An improved farmland fertility algorithm for many-objective optimization problems

Recent studies on many-objective optimization problems (MaOPs) have tended to employ some promising evolutionary algorithms with excellent convergence accuracy and speed. However, difficulties in scalability upon MaOPs including the selection of leaders, etc., are encountered because the most evolutionary algorithms are proposed for single-objective optimization. To further improve the performance of many-objective evolutionary algorithms in solving MaOPs when the number of the objectives increases, this paper proposes a many-objective optimization algorithm based on the improved Farmland Fertility algorithm (MOIFF). In MOIFF, a novel bio-inspired meta heuristic method proposed in 2018, called Farmland Fertility algorithm (FF), is employed to serve as the optimization strategy. In order to handle MaOPs effectively, FF has been tailored from the following aspects. An individual fitness assessment approach based on cumulative ranking value has been proposed to distinguish the quality of each individual; a novel method based on individual cumulative ranking value to constitute and update the global memory and local memory of each individual is proposed, and a hybrid subspace search and full space search method has been designed to update individuals in the stages of soil optimization and soil fusion. In addition, adaptive environmental selection has been proposed. Finally, MOIFF is compared with four state-of-the art many-objective evolutionary algorithms on many test problems with various characteristics, including the DTLZ and WFG test suites. Experimental results demonstrate that the proposed algorithm has competitive convergence and diversity on MaOPs.


Many-objective processing approaches
Evolution strategy
(1) Farmland Fertility algorithm (FF) as a novel bio-inspired meta heuristic method proposed in 2018, is employed to serve as the optimization strategy of MOIFF. FF performs better than many well-known meta-heuristic methods (including GA, DE, PSO, and ABC) in terms of convergence accuracy, stability, and speed FF. An improved FF algorithm (IFF) has been proposed in 2020 25 . However, FF and IFF are designed for solving complex single-objective optimization problems. In order to handle MaOPs effectively, FF has been tailored from the following aspects in this study: First, we propose a novel individual fitness assessment approach based on the cumulative ranking value to distinguish the advantages and disadvantages of each individual in MaOPs. Second, according to the characteristics of MaOPs, we proposed a novel method based on individual cumulative ranking value to constitute and update the global memory and local memory of each individual, and propose a hybrid search mode combining subspace search and full space search to update individuals at the stages of soil optimization and soil fusion. (2) The experimental results have proved the dual aggregation functions-based environmental selection is a representative and promising many-objective processing approach. However, satisfactory diversity is hard to obtain, since offspring individuals are selected randomly and some holes may appear in Pareto front (PF). We propose a novel adaptive environmental selection method to address these issues. It not only avoids the blindness of random selection but also satisfies the requirements of convergence and diversity of MaOEAs at different stages of algorithm evolution.
Finally, the proposed MOIFF is compared with four state-of-the art many-objective evolutionary algorithms on many test problems with various characteristics, including DTLZ and WFG test suites. Experimental results demonstrate that the proposed algorithm has competitive convergence and diversity on MaOPs.
The rest structure of the paper is organized as follows: "Methods" introduces the relevant theoretical knowledge, including the basic principles of the original FF and a representative many-objective processing approach; "Proposed method" describes the innovation, the principle, procedures, and detailed operations of our proposed MOIFF; "Results and discussion" compares the performance of MOIFF against four state-of-the art manyobjective evolutionary algorithms; "Conclusions" concludes the full text and points out the issues to be studied in future.

Methods
Farmland fertility algorithm. In the real world, farmers apply different fertilizers to farmlands with different soil qualities. By simulating the above behavior, Farmland Fertility algorithm (FF) has been proposed to handle single-objective optimization problems. In FF, the fertilization schemes and soil qualities of farmlands are equivalent to individuals and their fitness values, respectively. The section of the farmland with the worst soil quality selected the best fertilization scheme, and for the other sections of the farmland, the fertilization schemes are randomly selected. Finally, the soil quality of farmland can be effectively improved through continuous improvements in fertilization schemes. The pseudo-code of FF is shown in Algorithm 1.

Proposed method
We proposed a many-objective optimization algorithm based on an improved Farmland Fertility algorithm (MOIFF) to improve the convergence and diversity of MaEAs. FF is employed to serve as the main evolutionary strategy of MOIFF, and dual aggregation functions-based environmental selection with the VOL and KNEE functions is improved as the multi-objective processing approach of MOIFF. The basic procedure of the proposed MOIFF is similar to those of most decomposition-based MaEAs. The pseudo-code of MOIFF is shown in Algorithm 3.   www.nature.com/scientificreports/ value, recorded as s_sort(i). Obviously, the smaller the cumulative sorting value s_sort(i) of an individual, the better the individual. Figure 1 displays an example of handling two-objective DTLZ1 to further illustrate the effectiveness of the above fitness assessment approach based on cumulative ranking value. The final distribution of the solutions shows that the individuals with the first five cumulative ranking values are distributed close to the true Pareto front (PF); the larger the cumulative ranking value of the individual is, the father away from the true PF. The above finding shows that our proposed novel individual fitness assessment approach based on the cumulative ranking value is effective.
Mechanism of updating global memory and local memory. The decomposition-based MaEAs decompose a many-objective optimization problem into a number of single-objective sub-problems by an even set of weight vectors. Each sub-problem is defined by a weight vector, and if weight vectors are adjacent to each other, the optimal solutions of the sub-problems associated with them are also very close. Therefore, with the extension of evolution, the individuals associated with neighborhood weight vectors are similar to some extent. Obviously, each individual and its neighboring individuals form a region naturally; thus, dividing each individual into regions like FF is not necessary. However, a new mechanism must be proposed to compose and update local memory and global memory in MaOPs. Basing from the above analysis, we proposed the following mechanism based on individual cumulative sorting values to compose and update local memory and global memory.
(1) Mechanism of composing and updating global memory At each iteration, M Global individuals with the smallest s_sort value in the current population are selected and stored them in the global memory directly.
(2) Mechanism of composing and updating local memory Unlike FF, each local memory is assigned to each individual rather than each region in MOIFF. That is, the number of local memory is equal to the number of individuals. The local memory of each individual is updated with the help of its neighbor population (shown in Fig. 2) as follows. First, for each individual X i , the weight vector associated with it is determined, and then the T weight vectors closest to the associated weight vector are selected as the neighbor weight vectors. Second, individuals associated with each neighbor weight vector are identified to form the neighbor population of X i . Finally, M Local individuals with the smaller cumulative sorting values in the neighbor population of X i are selected as the local memory of X i directly.
In Fig. 2, dotted lines represent the true PF; w1, w2, and w3 represent the different weight vectors; and x1, x2, x3, and x4 represent four individuals in the current population, where x1 is associated with vector w1, x2 and x3 with weight vector w2, and x4 with weight vector w3. For the individual x1, w2 and w3 are the neighbor weight vectors of the associated weight vector w1 with it. Individuals x2 and x3 are associated with weight vector w2, and x4 is associated with weight vector w3. Individuals x2, x3, and x4 compose the neighbor population of individual x1.
New individual-updating mechanism for soil optimization. To handle MaOPs, we propose a novel individual-updating mechanism for soil optimization in view of the characteristics of decomposition-based MaEAs, described as Eq. (13). The main motivations and ideas are as follows: to facilitate the convergence and promote the diversity of MOIFF, individuals with the poorer convergence should learn from the excellent individuals in the neighborhood, and the others should explore new locations by further communication with other individuals different from itself.  (best) and X i_ML (others) represent the best individual and another random individual besides the best individual in X i 's local memory, respectively; and k is a constant, generally, k = 4. The above individual-updating mechanism proposed for soil optimization has the following advantages. On the one hand, the N/k individuals with poor convergence and diversity learn from the excellent individuals in their local memory randomly, which provides the direction of evolution within the neighborhood and maintains the diversity of the whole population. On the other hand, others learn from other random individual besides themself, which favors exploration of new locations, and can make some weight vectors without associated individuals may be associated with some individuals. Therefore, the diversity of MOIFF will be improved.
A new individual-updating mechanism for soil fusion. As shown in Eq. (5), in the soil fusion stage of FF, all individuals only learn from the best individual in the global memory or local memory, which improves the convergence speed of FF to a certain extent. Unfortunately, it is unsuitable for decomposition-based MaEAs. Therefore, we propose a new individual-updating method for soil fusion, described as Eqs. (14) and (15). The main motivations and ideas are as follows: in the soil fusion stage, only the individual in the global memory and the best individual in each local memory are exploited many times rather than all the individuals, and minor perturbations are made around them. However, considering that the number of these individuals is small, i.e., the evolutionary information is very limited, and each individual has carried some evolutionary information related to the convergence of corresponding sub-problems during iterations. Therefore, each excellent individual should be updated by the minor perturbations around themselves and add some evolutionary information of other individuals.
where, X MG (random) represents a random individual selected from global memory, X i_ML (best) represents the best individual stored in X i 's local memory, CR is the crossover probability, and generally, CR ∈ [0.6, 0.8].
Adaptive environmental selection. As described in Algorithm 2, dual aggregation functions-based environmental selection in DECAL consists of the two following steps: Step 1, for each weight vector, select the individual with the best VOL function value and the individual with the best KNEE function value from all the associated individuals with it respectively; Step 2, for each weight vector, randomly select one from the individuals identified in step 1 as an individual of the offspring population. Obviously, some weight vectors may not be associated with any individual. Therefore, the number of individuals in the offspring population may be smaller than the initial size of individuals, and satisfactory diversity is hard to obtain because some holes may appear in PF. In addition, the random selection in step 2 has a certain blindness, which does not satisfy the requirements of convergence and diversity of many-objective optimization algorithms at different stages of algorithm evolution.
To address these above issues, this article proposes a new adaptive environment selection, as shown in Algorithm 4. www.nature.com/scientificreports/ As shown in Algorithm 4, the adaptive environment selection proposed in this paper has the following advantages: First, when the size of the offspring population is less than the initial setting number of weight vectors, we will select some individuals into the offspring population, and vice versa. Finally, the size of the offspring population obtained by adaptive environment selection is equal to the initial setting number of weight vectors, which further guarantees that the PF is even distributed; Second, at the begging of iterations, the offspring population obtained by adaptive environment selection contains many individuals with better VOL fitness. At the end of iterations, the offspring population contains many individuals with better KNEE fitness. It not only avoids the www.nature.com/scientificreports/ blindness of random selection but also satisfies the requirements of convergence and diversity of many-objective optimization algorithms at different stages of algorithm evolution.
Hybrid subspace-full space search mode. For MaOPs, at the begging of iteration, the set of all the solutions is commonly far from PS. If each individual performs a complete search in the D-dimensional search space, called the full-space search mode, then the individual can obtain some promising evolutionary directions from such huge searching space. As the iteration progresses, the individuals are gradually close to the true PS. At this moment, some individuals have achieved the global optimum in most dimensions, whereas only a few dimensions are far from the global optimum. In this case, if individuals still adopt the full-space search mode of FF to evolve, the good dimensions may change greatly, which may cause the population to deviate from the excellent evolution directions, and the convergence of algorithm may be slow or fail to obtain the true PF. The current individuals are better suited for small-scale searches. Thus, the above-mentioned issues can be easily addressed, if individuals do not adopt the full-space search mode but only search on a dimension, called subspace search. In summary, the full-space search mode refers to all dimensions of an individual being updated in accordance with the established search strategy in the stages of soil optimization and soil fusion, and the subspace search mode means that during the stages of soil optimization and soil fusion, only one dimension of an individual is updated in accordance with the established search strategy, while the other dimensions are unchanged. The subspace search mode updates only one dimension of individual at one iteration. Thus, if individuals adopt the subspace search mode to update themselves for many iterations, the convergence of the algorithm may be slow down instead. Therefore, when the subspace search mode does not obtain a better PF, the full space search mode should be used again for further search. That is, the full space search mode and subspace search mode should alternate. Basing from the above ideas, we propose a hybrid subspace-full space search mode. The conversion condition between the subspace and full-space search mode are as follows.
Conversion condition between subspace and full-space search modes. As shown in "Novel individual fitness assessment approach based on cumulative ranking value", for each weight vector, the associated sub-problem is solved better, as the smallest s_sort associated with it (denoted as pbest) is smaller. As the iteration progresses, the change in pbest can be used to judge whether the evolution slow down, fail to update the individual into a   Table 4. Setting of parameters in each algorithms. www.nature.com/scientificreports/ better one, and fall into a local optimum. Considering the above, we take the change in pbest as the conversion condition between the subspace search mode and full-space search mode as follows: First, initialize parameters including c = 0, c1, and c2, where c1 is used to determine whether pbest has changed before and after the iteration, and c2 is used to determine the number of iterations that pbest remains unchanged. Second, for each weight vector, calculate the Euclidean distance Dis(i) between pbest i obtained at this iteration and the last iteration and then calculate the mean (denoted as average_Dis) of all Dis(i), as shown in Eq. (16). If average_Dis is less than c1, then c = c + 1; otherwise, c = 0. When c is equal to c2, the search mode is converted, i.e., the full space search mode is converted to the sub-space search mode, and vice versa.

Scientific Reports
where, pbest d i (j) represents the j-th dimension of the smallest s_sort associated with the i-th weight vector (W i ) at the d-th iteration.
Description of subspace search mode. When the subspace search mode is adopted, if the updated dimension is selected in accordance with the order of dimensions, it can better guarantee that each dimension can search more finely during the limited iterations compared with the updated dimension selected randomly. Therefore, in our proposed subspace search mode, when converting to the subspace search mode, the updated dimension of each individual is selected in accordance with the order of dimensions at each iteration, which is similar to the method proposed in 25 , detailed as follows. At the last iteration, dimension d is selected to update, and at this iteration, dimension d + 1 is selected to update. When all the dimensions have been updated by adopting the subspace search mode, unlike the method proposed in 25 , even if they fail to meet the conversion condition between the subspace and full-space search modes, all individuals adopt the full-space search mode in the next iteration.
Suppose that the flag-th dimension needs to be updated, the subspace search modes for soil optimization and soil fusion are as follows.
The subspace search mode for soil optimization and soil fusion is shown in Eq. (17).
(18) X inew,flag = X MG,flag (random) × randn, Q > rand X i_ML,flag (best) × randn, else.  26  Algorithm and parameter. In order to verify the validity of MOIFF proposed in this paper, we consider four state-of-the-art many-objective evolutionary algorithms, including NSGA III 28 , MOEA/D 23 , RVEA 9 , and ARMOEA 29 . To ensure the fairness of the comparison, the population size and the termination condition of each algorithm are consistent on the same test instance. The population size N is equal to the size of the reference vectors, which are used for different numbers of objectives and are summarized in Table 2, where H 1 and H 2 are the number of divisions of the boundary layer and inside layer, respectively. The termination condition of each algorithm is the maximum number of fitness estimations (MFE), summarized in Table 3. In addition, other parameters used in each algorithm are summarized from the original literature, as shown in Table 4.

Results on DTLZ test suite.
In this section, we conduct three experiments to verify the validity of MOIFF proposed in this paper on the DTLZ test suite. The first one is verification of the effectiveness of MOIFF in convergence accuracy. The second one is verification of the effectiveness of MOIFF in convergence speed. The third one is verification of the effectiveness of MOIFF in convergence stability.
(1) Verification of the effectiveness of MOIFF in convergence accuracy In our empirical studies, IGD and HV are employed to evaluate the performance of each algorithm simultaneously. Each algorithm runs 30 times independently on each test instance to avoid the unfavorable effect of the algorithm evaluation caused by the randomness of a single operation. Tables 5 and 6 show the average and standard deviation of the IGD and HV values over 30 independent runs for the five compared MaOEAs, respectively, where the best average among the five compared MaOEAs is highlighted in bold. In addition, to test the differences for statistical significance, the Wilcoxon rank sum test with a 5% significance level is performed between MOIFF and each of the compared algorithms over each test instance. Symbols "+", "−" and "=" indicate that the compared algorithm performs significantly better than, worse than, and equivalent to MOIFF in the corresponding column, respectively. The Friedman rank-sum test is performed on the data of Tables 5  and 6 to analyze the overall average performance of these above algorithms. The results are shown in Table 7,    www.nature.com/scientificreports/ where "avg.rank" represents the average rank of each algorithm, "rank" is the overall rank of five algorithms in average performance. Basing from the IGD results of DTLZ test instances shown in Table 5, we can find that the proposed MOIFF shows the best overall performance on DTLZ2 and DTLZ4 problems, compared with the four other MaOEAs. For the DTLZ1 problem, MOEA/D obtains the smallest IGD value on the fifteen-objective test instance, RVEA performs best on the three-objective test instance, while MOIFF works best on the five-, eight-, and ten-objective test instances. For DTLZ3, MOIFF has obvious advantages over the four other algorithms on the remaining test instances, and MOIFF is slightly worse than the four other algorithms on the three-objective test instance. For DTLZ5, NSGA-III achieves the best results on the three-objective test instance, whereas MOEA/D works best on the five-, eight-, ten-, and fifteen-objective test instances. The overall performance of MOIFF is significantly      Table 6. MOIFF can achieve the best results on all DTLZ instances except for ten-and fifteen-objective DTLZ instances. According to Table 7, we can see that, compared with the competitors, the overall performance of MOIFF over all test instances in terms of HV is the best.
(2) Verification of the effectiveness of MOIFF in convergence speed To compare the complexity of each algorithm more intuitively, Table 8 records the random respond time of each algorithm on each test instance, where the parameters of each algorithm are set as above.
Seen from the Table 8, we can find that RVEA needs the shortest run time on each test instance, the respond time of ARMOEA is longest on each test instance, and MOIFF needs the longer run time than NSGA-III and MOEAD on each test instance.
In order to intuitively compare the convergence process of each algorithm, Figs. 3, 4, 5 and 6 shows the iterative process curves of each algorithm, where parameters of each algorithm are set as above. We can find that MOIFF is excellent in the convergence speed.
(3) Verification of the effectiveness of MOIFF in convergence stability In order to compare the stability of each algorithm, we show the success rate of each algorithm in Table 9, where the success rate means the times of each algorithm reaching the preset IGD convergence accuracy in 30 independent experiments. Form Table 8, we can see that, MOIFF obtains the optimal success rate of 22 out of 35 test instances, which indicates that MOIFF is excellent in convergence stability.
To further visually compare the stability of each algorithm, Figs. 7 and 8 show the box plots of statistical values of IGD and HV obtained by the above five algorithms over 30 runs. Limited to the length of the paper, we only select the IGD values obtained by the five algorithms on the DTLZ2 and the HV values obtained by the five algorithms on the DTLZ3 to draw the box plots. We can see that MOIFF is the most stable compare to the other 4 algorithms for HV. For five-and ten-objective DTLZ3, the stability of IGD values obtained by IFF is slightly inferior to the other algorithms, but its IGD value is significantly best.

Results on WFG test suite. Comparison results of MOIFF with the four other MaOEAs in terms of IGD
values on the WFG test suite are listed in Table 10. MOIFF is significantly outperformed by the four other MOEAs on the WFG4, WFG5, WFG6, WFG7, WFG8 and WFG9 instances in terms of IGD. For WFG1, RVEA obtains the smallest IGD value on the fifteen-objective test instance, ARMOEA performs best on the eightobjective test instance, and MOIFF works best on the remaining test instances. For WFG2, RVEA performs best on the five-objective test instance, and MOIFF works best on the remaining test instances. For WFG3, except for the three-objective instance, MOIFF can achieve the best performance on the other test instances. MOEA/D could obtain the best results on any test instance.
The HV results of the WFG test instances are listed in Table 11. NSGA-III can obtain the best HV values on the three-, five-, and fifteen-objective WFG1 instances; the five-objective WFG2 instance; and the eight-objective WFG3 instance. MOEA/D only works best on the ten-objective WFG3 instance. RVEA performs best on the fifteen-objective WFG5 instance, and the ten-and fifteen-objective WFG9 instances. ARMOEA works best on the eight-and ten-objective WFG1 instances, eight-, ten-, and fifteen-objective WFG2 instances, fifteen-objective      www.nature.com/scientificreports/ WFG4 instance; the fifteen-objective WFG6 instance; the fifteen-objective WFG7 instance; and the eight-, ten-, and fifteen-objective WFG8 instances. Among the 45 test instances, MOIFF obtains the smallest HV values on 24 test instances. Figure 9 shows the parallel coordinates of the final non-dominated solutions obtained by these five algorithms on the five-objective WFG8 test instance. These plots clearly demonstrate that the PF obtained by MOIFF is close to the true PF and maintains a good distribution.

Conclusions
A novel algorithm named MOIFF is proposed in this paper for handling MaOPs to improve the comprehensive performance in terms of convergence and diversity. FF with excellent convergence performance is employed to serve as the optimization strategy of MOIFF. In order to handle MaOPs effectively, FF has been tailored from the following aspects in this paper. First, to distinguish the quality of each individual in MaOPs, we propose a novel individual fitness assessment approach based on cumulative ranking value. Second, considering the characteristics of MaOPs, we propose a novel method based on individual cumulative ranking value to constitute and update the global memory and local memory of each individual and a hybrid search mode combining subspace search and full space search to update individuals at the stages of soil optimization and soil fusion. In addition, we improve the dual aggregation function-based environmental selection. Finally, the results on the DTLZ and WFG test suites show that MOIFF has excellent convergence and diversity compared with four state-of-the art many-objective evolutionary algorithms.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.