# Improved Marine Predator Algorithm for Wireless Sensor Network Coverage Optimization Problem

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## Abstract

**:**

## 1. Introduction

- An improvement in the efficiency of a powerful metaheuristic method named MPA is investigated to solve the WSN coverage optimization problem.
- An improved marine predator algorithm (IMPA), combined with a dynamic inertia weight adjustment strategy and multi-elite random leading strategy, is developed to improve the ability of the standard MPA to handle complex problems. The performance of the IMPA is evaluated on 11 benchmark test functions and part of the CEC2014 test functions.
- The proposed IMPA is applied to solve the WSN coverage optimization problem. The results are compared with those of other metaheuristic algorithms and improved algorithms in the literature.

## 2. WSN Coverage Model

## 3. Improved Marine Predator Algorithm

#### 3.1. Standard Marine Predator Algorithm (MPA)

#### 3.2. Improved Marine Predator Algorithm (IMPA)

#### 3.2.1. Multi-Elite Random Leading Strategy

#### 3.2.2. Dynamic Inertia Weight Adjustment Strategy

#### 3.2.3. Detailed Steps for the Improved Marine Predator Algorithm

**Step 1:**- Randomly initialize the population and set relevant parameters, including population size N, dimension, maximum number of iterations ${T}_{\mathit{max}}$, FADs, etc.
**Step 2:**- Calculate the fitness value of each prey in the population and record the current optimal top three individual positions.
**Step 3:**- Update the nonlinear inertia weight factor w, and parameters such as parameters CF and ${R}_{B}$.
**Step 4:**- According to the iteration conditions, the prey position and the moving step are updated by Formulas (9), (11), (24) and (26), respectively.
**Step 5:**- According to Formulas (16)–(21), execute the multi-elite random leading strategy to update the individual position of the prey.
**Step 6:**- Considering the influence of FADs, use Formula (15) to further update the position and keep the optimal individual position.
**Step 7:**- Judge whether the algorithm meets the end condition; if so, the algorithm ends and outputs the optimal individual fitness value; otherwise, return to Step 2.

#### 3.2.4. Time Complexity Analysis of Improved Marine Predator Algorithm

## 4. The Simulation Results

#### 4.1. Experimental Environment and Parameter Setting

_{1}~f

_{5}are unimodal test functions, f

_{6}~f

_{10}are multimodal test functions, and f

_{11}is a fixed low-dimensional test function. The specific information of the benchmark function is shown in Table 1. The parameter settings of each comparison algorithm are shown in Table 2.

#### 4.2. Experimental Results and Analysis

_{1}~f

_{5}, the IMPA can solve the theoretical optimal value on the functions f

_{1}~f

_{4}. When solving the f

_{5}function, the IMPA easily falls into the same local extreme value space as the other algorithms, but the convergence accuracy and standard deviation are better than the other five algorithms. When solving multimodal functions f

_{6}~f

_{10}and fixed low-dimensional functions, the IMPA can solve the theoretical optimal value on functions f

_{6}and f

_{8}~f

_{10}. For functions f

_{7}and f

_{11}, although the IMPA fails to solve the theoretical optimal value, the solution accuracy is much higher than that of the other algorithms and the standard deviation is the smallest, indicating that the integration of the dynamic inertia weight adjustment strategy can balance the global exploration and local exploitation of the algorithm’s ability. It effectively pushes the predator to move towards the prey source, which improves the solution accuracy and stability of the algorithm. In addition, the average time consumed shown in Table 3 demonstrates that the average time consumption of the proposed algorithm is almost close to the standard MPA. Therefore, the time complexity of our proposed IMPA does not improve.

_{11}, the average fitness value obtained by the IMPA is similar to other algorithms, the IMPA has more prominent comprehensive optimization ability in general.

#### 4.3. Performance Analysis of Improved Algorithm in High Dimensions

_{1}, f

_{2}) and two multipeak test functions (f

_{7}, f

_{9}). The dimensionality is varied in 50-dimensional increments from 50 to 300 dimensions, and the mean and average rate of change of the algorithm’s solutions are used as evaluation metrics. To avoid the contingency of the experimental results, the proposed IMPA and the standard MPA were run independently 30 times, and the average value of the fitness with the function dimension was recorded. The parameter settings are the same as in Section 4.2, and the experimental results are shown in Table 4. The mathematical model for the average rate of change (ARC) is as follows:

_{1}, f

_{2}, and f

_{9}, the IMPA solution’s accuracy can maintain the theoretical optimum value of 0. However, with the increase of dimension, the solution accuracy of the standard MPA will increase significantly. For function f

_{7}, the IMPA fails to find the theoretical optimum of 0, but the average rate of change with increasing dimensionality is 0, which is much lower than the average rate of change of the standard MPA.

#### 4.4. Comparison with the Latest Improved Marine Predator Algorithm

_{1}~f

_{4}, the IMPA is far superior to the other two algorithms in terms of solution effect and stability. For function f

_{5}, because its shape is similar to that of a paraboloid, there are a large number of local extreme points, so neither the IMPA nor other algorithms can find the theoretical optimal value. However, the mean and standard deviation of the IMPA is one order of magnitude higher than those of the ODMPA and LEO-MPA and have a better ability to jump out of the local optimum. For the multi-peaked functions f

_{6}to f

_{8}, the IMPA is comparable to the other two algorithms. For solving the fixed-dimensional function f

_{11}, the standard deviation of the IMPA is slightly inferior to that of the LEO-MPA. In general, from the two indicators of the mean and standard deviation, the overall optimization performance of the IMPA is much higher than that of the other two improved algorithms, which further verifies the superiority of the IMPA.

#### 4.5. Experimental Analysis of CEC2014 Test Function

## 5. Coverage Optimization in Wireless Sensor Networks (WSNs)

#### 5.1. Comparison with Other Standard Metaheuristics

#### 5.2. Comparison with the Latest Improved Algorithm

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Partial convergence curves of the proposed IMPA and other metaheuristic algorithms on the benchmark test functions: (

**a**) Convergence curve of f

_{1}; (

**b**) Convergence curve of f

_{3}; (

**c**) Convergence curve of f

_{5}; (

**d**) Convergence curve of f

_{7}; (

**e**) Convergence curve of f

_{9}; (

**f**) Convergence curve of f

_{11}.

**Figure 4.**Function optimization with function dimension change curves: (

**a**) Change curves of f

_{1}; (

**b**) Change curves of f

_{2}; (

**c**) Change curves of f

_{7}; (

**d**) Change curves of f

_{9}.

**Figure 5.**Node coverage graph of each algorithm: (

**a**) Random deployment node coverage graph; (

**b**) EO-optimized node coverage graph; (

**c**) SCA-optimized node coverage graph; (

**d**) MPA-optimized node coverage graph; (

**e**) IMPA-optimized node coverage graph.

Fun No. | Name | Dim | Range | Optimal Value |
---|---|---|---|---|

f_{1} | Sphere | 30 | [−100, 100] | 0 |

f_{2} | Schwefel 2.22 | 30 | [−10, 10] | 0 |

f_{3} | Schwefel 1.2 | 30 | [−100, 100] | 0 |

f_{4} | Schwefel 2.21 | 30 | [−100, 100] | 0 |

f_{5} | Quartic | 30 | [−1.28, 1.28] | 0 |

f_{6} | Rastrigin | 30 | [−5.12, 5.12] | 0 |

f_{7} | Ackley | 30 | [−32, 32] | 0 |

f_{8} | Griewank | 30 | [−600, 600] | 0 |

f_{9} | Spline | 30 | [−10, 10] | 0 |

f_{10} | Schaffer | 30 | [−100, 100] | 0 |

f_{11} | Kowalik | 4 | [−5, 5] | 0.0003 |

Algorithm | Parameters |
---|---|

SCA [28] | f_{max} = 2.5, f_{min} = 0 |

TSA [29] | c_{1}, c_{2}, c_{3}∈(0, 1), P_{min} = 1, P_{max} = 4 |

MA [30] | g = 0.8, gdamp = 1, a_{1} = 1, a_{2} = 1.5, a_{3} = 1.5, dance = 5 |

EO [31] | a_{1} = 2, a_{2} = 1, GCP = 0.5 |

MPA [21] | FADs = 0.2, P = 0.5 |

IMPA | FADs = 0.2, P = 0.5 |

Fun No. | Index | EO | SCA | TSA | MA | MPA | IMPA |
---|---|---|---|---|---|---|---|

Mean | 4.58 × 10^{−41} | 3.65 | 3.01 × 10^{−21} | 2.09 × 10^{−6} | 8.62 × 10^{−23} | 0 | |

f_{1} | Std | 4.38 × 10^{−41} | 2.38 | 4.23 × 10^{−21} | 1.67 × 10^{−6} | 1.04 × 10^{−23} | 0 |

Time (s) | 0.24 | 0.14 | 1.17 | 2.56 | 0.42 | 0.44 | |

Mean | 9.06 × 10^{−24} | 3.60 × 10^{−2} | 2.18 × 10^{−13} | 7.38 × 10^{−3} | 2.32 × 10^{−13} | 0 | |

f_{2} | Std | 1.16 × 10^{−23} | 1.18 × 10^{−2} | 2.27 × 10^{−13} | 1.17 × 10^{−25} | 1.04 × 10^{−13} | 0 |

Time (s) | 0.27 | 0.16 | 2.11 | 2.08 | 0.45 | 0.46 | |

Mean | 8.75 × 10^{−11} | 4.66 × 10^{3} | 3.33 × 10^{−5} | 4.60 × 10^{3} | 1.63 × 10^{−4} | 0 | |

f_{3} | Std | 8.56 × 10^{−11} | 3.68 × 10^{3} | 4.82 × 10^{−5} | 1.87 × 10^{3} | 2.25 × 10^{−4} | 0 |

Time (s) | 0.88 | 0.74 | 2.83 | 3.74 | 1.61 | 1.73 | |

Mean | 4.92 × 10^{−10} | 2.98 × 10^{1} | 3.75 × 10^{−1} | 4.25 × 10^{1} | 2.29 × 10^{−9} | 0 | |

f_{4} | Std | 1.25 × 10^{−10} | 1.08 × 10^{1} | 2.88 × 10^{−1} | 8.48 | 3.10 × 10^{−10} | 0 |

Time (s) | 0.28 | 0.18 | 2.27 | 2.14 | 0.48 | 0.49 | |

Mean | 1.02 × 10^{−3} | 6.54 × 10^{−2} | 9.92 × 10^{−3} | 1.83 × 10^{−2} | 8.92 × 10^{−3} | 7.18 × 10^{−5} | |

f_{5} | Std | 9.48 × 10^{−3} | 6.84 × 10^{−2} | 1.62 × 10^{−3} | 8.92 × 10^{−3} | 8.14 × 10^{−3} | 2.00 × 10^{−5} |

Time (s) | 0.36 | 0.23 | 2.33 | 2.26 | 0.65 | 0.62 | |

Mean | 2.48 × 10^{−2} | 2.54 × 10^{1} | 1.90 × 10^{2} | 7.77 × 10^{1} | 0 | 0 | |

f_{6} | Std | 6.11 × 10^{−1} | 1.24 × 10^{1} | 3.22 × 10^{1} | 3.25 × 10^{1} | 0 | 0 |

Time (s) | 0.29 | 0.18 | 2.30 | 2.13 | 0.49 | 0.52 | |

Mean | 1.03 × 10^{−14} | 1.36 × 10^{1} | 3.10 | 2.07 | 2.27 × 10^{−12} | 8.88 × 10^{−16} | |

f_{7} | Std | 4.10 × 10^{−14} | 1.16 × 10^{1} | 1.22 × 10^{−1} | 6.72 × 10^{−1} | 1.17 × 10^{−12} | 0 |

Time (s) | 0.39 | 0.21 | 2.38 | 2.24 | 0.54 | 0.58 | |

Mean | 4.15 × 10^{−3} | 8.39 × 10^{−1} | 9.66 × 10^{−3} | 2.70 × 10^{−2} | 0 | 0 | |

f_{8} | Std | 6.61 × 10^{−3} | 4.55 × 10^{−1} | 8.59 × 10^{−3} | 1.61 × 10^{−2} | 0 | 0 |

Time (s) | 0.32 | 0.19 | 2.31 | 2.19 | 0.52 | 0.54 | |

Mean | 7.75 × 10^{−25} | 9.25 × 10^{−1} | 3.10 × 10^{1} | 5.76 × 10^{−5} | 1.33 × 10^{−13} | 0 | |

f_{9} | Std | 2.69 × 10^{−25} | 1.13 | 8.81 | 4.98 × 10^{−5} | 1.17 × 10^{−13} | 0 |

Time (s) | 0.28 | 0.16 | 2.28 | 2.05 | 0.46 | 0.48 | |

Mean | 1.19 × 10^{−3} | 9.72 × 10^{−4} | 9.78 × 10^{−3} | 8.72 × 10^{−3} | 9.73 × 10^{−4} | 0 | |

f_{10} | Std | 4.09 × 10^{−3} | 3.07 × 10^{−3} | 2.84 × 10^{−3} | 6.99 × 10^{−3} | 3.07 × 10^{−3} | 0 |

Time (s) | 0.27 | 0.12 | 0.24 | 1.98 | 0.35 | 0.36 | |

Mean | 7.30 × 10^{−3} | 7.92 × 10^{−4} | 7.11 × 10^{−3} | 7.36 × 10^{−4} | 3.07 × 10^{−4} | 3.07 × 10^{−4} | |

f_{11} | Std | 1.13 × 10^{−2} | 3.64 × 10^{−5} | 1.14 × 10^{−2} | 7.24 × 10^{−4} | 3.35 × 10^{−15} | 2.83 × 10^{−19} |

Time (s) | 0.28 | 0.09 | 0.21 | 1.95 | 0.29 | 0.31 |

**Table 4.**Comparison of IMPA and MPA on the mean value of optimization functions in different dimensions.

Fun No. | Algorithm | Dimension | Average Rate of Change (%) | |||||
---|---|---|---|---|---|---|---|---|

50 | 100 | 150 | 200 | 250 | 300 | |||

f_{1} | MPA | 4.19 × 10^{−21} | 1.04 × 10^{−19} | 1.32 × 10^{−18} | 2.37 × 10^{−18} | 7.81 × 10^{−18} | 2.51 × 10^{−17} | 1.02 × 10^{1} |

IMPA | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

f_{2} | MPA | 2.18 × 10^{−12} | 1.28 × 10^{−11} | 1.78 × 10^{−10} | 7.81 × 10^{−9} | 1.95 × 10^{−9} | 8.32 × 10^{−8} | 2.54 × 10^{1} |

IMPA | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

f_{7} | MPA | 5.98 × 10^{−12} | 4.04 × 10^{−11} | 8.79 × 10^{−11} | 1.17 × 10^{−10} | 2.08 × 10^{−10} | 3.10 × 10^{−10} | 2.13 |

IMPA | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 0 | |

f_{9} | MPA | 4.37 × 10^{−13} | 5.38 × 10^{−12} | 7.32 × 10^{−12} | 3.90 × 10^{−11} | 8.12 × 10^{−11} | 3.12 × 10^{−10} | 4.98 × 10^{1} |

IMPA | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Fun No. | Index | ODMPA [32] | LEO-MPA [33] | IMPA |
---|---|---|---|---|

f_{1} | Mean | 2.08 × 10^{−108} | 4.53 × 10^{−29} | 0 |

Std | 1.13 × 10^{−107} | 6.92 × 10^{−29} | 0 | |

f_{2} | Mean | 5.17 × 10^{−60} | 1.17 × 10^{−30} | 0 |

Std | 1.62 × 10^{−59} | 3.83 × 10^{−30} | 0 | |

f_{3} | Mean | 4.86 × 10^{−86} | 6.75 × 10^{−41} | 0 |

Std | 2.17 × 10^{−85} | 1.96 × 10^{−40} | 0 | |

f_{4} | Mean | 6.39 × 10^{−49} | 5.81 × 10^{−24} | 0 |

Std | 2.22 × 10^{−48} | 2.38 × 10^{−23} | 0 | |

f_{5} | Mean | 4.47 × 10^{−4} | 2.81 × 10^{−4} | 7.18 × 10^{−5} |

Std | 3.04 × 10^{−4} | 2.19 × 10^{−4} | 2.00 × 10^{−5} | |

f_{6} | Mean | 0 | 0 | 0 |

Std | 0 | 0 | 0 | |

f_{7} | Mean | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} |

Std | 0 | 0 | 0 | |

f_{8} | Mean | 0 | 8.22 × 10^{−4} | 0 |

Std | 0 | 2.53 × 10^{−3} | 0 | |

f_{9} | Mean | — | — | 0 |

Std | — | — | 0 | |

f_{10} | Mean | — | — | 0 |

Std | — | — | 0 | |

f_{11} | Mean | 3.07 × 10^{−4} | 3.07 × 10^{−4} | 3.07 × 10^{−4} |

Std | 9.13 × 10^{−18} | 1.35 × 10^{−19} | 2.83 × 10^{−19} |

Fun No. | Function Type | Function Name | Optimal Value |
---|---|---|---|

CEC03 | Unimodal Function | Rotated Discus Function | 300 |

CEC05 | Multimodal Function | Shifted and Rotated Ackley’s Function | 500 |

CEC08 | Multimodal Function | Shifted Rastrigin’s Function | 800 |

CEC11 | Multimodal Function | Shifted and Rotated Schwefel’s Function | 1100 |

CEC19 | Hybrid Function | Hybrid Function 3 (N = 4) | 1900 |

CEC20 | Hybrid Function | Hybrid Function 4 (N = 4) | 2000 |

CEC25 | Composition Function | Composition Function 3 (N = 3) | 2500 |

CEC27 | Composition Function | Composition Function 5 (N = 5) | 2700 |

CEC28 | Composition Function | Composition Function 6 (N = 5) | 2800 |

CEC30 | Composition Function | Composition Function 8 (N = 3) | 3000 |

Fun No. | Index | IMPA | MPA | HEGMPA [35] | TSA | EO | MA |
---|---|---|---|---|---|---|---|

CEC03 | Mean | 4.96 × 10^{2} | 3.61 × 10^{2} | 0 | 4.73 × 10^{4} | 9.31 × 10^{3} | 5.41 × 10^{5} |

Std | 6.28 × 10^{2} | 4.56 × 10^{2} | 0 | 1.03 × 10^{4} | 5.15 × 10^{3} | 4.64 × 10^{5} | |

CEC05 | Mean | 5.20 × 10^{2} | 5.21 × 10^{2} | 0 | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} |

Std | 4.20 × 10^{−2} | 8.10 × 10^{−2} | 0 | 6.03 × 10^{−2} | 6.00 × 10^{−2} | 5.95 × 10^{−2} | |

CEC08 | Mean | 8.65 × 10^{2} | 8.76 × 10^{2} | 1.35 × 10^{1} | 1.07 × 10^{3} | 8.71 × 10^{2} | 1.21 × 10^{3} |

Std | 5.77 | 1.68 × 10^{1} | 5.99 | 5.37 × 10^{1} | 1.64 × 10^{1} | 2.88 × 10^{1} | |

CEC11 | Mean | 3.55 × 10^{3} | 3.84 × 10^{3} | 2.46 × 10^{3} | 7.45 × 10^{3} | 4.76 × 10^{3} | 9.52 × 10^{3} |

Std | 2.46 × 10^{2} | 4.25 × 10^{2} | 4.13 × 10^{2} | 7.83 × 10^{2} | 7.47 × 10^{2} | 6.78 × 10^{2} | |

CEC19 | Mean | 1.91 × 10^{3} | 1.91 × 10^{3} | 2.09 | 2.17 × 10^{3} | 1.91 × 10^{3} | 2.42 × 10^{3} |

Std | 5.74 × 10^{−1} | 8.21 × 10^{−1} | 6.36 × 10^{−1} | 1.75 × 10^{2} | 6.83 × 10^{−1} | 8.71 × 10^{1} | |

CEC20 | Mean | 2.03 × 10^{3} | 2.05 × 10^{3} | 6.79 | 7.06 × 10^{4} | 1.47 × 10^{4} | 7.74 × 10^{5} |

Std | 1.37 × 10^{1} | 1.50 × 10^{1} | 2.45 | 7.49 × 10^{4} | 6.88 × 10^{3} | 9.04 × 10^{5} | |

CEC25 | Mean | 2.70 × 10^{3} | 2.61 × 10^{3} | 2.02 × 10^{2} | 2.73 × 10^{3} | 2.70 | 2.81 × 10^{3} |

Std | 0 | 3.95 × 10^{−3} | 5.08 × 10^{−1} | 1.27 × 10^{1} | 4.72 | 2.95 × 10^{2} | |

CEC27 | Mean | 2.90 × 10^{3} | 3.10 × 10^{3} | 3.08 × 10^{2} | 3.99 × 10^{3} | 3.23 × 10^{3} | 3.83 × 10^{3} |

Std | 0 | 2.10 | 3.37 × 10^{1} | 1.26 × 10^{2} | 5.54 × 10^{1} | 9.62 × 10^{1} | |

CEC28 | Mean | 3.00 × 10^{3} | 3.70 × 10^{3} | 7.37 × 10^{2} | 7.25 × 10^{3} | 3.81 × 10^{3} | 8.25 × 10^{3} |

Std | 0 | 4.95 × 10^{1} | 1.70 × 10^{2} | 1.10 × 10^{3} | 1.55 × 10^{2} | 9.28 × 10^{2} | |

CEC30 | Mean | 3.20 × 10^{3} | 6.18 × 10^{3} | 4.12 × 10^{2} | 2.28 × 10^{5} | 7.80 × 10^{3} | 2.71 × 10^{6} |

Std | 0 | 1.75 × 10^{3} | 4.33 × 10^{1} | 2.89 × 10^{5} | 1.70 × 10^{3} | 9.31 × 10^{6} |

Parameters | Value |
---|---|

Area | M = 100 × 100 m |

Sensor node | 40 |

Perceived radius | 5 m |

Communication radius | 10 m |

Maximum number of iterations | 500 |

Algorithm | Average Coverage Rate |
---|---|

Random deployment | 74.62% |

EO | 89.44% |

SCA | 76.88% |

MPA | 88.81% |

IMPA | 93.72% |

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## Share and Cite

**MDPI and ACS Style**

He, Q.; Lan, Z.; Zhang, D.; Yang, L.; Luo, S.
Improved Marine Predator Algorithm for Wireless Sensor Network Coverage Optimization Problem. *Sustainability* **2022**, *14*, 9944.
https://doi.org/10.3390/su14169944

**AMA Style**

He Q, Lan Z, Zhang D, Yang L, Luo S.
Improved Marine Predator Algorithm for Wireless Sensor Network Coverage Optimization Problem. *Sustainability*. 2022; 14(16):9944.
https://doi.org/10.3390/su14169944

**Chicago/Turabian Style**

He, Qing, Zhouxin Lan, Damin Zhang, Liu Yang, and Shihang Luo.
2022. "Improved Marine Predator Algorithm for Wireless Sensor Network Coverage Optimization Problem" *Sustainability* 14, no. 16: 9944.
https://doi.org/10.3390/su14169944