# Estimation of Multiple Parameters in Semitransparent Mediums Based on an Improved Grey Wolf Optimization Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. GWO Algorithm

#### 2.1. Original GWO Algorithm

**a**

_{ini}and

**a**

_{fin}are the beginning and ending values of the control parameters, respectively, generally take 2 and 0. t is the current number of iterations, and t

_{max}is the maximum number of iterations.

**a**is the convergence factor; the

**A**and

**C**both are coefficient vectors; r

_{1}and r

_{2}are random numbers within [0, 1].

**X**and

**X**

_{p}are the positions of the grey wolves and the prey, respectively. The distance

**D**between the grey wolves and the prey is defined in Equation (2), while the final position of the grey wolf

**X**(t + 1) is defined in Equation (3).

**D**

_{α},

**D**

_{β}, and

**D**

_{δ}in Equation (4). The current position of the α wolf, β wolf, and δ wolf are represented, respectively, by

**X**

_{α},

**X**

_{β}, and

**X**

_{δ}in Equation (4). The calculation formulas in Equation (5) respectively represent the advancement and direction of the ω wolf to the α wolf, β wolf, and δ wolf. The final position of the ω wolf is defined in Equation (6).

**a**gradually decreases, and the value of the

**A**(A

_{j}) also gradually decreases. The value of

**a**decreases linearly from 2 to 0, and the A

_{j}also varies within [−a, a]. When $\left|{A}_{j}\right|<1$, the prey is attacked by the grey wolves. In addition, the grey wolves are separated from each other when searching for prey and gather together when attacking prey. The grey wolf can be forced to separate from the prey by $\left|{A}_{j}\right|>1$, which emphasizes the role of exploration and strengthens the global search ability in the GWO algorithm. Moreover, the value of the

**C**(C

_{j}) is a random value between [0, 2] and can represent the random weight of the individual wolf’s position on the prey. C

_{j}> 1 means that the weight influence is significant, and conversely, the weight influence is small. The flow chart of the GWO algorithm is shown in Figure 3.

#### 2.2. Improved GWO Algorithm

**a**is replaced by the nonlinear convergence factor of the cosine curve to further balance the global search ability and the local search ability in the algorithm, which is calculated with the equation below. It is worth noting that for different algorithms, the selection of the nonlinear convergence factor is different.

#### 2.3. Performance Test of IGWO Algorithm

_{ini}and a

_{fin}are set to 2 and 0, respectively. The average best fitness values of 10 independent run times for the three test functions and the corresponding standard deviation are listed in Table 2. The iterative curves of the best fitness of the GWO algorithm and the IGWO algorithm for these three test functions are shown in Figure 5, Figure 6 and Figure 7.

## 3. Results and Discussion

**Y**

_{mea}and

**Y**

_{exact}indicate the measured value and exact value of the measured signals (simulated by the direct model), which served as the input for inverse analysis. ζ is a random variable of normal distribution with zero mean and unit standard deviation. σ

_{0}has a 99% confidence level for a γ measurement error and is defined as follows:

_{rel}is defined as follows:

#### 3.1. Inverse Estimation for Source Term Coefficients

_{L}bounded by two opaque, diffuse, and grey walls in semitransparent and grey media is shown in Figure 8. The left and right boundary walls are subjected to the first-type boundary condition with a constant temperature of T

_{w1}and T

_{w2}, respectively. For the case of only radiation without heat conduction, the radiative intensities can be expressed as follows [30]:

_{b1}and I

_{b2}denote the radiative intensities at the boundaries. μ is the directional cosine and μ = cosθ. We can assume that the source term S (τ) is represented by a polynomial of the variable τ

_{L}= 0.01 m, and the temperature of the boundary walls is assumed to be T

_{w1}= 1000 K and T

_{w2}= 1000 K in this paper. The source term coefficients are expressed as $a={\left[{a}_{1},{a}_{2},\dots ,{a}_{n}\right]}^{T}$. The objective function is defined as follows:

**X**in IGWO. The fitness function converges by updating X step by step (See Figure 4 for details). The radiative intensities are calculated by Equations (13)–(15) in each iteration (a detailed description of the finite volume method for solving the radiative transfer equation is clarified in Section 3.2). Then, the fitness function in each iteration step is obtained by Equation (16). The source term coefficients are obtained when the maximum number of iterations is reached. The parameters in the IGWO algorithm are listed in Table 3. We consider the two cases of the source term coefficients: $a={\left[3,14,-14\right]}^{\mathrm{T}}$, $a={\left[5,12,-12\right]}^{\mathrm{T}}$. The algorithms run independently 10 times in this study. The average values, standard deviation, and relative error of the two cases are shown in Table 4, and the estimated values of source term coefficients that vary with the number of runs are shown in Figure 9.

^{−2}%. Moreover, the biggest standard deviation is still smaller than 0.1. Therefore, the IGWO algorithm has a high estimating accuracy and good stability for the source term coefficients. When the measurement error increases, the estimating accuracy decreases, but it can still be seen as a good estimating accuracy. During the 10 runs, the estimated value of the source term coefficients fluctuated to a small extent by the IGWO algorithm when the measurement error γ = 0% in Figure 9 shows that the estimation of the source term coefficients is reliable and stable. Therefore, the inverse estimation for the source term coefficients by the IGWO algorithm in this paper is successful.

#### 3.2. Simultaneous Inverse Estimation of Refractive Index and Absorption Coefficient

#### 3.2.1. Physical Model

_{b}is the blackbody radiative intensity at the temperature of media Τ(τ), n, and κ

_{a}are the refractive index and the absorption coefficient, respectively. The energy transport process in semitransparent media can be described by an energy equation and a radiative transfer equation. The energy equation without internal heat generation can be written as [11]:

_{p}is the heat capacity. To obtain the refractive index and the absorption coefficient using an improved grey wolf optimization algorithm, the initial and boundary conditions are given as follows [11,30]:

_{laser}is the laser intensity. T

_{w1}and T

_{w2}are the temperatures of the left and right walls. T

_{s}is the initial temperature or ambient temperature, and ${h}_{\mathrm{w}1},{h}_{\mathrm{w}2}$ are convective heat transfer coefficients. Then, the ${q}_{\mathrm{w}1}^{\mathrm{r}},\hspace{0.33em}{q}_{\mathrm{w}2}^{\mathrm{r}}$ and $(\partial {q}^{r}/\partial \tau )$ can be expressed as follows [11,30]:

^{m}, g

^{m}, and h

^{m}are the cosines of each of the three axes. The integral scheme of Equation (23) based on the control volume is as follows [30]:

#### 3.2.2. Inverse Estimation of the Refractive Index and Absorption Coefficient

_{obj}is the value of the objective function and

**I**

_{est}and

**I**

_{mea}are the estimated parameters and measured parameters (in this work, the parameters denote the refractive index and absorption coefficient), respectively.

_{max}= 500 in the three cases of measurement errors. The relative error of refractive index is only 4.24 × 10

^{−4}% (the case of n = 2.36, κ

_{a}= 4.0 m

^{−1}) when measurement error γ = 0% in Table 7; therefore, the estimating accuracy is very high. If the population size is M = 100, the relative error of the refractive index will even drop to a lower level of 4.24 × 10

^{−5}% when measurement error γ = 0% in Figure 11. As the measurement errors increase, the estimating accuracy decreases. However, the estimation value is still acceptable. When the measurement error γ = 5%, the relative error of the absorption coefficient is 3.84 × 10

^{−1}% (the case of n = 3.50, κ

_{a}= 5.0 m

^{−1}), which is within a reasonable range. In addition, the estimated average values fluctuate more with the increasing measure errors in Table 8. All in all, the results show that the IGWO algorithm can estimate the absorption coefficient and refractive index with high accuracy and reliability even with noisy data.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Nomenclature | Greek symbols | ||

A | Coefficient vector | ${\kappa}_{s}$ | The scattering coefficient |

C | Coefficient vector | $\theta $ | The polar angle |

D | Distance between grey wolf and prey | $\kappa $ | Absorption coefficient |

X | Grey wolf position | $\tau $ | Optical thickness |

X_{p} | Prey position | $\epsilon $ | Stefan- The wall emissivity |

t | Number of iterations | $\sigma $ | Stefan-Boltzmann constant |

a | Convergence factor | $\lambda $ | Thermal conductivity |

${r}_{1}$ | Random number within [0, 1] | ${q}^{r}$ | Power density |

${r}_{2}$ | Random number within [0, 1] | ε_{rel} | The relative error |

$I$ | Radiative intensity | γ | The measurement error |

$n$ | Refractive index | Subscripts | |

$T$ | Temperature | mea | The measured value |

$q$ | Array radiative | est | The estimated value |

F_{obj} | Objective function | exact | The exact value |

Y | The estimated or exact values of refractive index, absorption coefficient, and source term | b | Black body |

$S$ | The source term | avg | Average value |

${a}_{n}$ | The source term coefficient | L_{2} | L_{2}-type norm |

M | The population size of grey wolf | i | The ith iteration |

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**Figure 11.**Estimation results of absorption coefficient and refractive index under different population sizes.

Function | Expression | Dimension | Space |
---|---|---|---|

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

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

Griewank | ${f}_{3}={\displaystyle {\sum}_{i=1}^{D}i{x}_{i}^{2}}$ | 30 | [−100, 100] |

Function | GWO | Iteration | IGWO | Iteration |
---|---|---|---|---|

F1 (Sphere) | $3.03\times {10}^{-33}\pm 3.69\times {10}^{-34}$ | 500 | $3.77\times {10}^{-50}\pm 3.69\times {10}^{-51}$ | 83 |

F2 (Rosenbrock) | $27.0121\pm 1.0631$ | 500 | $28.8766\pm 0.0623$ | 500 |

F3 (Griewank) | $2.15\times {10}^{-31}\pm 3.10\times {10}^{-32}$ | 500 | $2.45\times {10}^{-50}\pm 3.10\times {10}^{-51}$ | 86 |

Algorithm | a_{fin} | a_{ini} | t_{max} | M |
---|---|---|---|---|

IGWO | 0 | 2 | 500 | 100 |

Exact Value | Measure Error | a_{1} | ε_{rel} (%) | a_{2} | ε_{rel} (%) | a_{3} | ε_{rel} (%) |
---|---|---|---|---|---|---|---|

a_{1} = 3 | γ = 0% | 2.9998 ± 9.05 × 10^{−4} | 6.67 × 10^{−3} | 14.0010 ± 1.03 × 10^{−2} | 7.37 × 10^{−2} | −14.0016 ± 1.07 × 10^{−2} | 1.14 × 10^{−2} |

a_{2} = 14 | γ = 3% | 3.0003 ± 8.11 × 10^{−4} | 1.00 × 10^{−2} | 13.9979 ± 9.81 × 10^{−3} | 6.46 × 10^{−2} | −13.9981 ± 8.89 × 10^{−3} | 1.36 × 10^{−2} |

a_{3} = −14 | γ = 5% | 2.9995 ± 9.09 × 10^{−4} | 1.67 × 10^{−2} | 14.0039 ± 9.04 × 10^{−3} | 7.00 × 10^{−2} | −14.0039 ± 9.71 × 10^{−3} | 2.79 × 10^{−2} |

a_{1} = 5 | γ = 0% | 4.9999 ± 1.30 × 10^{−3} | 1.18 × 10^{−3} | 12.0008 ± 1.57 × 10^{−2} | 6.58 × 10^{−3} | −12.0010 ± 1.56 × 10^{−2} | 8.13 × 10^{−3} |

a_{2} = 12 | γ = 3% | 4.9997 ± 9.55 × 10^{−4} | 5.54 × 10^{−3} | 12.0033 ± 1.03 × 10^{−2} | 2.75 × 10^{−2} | −12.0032 ± 1.03 × 10^{−2} | 2.70 × 10^{−2} |

a_{3} = −12 | γ = 5% | 4.9997 ± 9.76 × 10^{−4} | 6.24 × 10^{−3} | 12.0046 ± 1.11 × 10^{−2} | 3.80 × 10^{−2} | −12.0034 ± 1.09 × 10^{−2} | 2.83 × 10^{−2} |

Physical Parameters | Symbol | Value |
---|---|---|

The thickness of the medium | τ_{L} | 0.01 m |

The heat capacity | ρc_{p} | 1.0 × 10^{7} J/(m^{3}·K) |

The thermal conductivity | λ | 0.7 W/(m·K) |

The convective heat transfer coefficient | ${h}_{\mathrm{w}1},{h}_{\mathrm{w}2}$ | 7.0 W/(m^{2}·K) |

Initial or ambient temperature | T_{s} | 300 K |

The laser intensity | q_{laser} | 500 W/m^{2} |

The wall emissivity | ${\epsilon}_{\mathrm{w}1},{\epsilon}_{\mathrm{w}2}$ | 1 |

Runs | Population Size = 30 | Population Size = 50 | Population Size = 100 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | |

1 | 2.36002 | 0.000847 | 4.00245 | 0.06125 | 2.36001 | 0.000424 | 3.99422 | 0.1445 | 2.36000 | 0 | 4.00102 | 0.0255 |

2 | 2.35994 | 0.002542 | 3.98836 | 0.291 | 2.36006 | 0.002542 | 4.01019 | 0.25475 | 2.36003 | 0.001271 | 4.00204 | 0.051 |

3 | 2.36009 | 0.003814 | 4.00454 | 0.1135 | 2.35990 | 0.004237 | 3.99224 | 0.194 | 2.36000 | 0 | 3.99930 | 0.0175 |

4 | 2.36012 | 0.005085 | 4.01512 | 0.378 | 2.36009 | 0.003814 | 3.99118 | 0.2205 | 2.35996 | 0.001695 | 3.99352 | 0.162 |

5 | 2.35997 | 0.001271 | 3.98440 | 0.39 | 2.35997 | 0.001271 | 3.99861 | 0.03475 | 2.36000 | 0 | 4.00135 | 0.03375 |

6 | 2.36006 | 0.002542 | 4.00759 | 0.18975 | 2.36002 | 0.000847 | 4.00334 | 0.0835 | 2.36001 | 0.000424 | 3.99957 | 0.01075 |

7 | 2.35998 | 0.000847 | 3.99912 | 0.022 | 2.36001 | 0.000424 | 3.99651 | 0.08725 | 2.35999 | 0.000424 | 3.99912 | 0.022 |

8 | 2.36000 | 0 | 4.00418 | 0.1045 | 2.35999 | 0.000424 | 3.99807 | 0.04825 | 2.36000 | 0 | 4.00124 | 0.031 |

9 | 2.36008 | 0.00339 | 4.01289 | 0.32225 | 2.35996 | 0.001695 | 3.99805 | 0.04875 | 2.35998 | 0.000847 | 3.99898 | 0.0255 |

10 | 2.35986 | 0.005932 | 3.99270 | 0.1825 | 2.35999 | 0.000424 | 3.99844 | 0.039 | 2.36002 | 0.000847 | 4.00045 | 0.01125 |

**Table 7.**Estimation results of absorption coefficient and refractive index under different measurement errors.

Exact Value | Measure Error | ε_{n} | ε_{rel} (%) | κ_{a} | ε_{rel} (%) |
---|---|---|---|---|---|

n = 2.36 κ _{a} = 4.0 | γ = 0% | $2.3600\pm 7.78\times {10}^{-5}$ | $4.24\times {10}^{-4}$ | $4.0011\pm 1.11\times {10}^{-2}$ | $2.75\times {10}^{-2}$ |

γ = 3% | $2.3610\pm 1.31\times {10}^{-2}$ | $4.24\times {10}^{-2}$ | $3.9965\pm 9.01\times {10}^{-3}$ | $8.75\times {10}^{-2}$ | |

γ = 5% | $2.3686\pm 1.28\times {10}^{-2}$ | $3.64\times {10}^{-1}$ | $3.9932\pm 3.81\times {10}^{-2}$ | $1.70\times {10}^{-1}$ | |

n = 3.50 κ _{a} = 5.0 | γ = 0% | $3.5001\pm 1.11\times {10}^{-4}$ | $1.57\times {10}^{-3}$ | $5.0038\pm 1.11\times {10}^{-2}$ | $7.61\times {10}^{-2}$ |

γ = 3% | $3.5036\pm 2.13\times {10}^{-2}$ | $1.03\times {10}^{-1}$ | $5.0067\pm 1.67\times {10}^{-2}$ | $1.33\times {10}^{-1}$ | |

γ = 5% | $3.4953\pm 3.65\times {10}^{-2}$ | $1.34\times {10}^{-1}$ | $4.9808\pm 1.96\times {10}^{-2}$ | $3.84\times {10}^{-1}$ |

Runs | Measure Error = 0% | Measure Error = 3% | Measure Error = 5% | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | |

1 | 2.36002 | 0.000847 | 4.00245 | 0.06125 | 2.35943 | 0.024153 | 3.99901 | 0.02475 | 2.36083 | 0.035169 | 3.97479 | 0.63025 |

2 | 2.35994 | 0.002542 | 3.98836 | 0.291 | 2.35652 | 0.147458 | 4.00351 | 0.08775 | 2.37942 | 0.822881 | 4.01223 | 0.30575 |

3 | 2.36009 | 0.003814 | 4.00454 | 0.1135 | 2.37278 | 0.541525 | 3.99553 | 0.11175 | 2.38106 | 0.892373 | 3.99321 | 0.16975 |

4 | 2.36012 | 0.005085 | 4.01512 | 0.378 | 2.35075 | 0.391949 | 3.99687 | 0.07825 | 2.39463 | 1.467373 | 4.01683 | 0.42075 |

5 | 2.35997 | 0.001271 | 3.98440 | 0.39 | 2.35259 | 0.313983 | 3.99507 | 0.12325 | 2.34236 | 0.747458 | 3.99695 | 0.07625 |

6 | 2.36006 | 0.002542 | 4.00759 | 0.18975 | 2.36087 | 0.036864 | 4.00166 | 0.0415 | 2.37423 | 0.602966 | 3.99618 | 0.0955 |

7 | 2.35998 | 0.000847 | 3.99912 | 0.022 | 2.36178 | 0.075424 | 3.99703 | 0.07425 | 2.36949 | 0.402119 | 4.04793 | 1.19825 |

8 | 2.36000 | 0 | 4.00418 | 0.1045 | 2.37068 | 0.452542 | 3.99690 | 0.0775 | 2.37773 | 0.751271 | 4.00493 | 0.12325 |

9 | 2.36008 | 0.00339 | 4.01289 | 0.32225 | 2.36673 | 0.285169 | 4.00650 | 0.1625 | 2.36561 | 0.237712 | 3.90035 | 2.49125 |

10 | 2.35986 | 0.005932 | 3.99270 | 0.1825 | 2.35481 | 0.219915 | 3.98811 | 0.29725 | 2.34342 | 0.702542 | 3.97317 | 0.67075 |

**Table 9.**Estimation results of absorption coefficient and refractive index with different thermal conductivities.

Exact Value | Thermal Conductivity | n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) |
---|---|---|---|---|---|

n = 2.36 κ _{a} = 4.0 | λ = 0.5 | $2.3600\pm 9.52\times {10}^{-5}$ | $6.36\times {10}^{-4}$ | $4.0008\pm 1.06\times {10}^{-2}$ | $1.97\times {10}^{-2}$ |

λ = 0.7 | $2.3600\pm 7.78\times {10}^{-5}$ | $4.24\times {10}^{-4}$ | $4.0011\pm 1.11\times {10}^{-2}$ | $2.75\times {10}^{-2}$ | |

λ = 0.9 | $2.3600\pm 9.23\times {10}^{-5}$ | $1.44\times {10}^{-3}$ | $4.0087\pm 2.31\times {10}^{-2}$ | $2.18\times {10}^{-1}$ | |

n = 3.50 κ _{a} = 5.0 | λ = 0.5 | $3.5000\pm 1.47\times {10}^{-4}$ | $1.26\times {10}^{-3}$ | $5.0036\pm 9.36\times {10}^{-3}$ | $7.22\times {10}^{-2}$ |

λ = 0.7 | $3.5001\pm 1.11\times {10}^{-4}$ | $1.57\times {10}^{-3}$ | $5.0038\pm 1.11\times {10}^{-2}$ | $7.61\times {10}^{-2}$ | |

λ = 0.9 | $3.4956\pm 1.41\times {10}^{-2}$ | $1.26\times {10}^{-1}$ | $5.0056\pm 3.94\times {10}^{-2}$ | $1.12\times {10}^{-1}$ |

Runs | Thermal Conductivity = 0.5 | Thermal Conductivity = 0.7 | Thermal Conductivity = 0.9 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | n | ε_{rel} (%) | κ_{a} | ε_{rel} (%) | |

1 | 2.36 | 0 | 4.00141 | 0.03525 | 2.36002 | 0.000847 | 4.00245 | 0.06125 | 2.36005 | 0.002119 | 3.99955 | 0.01125 |

2 | 2.36006 | 0.002542 | 4.00936 | 0.234 | 2.35994 | 0.002542 | 3.98836 | 0.291 | 2.36012 | 0.005085 | 3.98836 | 0.291 |

3 | 2.36003 | 0.001271 | 4.00454 | 0.1135 | 2.36009 | 0.003814 | 4.00454 | 0.1135 | 2.36016 | 0.00678 | 4.01749 | 0.43725 |

4 | 2.35996 | 0.001695 | 4.0048 | 0.12 | 2.36012 | 0.005085 | 4.01512 | 0.378 | 2.36022 | 0.009322 | 4.01512 | 0.378 |

5 | 2.35999 | 0.000424 | 3.99879 | 0.03025 | 2.35997 | 0.001271 | 3.98440 | 0.39 | 2.36012 | 0.005085 | 4.01273 | 0.31825 |

6 | 2.35992 | 0.00339 | 3.99398 | 0.1505 | 2.36006 | 0.002542 | 4.00759 | 0.18975 | 2.36013 | 0.005508 | 4.02681 | 0.67025 |

7 | 2.35998 | 0.000847 | 3.99912 | 0.022 | 2.35998 | 0.000847 | 3.99912 | 0.022 | 2.36007 | 0.002966 | 4.00884 | 0.221 |

8 | 2.36 | 0 | 4.00418 | 0.1045 | 2.36000 | 0 | 4.00418 | 0.1045 | 2.35983 | 0.007203 | 3.9611 | 0.9725 |

9 | 2.35998 | 0.000847 | 3.99766 | 0.0585 | 2.36008 | 0.00339 | 4.01289 | 0.32225 | 2.36008 | 0.00339 | 4.01289 | 0.32225 |

10 | 2.36 | 0 | 3.99965 | 0.00875 | 2.35986 | 0.005932 | 3.99270 | 0.1825 | 2.36019 | 0.008051 | 4.05203 | 1.30075 |

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## Share and Cite

**MDPI and ACS Style**

Li, K.; Xie, L.; Zhou, J.; Wu, X.; Ding, D.; Li, C.
Estimation of Multiple Parameters in Semitransparent Mediums Based on an Improved Grey Wolf Optimization Algorithm. *Processes* **2024**, *12*, 1445.
https://doi.org/10.3390/pr12071445

**AMA Style**

Li K, Xie L, Zhou J, Wu X, Ding D, Li C.
Estimation of Multiple Parameters in Semitransparent Mediums Based on an Improved Grey Wolf Optimization Algorithm. *Processes*. 2024; 12(7):1445.
https://doi.org/10.3390/pr12071445

**Chicago/Turabian Style**

Li, Kefu, Lang Xie, Jianhua Zhou, Xiaofang Wu, Ding Ding, and Caibin Li.
2024. "Estimation of Multiple Parameters in Semitransparent Mediums Based on an Improved Grey Wolf Optimization Algorithm" *Processes* 12, no. 7: 1445.
https://doi.org/10.3390/pr12071445