs that need to be planned. They are included in the scheduling problem as additional jobs that have given release dates and due dates. These maintenance jobs can mostly be performed only when a unit is empty and not in use. The optimization approach finds the best location for each maintenance job with the least impact on production throughput and, furthermore, modifies the batch recipes such that there will be a suitable break in the operation for the equipment that must be maintained. 

The solution developed is able to solve the scheduling problem very efficiently, resulting in good and realistic schedules. Of course, the solution quality depends to a great deal on how well the parameter estimation matches with the production process. More illustrations on the solution can be found in [5]. 

The scheduling problem can be decomposed into a core problem and a subproblem. 

The real-word case study considered here is the production of expandable polystyrene (EPS). Ten types of EPS are produced according to ten different recipes on a multiproduct plant which is essentially operated in batch mode. In this section, the multiproduct plant, the production process and the scheduling problem are presented. 

The EPS-production is driven by customer demands. The scheduling problem exhibits the following degrees of freedom, which may be discrete or continuous in... 

The scheduling problem is complicated by the fact that the coupled production of grain size fractions and the mixing in the finishing lines prohibit a fixed assignment of recipes to products. Furthermore, there is neither a fixed assignment of storage tanks nor of polymerization reactors to batch processes. 

The model of the scheduling problem is based on a discrete representation of time where each period i corresponds to one day. The scheduler assigns the number of batches x to be produced in each period. The capacity of the plant is constrained... 

The scheduling problem is subject to uncertainties in the demands. The demands di in period i are only known precisely after the period i. Thus, the production decision %s has to be made under uncertainty without knowing the demand exactly for the current and for later periods. Table 9.1 provides a model of the uncertain demands. The model consists of two possible outcomes of the demands for each period i d- and df. We assume a probability distribution with equal probabilities pj and pj for all outcomes. 

In order to investigate the performance of a deterministic online scheduler, we apply it to the example problem under demand uncertainty for three periods. The model of the scheduling problem used in the scheduler considers a prediction horizon of H = 2 periods. Only the current production decision Xi(ti) is applied... 

A very popular scheduling framework is based on mixed-integer programming. Herein, the scheduling problem is modeled in terms of variables and algebraic inequalities and solved by mathematical optimization techniques. In opposition to this well-established framework, a different approach is advocated in the paper by Alur and Dill [8] on timed automata (TA). 

We note that for many manoeuvring targets there may be no solution to the scheduling problem that satisfies the constraints. However, we have not been able to simulate a situation in which this happens. 

The remainder of this paper is organized as follows. In Section 2, we formulate the planning problem as a mixed-integer linear program. In Section 3, we present our schedule-generation scheme for the scheduling problem. In Section 4, we report the results of an experimental performance analysis that is based on a sample production process presented in Maravelias and Grossmann (2004). 

The following is the statement of the scheduling problem that can be solved using the methods of Bums and Carter (1985) ... 

Zhou et al. (1991) modified this approach by using a linear cost function emd concluded that this modification not only produced better results but also reduced network complexity. Other works related to using the Hopfield network for the scheduling problem include Zhang et ed. (1991) and Arizono et al. (1992). 

The objective of the scheduling problem is to minimize a performance indicator computed on the schedule. A schedule is the specification of a feasible sequence of starting (and waiting) times of operations for each job in each machine. 

For an existing plant, the scheduling problem involves a specification of the (1) product orders and recipes, (2) number and capacity of the equipment items, (3) a listing of the equipment items available for each task, (4) limitations on the shared resources (e.g., involving the usage of utilities and manpower), and (5) restrictions on the use of equipment due to operating or... 

Walker, R A. and Gunidhuri, S. (199S) Figh-Level Synthesis Introduction to the Scheduling Problem. IEEE Design Test of Computers, Vol 12, No 2. 

The scheduling problem is a cmcial problem in all industrial production systems. Scheduling of production serves essential roles which can be regarded as the indicator of overall production efficiency. With the intense competition of industries, suitable methods are needed in order to maximize the efficiency of the production management system to be able to compete in the industrial market. The common scheduling problem of production is exceedingly complicated, especially... 

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