In a continuum modeling approach of a multiphase flow, each transport phase is regarded as an individual pseudo-continuum fluid and all these “fluids” co-share the same space and time domains. Chapter 6 delineates the volume-averaging method to construct the pseudo-continuum fluids over which the volume-averaged Eulerian modeling approach is developed. The key concepts and formula of volume-averaged continuum modeling include the definitions of intrinsic and phase averages and their relationship; the volume-averaging theorems; the general form of volume-averaged transport equations and the individual equations of mass, momentum, and energy; the volume and mass balance conditions of all phases; the constitutive relations of the volume-averaged tensors by individual volume-averaged parameters; and the formulation of interfacial transport between phases. The effect of turbulence on phase transport is also handled via Reynolds decomposition and time-averaging over the volume-averaged equations. The detailed formulation of various turbulent transport coefficients and the salient behavior of the turbulence modulation from various interactions among phases are presented.

In the problems we have considered so far, we have either ignored the actual material consumption/production (sequential environments) or assumed that materials are consumed/produced at fixed proportions (network environments). There are problems, however, where the proportions in which materials are consumed can vary provided that some specifications are satisfied. This problem, which is termed multiperiod blending or simply blending, is fundamentally different from the ones discussed thus far because it leads to nonlinear models. There are two types of blending problems: (1) different streams/inputs are blended before they are processed/converted (process blending); and (2) streams/inputs are blended to produce final products (product blending). In Section 11.1, we introduce some preliminary concepts and a formal problem statement for product blending. In Section 11.2, we present two alternative formulations for product blending, and in Section 11.3, we present two approximate linear reformulations. We close, in Section 11.4, with a discussion of models for process blending. We focus on the equations necessary to account for the key new features of blending problems: (1) the selection of input materials and their blending in variable proportions, and (2) the requirement to satisfy given property specifications.

This chapter provides an overview of mixed-integer programming (MIP) modeling and solution methods.In Section 2.1, we present some preliminary concepts on optimization and mixed-integer programming. In Section 2.2, we discuss how binary variables can be used to model features commonly found in optimization problems. In Section 2.3, we present some basic MIP problems and models. Finally, in Section 2.4, we overview the basic approaches to solving MIP models and present some concepts regarding formulation tightness and decomposition methods.Finally, we discuss software tools for modeling and solving MIP models in Section 2.5.

In Chapter 1, we discussed how the market environment affects the scheduling of manufacturing facilities primarily because the volume and variability of product demand determine the regularity and frequency in which scheduling is performed. The production of high-volume products with relatively constant demand or high-volume intermediates can be based on demand forecasts, rather than specific orders. If the forecasts do not exhibit significant fluctuations over time, one approach is to generate a schedule that can be repeated periodically, to maintain a certain level of stock. The process of generating such schedules, which is the topic of the present chapter, is termed periodic scheduling. In Section 10.1, we use the single-unit environment to motivate the need for periodic scheduling, define relevant notation, and present some preliminary concepts and models. In Section 10.2, we use the single-stage environment to discuss some additional concepts and one formulation. Finally, in Section 10.3, we present periodic scheduling in network environments.We consider processes without any of the processing features discussed in Chapter 8.

In Chapter 1, we introduced the supply chain planning matrix and its different planning functions, discussed how scheduling fits within this matrix, and mentioned that integration across functions can lead to better solutions. Chemical production scheduling interacts directly with two functions: (1) production planning, and (2) process automation and control (though the latter are not typically defined as functions of the SC matrix). Integration with automation and control were discussed in Chapter 14. In the present chapter, we discuss the integration of production planning and scheduling. We start, in Section 15.1, with some preliminary concepts and motivation for the need to integrate planning with scheduling. In Section 15.2, we present a formulation for an introductory planning-scheduling problem. We continue, in Section 15.3, with an approach for more complex problems, both in single- and multiunit environments. Finally, in Section 15.4, we overview a general but also algorithmically more advanced approach that is applicable to any production environment. For simplicity, in Sections 15.2 and 15.3, we do not consider special processing features, such as complex storage policies and utility constraints. The method in Section 15.4 can in principle be applied to any facility with any processing feature.

In this chapter, we discuss problems in the single-stage or parallel-units environment. The problem statement is presented in Section 4.1. Three types of models are presented in Section 4.2 (sequence-based), Section 4.3 (continuous time grid-based), and Section 4.4 (discrete time grid-based). In Section 4.5, we present how batching decisions can be handled, and in Section 4.6 we discuss how the three types of models can be extended to handle a new feature, namely, general shared resources. Finally, in Section 4.7 we present extensions on the modeling of general resource constraints using discrete modeling of time. Building upon the material in Chapter 3, we illustrate how some of the modeling techniques introduced for single-unit problems can be extended to account for multiple units. Our goal is to outline some general ideas that the reader can apply to a wider range of problems.We focus on (1) problem features that are new, compared to the ones in single-unit problems (i.e., batching decisions and general shared resources); and (2) new modeling techniques that are necessary to account for these features.

The focus of the book so far has been on the development of models and solution methods to obtain high quality predicted schedules. While these two components are necessary towards the implementation of optimization-based scheduling methods, they are not sufficient by themselves. Specifically, the model must be solved repeatedly, in real time, taking into account new information and disturbances. The goal of the present chapter is to provide high-level understanding on how the optimization model should be used to obtain a real-time scheduling algorithm that yields high-quality implemented schedules.In Section 14.1, we motivate why repeated optimization is necessary, introduce necessary notation, and present the overall framework we use. In Section 14.2, we present a state-space model which offers a natural way to formulate the optimization model that is updated and solved in real time. In Section 14.3, we present the basic considerations and a general simulation-based framework for designing real-time scheduling algorithms, and, close, in Section 14.4, with a discussion on how integration with other functions can offer early feedback leading to faster recourse. We use models and examples based on network environments, but all the ideas and methods are directly applicable to problems in sequential environments.

In this chapter, we discuss scheduling in multipurpose environments, which are the most general sequential environments. As in single- and multistage facilities, the general problem is posed in terms of facility (e.g., number and capacity of units) and product (e.g., processing times) data, as well as raw material and resource availability (e.g., batch release times), and product demand (e.g., due times). If the batching problem is solved independently, however, then the problem can be expressed in terms of batches instead of products. This is the problem that we study in the present chapter. Models for the simultaneous batching and scheduling can be formulated using the ideas presented in Chapter 4. General shared resources and storage policies can be modeled using the techniques presented in e Chapters 4 and 5, respectively.

In this chapter, we discuss scheduling in multistage environments. The problem statement is presented in Section 5.1 and three types of models are presented in Section 5.2 (sequence-based), Section 5.3 (continuous grid-based), and Section 5.4 (discrete grid-based). In Section 5.5, we introduce an important new feature, namely, storage constraints. Again, we build upon the material covered in the previous chapters to model assignment and sequencing decisions, as well as other constraints such as release and due times.

In this chapter, we discuss models for the scheduling of continuous processes. We use the sets, subsets, and parameters introduced in Chapter 7 to represent a facility. To simplify the presentation, we consider problems with the following assumptions: (1) dedicated storage vessels; (2) no storage in processing units; (3) instantaneous and resource unconstrained material transfers; (4) no unit deterioration; and (5) demand that can be satisfied (no backlogs). We start, in Section 9.1 with some background and a discussion of the main differences between batch and continuous processing. In Section 9.2, we present the basic, STN-based, model; and we close, in Section 9.3, with numerous extensions including modeling for startups and shutdowns, transitions between tasks, and time delays.

In this chapter, we discuss how to model additional processing features that may be present in a chemical facility. To keep the presentation simple, we illustrate how models based on a common discrete grid can be modified to account for these features. Continuous time models can also be extended to account for most of these features, but often lead to more complex and/or nonlinear formulations. We start, in Section 8.1, with the modeling of material consumption and production during the execution of a batch. In Section 8.2, we discuss the modeling of complex material storage and transfer activities. In Section 8.3, we present how to account for unit and task setups and task families. Finally, in Section 8.4, we present how to model unit deterioration and maintenance tasks.

We discuss four solution methods for problems in general network production environments. After presenting some background and motivation, in Section 13.1, we cover (1) preprocessing and tightening methods, in Section 13.2; (2) reformulations, in Section 13.3; (3) an approach to formulate models that employ multiple discrete time grids, in Section 13.4; and (4) a three-stage algorithm that employs both a discrete and continuous time models, in Section 13.5. For simplicity, we do not consider shared utilities nor special processing features such as storage in processing units and multiple material transfers. The methods presented in Section 13.2 and Section 13.3 are applicable to both discrete and continuous time models, but to keep the presentation short, we apply them to discrete time models, though we comment on their application to their continuous counterparts. The reader can study each section, after Section 13.1, independently, that is, Section 13.2 is not prerequisite for Section 13.3, and so on.

The goal of the present chapter, as well as Chapter 13, is to illustrate how problem features can be exploited to develop more efficient models and/or specialized algorithms. We start, in the present chapter, with solution methods for problems in sequential environments. Specifically, we discuss four methods: (1) a decomposition approach, in Section 12.1; (2) preprocessing algorithms and tightening constraints, in Section 12.2; (3) a reformulation and tightening constraints based on time windows, in Section 12.3; and (4) a two-step algorithm, combining the advantages of discrete and continuous time models, in Section 12.4. While all presented methods can be applied to a wide range of problems, we present them for a subset of problems for the sake of brevity. Also, all methods can be applied to problems under different processing features, but to keep the presentation simple, we discuss problems with no shared utilities and no storage constraints.