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


In this section, we describe the model used to study and optimize the expected behavior of this process. Figure 2 illustrates a portion of a schedule for a typical timeline. The first goal was to determine the production capacity for the process. The major independent variable for plant operation was the cadence with which new cell culture lots were started. Simplistically, faster cadence should lead to higher production rates. Indeed, in the absence of process variability, batch failure, column repack, and other unexpected events, this would be true; if every activity in the plant went according to plan every time, we could calculate the maximum cadence that would allow successive batches to begin as soon as possible, subject to not violating any of the process constraints. Unfortunately, this is not a valid picture of the real world and certainly does not adequately describe a real biologics process. For example, the need to periodically repack the columns produces a problem in which the determination of maximum cadence cannot be determined without a realistic model that accounts for all process constraints. When process variability is considered, the situation becomes more interesting and complex. Proper analysis requires Monte Carlo methods involving the generation of hundreds or even thousands of timelines. Each timeline represents one possible outcome for a scheduling horizon. In this study, we generated timelines that spanned a full year of operation.

Using our approach, we were able to automatically generate schedules for a full year of operation in about 10 seconds on a desktop computer. These schedules honor all constraints in the UDM formulation and all process-specific constraints described above. Each timeline produced a different randomly generated set of time-volume-titer data and thus each timeline produced different results. Thus, when a timeline is generated with a given cadence, we can examine the results to calculate the plant capacity in terms of lots per year. By generating hundreds of timelines for a given cadence, we begin to get an accurate picture of how the plant would behave if the process were driven at that rate. In principle, the plant could be operated so that one batch completely exited the process before the next were started, which would also minimize dropped lots. In the limit of extremely low cadence, the batches move through the plant independently and the only dropped lots correspond to the random 1% batch failure rate. However, this makes very poor use of the capital investment.

As the cadence is increased, production increases as the batches follow one another more closely. The higher the cadence, the greater the probability of interaction or coupling between successive batches. An example of this coupling would be when batch k must wait at some point in the process because material from batch k–1 continues to occupy a storage vessel. Another example would be a batch waiting for column repack necessitated by preceding batches, as shown in Figure 3. Material must be held in production BioReactor-5 because downstream processing is held up while Column2 is repacked. Because we drive the system at a fixed cadence, the production rate is essentially fixed, assuming all of the batches finish successfully. For this reason, inter-batch coupling that produces a delay here and there does not reduce capacity. However, as cadence is increased, the process reaches a point at which interbatch coupling begins to cause dropped lots. In Figure 4, the repack of Column2 disrupts the normal flow of lots through the purification train and ColTank1 becomes a local bottleneck. Three lots later, the harvest material expires while waiting for Column1/ColTank1 and must be discarded. Here, because a 9-day bioreactor lot followed a 11-day lot, two lots must be harvested in rapid succession. However, ColTank1 is still catching up from the recent repack and thus the second Harvest lot cannot be sent to Column1 in time. This illustrates how process variability can give rise to rich behaviors not anticipated by a deterministic model.

Figure 5. Annual production (A) and dropped lots (B) as a function of cadence. Although the plant capacity with a cadence of 2.75 is higher than with a cadence of 3.0, the performance of the plant at 3.0 is considered more desirable.
We studied the process with cadences ranging from one lot every 2.75 days to one lot every 4 days. Even at the slowest cadence studied, dropped lots can occur. This is because of the coupling of batches caused by variable bioreactor process times (9–11 days), and the occasional column repack failure. As the cadence becomes more aggressive, the process becomes less tolerant of delays in the purification system, thus even successful column repacks begin to play a role. This effect is enhanced because processing higher titer batches requires more time on the columns and increases the frequency with which repacks must be run. Figure 5 illustrates annual production capacity as a function of cadence. Figure 5 also shows the number of lots in which bioreactor product was held in the reactor for more than 1 or 1.5 days. Although the plant capacity with a cadence of 2.75 is higher than with a cadence of 3.0 (Figure 5A), the performance of the plant at 3.0 is considered more desirable. At 2.75, the plant experiences a dropped lot rate of 2% and another 6% of the lots have been stored for more than 24 h in the bioreactor. Given the labor and material costs associated with creating a new batch, and the disruption to normal work flow caused by handling dropped lots, operation at this faster cadence is simply not economical.

The model also was used to investigate the effect of labor availability on plant performance. We defined high load and low load scenarios for staffing levels. For the high load case, we used a seed cadence of three days. For the low load case we used a cadence of four days accompanied by more conservative harvest projections and more aggressive batching of QC samples. These cases represent the extremes of expected operating conditions and serve to bracket the expected support area load. Labor levels represent the number of persons needed to staff production for 24/7 operations.

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