Continuous-time formulations

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. 

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

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. 

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... 

Jia, Z. and Ierapetritou, M. (2003) Refinery short-term scheduling using continuous time formulation crude-oil operations. 

A Novel Network-based Continuous-time Formulation for Process Scheduling... 

M.G. lerapetritou, C. Floudas, 1999. Effective Continuous-Time Formulation for Short-Term. scheduling III. Multiple intermediate due dates. 

The path-integral formulation of a quantum systems goes back to [37], and forms the basis of most QMC algorithms. Instead of following the historical route and discussing the Trotter-Suzuki (checkerboard) decomposition [38,39] for path integrals with discrete time steps At we will directly describe the continuous-time formulation used in modern codes. 

Castro, P., Barbosa-Povoa, A.P.F.D., Matos, H., 2001, An Improved RTN Continuous-Time Formulation for the Short-term Scheduling of Multipurpose Batch Plants, Ind. Eng. Chem. Res., 40,2059-2068. 

Different approaches may be used, but it turns out that the models become very large with growing number of orders. The specific problem considered may be modeled by using a continuous time formulation and the effectively solved by using decomposition methods. Care must be taken when choosing decomposition strategy. Two different decomposition approaches are used, one product based and one time based. Real world problems can be solved using both decomposition schemes. The solution time also depend on the choice of objective function. 

In contrast to the previous case study, this one (a) includes design variables within the search space, and (b) utilizes a continuous-time formulation. The problem formulation follows closely that described in Section 5.2.4. The study was presented originally in [29] and is based on a nonisothermal stirred tank reactor system analyzed in [30]. 

This chapter has focused on the use of controller parametrization in the integration of design and control. The focus here has been on the parametrization of linear controllers, although there have been promising recent developments in the parametrization of nonlinear controllers. As with the IMC approach, we seek to provide a performance limit independent of controller type, and indeed for stable systems they are structurally equivalent. Its application to operability analysis has been posed within an optimization framework, and as such is able to account for the simultaneous presence of all plant-inherent performance-limiting characteristics. Both discrete and continuous-time formulations have been presented, and its implementation for both controllability analysis and controllable plant design shown. Its application has also been illustrated through two case studies. 

Notice that for the actual period the proposed algorithm is considering the detailed scheduling, therefore disturbances can be contemplated as frequent as the time bucket utilized in the scheduling formulation. It is also important to point out that we are integrating the three standard hierarchical decision levels however, more decision levels may exist in an organization. In that case, the disturbances can be considered as frequent as the time bucket of the lower decision level in case of using discrete time SC formulations. Continuous time formulations should overcome this drawback. 

Castro, R, Barbosa-Povoa, A. R. F. D., Matos, H. (2001). An improved RTN continuous-time formulation for the short-term scheduling of multipurpose batch plants. Industrial and Engineering Chemistry Research, 40, 2059-2068. 

Sundaramoorthy, A., Karimi, 1. A. (2005). A simpler better slot-based continuous-time formulation for short-term scheduling in multipurpose batch plants. Chemical Engineering Science, 60,... 

---