Scheduling discrete time representation

The main challenge in short-term scheduling emanates from time domain representation, which eventually influences the number of binary variables and accuracy of the model. Contrary to continuous-time formulations, discrete-time formulations tend to be inaccurate and result in an explosive binary dimension. This justifies recent efforts in developing continuous-time models that are amenable to industrial size problems. 

The model is derived to take into consideration the possibility of multiple storage vessels which are dedicated to the storage of certain wastewater. The formulation shares some of the characteristics of the multiple contaminant model presented in the previous chapter. This is due to the fact that both formulations have roots in the scheduling methodology derived by Majozi and Zhu (2001). Furthermore, the uneven discretization of the time horizon is used as the time representation. 

The presented mathematical formulation is an extension of the scheduling model proposed by Majozi and Zhu (2001), which uses a state sequence network (SSN) representation. This formulation is based on an uneven discretization of time framework (see Chapter 2, Fig. 2.4) as shown in Fig. 10.1. A time point corresponds to the beginning of a particular task and is not necessarily equidistant from the preceding and the succeeding time points, as it encountered in discrete-time formulations. The... 

The continuous-time representation has the same ambition events are allowed to take place at any point in the continuous domain of time. This is carried out by introducing the concept of variable event time that can be defined either globally or for each unit. For this purpose, additional variables are necessary to determine the timings of events. Since a major fraetion of the inactive event-time interval assignments is eliminated with the continuoustime approach, the resulting mathematical models require less computational effort for their solution of the associated scheduling task involved in design/retrofit problems. However, the mathematical models involved in the continuous-time approach may embed more complicated structures compared with their discrete-time counterparts. 

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... 

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