December 2018, 8(4): 461-479. doi: 10.3934/naco.2018029

Weighted vertices optimizer (WVO): A novel metaheuristic optimization algorithm

Department of electrical and computer engineering, University of Tabriz, Tabriz, Iran

Received  August 2017 Revised  March 2018 Published  September 2018

This paper introduces a novel optimization algorithm that is based on the basic idea underlying the bisection root-finding method in mathematics. The bisection method is modified for use as an optimizer by weighting each agent or vertex, and the algorithm developed from this process is called the weighted vertices optimizer (WVO). For exploitation and exploration, both swarm intelligence and evolution strategy are used to improve the accuracy and speed of WVO, which is then compared with six other popular optimization algorithms. Results confirm the superiority of WVO in most of the test functions.

Citation: Soheil Dolatabadi. Weighted vertices optimizer (WVO): A novel metaheuristic optimization algorithm. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 461-479. doi: 10.3934/naco.2018029
References:
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M. Z. AliN. H. AwadP. N. Suganthan and R. G. Reynolds, A modified cultural algorithm with a balanced performance for the differential evolution frameworks, Knowledge-Based Systems, 111 (2016), 73-86. doi: 10.1016/j.knosys.2016.08.005.

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E. Atashpaz-Gargari and C. Lucas, Imperialist Competitive Algorithm: An algorithm for optimization inspired by imperialistic competition, IEEE Congress on Evolutionary Computation, Singapore, 2007. doi: 10.1109/CEC.2007.4425083.

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R. L. Burden and J. D. Faires, Numerical Analysis, 3rd edition, Prindle, Weber and Schmidt, 1985.

[4]

M. DorigoV. Maniezzo and A. Colorni, Ant system: optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 26 (1996), 29-41. doi: 10.1109/3477.484436.

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R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 1995. doi: 10.1109/MHS.1995.494215.

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L. J. Fogel, A. J. Owens and M. J. Walsh, Artificial Intelligence through Simulated Evolution, John Wiley and Sons, 1966. doi: 10.1109/9780470544600.ch7.

[7]

D. E. Goldberg and J. H. Holland, Genetic algorithms and machine learning, Machine Learning, 3 (1988), 95-99.

[8]

Z.-L. Gaing, A particle swarm optimization approach for optimum design of PID controller in AVR system, IEEE Transactions on Energy Conversion, 19 (2004), 384-391. doi: 10.1109/TEC.2003.821821.

[9]

Z. W. GeemJ. H. Kim and G. Loganathan, A New Heuristic Optimization Algorithm: Harmony Search, Simulation, 76 (2001), 60-68.

[10]

D. Karaboga and B. Basturk, Artificial Bee Colony (ABC) Optimization Algorithm for Solving Constrained Optimization Problems, International Fuzzy Systems Association World Congress, 2007. doi: 10.1007/s10898-007-9149-x.

[11]

E.-H. Kenane, F. Djahli and C. Dumond, A novel Modified Invasive Weeds Optimization for linear array antennas nulls control, 4th International Conference on Electrical Engineering (ICEE), Boumerdes, Algeria, 2015. doi: 10.1109/INTEE.2015.7416784.

[12]

J. Liang, P. Suganthan and K. Deb, Novel composition test functions for numerical global optimization, Swarm Intelligence Symposium, Pasadena, CA, USA, 2005. doi: 10.1109/SIS.2005.1501604.

[13]

A. Mehrabian and C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecological Informatics, 1 (2006), 355-366. doi: 10.1016/B978-0-12-416743-8.00001-4.

[14]

R. G. Reynolds, An Introduction to Cultural Algorithms, 3rd Annual Conference on Evolutionary Programming, 1994.

[15]

W. XiangM. AnY. LiR. He and J. Zhang, An improved global-best harmony search algorithm for faster optimization, Expert Systems with Applications, 41 (2014), 788-803. doi: 10.1016/j.eswa.2014.03.016.

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X.-S. Yang, Nature-Inspired Metaheuristic Algorithms: Second Edition, Luniver press, 2010.

[17]

A. E. M. Zavala, A. H. Aguirre and E. R. V. Diharce, Constrained optimization via particle evolutionary swarm optimization algorithm (PESO), 7th annual conference on Genetic and evolutionary computation, Washington DC, USA, 2005.

show all references

References:
[1]

M. Z. AliN. H. AwadP. N. Suganthan and R. G. Reynolds, A modified cultural algorithm with a balanced performance for the differential evolution frameworks, Knowledge-Based Systems, 111 (2016), 73-86. doi: 10.1016/j.knosys.2016.08.005.

[2]

E. Atashpaz-Gargari and C. Lucas, Imperialist Competitive Algorithm: An algorithm for optimization inspired by imperialistic competition, IEEE Congress on Evolutionary Computation, Singapore, 2007. doi: 10.1109/CEC.2007.4425083.

[3]

R. L. Burden and J. D. Faires, Numerical Analysis, 3rd edition, Prindle, Weber and Schmidt, 1985.

[4]

M. DorigoV. Maniezzo and A. Colorni, Ant system: optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 26 (1996), 29-41. doi: 10.1109/3477.484436.

[5]

R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 1995. doi: 10.1109/MHS.1995.494215.

[6]

L. J. Fogel, A. J. Owens and M. J. Walsh, Artificial Intelligence through Simulated Evolution, John Wiley and Sons, 1966. doi: 10.1109/9780470544600.ch7.

[7]

D. E. Goldberg and J. H. Holland, Genetic algorithms and machine learning, Machine Learning, 3 (1988), 95-99.

[8]

Z.-L. Gaing, A particle swarm optimization approach for optimum design of PID controller in AVR system, IEEE Transactions on Energy Conversion, 19 (2004), 384-391. doi: 10.1109/TEC.2003.821821.

[9]

Z. W. GeemJ. H. Kim and G. Loganathan, A New Heuristic Optimization Algorithm: Harmony Search, Simulation, 76 (2001), 60-68.

[10]

D. Karaboga and B. Basturk, Artificial Bee Colony (ABC) Optimization Algorithm for Solving Constrained Optimization Problems, International Fuzzy Systems Association World Congress, 2007. doi: 10.1007/s10898-007-9149-x.

[11]

E.-H. Kenane, F. Djahli and C. Dumond, A novel Modified Invasive Weeds Optimization for linear array antennas nulls control, 4th International Conference on Electrical Engineering (ICEE), Boumerdes, Algeria, 2015. doi: 10.1109/INTEE.2015.7416784.

[12]

J. Liang, P. Suganthan and K. Deb, Novel composition test functions for numerical global optimization, Swarm Intelligence Symposium, Pasadena, CA, USA, 2005. doi: 10.1109/SIS.2005.1501604.

[13]

A. Mehrabian and C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecological Informatics, 1 (2006), 355-366. doi: 10.1016/B978-0-12-416743-8.00001-4.

[14]

R. G. Reynolds, An Introduction to Cultural Algorithms, 3rd Annual Conference on Evolutionary Programming, 1994.

[15]

W. XiangM. AnY. LiR. He and J. Zhang, An improved global-best harmony search algorithm for faster optimization, Expert Systems with Applications, 41 (2014), 788-803. doi: 10.1016/j.eswa.2014.03.016.

[16]

X.-S. Yang, Nature-Inspired Metaheuristic Algorithms: Second Edition, Luniver press, 2010.

[17]

A. E. M. Zavala, A. H. Aguirre and E. R. V. Diharce, Constrained optimization via particle evolutionary swarm optimization algorithm (PESO), 7th annual conference on Genetic and evolutionary computation, Washington DC, USA, 2005.

Figure 1.  a) the bisection method b) the raw concept of WVO algorithm using two vertices
Figure 2.  the graphical description of proposed method for five vertices mode
Figure 3.  the flowchart of WVO algorithm
Figure 4.  2D plot of test functions
Figure 5.  3D sketch of Shurb's function (F1)
Figure 6.  the positions of WVO vertices in first iteration
Figure 7.  A) the positions of vertices in second iteration B)the positions of vertices in 5th iteration C)the positions of vertices in 9th iteration D)the positions of vertices in 13th iteration
Figure 8.  the cost value versus iteration
Figure 9.  cost value of F2 function in each iteration
Figure 10.  cost value of F3 function in each iteration
Figure 11.  cost value of F4 function in each iteration
Figure 12.  cost value of F5 function in each iteration
Figure 13.  cost value of F6 function in each iteration
Figure 14.  block diagram of AVR along with PID controller [17]
Figure 15.  cost value of each method for AVR's PID
Figure 16.  the step response of without PID controller and with optimized gains
Figure 17.  the logarithmic plot of cost function versus iteration for F5 function and different N${}_{V}$
Figure 18.  the logarithmic plot of cost function versus iteration for F6 function and different N${}_{V}$
Table 1.  parameters of WVO
C${}_{F}$ C${}_{B}$ C${}_{G}$ N${}_{V}$ V${}_{Speed}$ W${}_{GB}$ W${}_{GW}$
0.6 0.3 0.085 2 0.6 10 1
C${}_{F}$ C${}_{B}$ C${}_{G}$ N${}_{V}$ V${}_{Speed}$ W${}_{GB}$ W${}_{GW}$
0.6 0.3 0.085 2 0.6 10 1
Table 2.  benchmark optimization functions
ID Name Function Bound Global Min
F1 Shubert $(\sum^5_{i=1}{icos(\left(i+1\right)x_1+1)})$
$(\sum^5_{i=1}{icos(\left(i+1\right)x_2+1)})$
${\left[\text{-2.12.2.12}\right]}^{\text{2}}$ -186.7309
F2 Six-hump camel back $\left(4-2.1x^2_1+\frac{x^4_1}{3}\right)x^2_1 $$+x_1x_2+(-4+4x^2_2)x^2_2$ ${\left[\text{-5.5}\right]}^{\text{2}}$ -1.0316285
F3 Sphere $\sqrt{\sum^D_{i=1}{x^2_i}}$ ${\left[\text{-32.32}\right]}^{\text{10}}$ 0
F4 Ackley $-A\times exp\left(-0.02\sqrt{\frac{\sum^D_{i=1}{x^2_i}}{D}}\right)$ $-{\text{exp} \left(\frac{\sum^D_{i=1}{{\text{cos} \left(2\pi x_i\right)\ }}}{D}\right)\ }+A\ ;A=20$ ${\left[\text{-100.100}\right]}^{\text{10}}$ 0
F5 Griewank $1+\frac{1}{4000}\sum^D_{i=1}{x^2_i-\prod^D_{i=1}{\text{cos}\text{}(\frac{x_i}{\sqrt{i}})}}$ ${\left[\text{-600.600}\right]}^{\text{10}}$ 0
F6 Rastrigin $10D+\sum^D_{i=1}{(x^2_i-10\text{cos}\text{}(2\pi x_i))}$ ${\left[\text{-5.12.5.12}\right]}^{\text{10}}$ 0
ID Name Function Bound Global Min
F1 Shubert $(\sum^5_{i=1}{icos(\left(i+1\right)x_1+1)})$
$(\sum^5_{i=1}{icos(\left(i+1\right)x_2+1)})$
${\left[\text{-2.12.2.12}\right]}^{\text{2}}$ -186.7309
F2 Six-hump camel back $\left(4-2.1x^2_1+\frac{x^4_1}{3}\right)x^2_1 $$+x_1x_2+(-4+4x^2_2)x^2_2$ ${\left[\text{-5.5}\right]}^{\text{2}}$ -1.0316285
F3 Sphere $\sqrt{\sum^D_{i=1}{x^2_i}}$ ${\left[\text{-32.32}\right]}^{\text{10}}$ 0
F4 Ackley $-A\times exp\left(-0.02\sqrt{\frac{\sum^D_{i=1}{x^2_i}}{D}}\right)$ $-{\text{exp} \left(\frac{\sum^D_{i=1}{{\text{cos} \left(2\pi x_i\right)\ }}}{D}\right)\ }+A\ ;A=20$ ${\left[\text{-100.100}\right]}^{\text{10}}$ 0
F5 Griewank $1+\frac{1}{4000}\sum^D_{i=1}{x^2_i-\prod^D_{i=1}{\text{cos}\text{}(\frac{x_i}{\sqrt{i}})}}$ ${\left[\text{-600.600}\right]}^{\text{10}}$ 0
F6 Rastrigin $10D+\sum^D_{i=1}{(x^2_i-10\text{cos}\text{}(2\pi x_i))}$ ${\left[\text{-5.12.5.12}\right]}^{\text{10}}$ 0
Table 3.  cost value of each optimization algorithm in 78th iteration
Method Cost value
WVO 1.78 E-15
PSO 7.36 E-4
GA 9.17 E-5
IWO 2.157
HS 2.56E-3
CA 5.93E-6
mIWO 1.23 E-5
mHS 8.69 E-9
mCA 5.12 E-10
Method Cost value
WVO 1.78 E-15
PSO 7.36 E-4
GA 9.17 E-5
IWO 2.157
HS 2.56E-3
CA 5.93E-6
mIWO 1.23 E-5
mHS 8.69 E-9
mCA 5.12 E-10
Table 4.  the performance of each optimization method
WVO PSO GA IWO HS CA mIWO mHS mCA
F1 N1 13 45 27 184 88 47 132 44 23
B2 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309
R3 1 5 3 9 7 6 8 4 2
F2 N 19 50 24 54 60 24 39 45 20
B -1.03163 -1.03163 -1.03163 -1.03162 -1.03162 -1.03162 -1.03162 -1.03162 -1.03162
R 1 7 3 8 9 3 5 6 2
F3 N 77 1158 1500 1498 1501 1500 1382 1500 1500
B 1.71 E-58 0 7.64 E-128 2.43 E-6 2 E-10 1.89 E-143 1.13 E-12 3.12E-8 0
R 5 1 4 9 7 3 6 8 2
F4 N 67 233 199 198 1501 1336 173 1500 1363
B 8.88 E-16 4.44 E-15 20 20 6.2 E-5 20.29 6.13 E-3 2.52 E-8 1.23 E-4
R 1 2 7 8 4 9 6 3 5
F5 N 41 116 251 744 1501 468 632 432 321
B 0 0.09747 0 0.90271 1.52 E-8 20.25 2.38 E-3 1.58 E-32 9.78 E-2
R 1 7 2 8 4 9 5 3 6
F6 N 30 119 418 200 1501 1130 123 1245 1351
B 0 5.9697 0 8.9552 4.06 E-10 22.94947 3.25 E-9 3.15 E-21 4.65 E-3
R 1 7 2 8 4 9 5 3 6
$\boldsymbol{\mathit{\boldsymbol{\sum}}}$ R 1 5 2 9 6 8 6 4 3
   1N:Number of iteration - 2B:Best cost value - 3R:Rank
WVO PSO GA IWO HS CA mIWO mHS mCA
F1 N1 13 45 27 184 88 47 132 44 23
B2 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309
R3 1 5 3 9 7 6 8 4 2
F2 N 19 50 24 54 60 24 39 45 20
B -1.03163 -1.03163 -1.03163 -1.03162 -1.03162 -1.03162 -1.03162 -1.03162 -1.03162
R 1 7 3 8 9 3 5 6 2
F3 N 77 1158 1500 1498 1501 1500 1382 1500 1500
B 1.71 E-58 0 7.64 E-128 2.43 E-6 2 E-10 1.89 E-143 1.13 E-12 3.12E-8 0
R 5 1 4 9 7 3 6 8 2
F4 N 67 233 199 198 1501 1336 173 1500 1363
B 8.88 E-16 4.44 E-15 20 20 6.2 E-5 20.29 6.13 E-3 2.52 E-8 1.23 E-4
R 1 2 7 8 4 9 6 3 5
F5 N 41 116 251 744 1501 468 632 432 321
B 0 0.09747 0 0.90271 1.52 E-8 20.25 2.38 E-3 1.58 E-32 9.78 E-2
R 1 7 2 8 4 9 5 3 6
F6 N 30 119 418 200 1501 1130 123 1245 1351
B 0 5.9697 0 8.9552 4.06 E-10 22.94947 3.25 E-9 3.15 E-21 4.65 E-3
R 1 7 2 8 4 9 5 3 6
$\boldsymbol{\mathit{\boldsymbol{\sum}}}$ R 1 5 2 9 6 8 6 4 3
   1N:Number of iteration - 2B:Best cost value - 3R:Rank
Table 5.  the understudy composition functions [12]
CF1 CF2 CF3
$f_1, f_2, \dots , f_{10}=F5$ $f_{1-2}\left(x\right)=F4$
$f_{3-4}\left(x\right)=F6$
$f_{5-6}\left(x\right)=F7$
$f_{7-8}\left(x\right)=F5$
$f_{9-10}\left(x\right)=F3$
$f_{1-2}\left(x\right)=F6$
$f_{3-4}\left(x\right)=F7$
$f_{5-6}\left(x\right)=F5$
$f_{7-8}\left(x\right)=F4$
$f_{9-10}\left(x\right)=F3$
CF1 CF2 CF3
$f_1, f_2, \dots , f_{10}=F5$ $f_{1-2}\left(x\right)=F4$
$f_{3-4}\left(x\right)=F6$
$f_{5-6}\left(x\right)=F7$
$f_{7-8}\left(x\right)=F5$
$f_{9-10}\left(x\right)=F3$
$f_{1-2}\left(x\right)=F6$
$f_{3-4}\left(x\right)=F7$
$f_{5-6}\left(x\right)=F5$
$f_{7-8}\left(x\right)=F4$
$f_{9-10}\left(x\right)=F3$
Table 6.  results of optimization algorithms for three CFs
PSO [12] DE [12] GA WVO
CF1 Mean 1.7203 E2 1.4441 E2 1.3451 E2 1.1121 E2
Std. deviation 3.2869 E1 1.9401 E1 1.9142 E1 1.4232 E1
CF2 Mean 3.1430 E2 3.2486 E2 3.2314 E2 3.0021 E2
Std. deviation 2.0006 E1 1.4784 E1 1.8154 E1 1.6823 E1
CF3 Mean 8.3450 E1 1.0789 E1 7.5421 E1 3.8124 E1
Std. deviation 1.0111 E2 2.6040 E0 1.0512 E1 8.5412 E1
PSO [12] DE [12] GA WVO
CF1 Mean 1.7203 E2 1.4441 E2 1.3451 E2 1.1121 E2
Std. deviation 3.2869 E1 1.9401 E1 1.9142 E1 1.4232 E1
CF2 Mean 3.1430 E2 3.2486 E2 3.2314 E2 3.0021 E2
Std. deviation 2.0006 E1 1.4784 E1 1.8154 E1 1.6823 E1
CF3 Mean 8.3450 E1 1.0789 E1 7.5421 E1 3.8124 E1
Std. deviation 1.0111 E2 2.6040 E0 1.0512 E1 8.5412 E1
Table 7.  value of AVR's parameters [17]
Parameter value
K${}_{A}$ 10
${\tau _A}$ 0.1
K${}_{E}$ 1
${\tau _E}$ 0.4
K${}_{G}$ 1
${\tau _G}$ 1
K${}_{R}$ 1
${\tau _R}$ 0.01
Parameter value
K${}_{A}$ 10
${\tau _A}$ 0.1
K${}_{E}$ 1
${\tau _E}$ 0.4
K${}_{G}$ 1
${\tau _G}$ 1
K${}_{R}$ 1
${\tau _R}$ 0.01
Table 8.  obtained values and the result for each optimization method
KP KI KD RT ST
(sec)
OS
(%)
Final
error
(%)
Cost value
WVO 0.600518 0.41376 0.20136 0.3101 0.5013 0.0003 0 3.11706
PSO 0.600532 0.41386 0.20137 0.3141 0.5013 0.0017 0 3.11752
GA 0.610065 0.42965 0.20784 0.3226 0.5005 0.1522 0.018 3.17785
KP KI KD RT ST
(sec)
OS
(%)
Final
error
(%)
Cost value
WVO 0.600518 0.41376 0.20136 0.3101 0.5013 0.0003 0 3.11706
PSO 0.600532 0.41386 0.20137 0.3141 0.5013 0.0017 0 3.11752
GA 0.610065 0.42965 0.20784 0.3226 0.5005 0.1522 0.018 3.17785
Table 9.  effect of C${}_{F}$, C${}_{B}$, C${}_{G}$, W${}_{GB}$ and W${}_{GW}$ on performance of WVO
Function C${}_{F}$ The best cost Iteration C${}_{B}$ The best cost Iteration C${}_{G}$ The best cost Iteration W${}_{GB}$ The best cost Iteration W${}_{GW}$ The best cost Iteration
F5 0.2 3.12E-12 46 0.2 0 43 0.02 0 73 2 0 47 2 0 41
0.4 2.02E-19 42 0.4 0 41 0.04 0 45 5 0 43 5 0 45
0.6 0 42 0.6 0 42 0.06 0 45 10 0 42 10 0 47
0.8 0 42 0.8 2.31E-26 43 0.08 0 41 15 0 42 15 0 47
1 3.12E-30 44 1 8.64E-23 45 0.1 0 53 20 0 43 20 0 48
F6 0.2 2.31E-6 45 0.2 0 41 0.02 2.31E-26 45 2 0 45 2 0 43
0.4 0 42 0.4 0 42 0.04 0 43 5 0 43 5 0 43
0.6 0 42 0.6 1.12E-28 45 0.06 0 43 10 0 43 10 0 43
0.8 0 45 0.8 6.78E-25 49 0.08 0 41 15 0 44 15 0 45
1 0 45 1 1.32E-24 51 0.1 0 44 20 0 44 20 0 48
Function C${}_{F}$ The best cost Iteration C${}_{B}$ The best cost Iteration C${}_{G}$ The best cost Iteration W${}_{GB}$ The best cost Iteration W${}_{GW}$ The best cost Iteration
F5 0.2 3.12E-12 46 0.2 0 43 0.02 0 73 2 0 47 2 0 41
0.4 2.02E-19 42 0.4 0 41 0.04 0 45 5 0 43 5 0 45
0.6 0 42 0.6 0 42 0.06 0 45 10 0 42 10 0 47
0.8 0 42 0.8 2.31E-26 43 0.08 0 41 15 0 42 15 0 47
1 3.12E-30 44 1 8.64E-23 45 0.1 0 53 20 0 43 20 0 48
F6 0.2 2.31E-6 45 0.2 0 41 0.02 2.31E-26 45 2 0 45 2 0 43
0.4 0 42 0.4 0 42 0.04 0 43 5 0 43 5 0 43
0.6 0 42 0.6 1.12E-28 45 0.06 0 43 10 0 43 10 0 43
0.8 0 45 0.8 6.78E-25 49 0.08 0 41 15 0 44 15 0 45
1 0 45 1 1.32E-24 51 0.1 0 44 20 0 44 20 0 48
Table 10.  the effect of N${}_{V}$ value on speed of algorithm
Function N${}_{V}$ The best cost Iteration
F5 2 0 47
3 0 43
4 0 42
5 0 41
10 0 43
15 4.55E-8 116
F5 2 0 45
3 0 43
4 0 43
5 0 44
10 0 44
15 0.406497 116
Function N${}_{V}$ The best cost Iteration
F5 2 0 47
3 0 43
4 0 42
5 0 41
10 0 43
15 4.55E-8 116
F5 2 0 45
3 0 43
4 0 43
5 0 44
10 0 44
15 0.406497 116
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