Mathematical Programming for the Design and Analysis of a Biologics Facility - The use of mathematical programming methods for automated schedule generation. - BioPharm International


Mathematical Programming for the Design and Analysis of a Biologics Facility
The use of mathematical programming methods for automated schedule generation.

BioPharm International
Volume 23, Issue 2


Figure 4. Dropped lots engendered by column repacking
The storage of bioreactor product is a major factor in analyzing process behavior. Bioreactor product may be stored in the bioreactor itself while waiting for the centrifuge to become available. This happens when batches from two parallel bioreactors finish processing at nearly the same time. Although no hard limit was enforced in the core solver for the length of this storage, we did track bioreactor storage time for all lots. This can be used to determine the distribution of the hold time durations that will be encountered. This distribution can be used to determine the hold time that should be validated with the regulatory agency to facilitate manufacturing. Following the centrifuge and filter, the resulting material is collected in the harvest tank where it remains until it is fed to the purification stage. The storage duration in the harvest tank is subject to a strict time limit, approximately a day for this representative process. However, bioreactor batches that exceed this limit must be discarded, and are referred to dropped lots. To explore stochastic behavior and still obey causality, this harvest tank limit is not enforced by the core solver lest the solver introduce idle time before the start of a production bioreactor to noncausally avoid a dropped lot because of contention for the downstream purification stage. When deciding to start a production bioreactor, an operator cannot know that a particular batch will face a busy purification stage because of a confluence of future factors, such that delaying it will avoid a dropped lot. To ensure an accurate analysis, the customized solution algorithm does not act on future information either.

The dropped lot effect is a significant factor in determining expected plant capacity. There are a number of circumstances that can lead to a dropped lot. First, anytime the bioreactors produce an 11-day batch followed by a 9-day batch, the lots arrive at the purification stage in rapid succession, forcing storage of the second lot. Depending on the presence of other delaying factors, such as waiting for a chromatography column to be repacked, the storage duration for this lot may be too long, forcing a dropped lot. In addition, the presence of high harvest titers can cause processing delays. Batches with high concentrations of product take longer to process on the columns and can lead to unacceptable delays in processing subsequent batches. It is not easy to avoid dropping lots of high harvest batches because by the time the harvest titer of a lot is known, the subsequent lot may already have started. The processing of individual batches also may be delayed because of column repacks, repack failures or, in general, any unexpected events that reduce the capacity of the columns.


In addition to the mainline process, there are a number of auxiliary activities involved in plant operation. These consist of support functions like buffer and media preparation and quality control (QC) testing. We used a classic problem decomposition method to model the support activities (for an LP analogy see the Dantzig and Thapa paper).5 Limitations on labor and support activities were not allowed to affect main line capacity. The level of required support services was also considered, but it was assumed that these constraints would not be rate limiting. This assumption is justified because it suffices for the model to predict the required level of resource availability required at every point in the timeline for any particular process schedule. These data are then used to determine the level of support services that would be required to ensure that they would impose no limit on real production. This method was used in modeling such support services as buffer and media preparation, QC testing, and labor. Following the initial evaluation of plant capacity, a separate set of studies was performed in which constraints were added to the model, allowing the impact of limited support and staffing to be quantified. The model explicitly accounts for the fact that support activities can be rate limiting because the outcome of these activities (e.g., QC results) may influence downstream processing decisions. This model decomposition was carried out as follows: Materials were classified as either primary or secondary. Primary materials (those materials used in the main manufacturing model) were scheduled using the solver, ensuring all constraints were met. Secondary materials (manufacturing support areas) were included in the formulation equations, but their constraints were tracked rather than enforced during the main solution process. In practice, the subproblems are broken into independent models for buffer and media prep, and QC.

There are two primary benefits to this approach. First, the decomposition allows us to solve multiple related smaller problems instead of a single large one. This computational benefit is especially important as the number of support areas and their complexity grows. Second, the solution to the primary problem describes the behavior of the plant in the absence of support area restrictions. This aligned well with the industrial philosophy of first designing the manufacturing process and then sizing the support areas to fit the needs of manufacturing. Whether because of unanticipated factors or facility growth, support activities can become the de facto bottleneck of manufacturing operations. Initially, the model was solved with several constrained support areas. Later, it was resolved with fully unconstrained support areas. A comparison of results provided guidance into which support areas had a critical impact on manufacturing capacity.

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