The unit allocation constraints on equipment may be written as:
Equation 1 ensures that at most one task may begin at any time on any one piece of equipment. Equation 2 ensures that successive
tasks do not overlap on any piece of equipment.
Equation 3 ensures that equipment j can store at most one material at any one time.
Because some equipment can be used both for processing and storage, eq 4 is required to ensure that batch processing and storage
activities do not overlap.
The material balance constraints for material s at time t may be written as:
The UDM approach is easily extended to address other physical considerations.3 One advantage of being able to control the solution algorithm and UDM formulation is that an effective approach to stochastic
optimization can be implemented. Namely, multiple literal versions of a task with stochastic parameters can be incorporated
and the UDM formulation guarantees that only one of these versions is executed—just as if there were multiple alternate activities
to accomplish a certain process step. To be precise, consider a task TR that depends on stochastic parameters, for example,
titer. The UDM formulation and customized solution algorithm can easily accommodate introducing literal task and probability
tuples (TR1, p1), (TR2, p2),..., (TRk, pk), where the tuple probabilities for TR must sum to one. The solution algorithm can randomly choose among these versions,
according to the probabilities, to achieve a Monte Carlo sampling effect. The random selection of a particular literal version
of a stochastic task, TRi—one containing specific values for one or more stochastic parameters—forces the algorithm to accommodate
the consequences as the solution is completed. For example, a longer task instance may induce longer storage time or a high
titer batch may require a solution that uses multiple chromatographic separation cycles. Unlike discrete event simulation
approaches, Monte Carlo treatment of uncertainty can evolve over the timeline in arbitrary directions and not just left to
right. However, to be realistic, the solution algorithm cannot use information about the future behavior of a stochastic parameter,
like cycle time, to improve process behavior. This imposes constraints on the solution algorithm, hence the control of algorithm
details with full awareness of process physics through the RTN is necessary to implementing this approach.
Automatic Schedule Generation
The core VirtECS solver is able to automatically generate schedules that satisfy all of the above constraints and can be controlled
to solve subsets of constraints or variables in several ways to support algorithmic strategies implied by application physics.
The paper by Pekny describes the features required of the core solver in more detail and references earlier algorithm engineering
work underpinning the development of the core solver.5 For this paper, the performance of the core solver may be taken as a given, and we focus on the novel aspects of solving
the problem for a biologics facility that resides in the development of the outer algorithm. We will therefore treat the core
solver much as papers on branch and bound algorithms for MILP problems treat the underlying LP solver—we take its performance
as a given.
The purpose of the process-specific outer algorithm is two-fold. First, it is necessary to define the parameters within which
the core solver must work. This can mean specifically authorizing a restricted set of task-equipment-pairs (teps) and material-equipment-pairs
(meps) that the core solver is allowed to use for a particular demand. This allows the algorithm engineer to maintain tight
control over the search space of the core solver and still backtrack to relax suggestions that prove incorrect. The outer
algorithm must focus attention effectively for solution acceleration to be dramatic. A well-designed outer algorithm speeds
up the solution time because the core solver must search a smaller space and allows the user to ensure the solutions generated
will conform to the special constraints and degeneracy preferences of the process in question. In selecting which equipment
set to use for a particular demand, the outer algorithm weighs, for example, the relative advantages of load balancing on
parallel equipment versus avoiding transitions. This is one place where the underlying logic of experienced plant personnel
may be captured and codified to produce a high quality automated algorithm for a particular process. Second, the outer algorithm
must examine the results of the solution generated for a subset of the UDM formulation. This is essential to determine whether
remedial action must be taken because of the violation of some specialized process constraint that the core solver does not
explicitly address, but which is implied by the UDM formulation knowing the details of the process under study. In addition,
the outer algorithm has the opportunity to apply wholesale solution transformation by sliding task instances or adding additional
tasks to meet operational preferences in a way that exploits the power of the underlying UDM formulation. Examples of tasks
that may be treated in this exceptional fashion include CIP and column repacks, which are highly specific to biologics.