Towards Precision Fertilization: Multi-Strategy Grey Wolf Optimizer Based Model Evaluation and Yield Estimation

Precision fertilization is a major constraint in consistently balancing the contradiction between land resources, ecological environment, and population increase. Even more, it is a popular technology used to maintain sustainable development. Nitrogen (N), phosphorus (P), and potassium (K) are the main sources of nutrient income on farmland. The traditional fertilizer effect function cannot meet the conditional agrochemical theory’s conditional extremes because the soil is influenced by various factors and statistical errors in harvest and yield. In order to find more accurate scientific ratios, it has been proposed a multi-strategy-based grey wolf optimization algorithm (SLEGWO) to solve the fertilizer effect function in this paper, using the “3414” experimental field design scheme, taking the experimental field in Nongan County, Jilin Province as the experimental site to obtain experimental data, and using the residuals of the ternary fertilizer effect function of Nitrogen, phosphorus, and potassium as the target function. The experimental results showed that the SLEGWO algorithm could improve the fitting degree of the fertilizer effect equation and then reasonably predict the accurate fertilizer application ratio and improve the yield. It is a more accurate precision fertilization modeling method. It provides a new means to solve the problem of precision fertilizer and soil testing and fertilization.


Introduction
The ecosystem formed by "crop-soil-fertilizer" seems to continue indefinitely, but in each cycle, there is more or less natural loss, which needs to be replenished and controlled by human factors to continue the cycle [1][2][3][4][5][6]. Since its introduction in the 1980s, precision fertilization has been a significant constraint on balancing the contradiction between land resources, ecology, and population growth and a key technology for maintaining sustainable development [7][8][9][10][11]. Precision fertilization is based on soil testing, and field application trials and a comprehensive grasp of crop fertilization patterns, soil supply properties, and fertilizer effects are the primary means to ensure scientific fertilizer with yield increase, improve product quality, and food security while reducing environmental pollution and soil friendliness [12][13][14]. The growth of output depends on inputs, and crops need "food" to satisfy their growth. Plants need chemical elements, water, and carbon dioxide to synthesize organic matter under photosynthesis in sunlight, and fertilizers are essential "food" sponding improvement algorithms [70,117], such as enhanced comprehensive learning particle swarm optimization (GLOPSO) [118], chaotic moth-flame optimization (CMFO) [91], hybridizing grey wolf optimization (HGWO) [119], balanced whale optimization algorithm (BWOA) [120], double adaptive random spare reinforced whale optimization algorithm (RDWOA) [121], chaotic mutative moth-flame-inspired optimizer (CLSGMFO) [122], orthogonal learning sine cosine algorithm (OLSCA) [88], multi-strategy enhanced sine cosine algorithm (MSCA) [123], enhanced whale optimizer with associative learning (BM-WOA) [124], enhanced moth flame optimization (SMFO) [125], ant colony optimizer with random spare strategy and chaotic intensification strategy (RCACO) [126], etc.
These methods are widely used to solve the field of agricultural engineering optimization. Wang et al. [127] used a multi-objective chaotic particle swarm algorithm for water-saving crop planning to develop sustainable agriculture and soil resources. Saranya et al. [128] provided a crop plan optimization method using social spider optimization algorithms. Wu et al. [129] proposed an improved chaotic genetic algorithm for optimal reservoir scheduling. Amir Abbas et al. [130] proposed optimal route planning for farming operations based on an ant colony algorithm. Chagwiza et al. [88] proposed a mixed integer programming poultry feed ration optimization problem using the bat algorithm. Qazi et al. [131] proposed to solve the agricultural product scheduling problem using an improved particle swarm algorithm.
In this paper, we propose a multi-strategy improved grey wolf optimization (GWO) algorithm (SLEGWO) using combined with SMA foraging (SMA), levy flight (LF), oppositionbased learning (OBL), and greedy strategy (GS) to enhance the GWO algorithm. Unlike GWO, the command wolves are reduced, and only α and β wolves command the other wolves for foraging. Firstly, the initial α wolves are using OBL to accelerate the convergence to quality solutions. Secondly, the wolves are flown by LF and SMA mechanism to avoid getting into local optimum, enhancing the search balance. Finally, GS is used to fast convergence to the optimal solution. The proposed algorithm outperforms other competitors on 30 Classical functions and the CEC2014 test set. The SLEGWO proposed solving the nutrient equation coefficients and the highest yield (maximize fertilizer effect) in this paper. The established model is evaluated and compared with other swarm intelligence optimization algorithms using the decision coefficient R2. Experiments show that using the SLEGWO is a new feasible method that can improve the accuracy of soil measurement, better match the fertilizer application model, and ultimately provide a new computational tool for scientific fertilizer application decisions.
The rest of the paper is organized as follows. Chapter 2 introduces the improved multi-strategy grey wolf algorithm (SLEGWO). Chapter 3 compares the experiment of SLEGWO on Classical functions and CEC2014. Chapter 4 presents the precision fertilization dataset and the process and implementation of the 3414-fertilizer effect function model combined with SLEGWO, experimental results, and model evaluation. Chapter 5 presents a summary and future work.

GWO
Grey Wolf Optimizer (GWO) is a swarm intelligence optimization algorithm proposed in 2014 [132], and its performance has been the subject of analysis in many works, from clustering to global optimization [61,67,96,[133][134][135]. The algorithm was inspired by the prey hunting activity of grey wolves, which has strong convergence performance, few parameters, and easy implementation. It has been widely concerned by scholars in recent years, and it has been successfully applied to the fields of workshop schedule, parameter optimization, image classification, etc. The GWO can be regarded as an improvement of the firefly algorithm (FA). The firefly flies toward the individual due to itself, while the grey wolf has more demanding conditions and advances toward the top three of the group. The FA controls the search range by the step size, while the GWO directly defines the search range parameter A and makes A linearly decreasing. The structure of the GWO is simple, but it is not easy to improve. Several improvements only change the ratio of global search capability and local search capability, and the combined capability does not change much.
In GWO, the initial population should be divided into a number of categories, including alpha (α), beta (β), delta (δ), and omega (ω). The best wolves are considered α, β, and δ to help other wolves (ω) explore more favorable solution spaces.
In GWO, the wolves can identify the location of prey and encircle the process. Mathematically modeling this behavior, the equation is as follows.
where → A and → C are random coefficients; t is the number of iterations; → X(t) is the current position vector of the grey wolf; and → X p (t) is the position vector of the prey.
The calculation of → A and → C is shown below: where → a is decreasing from 2 to 0 as the local optimum is continuously searched and as the number of iterations increases; → r 1 and → r 2 are random numbers between [0, 1]. A wolf usually leads the hunting process. In a wolf pack, α has the highest rank in the pack. β ranks lower than α but higher than δ. in the algorithm, β and δ help α to determine the position of the pack and direct the ω wolves to hunt. So, the behavior is described by the following equation:

Opposition-Based Learning
Opposition-based learning (OBL) was proposed by Tizhoosh [136] in 2005, initially using opposites and later using approximate opposites and inverse approximate opposites. It is an improved mechanism widely used in evolutionary computation, which is designed so that an outcome opposite to the estimate is treated as the best possible outcome. When the GWO is initialized, a stochastic strategy is used. Then, in the process of random allocation of prey and food, suppose there are two opposing wolves; one of them is assumed to be the initial α wolf. The contrast learning is used, then the opposite one is selected as the α wolf, and the two wolves are compared, and the better one has been searched as the initial α wolf, which increases the accuracy of the selection of the α wolf and thus improves the convergence speed. Then there are: where → X OBL is the position of the opposite wolf in the search space, LB is the lower bound, UB is the upper bound, and → X α is the position of the α wolf. r 3 is a random vector within (0,1), and → X is the position vector of the initial random population.

Slime Mould Foraging
The slime mould algorithm (SMA), proposed by Li et al. (https://aliasgharheidari. com/SMA.html, accessed on 28 August 2021) [137] in 2020, is inspired by the diffusion and foraging behavior of slime mould, and mainly simulates the behavior and morphological changes of slime mould during the foraging process without modeling their approach, wrapping, and searching for food. SLEGWO mainly draws on SMA's foraging process. Firstly, it approaches the food according to the odor in the air; the higher the concentration of food, the stronger the bio-oscillator wave, the faster the cytoplasmic flow, and the thicker the mucilage venous tubules. A functional expression simulated this behavior with the following position update equation: where the equation for p is given as follows: where i ∈ 1, 2, . . . , n, S(i) denotes the fitness value of −−→

X(t)
and DF is the currently obtained best fitness value. The equation for → vb is given as follows: where condition denotes the top half of S(i) in the population, r denotes the random number in [0, 1], bF is the best fitness value obtained in the current iteration, wF denotes the worst fitness value obtained in the current iteration, and Smell Index denotes the sorted sequence of fitness values (in the minimum value problem in ascending order).
where → A 4 is calculated as follows.
where → A 4 is calculated in a similar way to → A 1 and → A 2 in GWO.

Levy Flight
Levy flight (LF), which is named after the French mathematician Paul Levy [138], refers to a random walk with a heavy-tailed probability distribution of step lengths.
where z denotes the variable and β shows an important Levy index to adjust the stability, and the β equation is updated with the following equation.
where r is a random value within (0, 1), and LF is used to update the distance of α and β wolves' position. Then we have the following equation: New update positions of α and β wolves were obtained according to LF. SLEGWO's LF-based stochastic decreasing operator β was combined with the wolf's equation of motion to increase the chance of exploration and exploitation.
where → X levy (t) is the position vector of the temporary wolf pack with the LF decision.

GS (Greedy Strategy)
According to the greedy strategy, the better positions → X levy (t) and → X SMA (t) among the resulting better positions based on SMA and LF can be selected as the best position vector of individuals in the next generation population according to the evaluation function.
This strategy helps SLEGWO to preserve the optimal solution and eliminate the suboptimal solutions.

Multi-Strategy Grey Wolf Optimizer (SLEGWO)
The proposed SLEGWO is based on an improvement of the GWO algorithm, reduced from three types of leader wolves to two types of leader wolves for command hunting. A random coefficient → A 4 and a random coefficient p in the SMA strategy similar to GWO is used for adjusting the execution strategy of SLEGWO. The integration of OBL can be used to accelerate the selection of the α wolf's high-quality solution in the initial stage, use the foraging mechanism of SMA and LF to keep SLEGWO balanced in exploration and detection performance, increase the possibility of jumping out of the local optimal solution while improving both exploration and detection. Finally, the GS is used to improve the quality of the optimal solution while accelerating the convergence speed. Figure 1 below shows the SLEGWO flowchart.

Multi-Strategy Grey Wolf Optimizer (SLEGWO)
The proposed SLEGWO is based on an improvement of the GWO algorithm, reduced from three types of leader wolves to two types of leader wolves for command hunting. A random coefficient 4 and a random coefficient in the SMA strategy similar to GWO is used for adjusting the execution strategy of SLEGWO. The integration of OBL can be used to accelerate the selection of the α wolf's high-quality solution in the initial stage, use the foraging mechanism of SMA and LF to keep SLEGWO balanced in exploration and detection performance, increase the possibility of jumping out of the local optimal solution while improving both exploration and detection. Finally, the GS is used to improve the quality of the optimal solution while accelerating the convergence speed. Figure  1 below shows the SLEGWO flowchart.

Experiments and Results for Benchmark Function
This chapter focuses on the comparison experiments between the proposed algorithm and other algorithms. In this paper, 23 single-mode and multi-mode classical benchmark functions and seven combined benchmark functions of CEC2014 are used to conduct

Experiments and Results for Benchmark Function
This chapter focuses on the comparison experiments between the proposed algorithm and other algorithms. In this paper, 23 single-mode and multi-mode classical benchmark functions and seven combined benchmark functions of CEC2014 are used to conduct unified experiments, expressed in Appendix A Table A3, presenting the benchmark function. There are six classical algorithms: GWO [132], MVO [107], WOA [93], SCA [139], SSA [110], MFO [140], and five improved grey wolf optimization algorithms: IGWO [100], HGWO [119], MEGWO [141], CAGWO [96], and RWGWO [142] that are compared to ensure the fairness of the experiments [143]. All experiments were coded on Matlab2018b. All experiments were performed using the same computer with a 3.40 GHz Intel®Core i7 processor and 16GB RAM. The population size was set to 30, and the maximum number of evaluations was set to 300,000. To make the experiments less affected by random conditions, the Wilcoxon signed-rank test [144] and the Freidman test [145] were also used to check the experimental results.

Benchmark Function Validation
The convergence curves of SLEGWO and other compared algorithms on unimodal, multimodal, and combinatorial functions with the number of evaluations set to 300,000 times are shown in Figure 2. From the results of the convergence curves, it was evident that the convergence is faster, and the convergence accuracy is better than other algorithms on F8, F21, F27, F28, F29, and F30. It is better than other algorithms because the positionbased learning strategy is carried out in the initial stage, which converges toward more high-quality solutions in the search space at the beginning of the population iteration. It is better to avoid falling into the local optimum, so it can be seen that the strategy used in this paper can effectively help converge to the optimal value quickly. Using the foraging mechanism of SMA and LF to keep SLEGWO improving both exploration and detection. Meanwhile, GS is helping to improve the quality of the optimal solution while accelerating the convergence speed. In general, SLEGWO can quickly approach the global optimal solution in the initial solution stage and converge extremely fast compared to other algorithms.

Comparison with Competitive Algorithms
In this part, SLEGWO is compared with 10 competitors on F1-F30 in Table 1, which contains the AVG and STD of the experimental results of SLEGWO and other algorithms. The 10 competitive optimizers are GWO, MVO, WOA, SCA, SSA, MFO, IGWO, HGWO, MEGWO, CAGWO, and RWGWO. Including AVG, STD, Table 2 shows the Mean, Rank, and result of the Wilcoxon sign rank test of experimental results and the results of the Freidman test.    According to the results shown in Table 1, SLEGWO works best. SLEGWO is the smallest on the average of 30 classical functions, which means that SLEGWO outperforms other improved algorithms in most benchmark functions. In addition, Table 2 shows the comparative results of the data analysis in Table 1 using the Wilcoxon signed-rank test and the Freidman test. The Mean indicates the result obtained from the analysis using the Freidman test, and the smaller the value of the Mean, the better the algorithm's performance. Meanwhile, where "+" represents that SLEGWO performs better than others, "-" represents that SLEGWO performs worse than others, and "=" represents that the performance of SLEGWO and others is equal. It can be seen that SLWGWO has the best performance among the 30 benchmark functions. The second ranking is MEGWO; the RWGWO, IGWO, CAGWO, and HGWO have relatively insignificant advantages. It can be concluded that SLEGWO still performs better than the improved algorithms proposed in recent years on most of the benchmark function

SLEGWO Precision Fertilization Model
For the various mineral nutrients required by plants, Nitrogen (N), phosphorus(P), and potassium (K) play an important role in improving crop yields. The soil is both the place for terrestrial plants to take root and a supplier of mineral nutrients, and it bears the heavy burden of providing various nutrients. Therefore, crops N, P, and K are all needed in high amounts in the soil and are usually available in agricultural soils in sufficient quantities for crop uptake. These three nutrients are needed in relatively high amounts and are the most deficient elements in the soil. Therefore, these three nutrients are often supplemented by the artificial fertilizer application for crop uptake and utilization, called the three elements of fertilizer. This chapter describes the process of implementing the SLEGEO-based three-element NPK precision fertilization method, the experimental environment, and the dataset.

SLEGWO and NPK Precision Fertilization Method
The flowchart of SLEGWO for a precise fertilizer model of NPK quadratic equation according to the maize test field in Nong'an country, Jilin Province, China, is shown in Figure 3. Using 3414 experimental schedules to obtain different yields of NPK at different levels, SLEGWO processed the data to obtain the ternary quadratic nonlinear equation. The polynomial coefficients of the equation are negative according to the constraints of the rule of diminishing returns of N, P, and K, and the quadratic term coefficients respond to the fact that an increase in N, P, and K at a certain level can increase the yield, but as the amount of N, P and K input exceeds the demand, it is instead a reduction in yield. The primary term coefficient responds to the parameter constraint of multiple conditions such as yield increase effect, and the equation coefficients of the fertilizer effect function are obtained by fitting using the swarm intelligence optimization method. Then, the maximum value, that is, the maximum yield of the crop, is obtained from the function model of the obtained equation coefficients. Finally, the results of the derived model are evaluated using the coefficient of determination R 2 .
primary term coefficient responds to the parameter constraint of multiple conditions such as yield increase effect, and the equation coefficients of the fertilizer effect function are obtained by fitting using the swarm intelligence optimization method. Then, the maximum value, that is, the maximum yield of the crop, is obtained from the function model of the obtained equation coefficients. Finally, the results of the derived model are evaluated using the coefficient of determination R 2 .

Experimental Environment
The following experiments are conducted under the Windows 10 operating system using MATLAB R2018b, using hardware platform configuration Intel® Core i7 processor 3.40GHz and 16GB RAM. To ensure the fairness of the experiments, all experiments are conducted under the conditions of equal parameter settings, the population number N is 30, the dimension of the objective function is 3, the maximum number of evaluations Max_iteration is set as 50,000 and followed by 30 parallel runs.

Experimental Environment
The following experiments are conducted under the Windows 10 operating system using MATLAB R2018b, using hardware platform configuration Intel®Core i7 processor 3.40 GHz and 16GB RAM. To ensure the fairness of the experiments, all experiments are conducted under the conditions of equal parameter settings, the population number N is 30, the dimension of the objective function is 3, the maximum number of evaluations Max_iteration is set as 50,000 and followed by 30 parallel runs.

Experimental Dataset
This paper used a maize test field in Nong'an County, Jilin Province [146] as the experimental site. The "3414" method was used as a fertilizer effect field experiment, where "3414" refers to 3 factors, 4 levels, and 14 different treatments of N, P, and K. Level 0 is no fertilizer application; level 2 is the optimal fertilizer application. Level 1 1 = level 2 ×0.5, level 3 = level 2 × 1.5 (over-fertilization). The area of each plot was 30 m 2 , no replication, and randomized. The experiments were based on the regional soil nutrient abundance index and the fertilizer nutrient application index to determine the relative optimum fertilizer application. Level 2 for N, P 2 O 5 , K 2 O at 180 kg/hm 2 , 75 kg/hm 2 , and 90 kg/hm 2 respectively. For fitting using the ternary quadratic fertilizer effect model [34], the equations used were: whereŷ 1 is the predicted value of the fertilizer effect function model; b 0 is the yield without fertilizer application, and b 1 , b 2 , b 3 , . . . , b 9 are the effect coefficients. Table 3 below shows the fertilizer use and yield at each plot of the experiment, where x 1 , x 2 , x 3 are the fertilizer application amounts of N, P, and K, and y is the actual yield. Based on the experimental data in Table 3, the experiments were conducted using the "3414" field experiment design and data.

Solution of Equation Coefficients
The fertilizer effect model, an n-dimensional space composed between crop yield y and the individual total nutrient influences x. According to the NPK fertilizer effect function Equation (31): Then, the ternary quadratic polynomial regression equation is change into a nine-element linear regression equation.
The residual function in the least square method is used as the objective function.
where N is 14 and y i is the true yield in the dataset. The residual function's minimum value is obtained to obtain better results using a shorter time. In the experiments of this section, the algorithm containing SLEGWO with the original GWO is applied to find the fertilizer effect function. The upper and lower bounds for the values of each coefficient are set as shown in Table 4. Table 4. The upper and lower limits of each coefficient.
Lower  Table 5 shows the values of each coefficient in the fertilizer effect function using SLEGWO, which has the better competitive performance in finding the minimum of the residual function. Appendix A Table A1 shows the results of the coefficients of the fertilizer equation by SLEGWO by random run 30 times. The coefficient of determination R 2 s used to evaluate the model. The R 2 can be used to test how well the model fits the sample data and takes values between 0 and 1. The closer the value of the R 2 is to 1, the better the model fits. The models with higher coefficients of determination are usually used in real-world problems. The formula for the coefficient of determination R 2 is shown below.
whereŷ i s the predicted value of the fertilizer effect function model; y is the average of the actual yield; and y i is the actual yield. Table 6 shows the values of the R 2 for the two kinds of fertilizer effect function models-SLEGWO and GWO. Table 6. The coefficients of determination of fertilizer effect function models.

Method
SLEGWO GWO R 2 0.9646 0.9645 Table 6 above shows that the fertilizer effect function obtained with SLEGWO is better than GWO.

Yield Estimation
The SLEGWO was used to obtain the maximum fertilizer effect function models yield of the crop. The objective function is the fertilizer effect residual function with dimension 3, corresponding to the fertilizer effect function model of nitrogen, phosphorus, and potassium fertilizer application, respectively. The upper and lower bounds for each dimension are d 1 ∈ [0, 300], d 2 ∈ [0, 120], and d 3 ∈ [0, 120], and the maximum number of iterations of the algorithm is 50,000 with a population size of 30. Table 7 lists the maximum crop yields and the corresponding NPK fertilizer applications according to SLEGWO and the other six algorithm models. Appendix A Table A2 expresses the result of nitrogen, phosphorus, potassium, and yield prediction by SLEGWO 30 times randomly. The above experiments demonstrate the superiority of SLEGWO over other comparative swarm intelligence optimization algorithms in solving the fertilizer effect function model. Swarm intelligence optimization has the advantage of internal constructs encapsulability and better portability than traditional methods and also has some advantages in the maximum yield obtained. It can be seen that GWO works better compared to other algorithms, so it is good to choose GWO as the improved base algorithm for the improved algorithm. Other optimization algorithms have no apparent advantages.

Discussions
The performance of the proposed GWO-based method is not limited to yield estimation, and it can also be tested based on other real-world applications, such as energy storage planning and scheduling [147], service ecosystem [148,149], image editing [150][151][152], epidemic prevention and control [153,154], social recommendation and QoS-aware service composition [155][156][157], active surveillance [158], large scale network analysis [159], spatial analysis [160], crop evapotranspiration prediction [161], control engineering [162,163], pedestrian dead reckoning [164] and evaluation of human lower limb motions [165]. The SLEGWO proposed is based on the improved GWO multi-strategy optimization method and it is applied to solve the fertilizer effect function, which is a new idea based on the traditional precision fertilizer application operation technology. It performs well in equation coefficient solving fitting and maximum yield solving. Exploring the method of combining swarm intelligence optimization algorithm with fertilizer effect function can help provide a new solution for precision agriculture. Since there are many uncertainties in the agricultural production process and the final criteria cannot be fully determined by a particular method, the swarm intelligent optimization method can be used to present multiple possibilities of validation results under multiple random conditions, which is more in line with the real needs than traditional validation methods such as regression.

Conclusions
In this paper, a multi-strategy grey wolf optimization algorithm (SLEGWO) is proposed. Using an opposition-based learning strategy increases the number of early highquality solutions, and the slime foraging and Levy flight strategies effectively avoid falling into local optima and increase the algorithm's ability to balance exploration and detection. The greedy selection strategy speeds up the final convergence to the optimal solution quickly. The SLEGWO algorithm outperforms other competing algorithms on both the classical function set and the CEC2014 function. Meanwhile, the SLEGWO algorithm applied to optimize the model for solving the fertilizer effect function in the maize NPK "3414" program obtained higher accuracy and more yield with good stability, which is an effective method to optimize the model for accurate prediction fertilizer application. It improves the scientific and scalability of the soil test and fertilizer application relationship model. However, since the constraints of the engineering problem are determined by the actual requirements and scenarios, the required constraints will increase when the algorithm is applied in practice. Therefore, the experimental results as well as the actual constraints may lead to deviations in the results but will not affect the application of the method.
In future research work, the SLEGWO algorithm will explore a library of pre-defined fertilization models with multiple model fits to address the scientific fertilization management needs of different regions and different needs. The SLEGWO algorithm will also be effectively used in more areas of agricultural engineering optimization problems, such as supply chain optimization problems, to improve the thematic research on agricultural engineering optimization problems and improve the yield and efficiency of agricultural products to create a cleaner agricultural practice.

Data Availability Statement:
The data involved in this study are all public data, which can be downloaded through public channels.

Acknowledgments:
We acknowledge the comments of the editor and anonymous reviewers that enhanced this research significantly. We also thank Ali Asghar Heidari (https://aliasgharheidari.com) for his help while working on this paper.

Appendix A
a ij x j− p ij