Multi-objective dynamic optimization study of fed-batch bio-reactor
Graphical abstract
Introduction
Dynamic Optimization (DO) is a tool for obtaining the optimal operating conditions that maximizes productivity in a fed-batch bio-reactor, thus reducing experimental costs (Pham, 1998). DO has been extensively applied to fed-batch bio-reactor operations. DO computes the optimum feed recipe such that nutrients are maintained into the bio-reactor to grow or synthesize the desired metabolite. DO guarantees both an optimal cell growth and a metabolite bio-synthesis, avoiding under and overfeeding of the substrate (Zhang et al., 2004). Researchers have used methods like two-point collocation, Iterative Dynamic Programming (IDP) (Luus, 2000), relaxed reduced space SQP strategy (Logsdon and Biegler, 1993), IDP with absolute error penalty functions (Dadebo and Mcauley, 1995), and adaptive direct multiple shooting (Assassa and Marquardt, 2014) for the solution of DO problems. Moreover, several evolutionary algorithms have also been used in the past two decades to solve DO such as Differential Evolution (DE) (Kapadi and Gudi, 2004), Iterative Ant-Colony Algorithm (Zhang et al., 2005), Particle Swam Optimization (PSO) with embedded region reduction strategy (Zhang et al., 2015a), Hybrid Improved GA (Sun et al., 2013), and Box-Complex GA (Patel and Padhiyar, 2015b) to name a few. Yield of the product is another important parameter apart from the productivity in the bio-reactors. Both, the yield and productivity are conflicting objectives and hence form a Multi Objective Optimization (MOO) problem. The solution of this MOO problem can be obtained in the form of a Pareto front. There exists dedicated articles on the multi-objective dynamic optimization study (Maiti et al., 2011, Sarkar and Modak, 2005, Mandal et al., 2005, Halsall-Whitney et al., 2003, Logist et al., 2009, Logist et al., 2010, Logist et al., 2013, Jadot et al., 1998, Patel and Padhiyar, 2015a). However, the batch operation time is not used as an objective function in these reported works.
One of the approaches of solving multi-objective optimization problems is to augment all the conflicting objectives using appropriate weights and solve the resulting Single Objective Optimization(SOO) problem. The major challenge with this approach is assigning the appropriate weights to the individual objectives. Further, this SOO has to be solved multiple times with different weighing factors to obtain the Pareto front, which does not guarantee distinct solution with distinct weights. Moreover, this approach also suffers with a disadvantage of missing the concave portions of a Pareto front (Das and Dennis, 1997). Another classical approach for solving MOO problems is to minimize one objective keeping the others as constraints. The disadvantage of this approach is the appropriate selection of the function to be minimized and defining the constraint limits. Logist et al. (2013) used a more efficient scalarization approach using ACADO multi-objective toolkit for solving the multi-objective optimization of dynamic optimization problems. The drawback of these approaches is that one SOO problem has to be solved for every Pareto solution point. Thus, numerous SOO problems are required to be solved for representing the entire Pareto front. However, the number of SOO problems to be solved can be reduced using interactive tools and visualization approaches (Sindhya et al., 2014, Vallerio et al., 2015). On the other hand, the whole population converges to the Pareto front in the population based Evolutionary Algorithms (EAs) in a single run. This feature of the EAs has attracted significant attention for multi-objective dynamic optimization applications in the past two decades (Deb, 2001, Coello et al., 2006, Sarkar and Modak, 2005, Mandal et al., 2005, Maiti et al., 2011, Zhang et al., 2015a).
Usually dynamic optimization problems for fed-batch reactors are solved by maximizing productivity and/or yield for the fixed fed-batch times. The optimum fed-batch time can be computed by solving multiple single objective dynamic optimization problems at different fed-batch time (Luus, 1994, Lopez et al., 2010). However, such an approach is computationally more expensive as the single objective dynamic optimization problems have to be solved numerous times. In this work, a novel multi-objective optimization problem formulation is proposed for maximizing the productivity/yield along with minimizing the fed-batch time as a computationally efficient alternative. To verify the proposed approach, we in this work formulate and solve five multi-objective dynamic optimization problems of the fed-batch bio-reactor for a popular case study of secreted protein production by an yeast strain (Park and Fred Ramirez, 1988). The five MOO problems considered in this work are: (1) maximizing productivity and yield, (2) maximizing productivity and minimizing fed-batch time, (3) maximizing yield and minimizing fed-batch time, (4) maximizing productivity and yield while minimizing fed-batch time, and (5) maximizing productivity and minimizing endpoint substrate concentration. The multi-objective dynamic optimization problems have been solved using control vector parameterization with Multi-Objective Differential Evolution (MODE) algorithm. Further, the MODE is used along with the recently proposed mesh sort algorithm (Patel and Padhiyar, 2015a) in this work. Mesh sort used in the current work is an update of its previous version (Patel and Padhiyar, 2015a) and hence is discussed with more detail for the sake of completeness.
The MODE algorithm with mesh sort is presented in the next section. The fed-batch process model for secreted protein production by a yeast strain is presented in the subsequent section. Before presenting the multi-objective optimization results, single objective optimization results and their comparison with the literature data have been shown. The multi-objective problems and their results are presented followed by the concluding remarks.
Section snippets
Multi-Objective Differential Evolution (MODE) with mesh sorting
Multi-Objective Optimization (MOO) is a class of optimization, which deals with multiple and conflicting objectives simultaneously. MOO problems with conflicting objectives will have a set of solutions, which are called Pareto optimal solutions. Evolutionary algorithms have gained significant attention for solving MOO problems in the past two decades. Non-dominated sorting, rank based sorting (Qu and Suganthan, 2010) and evolution with decomposition (Jiao et al., 2013, Zhao et al., 2012) are
Fed-batch process model
A model for the production of secreted protein in fed-batch bio-reactor was reported by Park and Fred Ramirez (1988), which is also studied for single objective dynamic optimization application (Sun et al., 2013). However no work is reported for MOO for this application. The process shows very low sensitivity of control profile on productivity, which causes computational difficulties. The process model (mass balance) for the production of secreted protein is reproduced here for the sake of
Single objective dynamic optimization of fed-batch bio-reactor
Fed-batch processes are of transient in nature and are generally modelled as a set of differential algebraic equations (DAEs). A dynamic optimization problem is formulated for obtaining optimal trajectory of manipulated variables (MVs) by minimizing an objective function satisfying these differential equations. MVs are discretized by finite number of points. The values of MVs are optimized at these discrete points, while the two successive points are joined by lower order polynomial function.
Multi-objective dynamic optimization for a fed-batch bio-reactor
There has been ample work on productivity maximization for a fixed batch time in the scientific literature. However, there is limited work on multi-objective dynamic optimization of such bio-reactors, which include simultaneous maximization of productivity and yield by manipulating the fed-batch time and the feed recipe (Sarkar and Modak, 2005, Maiti et al., 2011). In these studies, the fed-batch time has not been minimized. There is scant literature for obtaining the optimum fed-batch time.
Results and discussion
The MODE algorithm is implemented in MATLAB R2011a on Intel(R) Core(TM) i5-3210M @ 2.5 GHz, 2.00 GB RAM, Windows 7 computer configuration. The set of model ODEs are solved using ode45 subroutine with a relative tolerance of 10−5, and absolute tolerance of 10−6. MODE program for MOO developed uses DE/rand/1 crossover (1) and polynomial mutation (2) along with the fast mesh sort for the elite survival selection operators. The values of DE parameters used in this work are, CR = 1.0, F = 0.5, η = 20, and p
Conclusions
Five multi-objective dynamic optimization problems for a fed-batch bio-reactor application has been studied using a MODE algorithm in this work. The four objectives considered in this work are maximization of the productivity, maximization of the yield, minimization of the fed-batch operation time, and minimization of the endpoint substrate concentration. It has been observed that these four objectives are mutually conflicting and hence make good candidates for the MOO study. The Pareto
Acknowledgement
The authors acknowledge funding from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India, for financial assistance (No. EMR/2015/002038).
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