An Efficient Hybrid of an Ant Lion Optimizer and Genetic Algorithm for a Model Parameter Identification Problem

Abstract

The immense application of mathematical modeling for the improvement of bioprocesses determines model development as a topical field. Metaheuristic techniques, especially hybrid algorithms, have become a preferred tool in model parameter identification. In this study, two efficient algorithms, the ant lion optimizer (ALO), inspired by the interaction between antlions and ants in a trap, and the genetic algorithm (GA), influenced by evolution and the process of natural selection, have been hybridized for the first time. The novel ALO-GA hybrid aims to balance exploration and exploitation and significantly improve its global optimization ability. Firstly, to verify the effectiveness and superiority of the proposed work, the ALO-GA is compared with several state-of-the-art hybrid algorithms on a set of classical benchmark functions. Further, the efficiency of the ALO-GA is proved in the parameter identification of a model of an Escherichia coli MC4110 fed-batch cultivation process. The obtained results have been studied in contrast to the results of various metaheuristics employed for the same problem. Hybrids between the GA, the artificial bee colony (ABC) algorithm, the ant colony optimization (ACO) algorithm, and the firefly algorithm (FA) are considered. A series of statistical tests, parametric and nonparametric, are performed. Both numerical and statistical results clearly show that ALO-GA outperforms the other competing algorithms. The ALO-GA hybrid algorithm proposed here has achieved an improvement of 6.5% compared to the GA-ACO model, 7% compared to the ACO-FA model, and 7.8% compared to the ABC-GA model.

1. Introduction

Due to an insufficient understanding of the underlying phenomena, there are no available detailed models for many industrially relevant processes. Mathematical models, which could be incomplete and inaccurate to a certain degree, can still be very useful and effective in describing the processes that are essentials for control, optimization, or understanding. Modeling cultivation processes are a difficult and challenging task, due to their characteristics such as nonlinearity, time-varying parameters, and interdependent process variables, etc., [1,2,3]. Microorganisms have been highly regarded as biotechnological tools. Many valuable bacteria, yeasts, and fungi are widely spread in nature. However, the optimal conditions for their formation and growth in a natural environment are rarely found. Different models have been developed [4,5,6,7] due to their importance for process control, reduction in production costs, and improvement of product quality in cultivation. In [8], the authors concluded that the flexibility of the model allows optimizations of the bioreactor performance and further development of the process by adjusting its conditions and configurations. In their work, Du et al. [9] stated that, “…after collaborating with a range of disciplines to develop new mathematical and computational tools while enhancing our knowledge of biology, the evolution of fermentation process modeling could be limitless”. The authors of [10] performed process optimization by mathematical modeling. It was proved that the usage of co-cultures of different bacteria could enhance the process performance. In [11], a model for syngas fermentation was developed and the simulations indicated more than a 50% increase in bioethanol productivity. In [12], the developed model was employed in finding the best experimental conditions for achieving the maximum bioethanol concentration. The proposed model could be used to predict the bioethanol concentration under various conditions without performing real experimental measurements. The novel model proposed in [13] is particularly suitable to assist the rational design of cultivation processes because it can represent complex biochemistry in more detail. The generated model is robust and can be considered for control and optimization purposes. The above-mentioned works constitute a very small part of the existing investigations in the past two years.

This paper considers the modeling of a fed-batch cultivation process of the bacteria Escherichia coli—a host organism with great significance for recombinant protein production [14]. Often, the cultivation of such recombinant microorganisms is a means to produce pharmaceutical biochemicals such as interleukins, insulin, interferons, enzymes, and growth factors at a reasonable price. Some of the most recent works featuring mathematical models of E. coli are as follows: two E. coli outer membrane models were investigated using molecular dynamics simulations [15]; two E. coli strains with modified tryptophan models were reported [16]; the cytokine response of the cells of the biomimetic porcine urothelial model to different E. coli strains was studied [17]; an enhanced segment particle swarm optimization algorithm that can estimate the values of small-scale kinetic parameters was described and applied as a model system to the main metabolic network of E. coli [18]; and the kinetics of wild-type and recombinant broccoli myrosinases produced in E. coli were characterized in terms of the reaction conditions [19].

The choice of an appropriate optimization method for model parameter identification is of utmost importance when dealing with a complex model structure. Over the years, researchers have shown interest in finding optimal model parameter values aimed at the global optimization conditions. Mathematical models of various natural phenomena, the theory of physics and biology, animal and human nature, and even the rules of some games have inspired a great variety of metaheuristic algorithms [20,21]. Metaheuristic techniques, especially hybrid metaheuristic algorithms that show better computational efficiency in avoiding local optima, have become a preferred tool in the parameter identification of models of cultivation processes [1,18,22,23,24,25,26,27]. Hybrid metaheuristic algorithms are widely accepted and well-employed in real-life challenging problems. This is the primary motivation for applying a hybrid metaheuristic to the parameter identification problem considered here.

In this work, a hybrid between the ant lion optimizer (ALO) [28] and the genetic algorithm (GA) [29,30] is developed to improve the classic ALO performance and engage in the parameter identification of an E. coli cultivation process. Since the ALO suffers from local optima stagnation [31], this study intends to propose a new ALO hybrid that will lead to an enhanced efficiency and accuracy of the algorithm. A hybridization with the GA, which is well-established with its global optimum performance [32,33], is considered to overcome the local optima stagnation. Contrariwise, the GA drawback, which is that there is a lot of computational effort to converge to the optimum when the randomly generated GA initial population is quite distant from it, will be overcome by the hybridization with the ALO. The authors expect that the combination of these techniques will lead either to a better performance concerning the computational resources or, which is of greater interest to the authors, to a better model of the bacteria cultivation processes considered here in terms of lower objective function values, and correspondingly higher model accuracy.

According to the two comprehensive surveys [34,35] and to our best knowledge, the ALO has been successfully modified using a differential mutation operator [36] and an arithmetic crossover operation of the GA [37,38]. Our understanding is that this is the first comprehensive hybrid between the ALO and GA. Moreover, the ALO, classic or hybridized, has not been applied so far for the parameter identification of cultivation process models. These facts are the secondary motivation to present a hybrid ALO-GA metaheuristic.

The performance of the proposed ALO-GA has been compared with other existing hybrid techniques discussed in the Literature Review, using a classical set of benchmark functions. The ALO-GA has been applied for the parameter identification of a nonlinear mathematical model of a cultivation process. The optimization problem has been solved by utilizing real experimental data from an E. coli fed-batch cultivation. The ALO-GA results have been compared with already published results of different hybrid metaheuristic algorithms employed for the same model parameter identification problem.

An experimental analysis is usually used to assess whether an algorithm is regarded as better than other methods. This may not be a trivial task as it includes confirming whether this algorithm achieves a meaningful improvement in the objective of a given problem over the others. Statistical tests are widely adopted to evaluate the performance of a given method [39,40]. Therefore, the obtained numerical results in this research were analyzed by applying nonparametric tests, namely, the Friedman test [41,42,43], the Wilcoxon test [44], and its parametric equivalent, the paired t-test, the Kruskal–Wallis test [45], and its parametric equivalent, the one-way analysis of variance (ANOVA) test [46].

The main contributions of this work include the following:

• This work intends to propose a novel hybrid ALO-GA that can fully balance exploration and exploitation, and significantly improve the global optimization ability. The algorithm performance is validated on a set of classical benchmark functions.
• The ALO-GA hybrid has been applied for the first time for the parameter identification of a nonlinear mathematical model of an E. coli cultivation process. The outperformance of the ALO-GA has been confirmed. The algorithm provides better results than the recent hybrid metaheuristics applied to an E. coli model parameter identification.
• The better ALO-GA performance is further confirmed by applying two parametric and three nonparametric statistical tests. The results show that the ALO-GA statistically outperforms the other algorithms considered in the comparison.
• A new improved model of the fed-batch cultivation process of the strain E. coli MC4110 has been obtained. It is 6.5% better than the best model known so far. The developed mathematical model could be further used for process investigation and optimization based on process monitoring and control.

The rest of the paper is organized in the following order. A brief review of some of the recent hybrid metaheuristic algorithms is presented in Section 2. The considered E. coli fed-batch cultivation process and its mathematical model are given in Section 3. The proposed hybrid technique ALO-GA is described in Section 4. The obtained numerical results and the provided statistical analysis are presented and discussed in Section 5. Concluding remarks and some future research directions are presented in Section 6.

2. Literature Review

The ALO is a novel metaheuristic swarm-based approach [28] designed to emulate the hunting behavior of antlions in nature. Recent investigations show that various modifications and hybridizations of ALO have improved its exploration, exploitation, convergence, and ability to avoid local optima. An ALO algorithm with an adaptive boundary and optimal guidance was proposed in [47] to enhance the ALO convergence speed and global search ability. The ABLALO algorithm exhibited improved convergence speed, search accuracy, and robustness through the simulation results of the six benchmark functions’ optimizations. A novel opposition-based ALO (OB-L-ALO) was introduced in [48] to improve the performance of the original ALO by enhancing the exploration of the algorithm. The results of the OB-L-ALO application on a set of 27 benchmark problems confirmed the progress in the performance. In [36], an approach based on ALO and a differential mutation operator called ALO-DM was proposed. A differential mutation operator and a greedy strategy enhanced the population’s diversity. The simulation results demonstrated that ALO-DM is a feasible and effective hybrid algorithm. A novel ALO hybrid, namely HALO, was discussed in [37]. The ALO was hybridized with the arithmetic crossover operation of the GA to improve the exploration of the search space. The proposed algorithm provided effective and robust high-quality solutions. In [38], the HALO algorithm was compared to the spotted hyena algorithm. The authors confirmed that the hybridization had enhanced the convergence rate. A modified ALO (MALO) was considered in [49] and proved superior in instance reduction problems. The ALO performance was improved by adding a new parameter that was dependent on the step length of each ant while revising the antlion’s position. The results obtained by applying the algorithm to 23 benchmark functions for 500 iterations and 13 benchmark functions for 1000 iterations demonstrated that the proposed MALO algorithm could escape the local optima and provide a better convergence rate as compared to the basic ALO algorithm and state-of-the-art optimizers [49]. The authors point out the need to determine the best values of the algorithm parameters as a possible limitation of the method. The K-means algorithm was integrated with the ALO in [50] for optimal clustering. The results showed that the K-means-ALO performed better than the K-means and K-means-PSO algorithms. An improved ALO combined with a support vector regression (IALO-SVR) was introduced in [51]. The method exhibited a high prediction accuracy in battery health prognosis. In [31], the ALO was hybridized with a sine cosine algorithm approach. The EALO-SCA algorithm revealed a better tracking performance, especially for abrupt motion.

Other recently published hybrid metaheuristics are discussed next. All of them were tested on a set of classical benchmark functions. The obtained results have been further used in comparison with the results of the ALO-GA hybrid proposed here. To be easily identified, all discussed algorithms are summarized in Table 1.

In [21], a novel chaotic opposition-based learning-driven hybrid aquila optimizer (AO) and artificial rabbits optimization (ARO) algorithm called CHAOARO was introduced. To verify the superiority of the algorithm, CHAOARO was compared with the original AO and ARO on 23 classical benchmark functions and the IEEE CEC2019 test suite. CHAOARO demonstrated significantly better results than the other competitor methods in terms of solution accuracy, convergence speed, and robustness. The IHAOAVOA hybrid algorithm between AO and the African vultures optimization algorithm was proposed in [52]. The performance of IHAOAVOA was analyzed on 23 classical benchmark functions and the IEEE CEC2019 test suite. The numerical and statistical results showed that IHAOAVOA outperformed the competing algorithms. The algorithm’s computational cost was pointed out as a potential limitation. In [53], a chaotic salp swarm algorithm based on opposition-based learning (OCSSA) was presented. The algorithm performed better when tested on 28 benchmark functions with unimodal or multimodal characteristics. The grey wolf optimizer (GWO) was modified in [54] to improve the exploration and exploitation ability of the algorithm. The COGWO2D algorithm is a combination of the chaotic logistic map, the opposition-based learning, the differential evolution (DE), and the disruption operator. The comparison with other competing algorithms, based on the classical CEC2005 and the CEC2014 benchmark functions, showed the higher accuracy obtained by COGWO2D. The limitation of the algorithm pointed out by the authors was the time complexity, which should be reduced. A hybrid HGGWA embedding three genetic operators into the standard GWO was presented in [55]. The effectiveness of HGGWA was proved by the simulation results. In [56], a hybrid LGWO, an enhanced version of the GWO with Lévy flight, was introduced. The Lévy flight was employed as a special navigation solution for alpha, beta, and delta wolves. The performance of the LGWO appeared to be substantially increased. A new variant of PSO, namely APSO-PDC, was proposed in [57] based on an adaptive selection of particle roles, population diversity control, and an adaptive control of parameters. A More preferable searching accuracy and searching reliability of the algorithm were shown by comparison with other well-established PSO variants on 21 unimodal and multimodal functions. A modified marine predators algorithm, MMPA, was considered in [58]. A logistic opposition-based learning mechanism and effective self-adaptive updating methods (new position-updating rule, inertia weight coefficient, and nonlinear step size control parameter strategy) were introduced into the original algorithm. The algorithm performance was tested on 23 classical benchmark functions and the CEC 2020 test suite. The results demonstrated that the MMPA exhibited a superior performance. The authors of [59] suggested a framework for using an orthogonal crossover (OX) in DE variants. The numerical results demonstrated that the performance of the DE variants could be refined through OXDE hybrids. An improved version of the arithmetic optimization algorithm (AOA), called nAOA, was proposed in [60]. The exploratory ability of the AOA was enhanced with high-density values generated by the natural logarithm and exponential operators. The efficient performance of nAOA was studied on thirty benchmark functions and three engineering design benchmarks.

There are a few applications of hybrid metaheuristics for the parameter identification of cultivation process models. In [22], a hybrid between the ant colony optimization (ACO) and the firefly algorithm (ACO-FA) was proposed. The ACO-FA was applied for model identification of an E. coli fed-batch cultivation process. For the same identification problem, a hybrid between the GA and ACO was considered in [26]. The comparison results showed that the hybrid GA-ACO outperformed ACO-FA. Recently published research on a hybrid between the artificial bee colony (ABC) and GA [24] showed a better quality performance of the algorithm when it was compared to both the GA-ACO and ACO-FA. The model of the E. coli fed-batch cultivation obtained by the ABC-GA was of the highest quality.

According to the literature review, a comprehensive hybridization between the ALO and GA has not yet been applied. Moreover, the ALO, neither modified nor hybridized, has been implemented for the parameter identification of cultivation process models.

Table 1. Recently proposed hybrid approaches used as competing algorithms.

Hybrid Algorithm	Year	Used Approach
ABLALO [47]	2019	ALO with adaptive boundary and optimal guidance
OB-L-ALO [48]	2017	A novel opposition-based ALO
ALO-DM [36]	2018	ALO with differential mutation operator
HALO [37,38]	2019 and 2021	ALO with arithmetic crossover operation
MALO [49]	2022	ALO with a new algorithm parameter
K-means-ALO [50]	2019	ALO with integrated K-means clustering
IALO-SVR [51]	2019	ALO with support vector regression
EALO-SCA [31]	2020	ALO with a sine cosine algorithm approach
CHAOARO [21]	2022	Aquila optimizer and artificial rabbits optimization algorithm
IHAOAVOA [52]	2022	Aquila optimizer and African vultures optimization algorithm
ABC-GA [24]	2021	Artificial bee colony with genetic algorithm
ACO-FA [22]	2014	Ant colony optimization with firefly algorithm
GA-ACO [26]	2016	Ant colony optimization with genetic algorithm
OCSSA [53]	2020	Chaotic salp swarm algorithm based on opposition-based learning
COGWO2D [54]	2018	Grey wolf optimizer combined with a chaotic logistic map, opposition-based learning, differential evolution, and a disruption operator
HGGWA [55]	2019	Grey wolf optimizer with genetic operators
LGWO [56]	2022	Grey wolf optimizer with Lévy flight
APSO-PDC [57]	2019	Particle swarm optimization based on an adaptive selection of particle roles, population diversity control, and adaptive control of parameters
MMPA [58]	2021	Marine predators algorithm and logistic opposition-based learning mechanism and effective self-adaptive updating methods
OXDE [59]	2012	Orthogonal crossover and differential evolution variants
nAOA [60]	2021	Arithmetic optimization algorithm with the use of natural logarithm and exponential operators

Table 1. Recently proposed hybrid approaches used as competing algorithms.

Hybrid Algorithm	Year	Used Approach
ABLALO [47]	2019	ALO with adaptive boundary and optimal guidance
OB-L-ALO [48]	2017	A novel opposition-based ALO
ALO-DM [36]	2018	ALO with differential mutation operator
HALO [37,38]	2019 and 2021	ALO with arithmetic crossover operation
MALO [49]	2022	ALO with a new algorithm parameter
K-means-ALO [50]	2019	ALO with integrated K-means clustering
IALO-SVR [51]	2019	ALO with support vector regression
EALO-SCA [31]	2020	ALO with a sine cosine algorithm approach
CHAOARO [21]	2022	Aquila optimizer and artificial rabbits optimization algorithm
IHAOAVOA [52]	2022	Aquila optimizer and African vultures optimization algorithm
ABC-GA [24]	2021	Artificial bee colony with genetic algorithm
ACO-FA [22]	2014	Ant colony optimization with firefly algorithm
GA-ACO [26]	2016	Ant colony optimization with genetic algorithm
OCSSA [53]	2020	Chaotic salp swarm algorithm based on opposition-based learning
COGWO2D [54]	2018	Grey wolf optimizer combined with a chaotic logistic map, opposition-based learning, differential evolution, and a disruption operator
HGGWA [55]	2019	Grey wolf optimizer with genetic operators
LGWO [56]	2022	Grey wolf optimizer with Lévy flight
APSO-PDC [57]	2019	Particle swarm optimization based on an adaptive selection of particle roles, population diversity control, and adaptive control of parameters
MMPA [58]	2021	Marine predators algorithm and logistic opposition-based learning mechanism and effective self-adaptive updating methods
OXDE [59]	2012	Orthogonal crossover and differential evolution variants
nAOA [60]	2021	Arithmetic optimization algorithm with the use of natural logarithm and exponential operators

3. Escherichia coli Fed-Batch Cultivation Process

The experimental data sets of E. coli MC4110 were obtained by a fed-batch cultivation process carried out at the Institute of Technical Chemistry, University of Hannover, Germany [61]. A full description of the cultivation process conditions and experimental data is given in [22].

A system of nonlinear differential equations was introduced following the application of the general state space dynamical model to the fed-batch cultivation process of E. coli: [22]:

$$\frac{dX}{dt}={\mu }_{max}\frac{S}{S+k_S}X-\frac{F}{V}X,$$

(1)

$$\frac{dS}{dt}=\frac{{\mu }_{max}}{Y_{S/X}}\frac{S}{S+k_S}X+\frac{F}{V}\left(S_{in}-S\right),$$

(2)

$$\frac{dV}{dt}=F^{}.$$

(3)

where X is the concentration of the biomass measured in [g·L⁻¹]; S denotes the substrate concentration (glucose), [g·L⁻¹]; F denotes the feeding rate, [h⁻¹]; V is the volume of the bioreactor, [l]; $S_{in}$ is designated for the initial glucose concentration in the feeding solution, [g·L⁻¹]; ${\mu }_{max}$ marks the maximum growth rate, [h⁻¹]; $k_S$ is the saturation constant, [g·L⁻¹]; and the mass of cells formed per unit mass of consumed substrate is expressed by the yield coefficient $Y_{S/X}$, [g·g⁻¹].

The model is based on the following a priori assumptions [22]: (1) the bioreactor is completely mixed; (2) the main product is the biomass; (3) the oxidative consumption of glucose is expressed by the Monod kinetics; (4) the growth conditions are considered balanced, since there is no significant diversity in the elemental composition of biomass when the growth rate and substrate consumption are changed; (5) and the parameters (e.g., temperature, pH, and pO₂) are controlled at their constant set points.

According to [22], the initial conditions of the process are as follows:

t₀ = 6.68 h, X(t₀) = 1.25 g·L⁻¹, S(t₀) = 0.8 g·L⁻¹, and $S_{in}$ = 100 g·L⁻¹.

4. Hybrid Ant Lion Optimizer-Genetic Algorithm (ALO-GA)

The ALO is a novel swarm-based metaheuristic approach [28] that mimics the hunting behavior of antlions in nature. The hunting process is carried out in five important phases: a random walk, building a trap, trapping ants, sliding the ants toward the antlion, catching the prey, and rebuilding the trap. The ALOs phases are repeated for a certain number of iterations T. The algorithm presented in detail in [28] is adopted here.

Ants search through the parameter space using a random walk. They are guided to promising regions by the antlions. The initial populations of ants and antlions are generated randomly based on the following equation:

$$x=rand\times \left(Ub-Lb\right)+Lb,$$

(4)

where $Lb$ and $Ub$ signify, respectively, the lower and upper bounds of the problem parameters. The location of each ant and antlion in a D-dimensional search space is represented as a vector of size D.

The building of the trap is simulated with the choice of an antlion by the roulette wheel selection (RWS). Antlions with better fitness evaluations are more likely to be selected and to catch ants.

The random walk of each ant at each iteration of the algorithm is modelled as follows:

$$X\left(t\right)=\left[0, cumsum\left(2r\left(t_1\right)-1\right), \dots , cumsum\left(2r\left(t_{\mathrm{T}}\right)-1\right)\right],$$

(5)

where $cumsum$ is the cumulative sum; T is the maximum number of iterations; and r(t) is 0 if a generated $rand\le 0.5$, and 1, if not. To guarantee that the position of the ants will remain inside the search space, it is normalized using the upper and lower bounds of the search space:

$$X_i^t=\frac{\left(X_i^t-a_i\right)\times \left(b_i-c_i^t\right)}{\left(d_i^t-a_i\right)}+c_i$$

(6)

where $a_i$ and $b_i$ are, respectively, the minimum and maximum of the random walk of the ith variable; $c_i^t$ and $d_i^t$ denote the minimum and maximum of the ith variable at the tth iteration.

Trapping the ants in the antlions’ pits is modelled by the influence of the antlions on the ants’ movement:

$$c_i=antlion+c_i$$

(7)

$$d_i=antlion+d_i$$

(8)

Two antlions have an impact on the random walk of each ant. The first one is selected by the RWS. The second is the elite antlion with the best fitness evaluation.

On each iteration, the positions of the ants are changed in the following manner [28]:

$$An{t}_i^t=\frac{R_A^t+R_E^t}{2}.$$

(9)

$An{t}_i^t$ is the position of the ith ant at the tth iteration. The random walk around a selected antlion at the tth iteration is denoted as $R_A^t$. $R_E^t$ is the random walk around the elite at the tth iteration.

Sliding the ants toward the trap, or converging to an antlion, is represented by decreasing the boundaries of the following variables:

$$c^t=c^t/I$$

(10)

$$d^t=d^t/I,$$

(11)

where $I$ is the shrinking factor. For the first 10% of all iterations, $I=1$. For the rest of the iterations, $I={10}^k\frac{t}{T}$. The constant $k$ depends on the number of the current iteration, t, according to

Table 2. Dependence of the constant k on the current iteration.

$\mathbf{Current} \mathbf{Iteration} \mathit{t}$	$\mathbf{Constant} \mathit{k}$
>10% T	2
>50% T	3
>75% T	4
>90% T	5
>95% T	6

:

The final phase at each iteration is catching the prey and rebuilding the trap. When an ant becomes fitter than an antlion, the antlion takes the position of the ant.

The ALO produces the initial population of the GA for n executions. The GA improves the population over several generations. Parents are selected using the RWS based on the evaluations of the individuals. The new generation is formed by applying two genetic operators to the parents.

The new individual $z=\left\{z_j| j\in \left[1;D\right]\right\}$ is a result of the recombination of two parents x and y:

$$z_j=x_j+{\alpha }_j\left(y_j-x_j\right),$$

(12)

where ${\alpha }_j\in \left[-0.25;1.25\right]$ is selected with uniform probability.

Each element of the individual $z$ mutates with a probability inversely proportional to D:

$$z_j=x_j+sign.r_j.\delta ,$$

(13)

where $sign=\left\{-1,+1\right\}$; $r_j=\rho \left(U{b}_j-L{b}_j\right)$; $\rho \in \left[0.1;0.5\right]$ is the mutation range; and $\delta ={{\displaystyle \sum }}_{i=0}^{k-1}{\phi }_i{2}^{i-1}$, ${\phi }_i\in \left\{0, 1\right\}$ is from a Bernoulli probability distribution, and k ∈ [4; 20] is the mutation precision related to the minimum step size and the distribution of mutation steps in the mutation range.

The next generation includes offspring with a better fitness evaluation. A new generation is produced until the maximum number of generations specified as MaxGen is reached. The best individual in the last generation of the GA is the solution of the ALO-GA hybrid to the problem. According to [31], there is a higher probability for the RWS method to choose the elite. This can lead to falling into a local optimum. A hybridization with the GA [29,30] is proposed to avoid this limitation of the ALO algorithm. On the other hand, one drawback of the GA is that convergence to the optimum may take a lot of computational effort when the randomly generated initial population is quite distant from it. The collaborative combination of the two algorithms has a positive effect on this constraint as well. The ALO runs for a few iterations. As a result, the population gradually converges toward the optimum. The final population of the ALO is a much better starting point for the GA than any random one. Subsequently, the computational resources required by the GA to obtain the final solution are reduced. The genetic evolution of the ALO aims to provide a good balance between exploration and exploitation.

The ALO-GA pseudo-code is presented below (Algorithm 1). The hybrid ALO-GA is developed based on the ALO algorithm presented in [28] and the GA implemented in [24].

Algorithm 1: Pseudo-code of ALO-GA

1: begin 2:        Define the input parameters for both ALO and GA 3:        Define the parameters of the problem under consideration 4:        % Start ALO 5:      Generate randomly the initial populations of ants and antlions 6:      Calculate the corresponding fitness estimations 7:      Find the best antlion and adopt it as the elite (determined optimum) 8:      for j:= 1 to the size of initial population Pop0 9:            while the end criterion is not satisfied 10:                  for each ant 11:                        Select an antlion using the roulette wheel selection 12:                        Update the parameters of the random walk 13:                        Create a random walk and normalize it 14:                        Update the position of the ant 15:                end for 16:                Evaluate all ants 17:                Replace an antlion with its corresponding ant if it becomes fitter 18:                Update the elite if an antlion becomes fitter 19:              end while 20:              Memorize the best solution for the current iteration in Pop0 21:            end for 22:            % Start GA 23:            Set the initial population Pop0 to the set of best solutions generated by ALO 24:            Calculate the value of the fitness function for each individual in Pop0 25:            for j:= 1 to MaxGeneration 26:                Select individuals Popi from the current population Popi−1 in a way that gives an advantage to better individuals 27:                Perform crossover with probability pc 28:                Perform mutation with probability pm 29:                Calculate the value of the fitness function for each individual in Popi 30:            end for 31:            Rank the solutions, find the current best, and memorize it 32: end begin

The flowchart of the ALO-GA hybrid algorithm, according to the presented pseudo-code (Algorithm 1), is given in Figure 1.

The two algorithms ALO and GA are applied consequently in the ALO-GA hybrid. Therefore, the computational complexity of the hybrid can be estimated in the following way.

Let O(f) be the computational complexity of the assessment of the fitness function.

The computational complexity of the ALO depends on the number of iterations T, the size of the population Np, and the complexity of the fitness function.

The generation of a random population of ants and antlions has a computational complexity of O(Np × D). The evaluation of the population is estimated as O(Np × f). Sorting the search agents according to their fitness evaluations and selecting the elite antlion has a complexity of O(Np × log Np).

The ALO algorithm runs for T iterations. The selection of an antlion at each iteration is estimated as O(Np). The computational complexity of the random walk around an antlion is O(Np × D). Updating the positions of the ants is executed with a complexity of O(Np). The evaluation of these new positions is O(Np × f).

The computational complexity of the ALO can be assessed finally as O(T × Np × (D + f)).

The ALO is required to generate the initial population of GA. Therefore, the ALO is executed n times, where n is the number of individuals in the GA. The computational complexity of the GA can be estimated as O(MaxGen × n × f).

The overall computational complexity of the ALO-GA hybrid is O((MaxGen × f + Np × T × (D + f)) × n). The computational complexity of the method is sacrificed for the sake of its computational precision. This can be regarded as a limitation of the ALO-GA hybrid. The MaxGen, n, Np, and T used in the hybrid, however, are significantly smaller than those usually exploited in standard ALO and GA algorithms. Since the initial position of the GA is not random but rather directed to the optimum, the GA requires far fewer generations to come to a final solution.

5. Results and Discussion

5.1. ALO-GA Hybrid Algorithm Performance on Test Functions

The proposed algorithm ALO-GA has been tested on several well-known benchmark functions listed in

Table 3. Benchmark optimization problems.

Function	Definition	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
Sphere, UM	$F1\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^d}{x}_i^2$	[−100, 100]	0
Schwefel’s 2.22, UM	$F2\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^d}\left|x_i\right|+{\displaystyle {\displaystyle \prod }_{i=1}^d}\left|x_i\right|$	[−100, 100]	0
Schwefel’s 2.21, UM	$F4\left(x\right)=\underset{i}{\max }{\left\{\left|x_i\right|,1\le i\le d\right\}}_{}$	[−100, 100]	0
Rosenbrock, UM	$F5\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^{d-1}}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_{i+1}-1\right)}^2\right]$	[−200, 200]	0
Quartic, UM	$F7\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^d}i{x}_i^4+random\left[0, 1\right)$	[−1.28, 1.28]	0
Rastrigin, MM	$F9\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^d}{\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]}_{}^{}$	[−5.12, 5.12]	0
Ackley, MM	$F10\left(x\right)=-20\exp \left(-0.2\sqrt{\frac{1}{d}{\displaystyle {\displaystyle \sum }_{i=1}^d}{x}_i^2}\right)-\exp \left(\frac{1}{d}{\displaystyle {\displaystyle \sum }_{i=1}^d}\cos \left(2\pi x_i\right)\right)+20+e$	[−32, 32]	0
Grienwank, MM	$F11\left(x\right)=\frac{1}{4000}{\displaystyle {\displaystyle \sum }_{i=1}^d}{x}_i^2-{\displaystyle {\displaystyle \prod }_{i=1}^d}\cos \left(\frac{x_i}{\sqrt{i}}\right)+1$	[−600, 600]	0

. These optimization problems fell into two groups: unimodal (UM) (F1, F2, F4, F5, and F7) and multimodal (MM) (F9–F11). All benchmark functions are considered with a dimension of 30.

The exploitation properties of an algorithm can be evaluated through the unimodal functions (F1, F2, F4, F5, and F7), which have only one global optimum. The exploration properties of an algorithm as well as its ability to avoid local optima are tested on multimodal functions (F9–F11).

The qualitative analysis results (search history metric) of the proposed ALO-GA approach in solving the considered benchmark functions are presented in Figure 2. The purpose of this metric is to display the position of the solutions in the search space during the iterations of the algorithm, thus tracking the exploration of the parameter space.

The search history diagrams show that ALO-GA has scanned the search space well, both globally and locally, and after identifying the main optimal area, the method converges toward the optimal solution with a high convergence rate.

To verify the performance of the ALO-GA, the obtained results are compared with some hybrid metaheuristic algorithms proposed in the literature, which are as follows:

(i)

Evolution-based algorithms (COGWO2D [54], HGGWA [55], and OXDE [59]);

(ii)

Physics-based algorithms (nAOA [60]);

(iii)

Swarm-based algorithms (CHAOARO [21], OCSSA [53], IHAOAVOA [52], APSO-PDC [57], MMPA [58], ABLALO [47], MALO [49], and OB-L-ALO [48]).

The obtained results (the mean value and the standard deviation (SD)) are presented in

Table 4. Results for the benchmark functions for different hybrid algorithms.

$\mathit{F}\mathit{n}$		ALO-GA	MALO, [46]	OB-L-ALO, [45]	ABLALO, [44]	CHAOARO, [21]	OCSSA, [51]	nAOA, [58]	COGWO2D, [52]	IHAOAVOA, [50]	APSO-PDC, [55]	MMPA, [56]	HGGWA, [53]	OXDE, [57]
$F1$	Mean	0.00 × 100	0.00 × 100	1.09 × 10−11	4.26 × 10−18	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	4.85 × 10−53	1.58 × 10−16
	SD	0.00×100	0.00 × 100	3.86 × 10−11	2.74 × 10−18	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	2.41 × 10−52	1.41 × 10−16
	Rank	1	1	5	3	1	1	1	1	1	1	1	2	4
$F2$	Mean	0.00 × 100	0.00 × 100	1.85 × 10−18	9.88 × 10−2	0.00 × 100	3.27 × 10−199	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	4.01 × 10−33	4.38 × 10−12
	SD	0.00 × 100	0.00 × 100	1.01 × 10−17	5.57 × 10−1	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	4.34 × 10−33	1.93 × 10−12
	Rank	1	1	4	6	1	2	1	1	1	1	1	3	5
$F4$	Mean	0.00 × 100	0.00 × 100	4.09 × 10−6	no data	0.00 × 100	4.39 × 10−210	0.00 × 100	0.00 × 100	0.00 × 100	5.87 × 10−87	0.00 × 100	4.55 × 101	3.69 × 101
	SD	0.00 × 100	0.00 × 100	9.62 × 10−6	no data	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	9.59 × 10−87	0.00 × 100	1.49 × 100	0.96 × 100
	Rank	1	1	4	−	1	2	1	1	1	3	1	6	5
$F5$	Mean	1.23 × 10−6	5.31 × 10−4	5.46 × 10−4	9.41 × 10−3	2.66 × 10−3	1.19 × 101	4.61 × 100	2.70 × 101	5.83 × 10−7	1.36 × 101	9.65 × 101	6.82 × 101	1.59 × 10−1
	SD	1.49 × 10−6	8.98 × 10−4	1.60 × 10−3	5.02 × 10−3	4.89 × 10−4	8.55 × 100	2.59 × 10−1	4.06 × 100	9.72 × 10−7	1.20 × 10−1	3.96 × 10−1	1.25 × 10−1	7.79 × 10−1
	Rank	2	3	4	6	5	9	8	11	1	10	13	12	7
$F7$	Mean	4.13 × 10−9	1.83 × 10−4	2.85 × 10−4	no data	8.52 × 10−5	1.62 × 10−4	4.86 × 10−5	8.99 × 10−5	3.22 × 10−5	5.21 × 10−16	3.81 × 10−5	2.34 × 10−10	2.95 × 10−3
	SD	3.25 × 10−9	1.65 × 10−4	2.49 × 10−4	no data	8.29 × 10−5	1.60 × 10−4	4.12 × 10−5	9.79 × 10−5	2.53 × 10−5	9.17 × 10−16	2.36 × 10−5	3.17 × 10−10	1.32 × 10−3
	Rank	3	10	11	−	7	9	6	8	4	1	5	2	12
$F9$	Mean	0.00 × 100	0.00 × 100	3.01 × 10−9	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	4.06 × 100
	SD	0.00 × 100	0.00 × 100	4.82 × 10−9	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	1.95 × 100
	Rank	1	1	2	1	1	1	1	1	1	1	1	1	3
$F10$	Mean	3.22 × 10−18	8.88 × 10−16	2.10 × 10−5	no data	8.88 × 10−16	1.53 × 10−15	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	1.92 × 10−18	8.88 × 10−16	2.15 × 10−15	2.99 × 10−9
	SD	3.79 × 10−18	0.00 × 100	1.46 × 10−5	no data	0.00 × 100	1.42 × 10−15	0.00 × 100	0.00 × 100	0.00 × 100	3.24 × 10−18	0.00 × 100	3.48 × 10−15	1.54 × 10−9
	Rank	2	3	7	−	3	4	3	3	3	1	3	5	6
$F11$	Mean	0.00 × 100	0.00 × 100	6.32 × 10−9	1.11 × 10−16	0.00 × 100	0.00 × 100	2.49 × 10−4	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	1.48 × 10−3
	SD	0.00 × 100	0.00 × 100	1.27 × 10−8	1.92 × 10−16	0.00 × 100	0.00 × 100	1.37 × 10−3	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	3.02 × 10−3
	Rank	1	1	3	2	1	1	4	1	1	1	1	1	5
Total Rank		1	5	10	11	4	8	6	8	2	3	7	9	12

The best results are highlighted in bold.

. The performance of the compared hybrid algorithms is ranked for each benchmark function. The final ranking based on the results of all benchmark functions is also calculated.

According to Table 4, ALO-GA produced the best mean value for five of the eight benchmark functions: three best results of the five considered UM functions and the two best results of the three considered MM functions, respectively. For the unimodal functions F5 (Rosenbrock) and F7 (Quartic), the best mean values were obtained by the hybrids IHAOAVOA [52] and APSO-PDC [57], respectively. APSO-PDC obtained the best mean value in the case of the multimodal function F10, too. The hybrids IHAOAVOA and APSO-PDC are ranked in second and third place, respectively. The fourth place is for the CHAOARO hybrid, followed by the MALO algorithm.

The results show that the ALO-GA hybrid exhibits a better performance compared to some existing ALO hybrid algorithms [47,48,49], as well as compared to the other nine hybrid techniques [21,52,53,54,55,57,58,59,60]. The presented search history diagrams (Figure 2) and the performance comparison (Table 4) demonstrate that ABC-GA achieved a good balance between the algorithm’s exploration and exploitation.

5.2. Parameter Identification of E. coli MC4110 Fed-Batch Cultivation Process

The parameter identification of the nonlinear mathematical model (Equations (1)–(3)) of the E. coli MC4110 fed-batch cultivation process performed using the ALO-GA hybrid algorithm is described below.

5.2.1. Simulation Setup

The proposed mathematical model consists of a set of three ODEs (Equations (1)–(3)) with two dependent state variables x = [$X S$] and three unknown parameters $p=\left[{\mu }_{max} k_S Y_{S/X}\right]$.

The population of the ALO-GA hybrid consists of a set of individuals that are possible solutions to the considered problem. The solutions are represented as vectors with elements corresponding to the unknown model parameters. Based on the authors’ expertise [23,62,63], the ranges of these parameters are estimated as the following:

$$0.01 \le {\mu }_{max} \le 0.8; 0.001 \le k_S \le 1; 0.01 \le Y_{SX} \le 10.$$

(14)

The initial set of solutions is generated based on Equation (4). The solutions are later improved according to the specifics of the nature-inspired algorithms involved in the ALO-GA hybrid while observing the limitations in the values of the model parameters (Equation (14)). If necessary, the new solutions are additionally pushed back within the defined bounds.

The calculations are performed on Intel^® Core™i7-8700 CPU @ 3.20 GHz, 3192 MHz, 32 GB Memory (RAM), with a Windows 10 pro (64 bit) operating system. The considered competing algorithms are implemented in Matlab R2019a, based on [22,24,26]. The implementation of the ALO-GA hybrid technique follows the above-presented pseudo-code and flowchart in Figure 1. The mathematical model of E. coli is created in the Simulink R2019a environment. The solver options are the fixed-step size of 0.01 and ode4 (Runge–Kutta) with TIMESPAN = [6.68 11.57].

ALO has only two main parameters, namely the population size Np and the number of iterations T [28,31]. The values of these input parameters were determined with several preliminary tests. The characteristics of the specific identification problem solved here were also taken into consideration while defining the ALO-GA parameters. Finally, the parameter values were set as Np = 50 and T = 50. The constant $k$ depended on the number of the current iteration t according to Table 2. A small population (Np = 50) was engaged in the beginning. The set of initial solutions for the GA was generated by the ALO for only 50 iterations. The main GA parameters were set as follows: number of generations—100; population size—50 individuals; crossover rate—0.7; mutation rate—0.1; generation gap—0.97. The following GA operators were chosen: RWS as a selection function; recombination extended intermediate as a recombination function; and a real-value mutation like the breeder genetic algorithm [64] as a mutation function.

A series of parameter identification procedures of the model Equations (1)–(3) using ALO-GA were performed. Due to the stochastic characteristics of the applied algorithms, 30 runs were conducted. The model parameter’s vector, namely $p=\left[{\mu }_{max} k_S Y_{S/X}\right]$, was estimated based on the objective function J:

$$J={\displaystyle \sum }_{i=1}^n{\left\{\left[X_{exp}\left(i\right)-X_{mod}\left(i\right)\right]\right\}}^2+{\left\{\left[S_{exp}\left(i\right)-S_{mod}\left(i\right)\right]\right\}}^2\to min,$$

(15)

where n is the length of the data vector for the state variables $X$ and $S$; the existing experimental data are denoted as $X_{exp}$ and $S_{exp}$; and the model predictions for a given set of parameters are marked as $X_{mod}$ and $S_{mod}$.

5.2.2. Numerical Results

The results of the application of the novel ALO-GA hybrid algorithm to the parameter identification of an E. coli MC4110 cultivation process model are presented in

Table 5. Comparison of the objective function values obtained by ALO with published results.

Algorithm	Objective Function, J
	Mean	Best	Worst	SD
ABC-GA [24]	4.5462	4.4403	4.6127	0.04117
GA-ACO [26]	4.5706	4.3803	4.6949	0.06325
ACO-FA [22]	4.5195	4.4013	4.6803	0.05914
ALO-GA	4.3164	4.1128	4.5358	0.07114

The best results are highlighted in bold.

and

Table 6. Comparison of the parameter estimates obtained by ALO with the published results.

Algorithm	Model Parameters Estimates
	${\mathit{\mu }}_{\mathit{m}\mathit{a}\mathit{x}}, \left[{\mathbf{h}}^{-1}\right]$	SD	${\mathit{k}}_{\mathit{S}}, [\mathbf{g}·{\mathbf{L}}^{-1}]$	SD	${\mathit{Y}}_{\mathit{S}/\mathit{X}}, [\mathbf{g}·{\mathbf{g}}^{-1}]$	SD
ABC-GA [24]	0.4909	0.0039	0.0127	0.00072	2.0211	0.0011
GA-ACO [26]	0.4946	0.0121	0.0123	0.0020	2.0204	0.0025
ACO-FA [22]	0.4824	0.0110	0.0114	0.0019	2.0206	0.0021
ALO-GA	0.5022	0.0093	0.0146	0.0018	2.0182	0.0019

The best results are highlighted in bold.

. The mean, the best, and the worst results for the criteria value J of the performed 30 runs and the obtained SD are summarized in Table 5. The estimated model parameters with their SD are listed in Table 6. The algorithm convergence curves of the 30 runs and 150 total iterations are shown in Figure 3.

To show the superiority of the ALO-GA hybrid metaheuristic algorithm proposed here, the obtained results are compared to already-known outcomes for the same problem. The following published hybrid algorithms are considered: ABC-GA [24], GA-ACO [26], and ACO-FA [22]. The most accurate mathematical model of the cultivation process of E. coli MC4110 so far was obtained by GA-ACO with J_best = 4.3803.

Semi-log scale plot of convergence curves shows that the hybrid converges for the 150 iterations very well.

All results presented in Table 5 show that the ALO-GA hybrid algorithm finds the highest quality solution, e.g., the solution with the smallest objective function value (J = 4.1128), and improves the accuracy of the model presented in [22]. According to the mean J values, the ALO-GA model is 4.7% better than ACO-FA, 5.3% better than the ABC-GA model, and 5.9% better than the GA-ACO model. Considering the best-obtained J value, the GA-ACO model is improved by 6.5%. The improvement of the ACO-FA model is 7% and 7.8% when the ALO-GA model is compared to the ABC-GA model. Despite these good results, the observed SD of ALO-GA has the largest value compared to the SD values of other hybrid metaheuristics. Different possibilities for improving this performance of the algorithm can be investigated in future research.

The model parameter estimates summarized in Table 6 are adequate and correspond to their physical meaning [65,66,67]. The obtained SD values indicate the good performance of the applied algorithms. The results show that the model estimates are very close. However, there is a statistically significant difference between them, which will be shown below. It can be concluded that the best model is obtained by ALO-GA.

The predictions for the E. coli fed-batch cultivation process variables (biomass and substrate) acquired by exploiting the estimated model parameters in Table 6 are represented graphically in Figure 4. They are examined in contrast to the measured real experimental data. The obtained results are compared to the results of ABC-GA, GA-ACO, and ACO-FA.

To show the superior performance of the ALO-GA hybrid, a zoomed perspective of the graphical results is also presented. The graphical comparisons show that the ALO-GA model describes the experimental data superbly, following the trend of the process dynamics. The ALO-GA model dynamics are the closest to the dynamics of the experimental data.

5.2.3. Statistical Analysis

Two classes of statistical procedures are employed in the statistical analysis: parametric and nonparametric. Experiments are usually analyzed with parametric tests. Generally, they have more statistical power than their nonparametric equivalents [68] and are more likely to detect significant differences when such do exist. When it comes to analyzing the performance of stochastic algorithms, the assumptions of parametric tests are most probably violated [43]. Nonparametric statistical procedures are a practical tool when previous assumptions cannot be satisfied [39]. Nonparametric tests, namely, the Friedman test, the Wilcoxon test, and its parametric equivalent, the paired t-test, the Kruskal–Wallis test, and its parametric equivalent, the one-way analysis of variance (ANOVA) test, are applied here to obtain numerical data and to compare the performance of the algorithms. The tests are performed in Matlab, using the following functions: Friedman test—‘friedman’; Wilcoxon test—‘ranksum’; Paired t-test—‘ttest2′; Kruskal–Wallis test—‘kruskalwallis’; ANOVA—‘anova1′. The tests are applied to the results of the objective function J obtained by the 30 runs of each algorithm. The results are summarized in

Table 7. Results from statistical tests.

Friedman Test
‘Source’	‘SS’	‘df’	‘MS’	‘Chi-sq’	‘Prob > Chi-s’
‘Columns’	99.1333	3	33.0444	59.4800	7.5916 × 10−13
‘Error’	50.8667	87	0.5847	[]	[]
‘Total’	150	119	[]	[]	[]
meanranks	[3.1333 3.3667 2.4667 1.0333] for [ABC-GA GA-ACO ACO-FA ALO-GA]
sigma	1.2910
Wilcoxon test
ALO-GA vs.	p-value	H	STATS
			zval	ranksum
ABC-GA	1.0937 × 10−10	1	−6.4534	478
GA-ACO	1.4643 × 10−10	1	−6.4090	481
ACO-FA	2.8700 × 10−10	1	−6.3056	488
Paired t-test
ALO-GA vs.	p-value	H	ci	STATS
				tstat	df
ABC-GA	3.2061 × 10−17	1	−0.2753, −0.1844	−13.6293	43.0670
GA-ACO	6.9515 × 10−17	1	−0.3121, −0.1963	−11.6985	57.8919
ACO-FA	7.3148 × 10−15	1	−0.2545, −0.1518	−10.5496	55.3599
Kruskal–Wallis test
‘Source’	‘SS’	‘df’	‘MS’	‘Chi-sq’	‘Prob > Chi-s’
‘Columns’	8.2122 × 104	3	2.7374 × 104	67.8698	1.2199 × 10−14
‘Error’	6.1867 × 104	116	533.3394	[]	[]
‘Total’	1.4399 × 105	119	[]	[]	[]
meanranks	[76.1667 85.2333 63.3667 17.2333] for [ABC-GA GA-ACO ACO-FA ALO-GA]
sumt	6
ANOVA
‘Source’	‘SS’	‘df’	‘MS’	‘F’	‘Prob > F’
‘Columns’	1.2197	3	0.4066	80.241	3.6030 × 10−28
‘Error’	0.5878	116	0.0051	[]	[]
‘Total’	1.8075	119	[]	[]	[]
means	[4.5462 4.5706 4.5195 4.3164] for [ABC-GA GA-ACO ACO-FA ALO-GA]
df	116
s	0.0712

Statistical differences exist with a significance level α = 0.05

.

The statistical results presented in Table 7 reveal that there are statistical differences between the ALO-GA hybrid and the competing hybrid algorithms—GA-ACO, ACO-FA, and ABC-GA. All tests show significantly lower p-values than 0.05. Thus, the null hypothesis that the mean of the objective function is the same for all four algorithms can be rejected. This is sufficient evidence to conclude that the results obtained by GA-ACO, ACO-FA, ABC-GA, and ALO-GA are statistically significant. Since the mean J value of ALO-GA is the lowest compared to the other observed mean values, it may be concluded that the proposed hybrid metaheuristic outperforms all considered algorithms in the case of parameter identification of an E. coli cultivation model (Equations (1)–(3)).

The performance of the investigated hybrid algorithms is statistically evaluated by a comparison of the acquired mean values, SD, and the median of the estimated model parameters and the obtained objective function value J. To visualize the summary statistics, box plot diagrams are presented in Figure 5.

The proposed hybrid algorithm evidently outperforms the ABC-GA, GA-ACO, and ACO-FA metaheuristics. The collaborative combination between the ALO and GA improves the ALOs performance. Its source of exploitation is the selection according to the fitness of the solutions, while the mutation and crossover operators are the sources of exploration [69,70]. According to [71], recombination is quite useful in exploitation when the necessary diversity of the population is somehow ensured. Such diversity in the GA is achieved through higher mutation rates as in the proposed hybrid in the case of E. coli modeling.

6. Conclusions and Future Research Directions

A novel ALO-GA hybrid metaheuristic algorithm is proposed for parameter identification of a nonlinear mathematical model of the E. coli MC4110 fed-batch cultivation process. The collaboration between the ALO algorithm and GA in the proposed hybrid aims to balance exploration and exploitation properly.

The proposed hybrid is examined in comparison to a set of classical benchmark functions (unimodal and multimodal) and other hybrids of population-based metaheuristic algorithms inspired by nature. The ABC-GA, GA-ACO, and ACO-FA algorithms were chosen as they had been previously applied to the same parameter identification problem. The ALO-GA hybrid shows the best performance based on the final ranking in comparison to the 12 hybrid algorithms applied for the considered benchmark function. The numerical simulation results for the parameter identification of an E. coli MC4110 cultivation process model show that the ALO-GA hybrid algorithm surpasses the other competing algorithms. The statistical analysis based on the Friedman test, Wilcoxon test, Kruskal–Wallis test, paired t-test, and ANOVA confirms the obtained results. The results observed for the best objective function J value reveal an improvement of 6.5% compared to the GA-ACO model, 7% compared to the ACO-FA model, and 7.8% compared to the ABC-GA model. The outcomes are encouraging, and they demonstrate the high performance of the novel ACO-GA hybrid metaheuristic algorithm in the model identification problem of an E. coli MC4110 fed-batch cultivation process. In the future, the algorithm can be applied to designing more complex nonlinear mathematical models of cultivation processes.

Various new hybrid algorithms can be further developed for this topical field. The coyote optimization algorithm [72], the water cycle algorithm [73], the grey wolf optimization [74], etc., can be hybridized to enhance the ALO performance and propose a powerful technique for the nonlinear model parameter identification problem.