a smooth formulation for maximum on-specification production in campaign continuous manufacturing

Campaign continuous manufacturing (ccm), i#e# continuous processes with a short operational window, is gaining attention in the pharmaceutical industry as an alternative to the traditional batch-wise manufacturing# a major challenge in ccm is how to meet stringent product quality specifications during the operation, in which the startup and shutdown phases constitute a large portion of the process, making the quality specifications even harder to meet# the optimal operation of ccm was discussed in a recent publication, where an optimization problem for maximizing the on-specification production was also formulated# however, the formulation was potentially nondifferentiable due to its hybrid discrete-continuous nature, and required rather complicated remedies to ensure differentiability before the problem could be solved by gradient-based optimization algorithms# in this work, the same problem of maximizing on-specification production in ccm is addressed by proposing a much simpler formulation, in which the discrete events are approximated by smooth functions# therefore, the proposed formulation lends itself to conventional gradient-based algorithms, and can solve optimal ccm problems reliably as demonstrated in the case studies#