Scheduling horizon

In the polymerization stage, the number of events, i.e., of the polymerization starts, N, is given and the events are identified by the index n = 1... N. Start times of polymerizations are represented by continuous variables in e [0, H] in with H denoting the given scheduling horizon. As an initial condition, the first polymerization is defined to start at tn= i = t°. 

Due to the fact that one might not want to use all ERP business data objects in the APS system, an integration model is used to define for which plants, resources, materials, customers, etc., the integration interface should be active. The APS system also allows to maintain additional master data or to modify and enhance master data that came from the ERP system. Sometimes there are also additional types of master data (e.g. resource setup matrix) or information fields to master data (e.g., scheduling horizon) that really have no analogy in the ERP system and thus have to be maintained directly in the APS system. 

The amount of material in state s should be equal to the demand at the end of the scheduling horizon ... 

In this example, there are two reactors in which reactions 1, 2 and 3 can be performed. Equal mean reaction times for the different reactions in each of the reactors imply similar performances for the reactors. The overall process consists of four units, i.e. heater, reactor 1, reactor 2 and separator. In order to handle the usage of feed C in two distinct reactions, i.e. reactions 1 and 3, different states were assigned to each of the streams of feed C, i.e. states, v3 and s4, respectively. In this example, scheduling is performed over an 8-h time horizon. It should be noted that reactors 1 and 2 are suitable for performing reactions 1, 2 and 3, which implies that constraint (2.13) is crucial. Constraints that exhibit similar structure to those presented in example 1 are not repeated. 

Task Determine the schedule that corresponds to maximum throughput over a 12 h time horizon... 

A mathematical formulation based on uneven discretization of the time horizon for the reduction of freshwater utilization and wastewater production in batch processes has been developed. The formulation, which is founded on the exploitation of water reuse and recycle opportunities within one or more processes with a common single contaminant, is applicable to both multipurpose and multiproduct batch facilities. The main advantages of the formulation are its ability to capture the essence of time with relative exactness, adaptability to various performance indices (objective functions) and its structure that renders it solvable within a reasonable CPU time. Capturing the essence of time sets this formulation apart from most published methods in the field of batch process integration. The latter are based on the assumption that scheduling of the entire process is known a priori, thereby specifying the start and/or end times for the operations of interest. This assumption is not necessary in the model presented in this chapter, since water reuse/recycle opportunities can be explored within a broader scheduling framework. In this instance, only duration rather start/end time is necessary. Moreover, the removal of this assumption allows problem analysis to be performed over an unlimited time horizon. The specification of start and end times invariably sets limitations on the time horizon over which water reuse/recycle opportunities can be explored. In the four scenarios explored in... 

Constraints (6.42), (6.43) and (6.44) deal with the scheduling aspects of two streams leaving the storage vessel. Constraints (6.42) ensures that streams leaving the storage vessel at later time points correspond to a later absolute time within the time horizon. Constraints (6.43) and (6.44) ensure that if two water streams are leaving the storage vessel at the same time point, both streams leave at the same time in the time horizon. 

The final group of sequencing and scheduling constraints comprise of feasibility constraints and time horizon constraints. 

Each water using operation in the time horizon has to be scheduled accordingly, within the overall framework of operation scheduling. This is captured in constraints... 

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 constraints that comprise the scheduling module of the model are divided into four groups, namely, task scheduling, direct recycle/reuse scheduling, storage scheduling and time horizon constraints. 

Scheduling constraints have to be derived to account for the timing of multiple streams leaving a storage vessel. Constraint (7.36) ensures that water leaving a storage vessel at a later time point does so at a later absolute time in the time horizon. Constraints (7.37) and (7.38) ensure that the time at which two streams leave a storage vessel at a time point corresponds to the same time for each. 

The final constraints necessary to complete the scheduling model are constraints that ensure all operations occur within the time horizon. These are given in constraints... 

One would notice that storage vessel two is not required in the schedule. This is due to the fact that unit 3 recycles water to itself throughout the time horizon and unit 2 recycles water once in the time horizon. Water is reused through storage vessel one only once in the time horizon. 

The first scheduling constraints considered are binary variable constraints governing wastewater reuse. These constraints ensure the correct reuse of water, since water is only reused in distinct amounts at certain points in the time horizon. 

Scheduling constraints also have to be derived to ensure the time at which wastewater reuse occurs is correct within the time horizon. 

Constraints (8.38) - (8.40) are constraints that deal with the scheduling of streams to and from a storage vessel. If water leaves a storage vessel at a time point after the time point at which the water entered the vessel, then the time at which this happens must occur at a later absolute time in the time horizon. This is given in constraint (8.38). The time at which a stream leaves a storage vessel and the time at which water enters a storage vessel must coincide, provided the two streams enter at the same time point. This is ensured through constraints (8.39) and (8.40). 

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