An Evolutionary Algorithm: An Enhancement of Binary Tournament Selection for Fish Feed Formulation

algorithm. Hence, based on the comparative study, the fndings of the enhanced selection operator towards the EA have been convinced and accepted in terms of better cost and fulflling constraints requirements.

Te selection operator can explore the search space efciently and efectively. Te role of selection is to distinguish among individuals based on their quality and, in particular, to allow the better individuals to become parents of the next generation [13]; Eiben and Smith; 2015; [6-8, 10, 14-16]. Te selection operator gives preference to better individuals for the next generation and prohibits the worst-ft individuals into the next generation [10,[16][17][18]. As the generation pass, the members of the population should get ftter and ftter [2,4,10,15,17,19].
Since the selection operator has advantages in many aspects, many landscapes, especially in practical management applications, have increasingly drawn the attention of researchers from a variety of backgrounds. Of note, much of the research in this landscape has focused on the EA selection operator phase because of the need to combine vector performance gauges with the scalar method that existing EA rewards independent capabilities. Te earliest pioneering studies on selection operators emerge in the early 1990s [20,21]. Subsequently, a few diferent selection operator employments were suggested from the years 1995 to 1996 [22][23][24][25]. After that, these diferent selection operator approaches were fruitfully practical to numerous problems [13,18,[26][27][28][29][30][31][32][33][34].
For instance, Roulette wheel selection has been applied to berth scheduling problems [35,36], shrimp diet formulation [8,37], and vehicle routing problems [38,39]. Meanwhile, tournament selection has also been used in berth scheduling problems [35,40] and fsh feed formulation [41,42]. Whereas binary tournament (BT) selection has been deployed in berth scheduling problems [35], shrimp diet formulation [6,37], vehicle routing problems [35,39], and travelling salesman problem [43][44][45]. Consequently, this study aims to shed light on the performance of the selection operator to solve real-life grouper fsh feed formulation problem as a case study for unlocking the frontier in selection operator in EA research on formulating the fsh feed. Te standard grouper fsh feed formulation is an experimental design that involves many trial-and-error eforts, which is time-consuming. Te feed formulation problem has been reported as a nondeterministic polynomial, NP-hard problem [46][47][48] due to its complex nature. It has been revealed that studies related to animal feed formulation modelling are generally still limited (Sahman et al., 2009 [49], and similar studies on aquaculture, including grouper, are even more limited [7,42]. Terefore, it is essential to develop grouper feed formulations that prioritize necessary ingredients and nutrients. Hence, this study aims to develop a novelty selection operator in EA, applied in formulating the fsh feed for grouper. Subsequently, the following section reviews the literature to highlight the advantages of EA and its operators. It is then followed by the methodology that discusses the application of fsh feed formulation in EA. Experimental results and conclusions are drawn in the last section.
First and foremost, the argument is that this approach is simple, robust to change circumstances, and fexible [1,19,56,58,60]. EAs are easy to apply and often provide satisfactory solutions compared with other global optimization techniques. According to Fogel [1]; Burke and Kendall [69]; Grosan and Abraham [60]; Rahman [6]; Abd Rahman et al. [7]; Rahman et al. [8]; Hien and Gillis [70]; Bashab et al. [71]; and Ojstersek et al. [61]; EAs can be applied to problems where heuristics solutions lead to unsatisfactory results due to EA has global search characteristics. Terefore, EAs have been widely used for practical problem-solving, even in feed formulation.
Second, Banzhaf et al. [2] are in similar judgment to Fogel [1] and Grosan and Abraham [60]; Simon [5] and Zhou et al. [58] that the aim of choosing EAs is EAs easy in solving combinatorial optimization problems or learning tasks in computer science compare with other conventional methods. It is because EA has two powerful elements of exploration and exploitation in the search process of EA. Tus, it can be concluded that EAs can solve combinatorial optimization problems as claimed by Fogel [1]; Schwefel [84]; Banzhaf et al. [2]; Grosan and Abraham [60]; Kumar and Singh [85]; Şahman et al. [86]; Simon [5]; Eiben and Smith (2015); and Zhou et al. [58].
In conclusion, EA beneft over other methods in terms of searching lies in its fexibility and ability to adapt to existing tasks [1,19,56,58,60], global search characteristics [14,19,50], and robust performance [19,56,58,60,66,83]. EA can be declared as a general adaptable concept to solve complex optimization problems rather than ready-to-use algorithms.
Due to the advantages of EA, EA is applied mainly in many landscapes for solving problems related to management, optimization, scheduling [87], and so forth [57]. Such problems are wherever applicable, like production and so on. Te advantages of EA are due to the strength of the selection operator. In this selection stage, EA does not generate new chromosomes. Te selection operator prohibits the worst-ft individuals into the next generation [10,[16][17][18]. Te selection work is similar to the search path in the search space. In other words, deliberate parent selection is made by considering selection pressures. Logically, a higher selection intensity can be raised, which might be an advantage for a large population [14,88,89]. While in low diversity, high selection pressures can result in rapid convergence (rapid fall to local optimum).
In addition, considering their mating strategies, the question of how dissimilar the chosen parents are has been ignored. In other words, diferent pairs of parents have the potential to produce additional ofspring than similarlooking parents, or vice versa. Tis is crucial to the exploration of the selection operator because it undermines the fexible reproduction operator, as numerous studies classify the preferred operator as an exploitation strategy [10,14,26,32,65,90,91]. Nevertheless, some studies ignore its importance; they copy a few individuals into a mating pool without a selection strategy [14,32,92,93]. Indeed, the selection operator reacts to give space for permutation potential in the reproduction operator. Controlling or managing the search for diversity provides space to comprehend and improve selection operators.
Several researchers applied new diferent selection operators. Goldberg [20] applied the invention of the Boltzmann tournament selection in Pascal applications. His results have proven ready for practical to parallel hardware using this Boltzmann tournament selection approach. Te Boltzmann tournament selection process was obtained and executed to provide a stable distribution crossways space and time in the population structures that proved close to Boltzmann. Te Boltzmann tournament selection is applied using the concept of proportionate selection schemes. Distribution mechanism by imposing a group of individuals sharing restricted resources. Te Boltzmann tournament selection executes by imposing individuals to contest their potential. Te recovery mechanism of this fnal mechanism is unclear until this assessment of diferent individuals leads to more competition among undesirable individuals when selected randomly and uniformly from existing distributions. Goldberg and Deb [21] conducted a comparative study on the performance of proportionate reproduction, ranking selection, BT selection, and genitor selection. Proportionate reproduction chooses a group of individuals for birth based on objective function values. While the perception of ranking selection is straightforward, where the population is sorted accordingly based on best to worst, performance the proportionate selection based on the assignment function given. Te third idea of BT selection is simple; the best individual from a group of random individuals. Whereas, the last concept of genitor selection is difcult due to work one by one compared to concept generational, and the only one worst individual for replacement. BT selection is preferred for computation time due to the better timing complexity. In terms of growth ratio, genitor selection revealed a higher growth ratio than other selections over generations.
In years 1995 to 1996, Blickle and Tiele [22] as well as Blickle and Tiele [23] also conducted a similar comparative study with Goldberg and Deb [21] regards performance on ftness distributions for ranking selection operator, BT selection operator, truncation selection operator and lastly, exponential ranking selection operator. Te idea of the frst two selection operators is similar to the concept of Goldberg and Deb [21]. Te latter concepts of a truncation selection operator and lastly an exponential ranking selection operator that are merely the fraction of best individuals are chosen while exponential ranking selection ranked weighted individuals exponentially. Tey claimed that the concept of this study is not new, but the fnding of the ranking selection operator and BT selection operator is verifed in the expected ftness distribution. Hence, the result enables understanding the single methodology aspects independently and isolated.
Chakraborty et al. [24] studied four types of selection pressure for population-elitist selection operator, linear ranking selection operator, BT Selection operator, ftnessproportionate selection operator, and genitor selection operator. Tey investigated the possible values that can be achieved, probably change of utmost importance, probably highest maximum value and lastly, probabilities take overtime distribution. Te fnding shows that the boundary for linear ranking selection operator, BT selection operator, and ftness-proportionate selection operator match genetic drift. Still, the genitor selection operator is remarkably higher than the three operators above. In conclusion, the genitor selection operator obtained the most arduous push among other selection operators. For the change of ultimate values, the probability of BT Selection operator and linear ranking selection operator is always less than one. In contrast, the item highest value linear ranking selection operator and BT selection operator behave similarly, which is 0.8 selection rate. While item probabilities take over time distribution for the linear ranking selection operator, the linear ranking selection operator, and BT selection operator are almost time-homogeneous.
Miller and Goldberg [25] conducted similar studies to Chakraborty et al. [24] to give practitioners a rational approach to the impact of diferent noise levels, including Complexity human error, sampling error, and knowledge uncertainty which can afect the accuracy of ftness functions. Results found that proportionate selection never reaches absolute convergence. Tournament selection is the best selection because it can act faster to estimate the convergence time in small, medium, and large noisy environments.
Consequently, many studies explore tournament selection due to larger tournament size that can be used to increase competition between individuals and increased Pareto-approximation quality [32,94,95]. Moreover, BT selection is a better size option for selection. [21,32]. Furthermore, BT selection makes perfect sense when solving unimodal problems [20,32,96]. In addition, BT selection with replacement is better in achieving the best solution quality with low computational time [20,[96][97][98][99]. Additionally, the correspondence of BT selection in the expected ftness distribution is proven [23,32]. Likewise, the complexity of the tournament selection is lower than the complexity of other selections, such as the ranking selection [32,100], and the selective pressure is higher, which allows us to measure whether each crossover can keep the population diversity [21,32]. Tus, it can be revealed that BT selection is suitable because it is more efcient in time complexity. Due to the BT selection's strength, it is possible to explore this technique further. Tus, for practical management, it has sparked the interest to research on BT selection operator of EA since there is no work to include the standard deviation into BT selection to fll up the gap in fsh feed formulation.

Materials and Methods
Te method of EA has been discussed in those above and convinced us that BT selection could be appropriate guidance and help to unlock frontiers for solving fsh feed formulation. Te SD tournament selection is considered novel since, to the best of our knowledge, there is no work on this kind of computation for standard deviation into BT selection that has been reported for EA, especially in the feld of feed formulation. BTselection is the component in the EA, and it is higher-order than heuristic; in other words, it means higher than the neighborhood. BT selection is embedded into this study, which is the key contribution to constructing an improved selection operator based on the BT selection and standard deviation. From the perspective of a selection procedure or operator, a tournament selection has been proven efective by Sahman et al. [5]; which was represented by real-valued alleles in the chromosome of the EA. Furthermore, this triggered utilizing the BTselection operator as Back et al. [19] and Hussain and Muhammad [32]. A possible improvement on the BT Selection operator conisders the use of standard deviation in the tournament decision.
Before modeling the EA, information related to fsh feed is collected. It included the background of the fsh feed formulation and the fsh feed formulation problems, which guided the direction and motivation of the study. Te information on the composition of the nutrients in the contents of 100 kg feed ingredients used in the grouper diet formulation applied in this research was taken from sources such as National Research Council [101], Holt [102]; Davis [103]; and Ali [104]. Meanwhile, the nutritional needs for the growth of the grouper fsh feed formulation were taken from 30 manufacturers and experts in this feld. Te ingredients are chosen according to economic status, appropriateness for the digestive system of the grouper fsh, and nutritional value needed based on experts' suggestions. Tus, the list of ingredients and its range, as suggested, is depicted in Table 1.
On top of ingredients, nutrients are also essential in feed formulation. Te nutrient requirements needed by grouper fsh are crude protein, crude fat, and crude fber, which are precalculated with the tolerance limits carried out with one sample T-test hypothesis testing and Pearson correlation. Te signifcance of these nutrients has been tested and analyzed using coefcient of variation (CV), where 16 nutrients elements, including moisture, were considered initially, as exhibited in Table 2. As a result, 15 nutrients are signifcant (i.e., p < 0.05) with the value of each respective CV is less than 1 (i.e., CV < 1). Tese nutrients are crude protein, crude fat, crude fber, crude ash, phosphorus, calcium, arginine, histidine, isoleucine, leucine, lysine, methionine, phenylalanine, threonine, and valine. Tis research study does not considered moisture because its CV is greater than 1. Te rationale is that the smaller CV value indicates that the other 15 nutrients are less dispersed than moisture. Terefore, the 15 nutrients are a good indication to be included in this research. Furthermore, these nutrients positively afect fsh's growth in terms of weight (kg) or length (cm), as agreed by the consulted experts.
Tese listed nutrients are essential and crucial for the growth of the grouper fsh. Terefore, they need to be included in this researcher's list of essential nutrients. Tis is based on the recommendation by Muhammadar et al. [105] and the experts. Subsequently, the fnal list of nutrients, together with its minimum and maximum percentages required in the formulation of grouper feed, is given in Table 3.
Hence, all of those grouper above feed information is stored in the database of a prototype. Ten, the EA model was developed by vb.net, giving a user-friendly, extensible, and stretchable framework for tuning the metaheuristicrelated parameters and increasing the quality solution. Te purpose is to test the proposed solution's feasibility and the practicality of using enhanced EA in solving a real-world grouper fsh feed formulation. Subsequently, the next section is to be discussed how the requirements and constraints are constructed into the modelling of EA.

Application of EA in Fish Feed Formulation
All requirements and constraints are modeled and applied to the fsh feed formulation problem using the sophisticated EA method. Te modeling of EA is developed based on the titles discussed in the following subsections.

Objective Function.
Te objective function or ftness function value is taken as the minimum summation of the weight for each ingredient multiplied by its cost per kg, penalty value for the extra weight of ingredients and penalty value for nutrients that are out of its acceptable range, as formulated, where s i is the cumulative cost, nutrient penalty, penalty value for weight of jth ingredient for each chromosome, i or possible feed solution. Z k is the penalty value for k th nutrient, Dj is the 100,000 constant number for cumulative cost, pj is the penalty value for weight of jth ingredient not in the specifed range.
δ v � 1, if computed total weight for all ingredients, v for each chromosome, 0, Otherwise.  Table 3, where E k ≤ N k ≤ F k , E k is the lower boundary of required percentage for k th nutrient in all ingredients, F k is the upper boundary of required percentage for k th nutrient in all ingredients, and N k is the range of required percentage for k th nutrient in all ingredients. Te total weight of 100 kg must be fulflled, and the range between individual animal-based ingredients and total plant-based ingredients should be 0-100 and 0-40, respectively. Te range between total animal-based ingredients and total plant-based ingredients should be 40-100 and 0-60, respectively.

Penalty Values.
Penalty values are calculated for each ingredient wherever each of the total nutrients is not within its acceptable range (refer to Table 3). Penalty values for each nutrient are calculated based on Z-scores using the formula, Z i � x i − μ i /σ i , where x i is the observed value, μ i is the mean of the sample, and σ i is the standard deviation of the sample. Te total weight of all ingredients is equal to 100 kg. Tus, another penalty which is ingredient weight penalty values also added to the total penalty. Apart from the constraints, the ingredient weight penalty is calculated based on the diference between the extra weight in kilogram (kg); the maximum required weight of the total weight of all ingredients. Te ingredient weight penalty using formula (3) where p j is the penalty value for weight of j th ingredient not in the specifed range. J � 1, 2, . . ., J where J is the total number of ingredients considered.

Costs. Te lowest cost of all ingredients was chosen in this study, where penalty values are zero or the lowest values.
Costs are calculated by multiplying the weight for each ingredient multiply with its price value per kilogram of that feed. Te cost of each ingredient is calculated using the following formula: where C j is the cost of each ingredient j per kilogram. w j is the weight of the j th ingredient in kilogram, J = 1, 2, . . ., J where J is the total number of ingredients considered.

Structure of Proposed EA.
Once the requirements and constraints are constructed into mathematical formulations, the modeling of EA is started by generating a population. Te whole EA model is shown in Figure 1.
Creating the initial solution for this proposed SR-SD-EA is based on semirandom initialization (SR), whereby the total weight of 100 kg must be fulflled, and the range between individual animal-based ingredients and total plant-based ingredients should be satisfed. Te SD tournament selection used is similar to the BT selection operator, where the concept of the standard deviation of a sample is adapted in the BT selection. Te pseudocode for the BT selection and SD tournament selection is displayed in Figures 2 and 3. Both fgures clearly show that SD tournament selection incorporated the concept of standard deviation into BT selection, where the comparison between both pairs of chromosomes is made.
Te process of EA continues with implementing the onepoint crossover, boundary-based mutation and steady-state reproduction operators. Te stopping criterion used in this SR-SD-EA is based on the number of generations. Te fnal best-so-far chromosome or solution for the grouper fsh feed formulation problem is achieved after a certain number of generations, the 200 th generation. Tis new SD tournament selection operator shall be evaluated by comparing its performance with an existing similar selection operator, which is the BT selection operator. Consequently, the results of the proposed model are discussed in the following section.

Experimental Results
In this section, the experimentation results of the proposed SR-SD-EA models are discussed to ascertain the model's performance. Te model was tested using semirandom (SR) initialization operator, one-point crossover, BM and steadystate operator in the regeneration procedure. In running the EAs, a specifed number of generations, g � 200, is used as the stopping criterion. Tis number is suggested since the best-so-far solutions for several preceding generations before g � 200 have shown a plateau trend. Te best-so-far solutions or chromosomes obtained at each generation are taken as the best solution. Each operator was tested using several values to fnd the best parameter values to boost the EA performance. Eventually, each EA experimentation was carried out for 30 independent runs. Finally, the best  Table 4, which is suggested for further analysis. Based on the experimentation results, the population size of 90, crossover rate of 0.8 and mutation rate of 0.5 are used as the parameters in the model comparison of the EA discussed in the following subsections.

Results of EA with a Proposed Selection
Operator. In this research, the proposed method SR-SD-EA in the fsh formulation was tested on grouper fsh selected. Te performance of SR-SD-EA is presented in Figure 4.
In this Figure 4, at the 10 th generation, the best-so-far ftness value for the SR-SD-EA with a population size of 90 is 1533.966. Subsequently, it dropped to 1176.128 in the 20 th generation. In the 30 th generation, the best-so-far ftness value continues decreasing to 764.663. Tere is a fuctuation Step for choosing parents 1 and 2:  Step for choosing parents 1 and 2: For chromosome i = 1 to population, I Do while chromosome i ≤ 2 Choose two chromosomes randomly (with replacement) from the population, I Compute fitness for both chromosomes Compute standard deviation on these two chromosomes (pair 1) End while Do while chromosome i ≤ 2 Choose two chromosomes randomly (with replacement) from the population, I Compute fitness for both chromosomes Compute standard deviation on these two chromosomes (pair 2) End while If the standard deviation of pair 1 > standard deviation of pair 2 Then select pair 1 as parent 1 and parent 2, otherwise, select pair 2 as parent 1 and parent 2 End if End For  A sample solution with less than fourteen ingredients was selected with a total weight of 100 kg, as shown in Table 5. Te nutrient value of each solution attained from the SR-SD-EA model is shown in Table 6.

Comparison of the Selection Operator.
In this experiment, a modifed selection operator based on the concepts of BT selection and the standard deviation is proposed and is known as the binary-standard deviation (SD) tournament selection operator. Te description of this proposed SD tournament selection operator has been discussed in section Materials and Methods. Tis unique SD tournament selection operator is incorporated in the whole process of an EA to evaluate its performance and compare it with the existing similar selection operator, that is, the BT selection operator, which is also incorporated in an EA. Te BT selection operator has also been discussed in Materials and Methods. Te resulting EAs in this experiment are the SR-SD-EA and SR-BT-EA.
For consistency purposes, the standard parameters to be used in the experimentation are population size of 90, crossover rate of 0.8, and mutation rate of 0.5, as decided in Table 4. Tese parameters are applied in both SR-SD-EA and SR-BT-EA for comparison purposes. Furthermore, SR-BT-EA and SR-SD-EA applied common operators, such as the SR initialization operator, one-point crossover, BM operator, and elitism strategy in the steady-state regeneration operator. A specifed number of generations, g � 200, is used as the stopping criterion. Tis number is suggested since the best-so-far solutions for several preceding generations before g � 200 have shown a plateau trend. Te best-so-far solutions or chromosomes obtained at each generation are taken as the best solution. Eventually, each SR-BT-EA and SR-SD-EA was carried out for 30 independent runs for further experimentation and analysis.
In conclusion, for a fair selection operator comparison, two selection procedures were experimented with and thus suggesting the best selection operator to be used in the proposed model. Te frst procedure employed was the BT Selection operator, while the second was the SD tournament selection operator. Te tournament selection operation was recommended by Sahman et al. [5] to fnd a potential solution in the selection stage. Te proposed SD tournament selection operator is adopted based on the concept of statistical measurement. Standard deviation is a relevant and most-used measure of dispersion [106,107]. For experimentation purposes, the EA model was implemented with the same operator as one-point crossover, boundary mutation, steady-state replacement strategy, and fxed parameters (e.g., 200 generations, 90 population size and 30 runs). In the model evaluation, comparison on the existing selection operator with the newly proposed operator has been made in terms of the best-so-far ftness, average best-so-far ftness, standard deviation, and average run time of the sample (milliseconds). Subsequence, Figure 5 presents the performance graph of the SR-BT-EA model and SR-SD-EA model.
In this Figure 5, the initial best-so-far ftness values for SR-SD-EA and SR-BT-EA are 1533.966 and 1626.068, respectively. Both SR-SD-EA and SR-BT-EA started to show a plateau trend at 100 th generation. However, the SR-SD-EA has the lowest ftness value at 495.289, while the SR-BT-EA has the lowest ftness value at 507.359. A sample solution with fourteen ingredients from the SR-BT-EA model and SR-SD-EA were selected with the total weight of 100 kg shown in Table 7. Nutrient values of each solution yielded0 from the SR-BT-EA model and SR-SD-EA model are revealed in Table 8.
In the SR-SD-EA, only nine ingredients have been selected in the fnal feed mixed, where the ingredients that are not recommended are algae meal, wheat four, meat and bone meal, soybean oil, and imported fshmeal. While in the SR-BT-EA model, twelve ingredients have been selected in the fnal feed mixed, where the ingredients that are not recommended are algae meal and meat and bone meal. Both SR-SD-EA and SR-BT-EA satisfying constraints involved total weight of 100 kg, individual animal-based ingredients and total plant-based ingredients in the range of 0-100 and 0-40, respectively, as well as total animal-based ingredients and total plant-based ingredients in the range of 40-100 and 0-60, respectively.
Te best-so-far feed formulation solution obtained from SR-SD-EA has successfully fulflled all nutrients requirements. Tis is evidenced in Table 8 with all individual percentages of nutrients are within the minimum and maximum ranges. Table 9 shows that SR-SD-EA was able to gain a fairly better result than model SR-BT-EA in terms of best so far to obtain cost minimization. Even though the standard deviation value of the best-so-far ftness for SR-SD-EA is fairly larger than that of the SR-BT-EA but, looking from the positive perspective it can be considered good since there is potential exploration to obtain much lower best-so-far ftness value. Tis is in line with Delmas and Liu [108] who emphasized that the larger standard deviation refects that there would be a high chance of exploration that can take place, which leads to lower ftness. Furthermore, the average system run time of the SR-SD-EA needed approximately 479742.667 milliseconds which equivalent to 8.00 minutes of computation time, while the average system run time of the SR-BT-EA is 524842.600 milliseconds which equivalent to 8.75 minutes. Tus, based on this premise of better performance, the binary-standard deviation (SD) tournament Selection operator is a suitable and recommended to be applied in this feed formulation.
Te results of T-test analysis for both models, SR-SD-EA and SR-BT-EA, are shown in Table 10 and Figure 6. Te mean for SR-SD-EA is lower than SR-BT-EA, with the value of 1125.164 and 1142.131, respectively. However, the results of the T-test for the diference between two independent means shown in Figure 6 indicated that there is no significant diference between both means. In other words, both methods are equally efective. Based on the value of standard deviations in Table 10, the SR-SD-EA model is considered to    10 Complexity have a fairly larger standard deviation than that of the SR-BT-EA. Looking from a positive perspective, it can be considered good since there is a potential exploration to obtain much lower best-so-far ftness value. Tis is in line with the results of the study by Delmas and Liu [108] that emphasized that the larger standard deviation refects that there would be a high chance of exploration that can take place, which leads to lower ftness. Tis shows that SR-SD-EA provides more reliable methods that represent a real problem. On top of that, the standard error term includes all information, such as sample size and allele frequencies, thus providing optimal performance. In conclusion, 95% confdent that the true mean is between and 1125.164 and 1142.131. Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean. Tus, based on this premise of better performance, the SR-SD-EA selection operator is a suitable and recommended to be applied in this feed formulation.

Conclusion
In conclusion, both the SR-BT-EA and the SR-SD-EA yield good and reasonable results in solving nonlinear cost optimization of a feed formulation problem. However, the performance of SR-SD-EA is better than the SR-BT-EA model. Furthermore, the SR-SD-EA model produces quicker results with same penalties 135.719 in the solution of nonlinear problems when compared with the SR-BT-EA model. Tis efort has convinced us that this selection operator has shown the potential of exploring and exploiting the function as the alternative solution, thus enhancing the method for complex grouper fsh feed formulating. Potentially, future works on other operators of the EA can be   Figure 6: T-test for diference between two independent means. Complexity explored and consequently inaugurate a widespread research space for the growth and impact of the grouper fsh feed formulation [22,23,32,109].

Data Availability
Data were collected via primary data.

Conflicts of Interest
Te authors declare that they have no conficts of interest.