Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2017, Article ID 7053710, 12 pages
https://doi.org/10.1155/2017/7053710
Research Article

Shrimp Feed Formulation via Evolutionary Algorithm with Power Heuristics for Handling Constraints

1Department of Decision Science, School of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia
2University of Nottingham Malaysia Campus, Semenyih, Malaysia
3University of Nottingham, Nottingham, UK
4Mariculture Research Centre, Bukit Malut, 07000 Langkawi, Malaysia
5School of Computing, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia

Correspondence should be addressed to Rosshairy Abd. Rahman; ym.ude.muu@yriahs

Received 30 May 2017; Revised 26 August 2017; Accepted 2 November 2017; Published 26 November 2017

Academic Editor: Michele Scarpiniti

Copyright © 2017 Rosshairy Abd. Rahman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Formulating feed for shrimps represents a challenge to farmers and industry partners. Most previous studies selected from only a small number of ingredients due to cost pressures, even though hundreds of potential ingredients could be used in the shrimp feed mix. Even with a limited number of ingredients, the best combination of the most appropriate ingredients is still difficult to obtain due to various constraint requirements, such as nutrition value and cost. This paper proposes a new operator which we call Power Heuristics, as part of an Evolutionary Algorithm (EA), which acts as a constraint handling technique for the shrimp feed or diet formulation. The operator is able to choose and discard certain ingredients by utilising a specialized search mechanism. The aim is to achieve the most appropriate combination of ingredients. Power Heuristics are embedded in the EA at the early stage of a semirandom initialization procedure. The resulting combination of ingredients, after fulfilling all the necessary constraints, shows that this operator is useful in discarding inappropriate ingredients when a crucial constraint is violated.

1. Introduction

Shrimps are crustaceans and are among the main aquatic organisms being farmed. It contributes to the fast growing aquaculture industry in the food-production sector [1]. Feed has been identified as the most expensive component of the total cost when farming shrimps [25]. Providing a better shrimp feed, with minimum cost, could help farmers reduce costs and increase profits.

Shrimps require several nutrients for healthy growth. Shrimp nutrients are a complex subject because the nutritional requirements change at each stage of its life cycle (i.e., larval, nursery, juvenile, and adult). Thus, shrimp feed must be specifically formulated for different stages of its life [6]. Juvenile shrimps require higher nutritional values, especially protein, than shrimps in other life stages. For this reason, most previous studies have focused on juvenile shrimps [713]. This study also focusses on formulating feed for juvenile shrimps.

The quality of shrimp feed depends on two factors: nutrients and ingredients. Shrimps require specific nutritional requirements such as protein, lipid, ash, and fibre [14]. Protein is the most expensive nutrient source, comprising two different types (crude protein and amino acid). There are 22 amino acids that are commonly found in proteins [15], but shrimps require only ten essential amino acids (EAA), in a specific ratio, to achieve its optimal growth [15]. Only a small number of studies have taken into account the value of amino acids [16] due to a lack of information on the requirement of total amino acids for optimal growth of juvenile shrimp. The crude protein level, while maintaining ideal ratios of EAA, will increase growth [17]. In this paper, we take into account the requirement of amino acid in formulating shrimp feed. However, since the real amino acid requirement for shrimps is unknown, we estimate the amino acid requirement based on expert opinion, the scientific literature, and commercial feed data.

It is important to take into account ingredients, to ensure that a shrimp’s diet contains appropriate nutrients and has an attractive physical appearance to entice shrimps [18]. Fish meal is one of the most expensive ingredients and is also a primary source of protein [19]. For this reason, it is important for fish meal to be included in the shrimp diet.

This paper presents a feed formulation problem for juvenile shrimps, which is motivated by real world requirements, as experienced by the Department of Fisheries (DOF), Malaysia. The problem dataset has several new constraints in addition to those currently experienced by the DOF. Previous studies [20, 21] have obtained a ratio of ingredients based on a set of preselected ingredients. However, the combination of some ingredients is sometimes unable to return a feasible solution as each ingredient has a minimum quantity requirement. Therefore, in order to choose only the appropriate ingredients, a semirandom initialization operation [21] acts as a way of precontrolling the nutrient requirement. Our work utilises this semirandom initialization, improving upon it with a new alternative initialization operator based on the work in [22]. Our improvement differs from [22] in the computation of obtaining new ingredient values within the Evolutionary Algorithm (EA).

The organization of the rest of the paper is as follows. In Section 2, we describe the feed or diet formulation problem and present related work. In Section 3, the EA methodology for addressing the DOF feed formulation problem is described. In Section 4, we present the mathematical formulation of the problem, while in Section 5 we propose an EA with Power Heuristics as a strategy to improve the solution methodology for the feed formulation problem. In Section 6, comparisons between the variations of the EA are presented, which demonstrates the effectiveness of the proposed method. The final section concludes the paper and provides suggested future work directions.

2. Feed Formulation Problem

A feed formulation problem can be categorized in two forms which address humans and farmed animals. The term “menu planning problem” normally relates to methods used for planning menus for the human diet [23], and it can also be defined as the scheduling of meals associated with a person’s needs during a given time horizon [24]. In agricultural applications, the diet formulation problem is commonly known as the feed mix problem [25] or feed formulation [26, 27]. It involves a combination of several feed ingredients in specific quantities in order to satisfy nutritional requirements at a minimum cost [25] for the diet of farmed animals. Many studies [2831] have attempted to formulate animal diets using approaches to improve the quality of the feed. In this section we highlight past studies that have discussed the feed formulation problem involving animals. The approaches can be categorized into four classes: algebraic, optimization, heuristics, and integrated approaches [18]. The range of methodologies which have been used to address the feed formulation problem is presented in Figure 1. The abbreviations used are as follows: PS = Pearson’s Square, SAE = Simultaneous Algebraic Equation, LP = Linear Programming, GP = Goal Programming, CCP = Chance Constrained Programming, QP = Quadratic Programming, NLP = Nonlinear Programming, CH = Constructive Heuristics, and MH = Metaheuristics.

Figure 1: Classification of solution techniques for feed formulation problems as adapted and enhanced from Rahman (2014).
2.1. Algebraic Approaches

An algebraic approach is a method which involves mathematical calculations and is normally too computationally expensive. There are only two approaches that can be classified as algebraic approaches which have been applied to the feed formulation problem. These approaches are Pearson’s Square (PS) [26, 32, 33] and Simultaneous Algebraic Equations (SAE) [26, 33]. The limitation of PS is it can only balance one nutrient at a time, while SAE is computationally expensive when solving many nutrients and ingredients [34]. Therefore, these methods are not suitable for handling ingredient constraints.

2.2. Exact Approaches

Optimization involves maximizing or minimizing a function by systematically choosing the best value in a feasible region [35]. It is also defined as the process of attempting to find the best possible solution among all those available [36]. Optimization approaches, with regard to animal feed formulation, have included Linear Programming [28, 3762], Goal Programming [6376], Chance Constrained Programming [32, 7783], Quadratic Programming [84], and Nonlinear Programming [8587]. These methodologies were frequently used in various feed formulation problems, with LP dominating the studies. LP requires that all the constraints for the problem be exactly formulated, which can be problematic for some problems.

2.3. Heuristic and Metaheuristic Approaches

Heuristics are based on the concept of rule of thumb [88]. When applying a heuristic technique, the expectation is to achieve the best-so-far solution in a feasible region. In the case of feed formulation, the heuristic and metaheuristic approaches are discussed together in this section due to the limited number of studies being carried out using these approaches.

A constructive heuristic starts with an empty solution and gradually builds a feasible solution, usually by incrementally adding to the growing schedule, while respecting the problem constraints [89]. M. O. Afolayan and M. Afolayan [26] used a trial and error methodology for feed formulation for poultry. The formulation was done either manually or by using a spreadsheet. Ruohonen and Kettunen [90] proposed a mixture experiment method to improve on the experimental design (ED) method. This improved method successfully shortens time when compared to traditional ED. A few years later, Forster et al. [91] used the methodology of Ruohonen and Kettunen [90] to find an optimal diet for shrimps. This mixture experiment method is a good method for real situation testing. But of course, it requires a full system for shrimp growth with tank and expertise. In 2011, [92] proposed a search-based heuristic algorithm to solve several small-scale problems involving four to ten ingredients for swine feed formulation. However, to gain the optimal solution using this approach requires high computational time especially when there was a lot of ingredients and nutrients that need to be considered.

A metaheuristic is a high level algorithm that provides a set of strategies or procedures, which seek an optimal solution [93]. In relation to animal feed formulation, [20, 92] conducted studies using an Evolutionary Algorithm (EA), while [30] utilised Particle Swarm Optimization. Furuya et al. [20] solved nonlinear constraints involving ratios of ingredients for livestock feed. Their study showed that an EA is a promising methodology for the diet formulation problem as a feasible solution could be obtained even for a problem that had no apparent solution. They considered minimum and maximum ratios of ingredients, with many of the ingredients being free from any requirement or constraint.

Şahman et al. [21] produced a good solution for a cattle feed problem with a few constraints, which aimed to obtain a zero value for their EA penalty function. Şahman et al. [21] also worked on the feed formulation problem for poultry, which had many nutrient constraints. Altun and Şahman [30] explored a PSO (Particle Swarm Optimization) approach to solve nonlinear constraints of nutrients. In comparison to a Genetic Algorithm, the PSO solution is superior, with respect to achieving lower penalty values. However, both [21, 30] did not consider a ratio constraint in their animal feed formulation problems.

2.4. Integrated Approaches

Integrated approaches are the combination of two or more methodologies. In a feed formulation problem, most of the integrated techniques are based on LP [94102], with a few being search-based [103, 104].

2.5. Discussion

Considering the four approaches discussed above, an algebraic approach is not applicable since it is not able to cater for more than one nutrient. Optimization approaches are the favoured methodology in a diet formulation problem. However in determining the minimum cost feed, the linear or nonlinear constraints are increasingly complex and thus difficult to handle. In these situations, applications of standard linear or nonlinear programming techniques are both time consuming and inefficient [21]. Therefore, the EA as a metaheuristic has the potential of returning quality solutions which is the reason why it has been used by previous researchers. We hypothesize that an EA with enhanced search operators, such as in the initialization stage, will provide superior solutions even when new constraints are introduced.

3. Evolutionary Algorithm for the Feed Formulation Problem

An EA is a multistage algorithm, comprising initialization, selection of parents, crossover, and mutation. These operators are normally tuned in order to improve their performance. For example, new initialization procedures have been proposed by [105], selection by [106], crossover by [107], and mutation by [22].

A new mutation operator known as Power Mutation was introduced by [22] to cater for constrained problems with real number representation. It is an enhancement of uniform mutation and makes use of both lower and upper bounds of a constraint. The advantage of using each constraint’s boundaries is that we might get a number of possible solutions within that constraint’s range. Deep and Thakur [22] introduced Power Mutation with the aim of using an EA without any assistance from other constraint handling techniques. The idea of Power Mutation can be further extended for other operations such as the initialization.

In an animal feed formulation, [20] has pioneered the utilisation an EA, aiming to solve the nonlinear constraints which involve a ratio of ingredients. The study showed that an EA is able to produce a near optimal solution even for a problem that has no apparent solution. [20] considered the minimum and maximum values of each ingredient. However, almost all of the minimum values were considered as free values. Reference [21] then continued the work of [20] utilising a GA to achieve the least cost diet for livestock. Their experiments produced good quality solutions for this problem. However, [21] did not consider a ratio constraint in their study. Only these two studies utilise an EA for the animal feed formulation. Hence, this paper addresses both limitations in those two studies. We investigate an EA for the minimum and maximum values for nutrients and ingredients and take into account ratio constraints between nutrients.

Previous research for animal feed formulation using metaheuristics has attempted to get the appropriate mix of ingredients. For example, eight ingredients can be used in the formulation, and then a specific percentage of all ingredients are selected to be included in the formulation. Based on expert opinion, at most ten ingredients could be included in the formulation in order to obtain a low cost diet. However, it is very difficult to find the best ten ingredients with the optimal quantities to fulfill all specified nutrients with minimum cost. This paper investigates the use of an EA, to address the feed formulation problem. Table 1 summarizes the number of ingredients included in previous studies that have used heuristic approaches. Reference [26] did not mention the number of ingredients.

Table 1: Number of ingredients used in heuristic approach.

As shown in Table 1, the largest number of ingredients in previous studies was Furuya et al. [20], who considered 20 ingredients. However, [20] did not consider any restriction on the minimum and maximum values. All ingredients were considered when the final formulation was calculated. In other words, if ten ingredients were preselected at the beginning, the final formulation would include all ten ingredients with a minimum value based on the restriction. However, in the real world application of feed formulation, there are many ingredients that could be included. Due to the operational costs, only a combination of some ingredients will be considered. Therefore, this study introduces a filtering heuristic that is able to choose appropriate ingredients and remove some ingredients based on predefined constraints. Fourteen ingredients were chosen as this represents real data obtained from manufacturers and the Department of Fisheries, Malaysia.

4. Shrimp Feed Formulation Model

In the shrimp feed formulation problem, the aim is to satisfy all the nutritional needs of farmed shrimps at a minimum cost. The minimization problem considers 14 ingredients and 18 nutrients. The main aim of this shrimp feed formulation is to minimize the overall cost of the shrimp feed. In order to obtain the minimum cost, a penalty system is used. A penalty value is given if the ingredients and nutrients selected are unable to fulfill the constraints or requirements as discussed in the following subsection.

4.1. Objective Function and Constraints

As part of modelling the shrimp feed formulation problem, objective function and constraints are defined as follows:where is the cost of ingredient , equals the weight of the th ingredient, and is the total cost of the feed,

subject to(i)ingredients’ range based on maximum and minimum value as shown in Table 2:where is lower bound of ingredient i, is upper bound of ingredient , and equals the weight of the th ingredient;(ii)total ingredients weight iswhere is a total ingredients weight as predefined by the user in the user interface,the specific amount is fixed at 100 kg;(iii)single nutrients’ range based on maximum and minimum values as shown in Table 3:where is lower bound of total value of nutrient k, is upper bound of total value of nutrient k, and is total value of nutrient k in ingredient , ;(iv)combination of nutrients’ range based on maximum and minimum value as shown in Table 3:where is lower bound of combination nutrients i and j and is upper bound of combination nutrients i and j;(v)ratio of nutrients’ range based on maximum and minimum value as shown in Table 3:where is lower bound of ratio between nutrients i and j and is upper bound of ratio between nutrients i and j.

Table 2: A list of ingredients and its specific range.
Table 3: A list of nutrients and its specific range.
4.2. Ingredients and Nutrients

In this study, 14 ingredients and 18 nutrients are considered. The ingredients have minimum and maximum percentages that can be used. These values are shown in Table 2. The nutrient constraints are categorized into three types which are single, combination, and ratio. There are 16 nutrients that are classified as a single type, that is, crude protein, lipid, fibre, ash, calcium, phosphorus, and ten essential amino acids (EAAs). These EAAs are arginine, histidine, isoleucine, leucine, lysine, methionine, phenylalanine, threonine, tryptophan, and valine. These nutrients are shown in Table 3. For the combination types, there are two, which are combinations of methionine and cystine and also phenylalanine and tyrosine. A ratio of calcium and phosphorus is also taken into consideration. The minimum and maximum percentages and values given represent the lower and upper bounds for each of the ingredient and nutrients, respectively.

Based on our interviews with experts in the aquaculture industry and authority, shrimp can grow well if they get enough nutrients [14, 108]. In relation to nutrient levels, the best solution occurs when each of the nutrient’s constraints are fulfilled. The higher the penalty value given, the more the nutrient constraints that were unable to be satisfied. The minimum value is considered the best penalty value, which relates to the minimum feed cost achieved.

4.3. The Evolutionary Algorithm Model

The constraints function is embedded as part of the fitness function in our EA as shown in Algorithm 1.

Algorithm 1: The EA algorithm.

The EA draws from other established operators, such as the Roulette Wheel Selection [109, 110], one-point crossover [110], and Power Mutation (excerpted from [22]). However, in this study we highlight the use of a Power Heuristic, which is incorporated in the initialization stage of the EA to help obtain a good quality feasible solution for formulating shrimp feed mix with many ingredients.

5. Power Heuristics Algorithm for EA Initialization

The original idea of Power Heuristics came from a mutation operator known as Power Mutation, introduced by [22]. Power Heuristics is capable of deleting inappropriate ingredients and, thus, searches for the most suitable combination of ingredients. When too many preferred ingredients are candidates, nutrient constraints might be violated as each ingredient has its own restrictions. Hence, Power Heuristics are embedded in the early stage of the EA, to act as a filter against a poor combination of ingredients and search for better alternatives with minimal penalty values.

Power Heuristics uses a local search concept that searches around the neighbourhood of a solution. It begins by checking the feasibility of a potential solution. If the solution is infeasible, a new ingredient will be located that is within the range of one unit (kilogram) from the current ingredient. This mechanism is able to remove a few ingredients. Finally, the penalty value for the new solution is calculated. This mechanism is useful when inappropriate ingredients need to be removed in the case of an important constraint being violated; that is, the total ingredient weight exceeds the required amount. The incorporation of this operator can at least reduce the initial penalty value, thus increasing the possibility of locating a feasible solution. The algorithms for Power Heuristics and Power Mutation are presented in Algorithms 2 and 3, respectively.

Algorithm 2: Power Heuristics algorithm.
Algorithm 3: Power Mutation algorithm [22].

Power Heuristics is adapted from the Power Mutation operator as originally introduced by [22]. As the purpose of Power Mutation is to obtain a feasible solution, introducing a new weight for each ingredient in the mutation stage () is designed to include a lower and upper bound as in equations and .

Power Heuristics aims to reject some ingredients and then repair the solution to achieve a lower penalty value. The formula for getting an adjusted weight for each ingredient is designed so that the adjusted weight for the particular ingredient () can be either zero or slightly different from the original weight as shown in equation (refer to Algorithm 2). Most previous studies (e.g., [111115]), especially those that used experimental design approach, used less than ten ingredients for aquaculture feeds including shrimps. The reason was to cut down the operation costs such as ordering and storage [18].

In order to test the practicality of Power Heuristics, two initialization procedures were tested to obtain some insights and suggest the best initialization operator. The first procedure is a semirandom initialization operation which employed a formula using lower and upper bound values of the weight of each ingredient, as shown in (7). The second procedure also employed the same formula, but with the inclusion of Power Heuristics, as indicated in Algorithm 2, for filtering purposes.

For where is ingredients’ lower bound, is ingredients’ upper bound, and is the number of populations.

6. Results and Discussions

Semirandom initialization operation, adapted from [21], was implemented to find solutions at the initial stage so as to ensure that the weight of ingredients lies within the ingredients’ limitations. The experimentations between these two initialization operations were carried out and the results are shown in Table 4. The generation of the proposed EA model with a Power Heuristic operator (EA-PH) was carried out with certain established operators, which are Roulette Wheel Selection, One-Point Crossover, and Power Mutation.

Table 4: Performance of both models based on penalty value with different initialization procedures.

The evolutionary model with semirandom initialization operation (EA-SR) was used as a benchmark. Since this work is the first attempt of filtering some ingredients from the list, the only appropriate algorithm to be compared is by using EA-SR. Each model was run 30 times using the algorithm from Algorithm 1. Experimentations of EA with two initialization operators, that is, EA-PH and EA-SR, were conducted. These initialization operators were tested, while other EA operators remain the same. As in Table 4, the EA-SR provided an infeasible solution in every run, whereas the EA-PH model showed that no solution was infeasible. This experimentation shows that Power Heuristics allows unnecessary ingredients to be filtered out of the system. As a result, the total selected ingredients in the system might be less than fourteen. Indirectly, the requirements of ingredient’s weight and nutrient’s range can be fulfilled.

A sample solution for the shrimp feed formulation is shown each in Tables 5 and 6 as obtained from EA-SR and EA-PH, respectively. In comparison, the EA-SR solutions in Table 5 include all ingredients with a total weight of 124.4231 kg, while EA-PH solution in Table 6 shows that only eight ingredients were selected for the solution with a total weight of 100.0514 kg. The total price for this feed combination using EA-SR was MYR 297.34, and MYR 217.08 was the price for EA-PH. The results from this experiment conclude that Power Heuristics in the EA model is able to remove some inappropriate ingredients in searching for a feasible solution when many possible ingredients are under consideration.

Table 5: A sample solution obtained from the EA-SR model.
Table 6: A sample solution obtained from the EA-PH model.

In addition, the performance of the EA-PH model was statistically evaluated with , , , and . As confidence level of 95% using -test, and value being 0.0001, performance of the EA-PH model is significantly better than that of the EA-SR model. Therefore, we can conclude that the performance of EA-PH is better than EA-SR due to the significantly less penalty value obtained.

Regarding the ingredients, selecting them is based on the nutritional values obtained in the shrimp feed mix, whether fish meal is included or not. The underlying rationale for this selection is that the price of fish meal is relatively high. This is not a surprise since fish meal is high in demand and thus, many researchers [116, 117] have tried to substitute fish meal with alternative ingredients in their studies. The alternatives that could maintain the palatability of the shrimp feed such as krill meal [118, 119] and poultry by product meal [120] have been tested.

Many other combinations of ingredients could be obtained through this EA. When our proposed EA is run, a feasible and good solution or result is recommended by the embedded model in terms of the best combination of ingredients that fulfill the quality and cost requirements. The solution helps to reduce the number of tests that would otherwise have to be carried out considering a large number of combinations of ingredients. A potential extension to this research could be to use the solution from this research as input to an ED approach to confirm how the shrimp will perform. These approaches could complement each other to obtain the best shrimp feed.

7. Conclusions and Future Work

A shrimp feed formulation problem was explored in this study using an EA strategy. A heuristic procedure known as Power Heuristics was incorporated at the initialization stage in an EA model to explore the neighbourhood area when the initial solution was infeasible. The function of this Power Heuristic is to filter unsuitable ingredients and remove it from the EA computation. The heuristic operator is capable of filtering some combinations of ingredients from a selected database of choices, which could lead to potentially poor solutions. It works well if many options exist in the search space, which is suitable for a large problem like an animal diet formulation with many possible ingredients being considered. In future, many new avenues can be explored, such as improving the integration of Power Heuristics with the EA or new integration with other population based techniques with various ingredients, as hundreds of possible ingredients have been tested in the literature.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Research on the work reported in this paper was supported by Universiti Utara Malaysia and the Ministry of Higher Education Malaysia under the Fundamental Research Grant Scheme (FRGS) number 12817. The authors are grateful and would like to thank the parties for the financial support received during the research period and able to produce this piece of work.

References

  1. FAO, “Fisheries and Aquaculture Department,” 2014, http://www.fao.org/docrep/016/i2727e/i2727e.pdf.Last.
  2. K. Chin, “The feedmilling industry, the assessment of product quality and the economics of shrimp feeds. Report of the workshop on shrimp and finfish feed development. Johore Bahru,” Malaysia, pp. 25–29, 1988. View at Google Scholar
  3. C. Lim and D. M. Akiyama, “Chapter 6: Nutrient requirement of penaeid shrimp,” in Nutrition and utilization technology in aquaculture, pp. 60–73, AOCS Press, 1995. View at Google Scholar
  4. Y. C. Shang, P. Leung, and B.-H. Ling, “Comparative economics of shrimp farming in Asia,” Aquaculture, vol. 164, no. 1-4, pp. 183–200, 1998. View at Publisher · View at Google Scholar · View at Scopus
  5. J. W. Hertrampf, P. K. Biswal, and S. K. Mishra, “A revolutionary concept for shrimp feeding,” http://www.allaboutfeed.net/Process-Management/Management/2005/6/A-revolutionary-concept-for-shrimp-feeding-AAF010209W/.
  6. J. M. Fox, G. D. Treece, and D. R. Sanchez, “Shrimp nutrition and feed management,” in Methods for improving shrimp farming in Central America, M. C. Haws and C. E. Boyd, Eds., pp. 65–90, of Central America Press, Managua, Nicaragua:University, 2001. View at Google Scholar
  7. J. M. Fox, A. L. Lawrence, and E. Li-Chan, “Dietary requirement for lysine by juvenile penaeusvannamei using intact and free amino acid sources,” Aquaculture, vol. 131, pp. 279–290, 1995. View at Google Scholar
  8. A. G. J. Tacon, J. J. Cody, L. D. Conquest, S. Divakaran, I. P. Forster, and O. E. Decamp, “Effect of culture system on the nutrition and growth performance of Pacific white shrimp Litopenaeus vannamei (Boone) fed different diets,” Aquaculture Nutrition, vol. 8, no. 2, pp. 121–137, 2002. View at Publisher · View at Google Scholar · View at Scopus
  9. C. Pascual, E. Zenteno, G. Cuzon et al., “Litopenaeus vannamei juveniles energetic balance and immunological response to dietary proteins,” Aquaculture, vol. 239, no. 1-4, pp. 375–395, 2004. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Arnold, R. Smullen, M. Briggs, M. West, and B. Glencross, “The combined effect of feed frequency and ration size of diets with and without microbial biomass on the growth and feed conversion of juvenile Penaeus monodon,” Aquaculture Nutrition, vol. 22, no. 6, pp. 1340–1347, 2016. View at Publisher · View at Google Scholar · View at Scopus
  11. S. L. Zeng, W. Q. Long, L. X. Tian et al., “Effects of dietary aflatoxin B1 on growth performance, body composition, haematological parameters and histopathology of juvenile Pacific white shrimp (Litopenaeus vannamei),” Aquaculture Nutrition, vol. 22, no. 5, pp. 1152–1159, 2016. View at Publisher · View at Google Scholar · View at Scopus
  12. Y.-Y. Yu, W.-D. Chen, Y.-J. Liu, J. Niu, M. Chen, and L.-X. Tian, “Effect of different dietary levels of Gracilaria lemaneiformis dry power on growth performance, hematological parameters and intestinal structure of juvenile Pacific white shrimp (Litopenaeus vannamei),” Aquaculture, vol. 450, pp. 356–362, 2016. View at Publisher · View at Google Scholar · View at Scopus
  13. J. A. Venero, D. A. Davis, and D. B. Rouse, “Variable feed allowance with constant protein input for the pacific white shrimp Litopenaeus vannamei reared under semi-intensive conditions in tanks and ponds,” Aquaculture, vol. 269, no. 1-4, pp. 490–503, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. M. Jobling, “National Research Council (NRC): Nutrient requirements of fish and shrimp,” Aquaculture International, vol. 20, no. 3, pp. 601-602, 2012. View at Publisher · View at Google Scholar
  15. J. P. Lazo and D. A. Davis, “Ingredient and feed evaluation,” in Encyclopedia of Aquaculture, R. R. Stickney, Ed., pp. 453–463, Wiley-terscience Publication, Texas, Tx, USA, 2000. View at Google Scholar
  16. J. M. Fox, D. A. Davis, M. Wilson, and A. L. Lawrence, “Current status of amino acid requirement research with marine penaeid shrimp,” in Avances en Nutrición Acuícola VIII. VIII Simposium Internacional de Nutrición Acuícola, pp. 182–196, Nuevo León, 2006. View at Google Scholar
  17. G. M. Pesti, “Impact of dietary amino acid and crude protein levels in broiler feeds on biological performance,” Journal of Applied Poultry Research, vol. 18, no. 3, pp. 477–486, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. A. R. Rahman, Evolutionary algorithms with average crossover and power heuristics for aquaculture diet formulation [Ph.D. thesis], Universiti Utara Malaysia, 2014.
  19. L. E. Cruz-Suárez, M. Nieto-López, C. Guajardo-Barbosa, M. Tapia-Salazar, U. Scholz, and D. Ricque-Marie, “Replacement of fish meal with poultry by-product meal in practical diets for Litopenaeus vannamei, and digestibility of the tested ingredients and diets,” Aquaculture, vol. 272, no. 1-4, pp. 466–476, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. T. Furuya, T. Satake, and Y. Minami, “Evolutionary programming for mix design,” Computers and Electronics in Agriculture, vol. 18, no. 2-3, pp. 129–135, 1997. View at Publisher · View at Google Scholar · View at Scopus
  21. M. A. Şahman, M. Çunkaş, Ş. İnal, F. İnal, B. Coşkun, and U. Taşkiran, “Cost optimization of feed mixes by genetic algorithms,” Advances in Engineering Software, vol. 40, no. 10, pp. 965–974, 2009. View at Publisher · View at Google Scholar
  22. K. Deep and M. Thakur, “A new mutation operator for real coded genetic algorithms,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 211–230, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  23. E. Eckstein, “Communication to the Editor—Is the “Diet Problem” Identical to the “Menu Planning Problem”?” Management Science, vol. 16, no. 9, pp. 527-528, 1970. View at Publisher · View at Google Scholar
  24. J. L. Balintfy, G. T. Ross, P. Sinha, and A. A. Zoltners, “A mathematical programming system for preference and compatibility maximized menu planning and scheduling,” Mathematical Programming, vol. 15, no. 1, pp. 63–76, 1978. View at Publisher · View at Google Scholar · View at Scopus
  25. B. Render, J. Stair, M. E. Hanna, and R. M. Hanna, Quantitative analysis for management (, , ). Upper Saddle River, Pearson Education, Inc, New Jersey, NJ, USA, 9th edition, 2006.
  26. M. O. Afolayan and M. Afolayan, “Nigeria oriented poultry feed formulation software requirements,” Journal of Applied Sciences Research, vol. 4, no. 11, pp. 1596–1602, 2008. View at Google Scholar
  27. M. J. Villamide, R. Carobaño, I. Hernández et al., “Evolution of a feed formulation practice in amandatory course on animal production,” Procedia Social and Behavioral Sciences, vol. 1, pp. 1797–1801, 2009. View at Google Scholar
  28. M. K. D. K. Piyaratne, N. G. J. Dias, and N. S. B. M. Attapattu, “Linear model based software approach with ideal amino acid profiles for least-cost poultry ration formulation,” Information Technology Journal, vol. 11, no. 7, pp. 788–793, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. S. Thammaniwit and P. Charnsethikul, “Benders decomposition method for a large two-stage linear programming model,” International Transaction Journal of Engineering, vol. 4, pp. 253–268, 2013. View at Google Scholar
  30. A. A. Altun and M. A. Şahman, “Cost optimization of mixed feeds with the particle swarm optimization method,” Neural Computing and Applications, vol. 22, no. 2, pp. 383–390, 2013. View at Publisher · View at Google Scholar · View at Scopus
  31. A. R. Rahman and R. Ramli, “Average Concept of Crossover Operator,” in Proceedings of the International Conference on Management and Artificial Intelligent, 2013.
  32. W. B. Roush, T. Cravener, and F. Zhang, “Computer formulation observations and caveats,” Journal of Applied Poultry Research, vol. 5, no. 2, pp. 116–125, 1996. View at Publisher · View at Google Scholar · View at Scopus
  33. J. R. Gillespie and F. B. Flanders, Modern livestock and poultry production, Cengage Learning, Clifton Park, New York, USA, 8th edition, 2009.
  34. S. Ghosh, J. Ghosh, D. T. Pal, and R. Gupta, “Current concepts of feed formulation for livestock using mathematical modeling,” Animal Nutrition and Feed Technology, vol. 14, no. 1, pp. 205–223, 2014. View at Google Scholar · View at Scopus
  35. G. Zhang, J. Lu, and Y. Gao, Multi-level decision making models, methods and applications, Springer, methods and applications, 2015.
  36. E. K. Burke and G. Kendall, “Introduction,” in Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques, E. K. Burke and G. Kendall, Eds., p. 18, Springer, New York, NY, USA, 2005. View at Google Scholar
  37. F. V. Waugh, “The minimum-cost dairy feed,” Journal of Farm Economics, vol. 33, no. 3, pp. 299–310, 1951. View at Publisher · View at Google Scholar
  38. L. W. Swanson and J. G. Woodruff, “A sequential approach to the feed-mix problem a sequential approach to the feed-mix problem,” Operations Research, vol. 12, no. 1, pp. 89–109, 1964. View at Google Scholar
  39. H. Nott and J. F. Combs, “Data processing feed ingredient composition data,” Feed Stuffs, vol. 39, no. 41, pp. 21-22, 1967. View at Google Scholar
  40. S. A. Rahman and F. E. Bender, “Linear programming approximation of least-cost feed mixes with probability restrictions,” American Journal of Agricultural Economics, vol. 53, no. 4, pp. 612–618, 1971. View at Publisher · View at Google Scholar · View at Scopus
  41. G. M. Mohr, “The bulk constraint and computer formulations of least-cost feed mixes,” Review of Marketing and Agricultural Economics, vol. 40, no. 1, pp. 15–28, 1972. View at Google Scholar
  42. A. E. Chappell, “Linear programming cuts costs in production of animal feeds,” Operational Research Quarterly, vol. 25, no. 1, pp. 19–26, 1974. View at Publisher · View at Google Scholar · View at Scopus
  43. M. A. Barbieri and G. Cuzon, “Improved nutrient specification for linear programming of penaeid rations,” Aquaculture, vol. 19, no. 4, pp. 313–323, 1980. View at Publisher · View at Google Scholar · View at Scopus
  44. J. J. Glen, “A Mathematical Programming Approach to Beef Feedlot Optimization,” Management Science, vol. 26, no. 5, pp. 524–535, 1980. View at Publisher · View at Google Scholar
  45. C. Zioganas, The determination of viable, parity and optimum sizes of family-type sheep farms in the Epirus Region of Greece. Unpublished doctoral thesis [Ph.d. thesis], 1981.
  46. N. R. S. Pierre and W. R. Harvey, “Incorporation of uncertainty in composition of feeds into least-cost ration models,” Journal of Dairy Science, vol. 69, pp. 3051–3062, 1986. View at Publisher · View at Google Scholar
  47. H. C. De Kock and M. Sinclair, “Multi-mix feedstock problems on microcomputers,” Journal of the Operational Research Society, vol. 38, no. 7, pp. 585–590, 1987. View at Publisher · View at Google Scholar · View at Scopus
  48. A. G. Munford, “A microcomputer system for formulating animal diets which may involve liquid raw materials,” European Journal of Operational Research, vol. 41, no. 3, pp. 270–276, 1989. View at Publisher · View at Google Scholar · View at Scopus
  49. A. G. Munford, “Evaluating marginal costs associated with ratio and other constraints in linear programmes,” Journal of the Operational Research Society, vol. 40, no. 10, pp. 933–935, 1989. View at Publisher · View at Google Scholar · View at Scopus
  50. J. D. O'Connor, C. J. Sniffen, D. G. Fox, and R. A. Miligan, “Least cost dairy cattle ration formulation model based on the degradable protein system,” Journal of Dairy Science, vol. 72, pp. 2733–2745, 1989. View at Google Scholar
  51. D. M. Forsyth, “Chapter 5: Computer programming of beef cattle diet. In,” in Beef cattle feeding and nutrition, T. W. Perry and M. J. Cecava, Eds., p. 68, Academic Press, Inc, California, Ca, USA, 2nd edition, 1995. View at Google Scholar
  52. A. G. Munford, “The use of iterative linear programming in practical applications of animal diet formulation,” Mathematics and Computers in Simulation, vol. 42, no. 2-3, pp. 255–261, 1996. View at Publisher · View at Google Scholar · View at Scopus
  53. E. Thomson and J. Nolan, “UNEform: A powerful feed formulation spreadsheet suitable for teaching or on-farm formulation,” Animal Feed Science and Technology, vol. 91, no. 3-4, pp. 233–240, 2001. View at Publisher · View at Google Scholar · View at Scopus
  54. M. S. Htun, T. T. Thein, and P. Tin, Linear programming approach to diet problem for black tiger shrimp in shrimp aquaculture. Paper presented at the Information and Telecommunication Technologies, Yangon, Myanmar, Burma, 2005.
  55. T. O. S. Olorunfemi, “Linear programming approach to least-cost ration formulation for poults,” Information Technology Journal, vol. 6, no. 2, pp. 294–299, 2007. View at Publisher · View at Google Scholar · View at Scopus
  56. S. Chakeredza, F. K. Akinnifesi, O. C. Ajayi, G. Sileshi, S. Mngomba, and F. M. T. Gondwe, “A simple method of formulating least-cost diets for smallholder dairy production in sub-Saharan Africa,” African Journal of Biotechnology, vol. 7, no. 16, pp. 2925–2933, 2008. View at Google Scholar · View at Scopus
  57. E. Engelbrecht, Optimising animal diets at the Johannesburg zoo [Bachelor degree thesis], University of Pretoria, 2008.
  58. B. Al-Deseit, “Least-cost broiler ration formulation using linear programming technique,” Journal of Animal and Veterinary Advances, vol. 8, no. 7, pp. 1274–1278, 2009. View at Google Scholar · View at Scopus
  59. K. Oishi, H. Kumagai, and H. Hirooka, “Application of the modified feed formulation to optimize economic and environmental criteria in beef cattle fattening systems with food by-products,” Animal Feed Science and Technology, vol. 165, no. 1-2, pp. 38–50, 2011. View at Publisher · View at Google Scholar · View at Scopus
  60. I. U. Udo, C. B. Ndome, S. B. Ekanem, and P. E. Asuquo, “Application of linear programming technique in least-cost ration formulation for African catfish (Clarias gariepinus) in semi-intensive culture system in Nigeria,” Journal of Fisheries and Aquatic Sciences, vol. 6, no. 4, pp. 429–437, 2011. View at Publisher · View at Google Scholar · View at Scopus
  61. T. T. H. Nguyen, I. Bouvarel, P. Ponchant, and H. M. G. Van Der Werf, “Using environmental constraints to formulate low-impact poultry feeds,” Journal of Cleaner Production, vol. 28, pp. 215–224, 2012. View at Publisher · View at Google Scholar · View at Scopus
  62. L. E. Moraes, J. E. Wilen, P. H. Robinson, and J. G. Fadel, “A linear programming model to optimize diets in environmental policy scenarios,” Journal of Dairy Science, vol. 95, no. 3, pp. 1267–1282, 2012. View at Publisher · View at Google Scholar · View at Scopus
  63. T. Rehman and C. Romero, “Multiple-criteria decision-making techniques and their role in livestock ration formulation,” Agricultural Systems, vol. 15, no. 1, pp. 23–49, 1984. View at Publisher · View at Google Scholar · View at Scopus
  64. T. Rehman and C. Romero, “Goal programming with penalty functions and livestock ration formulation,” Agricultural Systems, vol. 23, no. 2, pp. 117–132, 1987. View at Publisher · View at Google Scholar · View at Scopus
  65. P. Lara, “An interactive multigoal programming model for determining livestock rations: An application to dairy cows in Andalusia, spain,” Journal of the Operational Research Society, vol. 43, no. 10, pp. 945–953, 1992. View at Publisher · View at Google Scholar · View at Scopus
  66. P. Lara, “Multiple objective fractional programming and livestock ration formulation: a case study for dairy cow diets in Spain,” Agricultural Systems, vol. 41, no. 3, pp. 321–334, 1993. View at Publisher · View at Google Scholar · View at Scopus
  67. P. Lara and C. Romero, “Relaxation of nutrient requirements on livestock rations through interactive multigoal programming,” Agricultural Systems, vol. 45, no. 4, pp. 443–453, 1994. View at Publisher · View at Google Scholar · View at Scopus
  68. K. Mitani and H. Nakayama, “A multiobjective diet planning support system using the satisficing trade-off method,” Journal of Multi-Criteria Decision Analysis, vol. 6, pp. 131–139, 1997. View at Google Scholar
  69. P. J. dit Bailleul, J. Rivest, F. Dubeau, and C. Pomar, “Reducing nitrogen excretion in pigs by modifying the traditional least-cost formulation algorithm,” Livestock Production Science, vol. 72, no. 3, pp. 199–211, 2001. View at Publisher · View at Google Scholar · View at Scopus
  70. P. R. Tozer and J. R. Stokes, “A multi-objective programming approach to feed ration balancing and nutrient management,” Agricultural Systems, vol. 67, no. 3, pp. 201–215, 2001. View at Publisher · View at Google Scholar · View at Scopus
  71. F. Zhang and W. B. Roush, “Multiple-objective (goal) programming model for feed formulation: an example for reducing nutrient variation,” Poultry Science, vol. 81, no. 2, pp. 182–192, 2002. View at Publisher · View at Google Scholar · View at Scopus
  72. C. Romero and T. Rehman, “Livestock ration formulation via goal programming with penalty functions,” in In Romero, vol. 11, pp. 149–161, In, 2003. View at Google Scholar
  73. C. Castrodeza, P. Lara, and T. Peña, “Multicriteria fractional model for feed formulation: Economic, nutritional and environmental criteria,” Agricultural Systems, vol. 86, no. 1, pp. 76–96, 2005. View at Publisher · View at Google Scholar · View at Scopus
  74. C. Pomar, F. Dubeau, M.-P. Létourneau-Montminy, C. Boucher, and P.-O. Julien, “Reducing phosphorus concentration in pig diets by adding an environmental objective to the traditional feed formulation algorithm,” Livestock Science, vol. 111, no. 1-2, pp. 16–27, 2007. View at Publisher · View at Google Scholar · View at Scopus
  75. T. Peña, P. Lara, and C. Castrodeza, “Multiobjective stochastic programming for feed formulation,” Journal of the Operational Research Society, vol. 60, no. 12, pp. 1738–1748, 2009. View at Publisher · View at Google Scholar · View at Scopus
  76. Z. Babić and T. Perić, “Optimization of livestock feed blend by use of goal programming,” International Journal of Production Economics, vol. 130, no. 2, pp. 218–223, 2011. View at Publisher · View at Google Scholar · View at Scopus
  77. W. Popp and C. V. D. van de Panne, “Minimum-cost cattle feed under probabilistic protein constraints,” Management Science, vol. 9, no. 3, pp. 405–430, 1963. View at Google Scholar
  78. G. M. Pesti and B. R. Miller, Animal feed formulation: Economics and computer applications, Chapman Hall, New York, 1993.
  79. G. M. Pesti and A. F. Seila, “The Use of an Electronic Spreadsheet to Solve Linear and Non-Linear "Stochastic" Feed Formulation Problems,” Journal of Applied Poultry Research, vol. 8, no. 1, pp. 110–121, 1999. View at Publisher · View at Google Scholar · View at Scopus
  80. P. R. Tozer, “Least-cost ration formulations for holstein dairy heifers by using linear and stochastic programming,” Journal of Dairy Science, vol. 83, no. 3, pp. 443–451, 2000. View at Publisher · View at Google Scholar · View at Scopus
  81. V. M. Katarkevich, A. N. Rubinov, S. A. Ryzhechkin, and T. Sh. Efendiev, “Compact holographic solid-state distributed-feedback laser,” Quantum Electronics A, vol. 24, no. 10, pp. 871–873, 1994. View at Publisher · View at Google Scholar
  82. W. B. Roush and T. L. Cravener, “Stochastic true digestible amino acid values,” Animal Feed Science and Technology, vol. 102, no. 1–4, pp. 225–239, 2002. View at Publisher · View at Google Scholar · View at Scopus
  83. I. U. Udo, C. B. Ndome, and P. E. Asuquo, “Use of stochastic programming in least-cost feed formulation for african catfish (Clarias gariepinus) in semi-intensive culture system in Nigeria,” Journal of Fisheries and Aquatic Science, vol. 6, no. 4, pp. 447–455, 2011. View at Publisher · View at Google Scholar · View at Scopus
  84. J. T. Chen, “Quadratic programming for least-cost feed formulations under probabilistic protein constraints,” American Journal of Agricultural Economics, vol. 55, no. 2, pp. 175–183, 1973. View at Publisher · View at Google Scholar
  85. V. R. Guevara, “Use of nonlinear programming to optimize performance response to energy density in broiler feed formulation,” Poultry Science, vol. 83, no. 2, pp. 147–151, 2004. View at Publisher · View at Google Scholar · View at Scopus
  86. P. Saxena and M. Chandra, “Animal diet formulation models: A review (1950-2010),” CAB Reviews: Perspectives in Agriculture, Veterinary Science, Nutrition and Natural Resources, vol. 6, article no. 57, pp. 1–9, 2011. View at Publisher · View at Google Scholar · View at Scopus
  87. P. Saxena, Animal diet formulation using nonlinear programming: an innovative approach, Germany:Lambert Academic Publishing, 2011.
  88. J. M. C. Hutchinson and G. Gigerenzer, “Simple heuristics and rules of thumb: Where psychologists and behavioural biologists might meet,” Behavioural Processes, vol. 69, no. 2, pp. 97–124, 2005. View at Publisher · View at Google Scholar · View at Scopus
  89. J. F. Oliveira and M. A. Carravilla, “Heuristics and Local Search,” Technical Report, 2009. View at Google Scholar
  90. K. Ruohonen and J. Kettunen, “Effective experimental designs for optimizing fish feeds,” Aquaculture Nutrition, vol. 10, no. 3, pp. 145–151, 2004. View at Publisher · View at Google Scholar · View at Scopus
  91. I. P. Forster, W. G. Dominy, A. L. Lawrence, F. L. Castille, and S. Patnaik, “Optimization of a research diet for the Pacific white shrimp, Litopenaeus vannamei, using mixture model methodology,” Aquaculture, vol. 298, no. 3-4, pp. 260–266, 2010. View at Publisher · View at Google Scholar · View at Scopus
  92. S. Pathumnakul, M. Ittiphalin, K. Piewthongngam, and S. Rujikietkumjorn, “Should feed mills go beyond traditional least cost formulation?” Computers and Electronics in Agriculture, vol. 75, no. 2, pp. 243–249, 2011. View at Publisher · View at Google Scholar · View at Scopus
  93. K. Sörensen and F. Glover, Encyclopedia of Operations Research and Management Science, K. Sörensen, F. Glover, and F. Metaheuristics, Eds., New York, 2013.
  94. J. J. Glen, “A linear programming model for an integrated crop and intensive beef production enterprise,” Journal of the Operational Research Society, vol. 37, no. 5, pp. 487–494, 1986. View at Publisher · View at Google Scholar · View at Scopus
  95. F. Polimeno, T. Rehman, H. Neal, and C. M. Yates, “Integrating the use of linear and dynamic programming methods for dairy cow diet formulation,” Journal of the Operational Research Society, vol. 50, no. 9, pp. 931–942, 1999. View at Publisher · View at Google Scholar · View at Scopus
  96. J. M. Cadenas, D. A. Pelta, H. R. Pelta, and J. L. Verdegay, “Application of fuzzy optimization to diet problems in Argentinean farms,” European Journal of Operational Research, vol. 158, no. 1, pp. 218–228, 2004. View at Publisher · View at Google Scholar · View at Scopus
  97. D. L. J. Alexander, P. C. H. Morel, and G. R. Wood, “Feeding strategies for maximising gross margin in pig production,” in Global Optimization: Scientific And Engineering Case Studies, pp. 33–43, Springer, New York, 2006. View at Google Scholar
  98. J. Žgajnar, L. Juvancic, S. Kavcic, J. Žgajnar, L. Juvančič, and S. Kavčič, “Spreadsheet tool for least-cost and nutrition balanced beef ration formulation,” in Proceedings of the Paper presented at the 16th Int. Symp. Animal Science Days, Strunjan, Slovenia, 2008.
  99. J. Žgajnar, L. Juvančič, and S. Kavčič, “Combination of linear and weighted goal programming with penalty function in optimisation of a daily dairy cow ration,” Agricultural Economics, vol. 55, no. 10, pp. 492–500, 2009. View at Google Scholar · View at Scopus
  100. J. Žgajnar, L. Juvancic, S. Kavcic, J. Žgajnar, L. Juvančič, and S. Kavčič, “Multi-goal pig ration formulation; mathematical optimization approach,” Agronomy Research, vol. 7, no. 2, pp. 775–782, 2009. View at Google Scholar
  101. D. Sirisatien, G. R. Wood, M. Dong, and P. C. H. Morel, “Two aspects of optimal diet determination for pig production: efficiency of solution and incorporation of cost variation,” Journal of Global Optimization, vol. 43, no. 2-3, pp. 249–261, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  102. S. Thammaniwit and P. Charnsethikul, “Benders decomposition method for a large two-stage linear programming model,” International Transaction Journal of Engineering, Management and Applied Sciences and Technologies, vol. 4, pp. 253–268.
  103. F.-C. Li and C.-X. Jin, “Study on fuzzy optimization methods based on principal operation and inequity degree,” Computers and Mathematics with Applications, vol. 56, no. 6, pp. 1545–1555, 2008. View at Publisher · View at Google Scholar · View at Scopus
  104. C. A. Poojari and B. Varghese, “Genetic algorithm based technique for solving chance constrained problems,” European Journal of Operational Research, vol. 185, no. 3, pp. 1128–1154, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  105. S. Rahnamayan, H. R. Tizhoosh, and M. M. A. Salama, “A novel population initialization method for accelerating evolutionary algorithms,” Computers & Mathematics with Applications, vol. 53, no. 10, pp. 1605–1614, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  106. R. Kumar and Jyotishree, “Blending Roulette Wheel Selection Rank Selection in Genetic Algorithms,” International Journal of Machine Learning and Computing, vol. 2, pp. 365–370.
  107. A. Shrestha and A. Mahmood, “Improving Genetic Algorithm with Fine-tuned Crossover and Scaled Architecture,” Journal of Mathematics, v.2016, Article ID, vol. 10, Article ID 4015845, 2016, http://dx.doi.org/10.1155/2016/pages. View at Google Scholar
  108. D. A. Davis, “Nutrient requirements of penaeid shrimp,” Global Aquaculture Advocate, pp. 72–74, 2005. View at Google Scholar
  109. K. Sastry, D. Goldberg, and G. Kendall, “Chapter 4: Genetic Algorithms,” in Introductory tutorials in optimization and decision support techniques, E. K. Burke and G. Kendall, Eds., pp. 93–117, Springer, New York, NY, USA, 2014. View at Google Scholar
  110. D. E. Goldberg, “Genetic algorithms in search, optimization, and machine learning,” Choice Reviews Online, vol. 27, no. 02, pp. 27-0936–27-0936, 1989. View at Publisher · View at Google Scholar
  111. C. Seenivasan, S. Radhakrishnan, T. Muralisankar, and P. Saravana Bhavan, “Bacillus subtilis on survival, growth, biochemical constituents and energy utilization of the freshwater prawn Macrobrachium rosenbergii post larvae,” Egyptian Journal of Aquatic Research, vol. 38, no. 3, pp. 195–203, 2012. View at Publisher · View at Google Scholar · View at Scopus
  112. S. Lin, X. Lin, Y. Yang, F. Li, and L. Luo, “Comparison of chelated zinc and zinc sulfate as zinc sources for growth and immune response of shrimp (Litopenaeus vannamei),” Aquaculture, vol. 406-407, pp. 79–84, 2013. View at Publisher · View at Google Scholar · View at Scopus
  113. T. Muralisankar, P. S. Bhavan, S. Radhakrishnan, C. Seenivasan, N. Manickam, and R. Shanthi, “Effects of dietary supplementation of fish and vegetable oils on the growth performance and muscle compositions of the freshwater prawn Macrobrachium rosenbergii,” The Journal of Basic & Applied Zoology, vol. 67, pp. 34–39, 2014. View at Google Scholar
  114. A. J. P. Nunes, M. V. C. Sá, C. L. Browdy, and M. Vazquez-Anon, “Practical supplementation of shrimp and fish feeds with crystalline amino acids,” Aquaculture, vol. 431, pp. 20–27, 2014. View at Publisher · View at Google Scholar · View at Scopus
  115. B. T. N. Kumar, H. S. Murthy, P. Patil, P. L. Doddamani, and R. Patil, “Enhanced immune response and resistance to white tail disease in chitin-diet fed freshwater prawn, Macrobrachium rosenbergii,” Aquaculture Reports, vol. 2, pp. 34–38, 2015. View at Publisher · View at Google Scholar · View at Scopus
  116. C. Hernández, M. A. Olvera-Novoa, K. Aguilar-Vejar, B. González-Rodríguez, and I. Abdo de la Parra, “Partial replacement of fish meal by porcine meat meal in practical diets for Pacific white shrimp (Litopenaeus vannamei),” Aquaculture, vol. 277, no. 3-4, pp. 244–250, 2008. View at Publisher · View at Google Scholar · View at Scopus
  117. D. M. Smith, G. L. Allan, K. C. Williams, and C. Barlow, Avances en Nutrición Acuícola V. Memorias del V Simposium Internacional de Nutrición Acuícola, Cruz-Sußrez., L. E. Ricque-Marie, D. Tapia-Salazar, M. Olvera-Novoa, and M. A. y Civera-Cerecedo, Eds., Mérida, Yucatán, México, 2000.
  118. L. Burri and J. P. Nunes, “Benefits of including krill meal into shrimp diets,” World Aquaculture, 2016. View at Google Scholar
  119. A. V. Suresh, K. P. Kumaraguru vasagam, and S. Nates, “Attractability and palatability of protein ingredients of aquatic and terrestrial animal origin, and their practical value for blue shrimp, Litopenaeus stylirostris fed diets formulated with high levels of poultry byproduct meal,” Aquaculture, vol. 319, no. 1-2, pp. 132–140, 2011. View at Publisher · View at Google Scholar · View at Scopus
  120. T. M. Samocha, D. A. Davis, I. P. Saoud, and K. DeBault, “Substitution of fish meal by co-extruded soybean poultry by-product meal in practical diets for the Pacific white shrimp, Litopenaeus vannamei,” Aquaculture, vol. 231, no. 1-4, pp. 197–203, 2004. View at Publisher · View at Google Scholar · View at Scopus