Task scheduling constraints

The task scheduling constraints used in this model are similar to those that have been presented in full in Chapter 2. However, the reader should note that the time variables in these constraints relate to water using operations only. These constraints will therefore not be discussed here, since the full description of the constraints can be found in the preceding chapters, particularly Chapter 2. However, these constraints are presented below. 

It is important to note the impact of the reactor on the resulting model. Since the reactor does not use water at all, water mass balances around the reactor are not required. The reactor can also be excluded from the reuse scheduling as the operation of the reactor does not directly affect the reuse of water. However, task scheduling constraints are still required for the reactor as are raw material and product mass balances. 

Following the task scheduling constraints are the constraints dealing with the scheduling of the direct recycle/reuse of wastewater. 

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. 

The direct recycle/reuse scheduling constraints are given in constraints (9.29), (9.30), (9.31), (9.32) and (9.33). Constraint (9.29) states that water can only be recycled/reused to a processing unit provided the unit is operating at that time point. However, the constraint also states that the recycle/reuse of water is not a prerequisite for the operation of a processing unit. Constraints (9.30) and (9.31) ensure that the time at which water is recycled/reused corresponds to the time at which the water is produced. Constraints (9.32) and (9.33) ensure that the time at which water is recycled/reused corresponds to the starting time of the task using the water. 

The final scheduling constraints are the feasibility constraints and time horizon constraints. Constraint (9.68) ensures that a processing unit can only process one task... 

The second problem concerns a collection of tasks that need to be performed. Each task has to be performed exactly once, and the tasks are to be performed in regularly allocated slots (e.g., hours). The only constraint is that certain tasks cannot be performed simultaneously. Again, the minimal number of slots required for the task scheduling is equal to the chromatic number of the graph whose vertices correspond to tasks, with two vertices coimected by an edge if and only if the corresponding tasks cannot be performed simultaneously. 

Code selection consists of three tasks scheduling, allocation and evaluation. The scheduling task packs microoperations into control states subject to the constraint that some criteria are satisfied. The allocation task allocates functional units, storage units and interconnection units, subject to the constraint that some criteria are satisfied. Allocation may be divided into a symbolic unit creation task and a symbolic unit to structural unit assignment task. The evaluation task analyzes a design to determine its quality. 

It may be useful to consider the program plan, at this stage of its development. as your company s ideal," within the limitations you have established for its scope. By first identifying what is needed, independent of the constraints of time or resources, you emphasize the tasks themselves as the substance of the plan—the core that drives decisions about resource allocations, rather than the other way around. (For example, see Figure 5-7) The result may prompt your team to think more creatively about schedule and resource requirements, as discussed in the following sections. 

The ability to schedule the tasks logically within the constraints of available resources. 

Casual constraints the precedence or required sequence of tasks and resource requirements. There are unexpected events, failures, and/or events that have not been included in the scheduling program. Off-specification batches occasionally disturb the schedule. 

Availability constraints the macroscopic limits on material resources and the availability or up-time of equipment. Availability of raw materials is an obvious constraint at scheduling. Obviously, no catalytic hydrogenation can be done if the catalyst is unavailable. Simultaneous operation of certain tasks is restricted by the limited availability of common utilities such as steam, electricity, or labour. The priority sequence in a product chain needs to be respected by ensuring that intermediate products are manufactured in time to be available when required by a batch of the consecutive product. 

In this chapter, state sequence network (SSN) representation has been presented. Based on this representation, a continuous-time formulation for scheduling of multipurpose batch processes is developed. This representation involves states only, which are characteristic of the units and tasks present in the process. Due to the elimination of tasks and units which are encountered in formulations based on the state task network (STN), the SSN based formulation leads to a much smaller number of binary variables and fewer constraints. This eventually leads to much shorter CPU times as substantiated by both the examples presented in this chapter. This advantage becomes more apparent as the problem size increases. In the second literature example, which involved a multipurpose plant producing two products, this formulation required 40 binary variables and gave a performance index of 1513.35, whilst other continuous-time formulations required between 48 (Ierapetritou and Floudas, 1998) and 147 binary variables (Zhang, 1995). 

Due to the nature of the process constraints have to be derived that capture the essence of time. The first of these considered deal with the scheduling of the tasks in a unit. 

The above constraints deal with the mass flows between the various units in a batch plant. They do not consider the timing of the streams, tasks and such. Therefore, further constraints have to be derived to ensure the correct sequencing and scheduling of the processes, streams and tasks. 

A simpler and general discrete time scheduling formulation can also be derived by means of the Resource Task Network concept proposed by Pantelides [10], The major advantage of the RTN formulation over the STN counterpart arises in some problems involving many identical pieces of equipment. In these cases, the RTN formulation introduces a single binary variable instead of the multiple variables used by the STN model. The RTN-based model also covers all the features at the column on discrete time in Table 8.1. In order to deal with different types of resources in a uniform way, this approach requires only three different classes of constraints in terms ofthree types of variables defining the task allocation, the batch size, and the resource availability. Briefly, this model reduces the batch scheduling problem to a simple resource balance problem carried out in each predefined time period. 

The top-down approach, which defines appropriate hierarchical coordination mechanisms between the different decision levels and decision structures at each level. These structures force constraints on lower operating levels and require heuristic decision rules for each task. Although this approach reduces the size and complexity of scheduling problems, it potentially introduces coordination problems. 

None of preventive maintenance planning models considers constraints on resources available in process plants, which include labor and materials (spare parts). For example, the maintenance work force, which is usually limited, cannot perform scheduled PM tasks for some equipments at scheduled PM time because of the need to repair other failed equipments. Such dynamic situations can not be handled by deterministic maintenance planning models or are not considered in published maintenance planning models that use Monte Carlo simulation tools. 

In model constraints given next, Q is a wrap-around operator (Shat et al., 1993), r, holds the duration of tasks in number of time intervals (5=8 h) and set K, gives the tasks belonging to chemical z. The objective function minimizes the total cost of the schedule in relative money units (r.m.u.). Eq 2 ensures that the volume handled by the task does not exceed the capacity of the vessel Vm. Eq 3 ensures that material production only occurs if the corresponding task is executed. The periodic schedule features exactly one batch of each chemical (eq 4). Eqs 5-6 are the excess resource balances. Eq 7 ensures that the start-up procedure does not require more units than those available. 

Project management tools offer possibilities to define time constraints like deadlines for tasks. But most of these tools do not provide any means to detect violations of constraints and they do not enforce actions to bring the project back on schedule. Such issues will have to be addressed by our developments. The planned management system will permanently calculate the current execution state of the design process and will compare it with the... 

---