# Smart Root Search (SRS): A Novel Nature-Inspired Search Algorithm

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.1.1. Plant Senses and Reactions

**Hydrotropism:**Plant survival depends on the ability of roots to find water in soil. Hence, plant root growth curves correspond to the moisture gradient (higher water potential) called hydrotropism [36,37].

**Nutrient tropism:**Nitrates and phosphates are considered the most important elements for plant growth [41]. Important developmental processes, such as lateral root (LR) and hair root (HR) formation as well as primary root (PR) elongation (length), are to a great extent sensitive to the nutrient concentration changes.

**Memory:**Although plants do not have a neural network, many studies show that they can recall some conditions, which suggests that plants exhibit memory. Ref. [47] addressed traumatic plant memories, related facts, and potential mechanisms. Stress factors make the plant impervious to subsequent exposures. This stress-related feature indicates that every plant has a memory capacity. In addition, plants also possess “stress memory” and “drought memory”. Surprisingly, the proportion of live biomass after a late drought is higher in plants that were exposed to drought earlier in their growing season contrasted with single-stressed plants [48].

**Electrical Impulse:**Plants also use a message-passing system [49]. Research on plants has shown that electrical communication plays a significant role in root-to-shoot contact in the plants under water stress. Furthermore, Ref. [50] showed that when one organ of a seedling is stimulated (i.e., the root region), a characteristic response (electrical stimulus) is produced and would be recorded upward in another organ from the stimulating area.

#### 1.1.2. Plant-Imitating Methods

## 2. Smart Root Search Algorithm

#### 2.1. Intelligence of Plant Root Growth

- (a)
- Roots grow in the direction of the nutrient sources in the soil.
- (b)
- Root growth accelerates and generates new branches and HRs depending on the nutrient concentration of the soil.
- (c)
- Each part of a root uses electrical impulses to send information about its current situation to the other parts.
- (d)
- Plants can memorize and respond to information.
- (e)
- Water stress states cause roots to dry up.

#### 2.2. SRS Algorithm

- SRS divides the search space into several subspaces and distributes the first generation of roots equally among them. This helps SRS to apply different search policies to different parts of the search space. Similar algorithms do not provide such functions in their standard versions.
- SRS-generated roots are immature upon germination but become mature after a few iterations. Thus, the algorithm can apply different search policies by using the same roots based on their age. In contrast, other algorithms use fixed exploratory policies during their execution.
- SRS utilizes an embedded local search mechanism applied by a group of roots called HRs.
- SRS utilizes a dynamic population size. This gives the SRS the capability to decrease the number of solutions in non-promising subspaces and to increase the number of solutions in promising areas.

#### 2.2.1. Parameter Initialization

#### 2.2.2. Dividing the Search Space

#### 2.2.3. Initialization of First Generation

#### 2.2.4. Evaluation of Roots

#### 2.2.5. Root Sorting and Ranking

#### 2.2.6. Root Growth

#### 2.2.6.1. Types of Roots

- Mature Roots

- II.
- Immature Roots

- III.
- HR

#### 2.2.6.2. Growth of Mature Roots

**Definition 1.**

- Velocity of Mature Roots

- II.
- Direction of Mature Roots

#### 2.2.6.3. Root Growth of Immature Roots

#### 2.2.6.4. Immature to Mature Transformation Mechanism

#### 2.2.7. Root Branching

#### 2.2.8. Root Drouth

**Definition 2.**

#### 2.2.9. HRG

- A set of the best mature roots (let us call them m) of every subspace will be picked by “Roulette Wheel Selection via Stochastic Acceptance” (RWSSA) [60] for generating plenty of HRs in their neighborhood regions.
- A random number l in the range (1, a) will be generated, where a is the neighborhood radius defined by the user based on the problem characteristics.
- For every selected mature root i in step 1, RWSSA will be used to generate a random number k in the range (1, D).
- k of D dimensions of root i will be selected randomly.
- For every selected dimension d in 3.a, two new roots, $HR{1}_{i}^{d}=\left({x}_{i}^{1},{x}_{i}^{2},\dots ,{x}_{i}^{d}+l,\dots ,{x}_{i}^{D}\right)$ and $HR{2}_{i}^{d}=\left({x}_{i}^{1},{x}_{i}^{2},\dots ,{x}_{i}^{d}-l,\dots ,{x}_{i}^{D}\right),$ will be generated, and their nitrate concentration needs will be calculated.

- If $HR{j}_{i}^{d}$ is one of the generated HRs with the greatest nitrate concentration value among the other HRs and their parents, then the parent will grow to reach the location of $HR{j}_{i}^{d}$.
- The generated HRs are no longer required and will dry immediately.

#### 2.2.10. Termination Criteria

#### 2.2.11. On the Convergence of the SRS

## 3. Experimental Test, Results and Discussion

#### 3.1. Experimental Tests: SRS vs. GA, PSO, Independent Component Analysis (ICA) and Differential Evolution (DE)

#### 3.1.1. Settings

#### 3.1.2. Benchmark Functions

#### 3.2. Obtained Results and Discussion

**Definition 3.**

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. An Example of Root Growth

**Table A1.**SRS Parameters and current properties of the investigated root in the root growth example.

SRS Parameters | The Investigated Root Current Properties | ||||
---|---|---|---|---|---|

Max velocity | NumCurrRoot | Dimension | ${glb}_{\_}{rank}_{i}$ | Current location ${x}_{i}\left(t\right)$ | Best closest ${x}_{best\_closest\text{}i}\left(t\right)$ |

10 | 50 | 6 | 36 | $\left(83,\text{}15,\text{}7,\text{}43,\text{}79,\text{}15\right)$ | $\left(35,\text{}93,\text{}66,\text{}17,\text{}21,\text{}4\right)$ |

## Appendix B. On the Time Complexity of the SRS

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**Figure 2.**The basic structure of an SRS root. The first row represents the structure while the second row explains what should be assigned to every element of the structure.

**Figure 3.**Types of roots [57].

**Figure 4.**An example of NC-based grouping affects parent root branching when locating in a (

**a**) promising part of search space, and (

**b**) unpromising part of search space.

**Figure 7.**Converges of SRS toward global optima; (

**a**) shows how search space gets divided into four subspaces; (

**b**) shows process of generating random solutions in subspaces and choosing optimal solution of each of them; (

**c**) demonstrates how number of solutions gets less in subspaces that could not find proper solutions while it gets more in promising subspaces; (

**d**,

**e**) represent the worst subspaces having no solution contrasted with the best subspace determined gradually; (

**f**) SRS found the global optima in the best subspace.

**Figure 12.**Dendrogram diagram plotted using Agglomerative hierarchical clustering results of comparative algorithms.

**Table 1.**Intelligent and optimization-averse behaviors of artificial root foraging model (ARFO) algorithm.

Optimization-Adversative Behaviors | Common Intelligent Behaviors |
---|---|

Increasing root length in promising main root areas | Using short-step movements/changes in promising areas to identify additional search locations |

Decreasing root length in non-promising LR areas | Using large-step movements/changes in non-promising areas to escape non-promising areas |

LR are exploited in non-promising areas | Exploitation occurs in promising areas |

Applying short-length branches causes very fast convergence | Avoiding fast convergence |

No chance for enhancing bad solutions | Bad solutions have more chances for enhancement |

Botany Terms | Optimization Terms |
---|---|

Soil | Search Space |

Plant Root Set | Solutions’ Vector |

Root | Solution |

Nitrate Concentration | Objective Function |

Location of the Highest Nitrate Concentration | Optimal Solution |

Growth Step | Iteration |

Hair Roots Germination | Local Search Operator |

Root Growth | Solution Movement |

Root Drouth | Solution Elimination |

Root Growth Speed | Velocity of Movement |

Branching | Solution Reproduction |

Immature Root | Limited-move Solution |

Growth Direction | Movement Coefficient Set |

Nutrients Concentration | Effects | ||
---|---|---|---|

Root Growth Speed | Hair Roots Density | Branching Density | |

High Nitrate | Nothing | Nothing | |

Low Phosphate | |||

Low Nitrate | Nothing | Nothing | |

High Phosphate |

Nutrients Concentration | Effects | ||
---|---|---|---|

Root Growth Speed | Hair Roots Density | Branching Density | |

High Nitrate & Low Phosphate | |||

Low Nitrate & High Phosphate |

Coefficient | Ratio of Group Roots to All | Score | |
---|---|---|---|

Group 1 | 1 | 1 × 6.66 = 6.66% | 5 |

Group 2 | 2 | 2 × 6. 66 = 13.33% | 4 |

Group 3 | 4 | 4 × 6. 66 = 26.66% | 3 |

Group 4 | 8 | 8 × 6. 66 = 53.33% | 2 |

Ns | NumMinRoot (Population Size) | NumMaxRoot | Vmax | MRA | Max_Penalty | Encourage_Value | Penalty_Rate | Mature_Age | |
---|---|---|---|---|---|---|---|---|---|

Unimodal Functions | 8 | 125 | 2000 | 0.33 * MDV | 20 | 10 | 2 | 0.75 | 4 |

Multimodal Functions | 8 | 125 | 2000 | 0.33 * MDV | 20 | 10 | 2 | 0.15, 0.25 | 15 |

No. | Function | D | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | Formulation |
---|---|---|---|---|---|

${f}_{1}$ | Cigar | 30 | [−100, 100] | 0 | ${x}_{1}^{2}+{10}^{6}{\displaystyle {\displaystyle \sum}_{i=2}^{D}}{x}_{i}^{2}$ |

${f}_{2}$ | Dixon-Price | 30 | [−10, 10] | 0 | ${\left({x}_{1}-1\right)}^{2}+{\displaystyle {\displaystyle \sum}_{i=2}^{D}}i{\left(2{x}_{i}^{2}-{x}_{i-1}\right)}^{2}$ |

${f}_{3}$ | Quartic | 30 | [−1.28, 1.28] | 0 | ${\displaystyle \sum}_{i=1}^{D}}i{x}_{i}^{4}+random\left[0,1\right)$ |

${f}_{4}$ | Rosenbrock | 30 | [−5, 5] | 0 | ${\displaystyle \sum}_{i=1}^{D-1}}100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2$ |

${f}_{5}$ | Schwefel 1.2 | 30 | [−100, 100] | 0 | ${\displaystyle \sum}_{i=1}^{D}}{\left({\displaystyle {\displaystyle \sum}_{j=1}^{i}}{x}_{j}\right)}^{2$ |

${f}_{6}$ | Schwefel 2.22 | 30 | [−100, 100] | 0 | ${\displaystyle \sum}_{i=1}^{D}}\left|{x}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{D}}\left|{x}_{i}\right|$ |

${f}_{7}$ | Schwefel 2.23 | 30 | [−10, 10] | 0 | ${\displaystyle \sum}_{i=1}^{D}}{x}_{i}^{10$ |

${f}_{8}$ | Sphere | 30 | [−100, 100] | 0 | ${\displaystyle \sum}_{i=1}^{D}}{x}_{i}^{2$ |

${f}_{9}$ | Step | 30 | [−100, 100] | 0 | ${\displaystyle \sum}_{i=1}^{D}}{\left(\lfloor {x}_{i}+0.5\rfloor \right)}^{2$ |

${f}_{10}$ | SumSquares | 30 | [−10, 10] | 0 | ${\displaystyle \sum}_{i=1}^{D}}i{x}_{i}^{2$ |

${f}_{11}$ | Trid 10 | 10 | $\left[-{D}^{2},{D}^{2}\right]$ | 0 | $210+{\displaystyle {\displaystyle \sum}_{i=1}^{D}}{\left({x}_{i}-1\right)}^{2}-{\displaystyle {\displaystyle \sum}_{i=2}^{D}}{x}_{i}{x}_{i-1}$ |

${f}_{12}$ | Zakharov | 10 | [−5, 10] | 0 | ${\displaystyle \sum}_{i=1}^{D}}{x}_{i}^{2}+{\left(0.5{\displaystyle {\displaystyle \sum}_{i=1}^{D}}i{x}_{i}\right)}^{2}+{\left(0.5{\displaystyle {\displaystyle \sum}_{i=1}^{D}}i{x}_{i}\right)}^{4$ |

No. | Function | D | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | Formulation |
---|---|---|---|---|---|

${f}_{13}$ | Ackley | 30 | [−32, 32] | 0 | $20+e-20{e}^{\left(\sqrt[-0.2]{\frac{1}{D}{{\displaystyle \sum}}_{i=1}^{D}{x}_{i}^{2}}\right)}-{e}^{\frac{1}{D}{{\displaystyle \sum}}_{i=1}^{D}\mathrm{cos}\left(2\pi {x}_{i}\right)}$ |

${f}_{14}$ | CosineMixture | 30 | [−500, 500] | 0 | $\mathrm{D}\times 1.643788341-0.1{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{D}}}\mathrm{cos}\left(5{\mathsf{\pi}\mathrm{x}}_{\mathrm{i}}\right)-{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{D}}}{\mathrm{x}}_{\mathrm{i}}^{2}$ |

${f}_{15}$ | Griewank | 30 | [−600, 600] | 0 | $\frac{1}{4000}\left({\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{D}}}{\mathrm{x}}_{\mathrm{i}}^{2}\right)-\left({\displaystyle {\displaystyle \prod}_{\mathrm{i}=1}^{\mathrm{D}}}\mathrm{cos}\left(\frac{{\mathrm{x}}_{\mathrm{i}}}{\sqrt{\mathrm{i}}}\right)\right)+1$ |

${f}_{16}$ | Perm | 4 | [−D, D] | 0 | ${\displaystyle \sum}_{i=1}^{D}}{\left({\displaystyle {\displaystyle \sum}_{j=1}^{D}}\left({j}^{i}+\beta \right)\left({\left(\frac{{x}_{j}}{j}\right)}^{i}-1\right)\right)}^{2},\text{}\left(\beta 0\right)$ |

${f}_{17}$ | Qing | 30 | [−10, 10] | 0 | ${\displaystyle \sum}_{i=1}^{D}}{\left({x}_{i}^{2}-i\right)}^{2$ |

${f}_{18}$ | Quintic | 30 | [−1, 1] | 0 | ${\displaystyle \sum}_{i=1}^{D}}\left|{x}_{i}^{5}-3{x}_{i}^{4}+4{x}_{i}^{3}+2{x}_{i}^{2}-10{x}_{i}-4\right|$ |

${f}_{19}$ | Rastrigin | 30 | [−5.12, 5.12] | 0 | ${\displaystyle \sum}_{i=1}^{D}}\left({x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10D\right)$ |

${f}_{20}$ | Schwefel | 30 | [−500, 500] | 0 | $D\times 418.9829+{\displaystyle {\displaystyle \sum}_{i=1}^{D}}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)$ |

${f}_{21}$ | Schwefel 2.25 | 30 | [0, 10] | 0 | ${\displaystyle \sum}_{i=2}^{D}}\left[{\left({x}_{i}-1\right)}^{2}+{\left({x}_{1}-{x}_{i}^{2}\right)}^{2}\right]$ |

${f}_{22}$ | Styblinski_Tang | 30 | [−5, 5] | 0 | $D\times 39.16599+\frac{1}{2}{\displaystyle {\displaystyle \sum}_{i=1}^{D}}\left({x}_{i}^{4}-16{x}_{i}^{2}+5{x}_{i}\right)$ |

${f}_{23}$ | Xin-She Yang 02 | 30 | [−2π, 2π] | 0 | $\frac{{{\displaystyle \sum}}_{i=1}^{D}\left|{x}_{i}\right|}{exp\left({{\displaystyle \sum}}_{i=1}^{D}\mathrm{sin}\left({x}_{i}^{2}\right)\right)}$ |

**Table 9.**Comparison of SRS with GA, PSO, ICA and DE on Unimodal test functions. All results have been averaged over 40 runs.

No. | Function | GA | PSO | ICA | DE | SRS | |
---|---|---|---|---|---|---|---|

${\mathit{f}}_{\mathit{1}}$ | Cigar | Mean | 6.62 × 10^{8} | 1.12 × 10^{9} | 8.44 × 10^{8} | 9.33 × 10^{8} | 1.11 × 10^{7} |

Std | 2.51 × 10^{8} | 1.5 × 10^{8} | 4.93 × 10^{8} | 1.37 × 10^{9} | 2.62 × 10^{6} | ||

Rank | 2 | 5 | 3 | 4 | 1 | ||

${\mathit{f}}_{\mathit{2}}$ | Dixon-Price | Mean | 677.43 | 276.84 | 2330.86 | 900.74 | 1.73 |

Std | 420.64 | 86.26 | 2929.02 | 1330.92 | 0.63 | ||

Rank | 3 | 2 | 5 | 4 | 1 | ||

${\mathit{f}}_{\mathit{3}}$ | Quartic | Mean | 18.88 | 11.94 | 11.03 | 9.53 | 14.003 |

Std | 2.91 | 0.78 | 0.64 | 0.29 | 1.4 | ||

Rank | 5 | 3 | 2 | 1 | 4 | ||

${\mathit{f}}_{\mathit{4}}$ | Rosenbrock | Mean | 528.93 | 437.82 | 545.02 | 258.45 | 25.55 |

Std | 315.32 | 106.66 | 418.31 | 139.21 | 2.85 | ||

Rank | 4 | 3 | 5 | 2 | 1 | ||

${\mathit{f}}_{\mathit{5}}$ | Schwefel 1.2 | Mean | 37,415.48 | 7301.03 | 1652.85 | 1109.23 | 883.58 |

Std | 9582.99 | 1350.01 | 520.67 | 730.18 | 204.17 | ||

Rank | 5 | 4 | 3 | 2 | 1 | ||

${\mathit{f}}_{\mathit{6}}$ | Schwefel 2.22 | Mean | 94.55 | 236.15 | 60.625 | 85 | 21 |

Std | 14.01 | 15.89 | 22.01 | 56.19 | 3.96 | ||

Rank | 4 | 5 | 2 | 3 | 1 | ||

${\mathit{f}}_{\mathit{7}}$ | Schwefel 2.23 | Mean | 25.05 | 7.48 | 5127.09 | 104,269.3 | 0 |

Std | 46.41 | 4.46 | 11,643.33 | 542,120.6 | 0 | ||

Rank | 3 | 2 | 4 | 5 | 1 | ||

${\mathit{f}}_{\mathit{8}}$ | Sphere | Mean | 885.05 | 3195.13 | 800.43 | 727.2 | 11.75 |

Std | 287.48 | 525.35 | 395.63 | 1287.79 | 2.8 | ||

Rank | 4 | 5 | 3 | 2 | 1 | ||

${\mathit{f}}_{\mathit{9}}$ | Step | Mean | 811.78 | 3227.32 | 921.43 | 760.93 | 11.78 |

Std | 315.9 | 399.79 | 569.69 | 1298.97 | 2.96 | ||

Rank | 3 | 5 | 4 | 2 | 1 | ||

${\mathit{f}}_{\mathit{10}}$ | SumSquares | Mean | 104.36 | 74.59 | 125.65 | 107.42 | 0.52 |

Std | 40.67 | 16.14 | 43.7 | 136 | 0.31 | ||

Rank | 3 | 2 | 5 | 4 | 1 | ||

${\mathit{f}}_{\mathit{11}}$ | Trid 10 | Mean | 454.98 | 22.78 | 133.88 | 95.05 | 2.88 |

Std | 444.18 | 13.52 | 68.36 | 215.89 | 8.8 | ||

Rank | 5 | 2 | 4 | 3 | 1 | ||

${\mathit{f}}_{\mathit{12}}$ | Zakharov | Mean | 58.9 | 0.032 | 2.31 | 1.28 | 0.003 |

Std | 27.98 | 0.01 | 2.04 | 3.31 | 0.002 | ||

Rank | 5 | 2 | 4 | 3 | 1 |

**Table 10.**Comparison of SRS with GA, PSO, ICA and DE on Multimodal test functions. All results have been averaged over 40 runs.

No. | Function | GA | PSO | ICA | DE | SRS | |
---|---|---|---|---|---|---|---|

${\mathit{f}}_{\mathit{13}}$ | Ackley | Mean | 7.88 | 5.64 | 8.67 | 10.43 | 0.64 |

Std | 0.79 | 0.36 | 1.21 | 1.47 | 0.16 | ||

Rank | 3 | 2 | 4 | 5 | 1 | ||

${\mathit{f}}_{\mathit{14}}$ | CosineMixture | Mean | 2.993 | 2.999 | 3.296 | 2.978 | 2.975 |

Std | 0.019 | 0.016 | 0.063 | 0.015 | 0 | ||

Rank | 3 | 4 | 5 | 2 | 1 | ||

${\mathit{f}}_{\mathit{15}}$ | Griewank | Mean | 6.43 | 10.61 | 13.38 | 6.7 | 0.99 |

Std | 1.65 | 1.83 | 6.27 | 4.16 | 0.05 | ||

Rank | 2 | 4 | 5 | 3 | 1 | ||

${\mathit{f}}_{\mathit{16}}$ | Perm | Mean | 4.941 | 0.057 | 0.286 | 0.292 | 0.012 |

Std | 10.35 | 0.075 | 0.19 | 1.16 | 0.017 | ||

Rank | 5 | 2 | 3 | 4 | 1 | ||

${\mathit{f}}_{\mathit{17}}$ | Qing | Mean | 3.29 × 10^{7} | 8.71 × 10^{7} | 3.02 × 10^{8} | 2.38 × 10^{8} | 2.67 × 10^{3} |

Std | 2.91 × 10^{7} | 3.50 × 10^{7} | 2.83 × 10^{8} | 4.47 × 10^{8} | 4.90 × 10^{2} | ||

Rank | 2 | 3 | 5 | 4 | 1 | ||

${\mathit{f}}_{\mathit{18}}$ | Quintic | Mean | 50.4 | 86.05 | 149.16 | 63.38 | 15.91 |

Std | 9.33 | 8.69 | 84.74 | 63.89 | 5.02 | ||

Rank | 2 | 4 | 5 | 3 | 1 | ||

${\mathit{f}}_{\mathit{19}}$ | Rastrigin | Mean | 46.04 | 132.34 | 56.02 | 60.66 | 39.92 |

Std | 9.03 | 9.21 | 13.7 | 16.68 | 6.69 | ||

Rank | 2 | 5 | 3 | 4 | 1 | ||

${\mathit{f}}_{\mathit{20}}$ | Schwefel | Mean | 1499.5 | 4553.8 | 7530.7 | 5204.3 | 4314.7 |

Std | 336.51 | 339.53 | 595.61 | 587.37 | 848.04 | ||

Rank | 1 | 3 | 5 | 4 | 2 | ||

${\mathit{f}}_{\mathit{21}}$ | Schwefel 2.25 | Mean | 1745.65 | 98,272.93 | 25,246.87 | 7037.74 | 754.08 |

Std | 637.03 | 17,093.3 | 7069.6 | 3546.7 | 523.5 | ||

Rank | 2 | 5 | 4 | 3 | 1 | ||

${\mathit{f}}_{\mathit{22}}$ | Styblinski-Tang | Mean | 24.49 | 63.27 | 408.48 | 220.37 | 137.1 |

Std | 5.437 | 14.68 | 37.13 | 38.46 | 26.71 | ||

Rank | 1 | 2 | 5 | 4 | 3 | ||

${\mathit{f}}_{\mathit{23}}$ | Xin-She Yang 02 | Mean | 9.15 × 10^{−12} | 8.88 × 10^{−10} | 7.41 × 10^{−10} | 2.67 × 10^{−11} | 4.33 × 10^{−12} |

Std | 1.64 × 10^{−12} | 5.91 × 10^{−10} | 9.15 × 10^{−10} | 2.60 × 10^{−11} | 5.72 × 10^{−13} | ||

Rank | 2 | 5 | 4 | 3 | 1 |

**Table 11.**The average of achieved rank, and percentage of reaching rank 1 by the comparative algorithms.

GA | PSO | ICA | DE | SRS | |
---|---|---|---|---|---|

Average Rank for Unimodal test functions | 3.83 | 3.33 | 3.66 | 2.91 | 1.25 |

Average Rank for Multimodal test functions | 2.27 | 3.54 | 4.36 | 3.54 | 1.27 |

Average Rank for all test functions | 3.09 | 3.43 | 4 | 3.22 | 1.26 |

Percentage of reaching rank 1 for Unimodal test functions | 0 | 0 | 0 | 8.33% | 91.67% |

Percentage of reaching rank 1 for Multimodal test functions | 18.18% | 0 | 0 | 0 | 81.81% |

Percentage of reaching rank 1 for all test functions | 8.7% | 0 | 0 | 4.35% | 86.96% |

No. | Function | Source of Variation | Sum of Squares (SS) | df | Mean Square (MS) | F | p-Value | F Criteria |
---|---|---|---|---|---|---|---|---|

${\mathit{f}}_{\mathit{1}}$ | Cigar | Between Groups | 3.29 × 10^{23} | 4 | 8.23 × 10^{22} | 1661.663 | 0.0000 | 2.372262 |

Within Groups | 1.33 × 10^{24} | 26,940 | 4.95 × 10^{19} | |||||

${\mathit{f}}_{\mathit{2}}$ | Dixon-Price | Between Groups | 8.68 × 10^{13} | 4 | 2.17 × 10^{13} | 1123.734 | 0.0000 | 2.372262 |

Within Groups | 5.21 × 10^{14} | 26,973 | 1.93 × 10^{10} | |||||

${\mathit{f}}_{\mathit{3}}$ | Quartic | Between Groups | 1,686,230 | 4 | 421,557.5 | 5294.365 | 0.0000 | 2.372261 |

Within Groups | 2,148,251 | 26,980 | 79.62382 | |||||

${\mathit{f}}_{\mathit{4}}$ | Rosenbrock | Between Groups | 1.63 × 10^{12} | 4 | 4.07 × 10^{11} | 1196.571 | 0.0000 | 2.372261 |

Within Groups | 9.17 × 10^{12} | 26,981 | 3.4 × 10^{8} | |||||

${\mathit{f}}_{\mathit{5}}$ | Schwefel 1.2 | Between Groups | 1.18 × 10^{13} | 4 | 2.95 × 10^{12} | 72,289.63 | 0.0000 | 2.372262 |

Within Groups | 1.1 × 10^{12} | 26,953 | 40,871,333 | |||||

${\mathit{f}}_{\mathit{6}}$ | Schwefel 2.22 | Between Groups | 2.45 × 10^{8} | 4 | 61,363,422 | 2792.806 | 0.0000 | 2.372263 |

Within Groups | 5.9 × 10^{8} | 26,859 | 21,971.96 | |||||

${\mathit{f}}_{\mathit{7}}$ | Schwefel 2.23 | Between Groups | 4.55082 × 10^{20} | 4 | 1.14 × 10^{20} | 652.9695 | 0.0000 | 2.372262 |

Within Groups | 4.69513 × 10^{21} | 26,947 | 1.74 × 10^{17} | |||||

${\mathit{f}}_{\mathit{8}}$ | Sphere | Between Groups | 4.45 × 10^{11} | 4 | 1.11 × 10^{11} | 1787.44 | 0.0000 | 2.372262 |

Within Groups | 1.68 × 10^{12} | 26,938 | 62,294,092 | |||||

${\mathit{f}}_{\mathit{9}}$ | Step | Between Groups | 4.03 × 10^{11} | 4 | 1.01 × 10^{11} | 1738.198 | 0.0000 | 2.372261 |

Within Groups | 1.57 × 10^{12} | 26,987 | 58,023,788 | |||||

${\mathit{f}}_{\mathit{10}}$ | SumSquares | Between Groups | 8.83 × 10^{9} | 4 | 2.21 × 10^{9} | 2154.738 | 0.0000 | 2.372262 |

Within Groups | 2.76 × 10^{10} | 26,945 | 1,024,844 | |||||

${\mathit{f}}_{\mathit{11}}$ | Trid 10 | Between Groups | 8.92 × 10^{9} | 4 | 2.23 × 10^{9} | 3561.234 | 0.0000 | 2.372262 |

Within Groups | 1.69 × 10^{10} | 26,954 | 626,415.4 | |||||

${\mathit{f}}_{\mathit{12}}$ | Zakharov | Between Groups | 48487053 | 4 | 12,121,763 | 330.2292 | 0.0000 | 2.372253 |

Within Groups | 1.02 × 10^{9} | 27,699 | 36,707.12 |

No. | Function | Source of Variation | Sum of Squares (SS) | df | Mean Square (MS) | F | p-Value | F Criteria |
---|---|---|---|---|---|---|---|---|

${\mathit{f}}_{\mathit{13}}$ | Ackley | Between Groups | 207,007 | 4 | 51,751.75 | 8462.465 | 0.0000 | 2.372261 |

Within Groups | 165,251.6 | 27,022 | 6.115446 | |||||

${\mathit{f}}_{\mathit{14}}$ | CosineMixture | Between Groups | 181.8325 | 4 | 45.45813 | 12,153.57 | 0.0000 | 2.372261 |

Within Groups | 100.9847 | 26,999 | 0.00374 | |||||

${\mathit{f}}_{\mathit{15}}$ | Griewank | Between Groups | 30,220,789 | 4 | 7,555,197 | 1710.236 | 0.0000 | 2.372262 |

Within Groups | 1.19 × 10^{8} | 26,970 | 4417.633 | |||||

${\mathit{f}}_{\mathit{16}}$ | Perm | Between Groups | 285,060.8 | 4 | 71,265.21 | 11,057.39 | 0.0000 | 2.372257 |

Within Groups | 176,497.1 | 27,385 | 6.445029 | |||||

${\mathit{f}}_{\mathit{17}}$ | Qing | Between Groups | 9.84 × 10^{23} | 4 | 2.46 × 10^{23} | 868.8985 | 0.0000 | 2.372261 |

Within Groups | 7.65 × 10^{24} | 27,008 | 2.83 × 10^{20} | |||||

${\mathit{f}}_{\mathit{18}}$ | Quintic | Between Groups | 1.3 × 10^{12} | 4 | 3.26 × 10^{11} | 797.0792 | 0.0000 | 2.372261 |

Within Groups | 1.1 × 10^{13} | 26,984 | 4.08 × 10^{8} | |||||

${\mathit{f}}_{\mathit{19}}$ | Rastrigin | Between Groups | 34,201,633 | 4 | 8,550,408 | 2962.64 | 0.0000 | 2.372261 |

Within Groups | 77,851,945 | 26,975 | 2886.078 | |||||

${\mathit{f}}_{\mathit{20}}$ | Schwefel | Between Groups | 2.68 × 10^{10} | 4 | 6.71 × 10^{9} | 4296.281 | 0.0000 | 2.372261 |

Within Groups | 4.21 × 10^{10} | 26,991 | 1,561,568 | |||||

${\mathit{f}}_{\mathit{21}}$ | Schwefel 2.25 | Between Groups | 2.1 × 10^{14} | 4 | 5.26 × 10^{13} | 2194.452 | 0.0000 | 2.372261 |

Within Groups | 6.46 × 10^{14} | 26,978 | 2.4 × 10^{10} | |||||

${\mathit{f}}_{\mathit{22}}$ | Styblinski-Tang | Between Groups | 1.86 × 10^{8} | 4 | 46,522,011 | 5330.031 | 0.0000 | 2.372261 |

Within Groups | 2.36 × 10^{8} | 27,001 | 8728.281 | |||||

${\mathit{f}}_{\mathit{23}}$ | Xin-She Yang 02 | Between Groups | 1.99 × 10^{−7} | 4 | 4.97 × 10^{−8} | 77.35719 | 0.0000 | 2.372261 |

Within Groups | 1.73 × 10^{−5} | 26,996 | 6.42 × 10^{−10} |

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**MDPI and ACS Style**

Naseri, N.K.; Sundararajan, E.A.; Ayob, M.; Jula, A.
Smart Root Search (SRS): A Novel Nature-Inspired Search Algorithm. *Symmetry* **2020**, *12*, 2025.
https://doi.org/10.3390/sym12122025

**AMA Style**

Naseri NK, Sundararajan EA, Ayob M, Jula A.
Smart Root Search (SRS): A Novel Nature-Inspired Search Algorithm. *Symmetry*. 2020; 12(12):2025.
https://doi.org/10.3390/sym12122025

**Chicago/Turabian Style**

Naseri, Narjes Khatoon, Elankovan A. Sundararajan, Masri Ayob, and Amin Jula.
2020. "Smart Root Search (SRS): A Novel Nature-Inspired Search Algorithm" *Symmetry* 12, no. 12: 2025.
https://doi.org/10.3390/sym12122025