An evolutionary strategy for fed-batch bioreactor optimization; concepts and performance

An evolutionary program, based on a real-code genetic algorithm (GA), is applied to calculate optimal control policies for bioreactors. The GA is used as a nonlinear optimizer in combination with simulation software and constraint handling procedures. A new class of GA-operators is introduced to obtain smooth control trajectories, which leads also to a drastic reduction in computational load. The proposed method is easy to understand and has no restrictions on the model type and structure. The performance and optimal trajectories obtained by the extended GA are compared with those calculated with two common methods: (i) dynamic programming, and (ii) a Hamiltonian based gradient algorithm. The GA proved to be a good and often superior alternative for solving optimal control problems.     

Optimal feeding strategies by adaptive mesh selection for fed-batch bioprocesses

The objective of this contribution is the design of optimal feeding strategies for fed-batch bioprocesses, where complex dynamic models with input and state constraints are present. For the solution of this dynamic optimization problem a transformation to a finite dimensional optimization problem is made using piecewise linear control profiles. The optimization of these profiles is performed by a sequential approach, that includes an ODE solver for the solution of the model ODE's. Further an adaptive mesh selection algorithm was investigated for an appropriate discretization of the control profiles. The implementation of the resulting optimal feeding profiles is shown for a process example, namely the production of nikkomycin by Streptomyces tendae. This implementation uses a hierarchical process control framework, that consists of components for process monitoring, state estimation, and trajectory control.     

Numerical Solution to Generalized Burgers'-Fisher Equation Using Exp-Function Method Hybridized with Heuristic Computation

In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers''-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers''-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.     

Evaluation of a particle swarm algorithm for biomechanical optimization

Optimization is frequently employed in biomechanics research to solve system identification problems, predict human movement, or estimate muscle or other internal forces that cannot be measured directly. Unfortunately, biomechanical optimization problems often possess multiple local minima, making it difficult to find the best solution. Furthermore, convergence in gradient-based algorithms can be affected by scaling to account for design variables with different length scales or units. In this study we evaluate a recently-developed version of the particle swarm optimization (PSO) algorithm to address these problems. The algorithm's global search capabilities were investigated using a suite of difficult analytical test problems, while its scale-independent nature was proven mathematically and verified using a biomechanical test problem. For comparison, all test problems were also solved with three off-the-shelf optimization algorithms--a global genetic algorithm (GA) and multistart gradient-based sequential quadratic programming (SQP) and quasi-Newton (BFGS) algorithms. For the analytical test problems, only the PSO algorithm was successful on the majority of the problems. When compared to previously published results for the same problems, PSO was more robust than a global simulated annealing algorithm but less robust than a different, more complex genetic algorithm. For the biomechanical test problem, only the PSO algorithm was insensitive to design variable scaling, with the GA algorithm being mildly sensitive and the SQP and BFGS algorithms being highly sensitive. The proposed PSO algorithm provides a new off-the-shelf global optimization option for difficult biomechanical problems, especially those utilizing design variables with different length scales or units.     

An Algorithm for operating a fed-batch fermentation using quasi-steady state

The determination of an optimum feeding profile of a fed-batch fermentation requires the solution of a singular optimum control problem, which is often complicated by changes in the process kinetics during the fermentation. The procedure of optimization may be sufficiently simple, if the feeding part of fermentation is carried out in the quasi-steady state. In this work an algorithm for operating a fed-batch fermentation using mentioned regime is offered. The algorithm supposes a periodical correction of the feeding strategy. Applying to fed-batch lysine fermentation demonstrate efficacy of this algorithm over frequently used strategies.     

A comparison of physical mapping algorithms based on the maximum likelihood model

MOTIVATION: Physical mapping of chromosomes using the maximum likelihood (ML) model is a problem of high computational complexity entailing both discrete optimization to recover the optimal probe order as well as continuous optimization to recover the optimal inter-probe spacings. In this paper, two versions of the genetic algorithm (GA) are proposed, one with heuristic crossover and deterministic replacement and the other with heuristic crossover and stochastic replacement, for the physical mapping problem under the maximum likelihood model. The genetic algorithms are compared with two other discrete optimization approaches, namely simulated annealing (SA) and large-step Markov chains (LSMC), in terms of solution quality and runtime efficiency. RESULTS: The physical mapping algorithms based on the GA, SA and LSMC have been tested using synthetic datasets and real datasets derived from cosmid libraries of the fungus Neurospora crassa. The GA, especially the version with heuristic crossover and stochastic replacement, is shown to consistently outperform the SA-based and LSMC-based physical mapping algorithms in terms of runtime and final solution quality. Experimental results on real datasets and simulated datasets are presented. Further improvements to the GA in the context of physical mapping under the maximum likelihood model are proposed. AVAILABILITY: The software is available upon request from the first author.     

Optimization of fed-batch bioreactor using neural network model

An algorithm using feedforward neural network model for determining optimal substrate feeding policies for fed-batch fermentation process is presented in this work. The algorithm involves developing the neural network model of the process using the sampled data. The trained neural network model in turn is used for optimization purposes. The advantages of this technique is that optimization can be achieved without detailed kinetic model of the process and the computation of gradient of objective function with respect to control variables is straightforward. The application of the technique is demonstrated with two examples, namely, production of secreted protein and invertase. The simulation results show that the discrete-time dynamics of fed-batch bioreactor can be satisfactorily approximated using a feedforward sigmoidal neural network. The optimal policies obtained with the neural network model agree reasonably well with the previously reported results.     

Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind

In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.     

用遗传算法优化流加培养的底物流加轨迹

遗传算法（Genetic Algorithm,GA)j是把生物进化论和遗传学原理应用于工程优化而创造出来的新的优化算法,在复杂问题的优化方面显示出了优良性能。近年来GA开始应用于发酵工程领域,本文介绍了应用GA优化流加培养流加轨迹的原理和方法。     

Fitting piecewise linear regression functions to biological responses

An iterative approach was achieved for fitting piecewise linear functions to nonrectilinear responses of biological variables. This algorithm is used to estimate the parameters of the two (or more) regression functions and the separation point(s) (thresholds, sensitivities) by statistical approximation. Although it is often unknown whether the response of a biological variable is adequately described by one rectilinear regression function or by piecewise linear regression function(s) with separation point(s), an F test is proposed to determine whether one regression line is the optimal fitted function. A FORTRAN-77 program has been developed for estimating the optimal parameters and the coordinates of the separation point(s). A few sets of data illustrating this kind of problem in the analysis of thermoregulation, osmoregulation, and the neuronal responses are discussed.     

Genetic algorithm with alphabet optimization

 In recent years the genetic algorithm (GA) was used successfully to solve many optimization problems. One of the most difficult questions of applying GA to a particular problem is that of coding. In this paper a scheme is derived to optimize one aspect of the coding in an automatic fashion. This is done by using a high cardinality alphabet and optimizing the meaning of the letters. The scheme is especially well suited in cases where a number of similar problems need to be solved. The use of the scheme is demonstrated with such a group of problems: the simplified problem of navigating a ‘robot’ in a ‘room.’ It is shown that for the sample problem family the proposed algorithm is superior to the canonical GA. Received: 26 August 1994/Accepted in revised form: 13 January 1995     

Minimal time control of fed-batch bioreactor with product inhibition

This paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green’s theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies.     

Optimization of feeding profile for baker's yeast production by dynamic programming

An optimal substrate feeding for an industrial scale fed-batch fermenter is determined through iterative dynamic programming in order to maximize the cell-mass production and to minimize the ethanol formation. An experimentally validated rigorous dynamic model comprises constraints in the optimization problem. A new objective function is proposed to accommodate the competing requirements of maximum yeast production and minimum ethanol formation. The objective function is maximized with iterative dynamic programming with respect to the sugar feed rate. Results prove the effectiveness of dynamic programming for solving such high-dimensional and nonlinear optimization problems, and the resulting optimal policy indicates that considerable increase in yeast production in fed-batch fermenters can be achieved while minimizing the undesired by-product, ethanol.     

Optimization of feeding profile for a fed-batch bioreactor by an evolutionary algorithm

The optimal feeding profile of a fed batch process was designed by means of an evolutionary algorithm. The algorithm chromosomes include the real-valued parameters of a profile function, defined by previous knowledge. Each chromosome is composed of the parameters that define the feeding profile: the feed rates, the singular arc parameters and the switching times between the profile states. The feed profile design was tested on a fed-batch process simulation. The accepted profiles were smooth and similar to those derived analytically in other studies. Two selection functions, roulette wheel and geometric ranking, were compared. In order to overcome the problem of model mismatches, a novel optimization scheme was carried out. During its operation the process was sampled, the model was updated and the optimization procedure was applied. The on-line optimization showed improvement in the objective function for relatively low sample times. Choosing the sampling frequencies depends on the process dynamics and the time required for the measurements and optimization. Further study on experiments of fed-batch process demonstrated the use of complex, non-differentiable model and produced improved process performances using the optimal feeding profile.     

A Genetic Algorithm Approach to the Scheduling of FMSs with Multiple Routes

Usually, most of the typical job shop scheduling approaches deal with the processing sequence of parts in a fixed routing condition. In this paper, we suggest a genetic algorithm (GA) to solve the job-sequencing problem for a production shop that is characterized by flexible routing and flexible machines. This means that all parts, of all part types, can be processed through alternative routings. Also, there can be several machines for each machine type. To solve these general scheduling problems, a genetic algorithm approach is proposed and the concepts of virtual and real operations are introduced. Chromosome coding and genetic operators of GAs are defined during the problem solving. A minimum weighted tardiness objective function is used to define code fitness, which is used for selecting species and producing a new generation of codes. Finally, several experimental results are given.     

A contribution to optimal control of fed-batch biochemical processes

The problem of looking for high efficient modern control strategies in fermentation technology is very urgent, nowdays. Particular attention should be paid to the processes in fed-batch mode. Both, optimal feedforward and feedback control approaches are suggested. A contribution is considered to have been made in the feedback control where continuous and discrete versions are treated as well. The control laws are carried out by a variation calculus problem and a polynomial pole placement synthesis solution, respectively. All the algorithms result in an optimal substrate feed rate profile. On the basis of recursive least squares identification of the model coefficients an adaptive discrete-time control strategy is proposed. Some satisfying simulation results are dealt with.     

Dynamic optimization of bioprocesses: efficient and robust numerical strategies

The dynamic optimization (open loop optimal control) of non-linear bioprocesses is considered in this contribution. These processes can be described by sets of non-linear differential and algebraic equations (DAEs), usually subject to constraints in the state and control variables. A review of the available solution techniques for this class of problems is presented, highlighting the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these systems. In order to surmount these difficulties, we present several alternative stochastic and hybrid techniques based on the control vector parameterization (CVP) approach. The CVP approach is a direct method which transforms the original problem into a non-linear programming (NLP) problem, which must be solved by a suitable (efficient and robust) solver. In particular, a hybrid technique uses a first global optimization phase followed by a fast second phase based on a local deterministic method, so it can handle the nonconvexity of many of these NLPs. The efficiency and robustness of these techniques is illustrated by solving several challenging case studies regarding the optimal control of fed-batch bioreactors and other bioprocesses. In order to fairly evaluate their advantages, a careful and critical comparison with several other direct approaches is provided. The results indicate that the two-phase hybrid approach presents the best compromise between robustness and efficiency.     

Effect of setpoint on L-lysine fermentation

In control of a bioprocess, setpoint of fed-batch fermentation processes has a great influence on both the cost and the operating efficiency. Determination of a setpoint depends on the system and objective function. This work investigates two operating conditions for a fed-batch culture of L-lysine production. One is to maintain the reducing sugar concentration (RSC) on a constant setpoint, whereas the other has a piecewise variation of the RSC setpoint. Productivity, yield, and a cost function are employed to evaluate the performance of different setpoints on the fermentation process. Constant setpoint which is commonly used in fed-batch culture is not the best approach in L-lysine fermentation. Piecewise variation of setpoint shows that the better policy is to set the RSC at a higher concentration in the early cell growth stage then to decrease the RSC to a lower level.     

A parameter optimization approach for the optimal control of large-scale musculoskeletal systems.

This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).     

蚂蚁群落优化算法在蛋白质折叠二维亲-疏水格点模型中的应用

氨基酸的亲疏水格点模型是研究蛋白质折叠的一种重要的简化模型,其优化问题是一个非确定型的多项式问题。采用蚂蚁群落优化算法对这一问题进行了研究,对测试数据的计算结果表明,在一定规模下,此算法能够有效地获得亲-疏水格点模型的最优解,其效率优于传统的Monte Carlo仿真等方法。