R.W.H. Sargent

A general algorithm for short-term scheduling of batch operations—I. MILP formulation

Abstract A general framework for handling a wide range of scheduling problems arising in multiproduct/multipurpose batch chemical plants is presented. Batch processes involving a variety of complexities are represented using a state-task network. The novel feature of this representation is that both the individual batch operations (“tasks”) and the feedstocks, intermediate and final products (“states”) are included explicitly as network nodes. Processes involving sharing of raw materials and intermediates, batch splitting and mixing and recycles of material, can be represented unambiguously as such networks. The short-term scheduling problem is formulated as a mixed integer linear program (MILP) based on a discrete time representation. Flexible equipment allocation, variable batchsizes and mixed intermediate storage policies involving both dedicated and multipurpose storage vessels are taken into account. Limited availability of raw materials, both at the start and during the time horizon of interest, is accommodated. Product deliveries may take place at any time during the horizon, and the amounts involved may be either fixed or variable. The use of utilities by the various tasks may vary over the task processing time, and may be constant or proportional to the batchsize. The availability and/or cost of utilities may vary over the time horizon of interest. The objective function is the maximization of a profit function involving the value of the products, and the cost of raw materials, utilities and material storage. The formulation may result in MILPs involving large numbers of binary variables. Issues pertaining to the efficient solution of these problems are discussed in Part II of this paper.

A general algorithm for short-term scheduling of batch operations—II. Computational issues

Abstract The first part of this paper (p. 211) presented a general mathematical framework for describing a wide variety of scheduling problems arising in multiproduct/multipurpose batch chemical plants. The problem is formulated as a large mixed integer linear programming model (MILP). We describe a variety of techniques that exploit the characteristics of the problem in order to reduce the amount of computation required. These include reformulation of some of the constraints, derivation of an alternative and much more compact linear programming relaxation of the MILP, and reduction of the non-integrality of the solutions of relaxed LPs through their a posteriori analysis. The combination of the three measures results in a significant improvement in computational performance without compromising the optimality of the solution obtained. A case study is presented to illustrate the applicability of the method to the scheduling of multipurpose plants under a variety of operational constraints.

The optimal operation of mixed production facilities—a general formulation and some approaches for the solution

Abstract This paper extends the concept of a “resource-task network”, described by Pantelides, to provide a unified mathematical formulation of the problem of determining the optimal operating conditions of a mixed production facility, comprising multipurpose plant for both batch and continuous operations. The formulation uses a variable event-time sequence common to all system events. This yields a large mixed integer nonlinear programming problem. For batch processes with fixed recipes the problem is linear and can be solved by existing techniques. Alternative approaches for solving the general problem are discussed, and some preliminary numerical results are presented.

The mathematical modelling of transient systems using differential-algebraic equations

Abstract The mathematical difficulties associated with a class of mixed systems of differential and algebraic equations are presented and an algorithm for dealing with them is described. Two classes of chemical engineering models which give rise to such systems, are identified. Also some problems arising from rigorous dynamic distillation models are analysed.

Optimal periodic scheduling of multipurpose batch plants

A rigorous mathematical programming framework for the scheduling of multipurpose batch plants operated in a cyclic mode is presented. The proposed formulation can deal with batch operations described by complex processing networks, involving shared intermediates, material recycles, and multiple processing routes to the same end-product or intermediate. Batch aggregation and splitting are also allowed. The formulation permits considerable flexibility in the utilisation of processing equipment and storage capacity, and accommodates problems with limited availability of utilities. The scheduling problem is formulated as a large mixed integer linear program (MILP). For a given cycle time, it is shown that it is sufficient for the formulation to characterize a single cycle of the periodic schedule despite the existence of tasks that span two successive cycles. The optimal cycle time is determined by solving a sequence of fixed cycle time problems. The MILP is solved by a branch-and-bound algorithm modified so as to avoid exploring branches that are cyclic permutations of others already fathomed. The resulting implementation permits the solution of problems of realistic size within reasonable computational effort. Several examples are used to illustrate the applicability of the algorithm.

Optimum design of heat exchanger networks

Abstract The optimum design of heat exchanger networks is considered in two stages. In the first, the optimum configuration for fixed values of the continuous variables is determined using an implicit enumeration algorithm. In the second stage the optimization of continuous variables of the network is performed solving a large scale nonlinear programme. The proposed method for discrete variables lessens considerably the computational effort for solving the combinatorial problem in the first stage. The results show the importance of optimizing the continuous variables of these networks.

A functional approach to process synthesis and its application to distillation systems

Following Douglass philosophy of a target-directed hierarchical approach to process design, we propose a general framework for systematic implicit enumeration and evaluation of feasible designs, with successive model refinement leading to a final detailed design. The process is represented by a state-task-network, and feasible networks are generated by assigning a purpose, or function, to each elementary task. The approach is applied to the synthesis of both simple and complex distillation systems, and two examples of homogeneous azeotropic systems are treated in some detail.

Computer generation of process models

The paper describes a prototype system for the automatic generation of a mathematical model, describing the dynamic behavior of a process system, from a purely physical description. A synthetic approach is adopted, in which models are built up using a library of physico-chemical laws. Essentially a process is modelled as a collection of interacting phases, with various kinds of interaction or transfer between them, together with additional relations expressing containment and control. A prototype package, at present limited to the generation of lumped-parameter models, has demonstrated the feasibility of the approach.

The optimal operation of mixed production facilities — Extensions and improvements

The paper presents some extensions and improvements of an earlier general formulation to deal with operational constraints and plants involving continuous operations. The concept of Unit Usability is introduced to give a unified formulation for a variety of operational constraints. A modified task model for continuous operations enables us to linearize cross-products of one integer variable and two continuous variables. These ideas are illustrated by application to a realistic case-study.

Dynamic behaviour of multi-component multi-stage systems. Numerical methods for the solution

Abstract The dynamic behaviour of multistage systems is described by large sets of non-linear first-order differential equations. These are usually linearized in order to obtain a solution, but it is here shown that this in general leads to inconsistency with the physical requirements of the problem. The extreme stability of the physical system confers properties on the equations which cause instability in most of the standard numerical methods of solution, and the consequent error swamps the effects of non-linearity. A step-by-step procedure is proposed which makes use of exponential functions and avoids this instability. The non-linear effects then determine the permissible step lengths, and study of the behaviour of the approximating solution results in a method of correcting for finite step length, which is confirmed by numerical experiments. Step length criteria are discussed, and comparison with the Kutta-Simpson process shows that at least in some cases the step length can be up to 128 times greater for a given accuracy in the results. A review of standard numerical procedures is included for general interest.