Continuous time

Figure C3.6.4(a) shows an experimental chaotic attractor reconstmcted from tire Br electrode potential, i.e. tire logaritlim of tire Br ion concentration, in tlie BZ reaction [F7]. Such reconstmction is defined, in principle, for continuous time t. However, in practice, data are recorded as a discrete time series of measurements (A (tj) / = 1,...

Markov model A mathematical model used in reliabihty analysis. For many safety apphcations, a discrete-state (e.g., working or failed), continuous-time model is used. The failed state may or may not be repairable. 

The continuous-time solution of the state equation is given in equation (8.47). If the time interval t — to) in this equation is T, the sampling time of a discrete-time system, then the discrete-time solution of the state equation can be written as... 

Continuous time series These data are either passed as... 

Majozi, T., Zhu, X.X., 2001. A novel continuous time MILP formulation for multipurpose batch plants. 1. Short-term scheduling. Ind. Eng. Chem. Res., 40(25) 5935-5949 Mignon, D., Hermia, J., 1993. Using BATCHES for modeling and optimizing the brewhouses of an industrial brewery. Comput. Chem. Eng., 17(Suppl.) S51-S56. 

Schilling, G., Pantehdes, C.C., 1996. A simple continuous-time process scheduling formulation and a novel solution algorithm. Comput. Chem. Eng., 20(Suppl.) S1221-1226 Umeda, T., Harada, T., Shiroko, K., 1979. A thermodynamic approach to the synthesis of heat integration systems in chemical processes. Comput. Chem. Eng., 3 273-282 Wang, Y.P., Smith, R., 1994. Wastewater minimization. Chem. Eng. Sci., 49(7) 981-1002... 

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. 

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). 

Ierapetritou, M.G., Floudas, C.A., 1998. Effective continuous-time formulation for short-term scheduling. 1. Multipurpose batch processes. Ind. Eng. Chem. Res., 37 4341-4359. 

Schilling, G., Pantelides, C.C., 1996. A simple continuous-time process scheduling formulation and a novel solution algorithm. Comput. Chem. Eng., 20(Suppl.) S1221-S1226. 

The models developed to take the PIS operational philosophy into account are detailed in this chapter. The models are based on the SSN and continuous time model developed by Majozi and Zhu (2001), as such their model is presented in full. Following this the additional constraints required to take the PIS operational philosophy into account are presented, after which, the necessary changes to constraints developed by Majozi and Zhu (2001) are presented. In order to test the scheduling implications of the developed model, two solution algorithms are developed and applied to an illustrative example. The final subsection of the chapter details the use of the PIS operational philosophy as the basis of operation to design batch facilities. This model is then applied to an illustrative example. All models were solved on an Intel Core 2 CPU, T7200 2 GHz processor with 1 GB of RAM, unless specifically stated. 

Majozi, T., 2006. Heat integration of multipurpose batch plants using a continuous-time formulation. Appl. Thermal Eng. J., 26 1369-1377... 

Fig. 11.1 Continuous time representation of the time horizon (Majozi, 2009)...

Schrodinger equation (Continued) time-dependent equation ... 

A standard continuous-time job-shop scheduling formulation [3] can be used to model the basic aspects of the production decisions, such as sequencing and assignment of jobs. Here, the key of the mathematical solution is to capture the durations of each processing step and to relate it to the amounts of material. Therefore, only a top-down approach will be presented to illustrate some main principles of the model. 

It is the objective of this paper to provide a comprehensive review of the state-of-the art of short-term batch scheduling. Our aim is to provide answers to the questions posed in the above paragraph. The paper is organized as follows. We first present a classification for scheduling problems of batch processes, as well as of the features that characterize the optimization models for scheduling. We then discuss representative MILP optimization approaches for general network and sequential batch plants, focusing on discrete and continuous-time models. Computational... 

In order to overcome the previous limitations and generate data-independent models, a wide variety of optimization approaches employ a continuous-time representation. In these formulations, timing decisions are explicitly represented as a set of continuous variables defining the exact times at which the events take place. 

In contrast to the discrete-time representation, continuous-time formulations are based on an extensive range of alternative event representations which are focused... 

The previous general continuous-time formulations are mostly oriented towards arbitrary network processes. On the other hand, different continuous-time formulations focused their attention on particular features of a wide variety of sequential processes. One of the first contributions following this direction is based on the concept of time slots, which stand for a set of predefined time intervals with unknown durations. The main idea is to postulate an appropriate number of time slots for each processing unit in order to allocate them to the batches to be processed. The definition of the number of time slots required is not a trivial decision and represents an important trade-offbetween optimality and computational performance. Other alternative approaches for sequential processes were developed based on the concept of batch precedence. Model variables defining the processing sequence of batch tasks are explicitly embedded into these formulations and, consequently,... 

Different continuous-time formulations were also developed based on the RTN concept initially proposed by Pantelides [10]. The work developed by Castro et al. [18] which has been improved by Castro et al. [19] falls into this group. Major assumptions of this approach are (1) processing units are considered individually, i.e., one resource is defined for each available unit, and (2) only one task can be... 

In the same way as in the previous STN-based continuous-time formulation, a set of global time points N is predefined where the first time point takes place at the beginning T1 = 0 whereas the last at the end of the time horizon of interest Tn = H. However, the main difference in comparison to the previous model arises in the definition of the allocation variable Winn which is equal to 1 whenever task i starts at time point n and finishes at or before time point n >n. In this way, the starting and finishing time points for a given task i are defined through only one set of binary variables. It should be noted that this definition on the one hand makes the model simpler and more compact, but on the other hand it significantly increases the number of constraints and variables to be defined. 

We can conclude that the continuous-time STN and RTN models based on the definition of global time points are quite general. They are capable of easily accommodating a variety of objective functions such as profit maximization or makespan minimization. However, events taking place during the time horizon such as multiple due dates and raw material receptions are more complex to implement given that the exact position of the time points is unknown. 

The previous general continuous-time formulations are mostly oriented towards general network processes. On the other hand, different continuous-time formulations focused their attention on the particular features of a wide variety of sequential... 

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