The moth-flame optimization (MFO) algorithm is a novel nature-inspired heuristic paradigm. The main inspiration of this algorithm is the navigation method of moths in nature called transverse orientation. Moths fly in night by maintaining a fixed angle with respect to the moon, a very effective mechanism for travelling in a straight line for long distances. However, these fancy insects are trapped in a spiral path around artificial lights. Aiming at the phenomenon that MFO algorithm has slow convergence and low precision, an improved version of MFO algorithm based on Lévy-flight strategy, which is named as LMFO, is proposed. Lévy-flight can increase the diversity of the population against premature convergence and make the algorithm jump out of local optimum more effectively. This approach is helpful to obtain a better trade-off between exploration and exploitation ability of MFO, thus, which can make LMFO faster and more robust than MFO. And a comparison with ABC, BA, GGSA, DA, PSOGSA, and MFO on 19 unconstrained benchmark functions and 2 constrained engineering design problems is tested. These results demonstrate the superior performance of LMFO.
Optimization is a process of finding the best possible solution(s) for a given problem. In real world, many problems can be viewed as optimization problems. Since the complexity of problems increases, the need for new optimization techniques becomes more evident than before. Over the past several decades, some kinds of methods have been proposed to solve optimization problems and have made great progress. For example, mathematical optimization techniques used to be the only tool for optimizing problems before the proposal of heuristic optimization techniques. However, these methods need to know the property of optimization problem, such as continuity or differentiability. In recent years, metaheuristic optimization algorithms have become more and more popular in optimization techniques. Some popular algorithms in this field are Genetic Algorithms (GA) [
Moth-flame optimization (MFO) [
We know that Lévy-flight [
The rest of the paper is organized as follows: Section
In this section, a background about the moth-flame optimization algorithm and Lévy-flight will be provided briefly.
Moth-flame optimization [
In the MFO algorithm, the set of moths is represented in a matrix
The MFO algorithm is a three-tuple that approximates the global optimal of the optimization problems and defined as follows:
The
The
With
while
end
After the initialization, the
Any types of spiral can be utilized here subject to the following conditions: Spiral’s initial point should start from the moth. Spiral’s final point should be the position of the flame. Fluctuation of the range of spiral should not exceed the search space.
Considering these points, a logarithmic spiral is defined for the MFO algorithm as follows:
Equation (
A question that may rise here is that the position updating in (
The gradual decrement in number of flames balances exploration and exploitation of the search space. After all, the general steps of the
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
As described in Algorithm
Note that the Quicksort method is utilized in MFO and the sort’s computational complexity is
Lévy-flight was originally introduced by the French mathematician in 1937 named Paul Lévy. Lévy-flight is a statistical description of motion that extends beyond the more traditional Brownian motion discovered over one hundred years earlier. A diverse range of both natural and artificial phenomena are now being described in terms of Lévy statistics [
Generally speaking, animals looking for food is random, from one place to another place. A large number of studies have shown that flight behavior of many animals and insects has demonstrated the typical characteristics of randomness. However, the choice of the direction relies only on a mathematical model [
In order to increase the diversity of population against premature convergence and accelerate the convergence speed, this paper proposes an improved Lévy-flight moth-flame optimization (LMFO) algorithm. Lévy-flight has the prominent properties to increase the diversity of population, sequentially, which can make the algorithm effectively jump out of the local optimum. In other words, this approach is beneficial to obtain a better trade-off between the exploration and exploitation ability of MFO. So, we let each moth perform once Lévy-flight using (
Formula (
To sum up, global search ability of the proposed algorithm is strengthened using random walk with Lévy-flight to eliminate the weakness of MFO, its being trapped in local minimum is prevented, and it is observed to give more successful results particularly for unimodal and multimodal benchmark functions. Because of these features, the proposed algorithm has potential to provide superior performance compared to MFO. In following section, all kinds of benchmark functions are hired to verify the effectiveness of the proposed algorithm. The main steps of Lévy-flight moth-flame optimization can be simply presented in Algorithm
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
All the algorithms are tested in MATLAB R2012a (7.14) and numerical experiment is set up on Intel Core (TM) i5-4590 Processor, 3.30 GHz, 4 GB RAM, running on Windows 7.
It is common in this field to benchmark the performance of algorithm on a set of mathematical functions with known global optimal. The same process is followed, in which nineteen standard benchmark functions are employed from the literature [
Unimodal benchmark functions.
Name | Function | Range | Dim |
|
---|---|---|---|---|
Sphere |
|
|
200 | 0 |
Schwefel’s 2.22 |
|
|
200 | 0 |
Schwefel’s 1.2 |
|
|
200 | 0 |
Schwefel’s 2.21 |
|
|
200 | 0 |
Rosenbrock |
|
|
200 | 0 |
Step |
|
|
200 | 0 |
Quartic |
|
|
200 | 0 |
X.S.Yang-7 |
|
|
200 | 0 |
Multimodal benchmark functions.
Name | Function | Range | Dim |
|
---|---|---|---|---|
Rastrigin |
|
|
200 | 0 |
Ackley |
|
|
200 | 0 |
Griewank |
|
|
200 | 0 |
Penalized 1 |
|
|
200 | 0 |
Penalized 2 |
|
|
200 | 0 |
Alpine |
|
|
200 | 0 |
Zakharov |
|
|
200 | 0 |
Fixed-dimension multimodal benchmark functions.
Name | Function | Range | Dim |
|
---|---|---|---|---|
Goldstein-Price |
|
|
2 | 3 |
Drop Wave |
|
|
2 | −1 |
Schaffer’s F6 |
|
|
2 | −1 |
Easom |
|
|
2 | −1 |
Heuristic algorithms are stochastic optimization techniques, and therefore they have to be run more than 10 times for generating meaningful statistical results. The best obtained solution in the last iteration is calculated as the metrics of performance. The same method is selected to generate and report the results over 30 independent runs. However, average and standard deviation only compare the overall performance of algorithms.
To explore the performance of the proposed LMFO algorithm, some of the recent and well-known algorithms in the literature are chosen: ABC [
In this paper, Best, Mean, Worst, and Std represent the optimal fitness value, mean fitness value, worst fitness value, and standard deviation, respectively. Experimental results are listed in Tables
Results of unimodal benchmark functions.
Benchmark function | Result | Algorithm | ||||||
---|---|---|---|---|---|---|---|---|
ABC | BA | GGSA | DA | PSOGSA | MFO | LMFO | ||
|
Best | 19896.78 | 365717.3 | 16278.78 | 1513.435 | 34835.45 | 140219.4 |
|
Worst | 36049.23 | 456284 | 24262.84 | 52511.62 | 142227.4 | 240000.7 |
|
|
Mean | 26677.04 | 414807.8 | 19224.04 | 24716.82 | 94735.12 | 185588.4 |
|
|
Std | 4640.383 | 20515.24 | 1852.355 | 11046.55 | 25051.92 | 24177.67 |
|
|
|
||||||||
|
Best | 108.0735 | 1.83 |
149.7263 | 26.27986 | 637.56 | 420.5512 |
|
Worst | 186.4672 | 3.9 |
189.8577 | 255.79923 | 8.06 |
722.4955 |
|
|
Mean | 140.2811 | 1.45 |
165.8829 | 137.88594 | 2.69 |
560.1173 |
|
|
Std | 14.5252 | 7.06 |
9.599018 | 56.674292 | 1.47 |
61.66749 |
|
|
|
||||||||
|
Best | 582404 | 823783.5 | 160277.3 | 69110.62 | 344077.3 | 462517.7 |
|
Worst | 920006.2 | 4617625 | 1020727 | 906356.8 | 891332.5 | 1094366 |
|
|
Mean | 757278.3 | 1572330 | 434895.9 | 342508.6 | 541491.3 | 772538.3 |
|
|
Std | 72837.78 | 837152.1 | 190865.2 | 203490.9 | 142131.3 | 170946.5 |
|
|
|
||||||||
|
Best | 94.265 | 87.48997 | 20.44273 | 27.0486 | 76.25401 | 95.26014 |
|
Worst | 97.40102 | 92.12282 | 31.30115 | 59.69617 | 98.74303 | 98.15999 |
|
|
Mean | 95.8655 | 90.09865 | 27.12328 | 41.97914 | 95.69831 | 97.03653 |
|
|
Std | 0.778229 | 1.330154 | 2.551274 | 7.80909 | 5.90858 | 0.687621 |
|
|
|
||||||||
|
Best | 16111660 | 88328339 | 2399953 | 4717141 | 2565682 | 3.61 |
|
Worst | 94041761 | 1.59 |
5103632 | 47976603 | 4.82 |
8.92 |
|
|
Mean | 41344878 | 1.33 |
3740872 | 18257995 | 1.35 |
6.14 |
|
|
Std | 17567952 | 18028179 | 643192.1 | 10933433 | 1.24 |
1.41 |
|
|
|
||||||||
|
Best | 17165.43 | 384155.3 | 14899 | 4422.437 | 33665.58 | 136678.5 |
|
Worst | 30743.79 | 453905.5 | 24873 | 63473.4 | 109123.6 | 228511.5 |
|
|
Mean | 23922.99 | 417903 | 19220.47 | 24167.16 | 76561.32 | 180669.8 |
|
|
Std | 3074.406 | 17001.78 | 2314.721 | 12441.1 | 19018.55 | 24671.19 |
|
|
|
||||||||
|
Best | 51.88092 | 0.400252 | 6.492429 | 9.23801 | 11.89278 | 1181.991 |
|
Worst | 299.2207 | 0.744656 | 14.77883 | 181.7351 | 71.66514 | 2686.255 |
|
|
Mean | 157.1676 | 0.551746 | 9.090199 | 52.06488 | 19.83666 | 1908.788 |
|
|
Std | 58.73575 | 0.072809 | 1.742479 | 39.34587 | 10.47516 | 360.7708 |
|
|
|
||||||||
|
Best | 181.8735 | 172.1645 | 28.45978 | 14.59629 | 130.6204 | 168.6843 |
|
Worst | 198.3101 | 200.1905 | 40.21054 | 80.16251 | 169.9275 | 202.2427 |
|
|
Mean | 190.2888 | 188.036 | 33.67195 | 43.38706 | 153.398 | 187.9161 |
|
|
Std | 4.168661 | 6.867959 | 2.232835 | 13.6498 | 10.91374 | 7.601635 |
|
Results of multimodal benchmark functions.
Benchmark function | Result | Algorithm | ||||||
---|---|---|---|---|---|---|---|---|
ABC | BA | GGSA | DA | PSOGSA | MFO | LMFO | ||
|
Best | 575.1608 | 1363.369 | 1563.422 | 730.92416 | 922.1832 | 1769.441 |
|
Worst | 709.019 | 1810.465 | 1867.616 | 1890.9545 | 1476.272 | 2125.69 |
|
|
Mean | 653.9316 | 1630.768 | 1752.838 | 1367.3899 | 1231.578 | 1951.426 |
|
|
Std | 36.46425 | 100.7203 | 81.02372 | 280.04565 | 117.1903 | 79.01827 |
|
|
|
||||||||
|
Best | 12.54258 | 19.20527 | 10.52907 | 8.666612 | 19.23309 | 19.92131 |
|
Worst | 14.8045 | 19.9564 | 11.89042 | 14.75436 | 19.96677 | 20.01897 |
|
|
Mean | 13.82113 | 19.73614 | 11.21024 | 11.88909 | 19.69719 | 19.95433 |
|
|
Std | 0.554162 | 0.260154 | 0.383654 | 1.453704 | 0.31107 | 0.019396 |
|
|
|
||||||||
|
Best | 142.2548 | 4290.404 | 137.6285 | 61.43179 | 654.9388 | 1229.64 |
|
Worst | 316.036 | 5283.435 | 221.9276 | 600.151 | 1631.547 | 2048.454 |
|
|
Mean | 225.3103 | 4997.492 | 171.5404 | 224.0744 | 1251.695 | 1543.04 |
|
|
Std | 51.75854 | 184.8135 | 16.55978 | 104.4042 | 204.6207 | 197.6001 |
|
|
|
||||||||
|
Best | 7393398 | 3.86 |
28.17774 | 1662.384 | 7043398 | 7.03 |
|
Worst | 1.67 |
6.85 |
77685.77 | 23172033 | 2.05 |
1.92 |
|
|
Mean | 58731093 | 5.59 |
11043.76 | 2516012 | 7.28 |
1.3 |
|
|
Std | 37177591 | 81914079 | 20355.03 | 4641204 | 5.15 |
3.01 |
|
|
|
||||||||
|
Best | 19464968 | 1.25 |
646700.5 | 2620391 | 26666517 | 1.68 |
|
Worst | 3.4 |
2.09 |
3319003 | 1.83 |
2.47 |
4.04 |
|
|
Mean | 1.53 |
1.72 |
1684844 | 27183053 | 9.73 |
2.62 |
|
|
Std | 82296565 | 1.89 |
724732.9 | 34759974 | 7.07 |
5.46 |
|
|
|
||||||||
|
Best | 47.28706 | 90.66079 | 103.3283 | 15.38694 | 58.10706 | 129.0595 |
|
Worst | 62.0743 | 168.7346 | 139.0044 | 197.7987 | 107.5327 | 216.9578 |
|
|
Mean | 53.55967 | 119.4603 | 118.7804 | 116.3619 | 80.96392 | 171.3074 |
|
|
Std | 3.438082 | 19.18661 | 8.52589 | 43.5411 | 12.93695 | 23.18654 |
|
|
|
||||||||
|
Best | 5373.354 | 4315.808 | 461.6698 | 1142.924 | 4411.244 | 5741.417 |
|
Worst | 6101.204 | 10167.76 | 3585.345 | 4914.844 | 9429.568 | 11241.25 |
|
|
Mean | 5810.789 | 5283 | 1327.234 | 3451.42 | 6873.222 | 8566.525 |
|
|
Std | 207.7704 | 1065.437 | 640.8766 | 1139.356 | 1451.294 | 1527.992 |
|
Results of fixed-dimension multimodal benchmark functions.
Benchmark function | Result | Algorithm | ||||||
---|---|---|---|---|---|---|---|---|
ABC | BA | GGSA | DA | PSOGSA | MFO | LMFO | ||
|
Best | 3.000547 | 3 | 3 | 3 | 3 |
|
3 |
Worst | 3.052848 | 84.00001 | 30 | 3.179715 | 84 |
|
3.000357 | |
Mean | 3.015989 | 15.6 | 7.144215 | 3.018584 | 8.4 |
|
3.000061 | |
Std | 0.015909 | 25.30177 | 9.250619 | 0.049376 | 20.55036 |
|
7.19 | |
|
||||||||
|
Best | −1 | −1 | −1 | −1 | −1 | −1 |
|
Worst | −0.98844 | −0.36913 | −0.93625 | −0.78575 | −0.93625 | −0.93625 |
|
|
Mean | −0.99731 | −0.71344 | −0.949 | −0.94932 | −0.9915 | −0.96175 |
|
|
Std | 0.003196 | 0.175563 | 0.025938 | 0.053641 | 0.022043 | 0.031767 |
|
|
|
||||||||
|
Best | −0.99909 | −0.99028 | −1 | −1 | −1 | −1 |
|
Worst | −0.98981 | −0.54822 | −0.99028 | −0.92181 | −0.99028 | −0.99028 |
|
|
Mean | −0.99095 | −0.71504 | −0.99286 | −0.98153 | −0.99093 | −0.99061 |
|
|
Std | 0.002023 | 0.126856 | 0.004349 | 0.02047 | 0.002465 | 0.001774 |
|
|
|
||||||||
|
Best | −1 | −1 | −1 | −1 |
|
|
−1 |
Worst | −0.99991 | −8.1 |
−8.1 |
−0.95227 |
|
|
−0.99918 | |
Mean | −0.99999 | −0.80002 | −0.92221 | −0.99656 |
|
|
−0.99973 | |
Std | 2.03 |
0.406805 | 0.257944 | 0.010494 |
|
|
0.000229 |
Due to the stochastic nature of the algorithms, statistical tests should be conducted to confirm the significance of the results [
Results of
|
|
|
|
|
|
|
| |
---|---|---|---|---|---|---|---|---|
ABC versus LMFO | 3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
BA versus LMFO | 3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
GGSA versus LMFO | 3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
DA versus LMFO | 3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
PSOGSA versus LMFO | 3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
MFO versus LMFO | 3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
3.02 |
Results of
|
|
|
|
|
|
| |
---|---|---|---|---|---|---|---|
ABC versus LMFO | 1.21 |
1.21 |
1.21 |
3.02 |
3.02 |
3.02 |
3.02 |
BA versus LMFO | 1.21 |
1.21 |
1.21 |
3.02 |
3.02 |
3.02 |
3.02 |
GGSA versus LMFO | 1.21 |
1.21 |
1.21 |
3.02 |
3.02 |
3.02 |
3.02 |
DA versus LMFO | 1.21 |
1.21 |
1.21 |
3.02 |
3.02 |
3.02 |
3.02 |
PSOGSA versus LMFO | 1.21 |
1.21 |
1.21 |
3.02 |
3.02 |
3.02 |
3.02 |
MFO versus LMFO | 1.21 |
1.21 |
1.21 |
3.02 |
3.02 |
3.02 |
3.02 |
Results of
|
|
|
| |
---|---|---|---|---|
ABC versus LMFO | 3.02 |
1.21 |
1.21 |
5.07 |
BA versus LMFO | 7.29 |
1.21 |
1.21 |
6.77 |
GGSA versus LMFO | 7.70 |
8.81 |
5.13 |
2.48 |
DA versus LMFO | 5.19 |
4.56 |
1.45 |
2.96 |
PSOGSA versus LMFO | 7.91 |
4.18 |
7.15 |
1.21 |
MFO versus LMFO | 2.56 |
6.89 |
1.17 |
1.21 |
The unimodal benchmarks functions have only one global minimum and there are no local minima for them. Therefore, these kinds of functions are very suitable for benchmarking the convergence capability of algorithms. According to the results of Table
Figures
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
In contrast to the unimodal benchmark functions, multimodal benchmark functions have many local minima with the number increasing exponentially with dimension. This makes them suitable for benchmarking the exploration ability of an algorithm. So, the final results are more important because these benchmark functions can reflect the ability of the algorithm to escape from poor local optima and obtain the global optimum. The statistical results of the algorithms on multimodal benchmark functions are presented in Table
Seen from Table
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
For fixed-dimension multimodal benchmark functions with only a few local minima, the dimensions of the multimodal benchmark functions are also small. Under such circumstances, it is difficult to judge the performance of individual algorithm. The major difference compared with multimodal functions is that fixed-dimension multimodal functions appear to be simpler because of their low dimensions and a smaller number of local minima. In this experiment, the results of Best, Worst, Mean, and Std values of fixed-dimension multimodal benchmark functions are summarized in Table
In addition, the convergence rate of LMFO on the fixed-dimension benchmark functions with 2-dim can be shown in Figures
The convergence curves for
The convergence curves for
The convergence curves for
The convergence curves for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Standard deviation for
Overall, the results from Tables
Since constraints are one of the major challenges in solving real problems and the main objective of designing the LMFO algorithm is to solve real problems, two constrained real engineering problems are employed in the next section to further investigate the performance of the MFO algorithm and provide a comprehensive study.
In this section, a set of two engineering problems (welded beam design and speed reducer design) is solved so as to further testify the performance of the proposed algorithm. There are some inequality constraints in real problems, so the LMFO algorithm should be capable of dealing with them during optimization. Several methods have been applied to handle constraints in the literature: penalty function, special operators, repaired algorithms, separation of objectives and constraints, and hybrid methods [
The objective is to evaluate the optimal fabrication cost of a welded beam as shown in Figure
Structure of welded beam design.
This problem has four variables that are thickness of weld (
Mirjalili tried to solve this problem using MFO [
Comparison results of the welded beam design problem.
Algorithm | Optimum variables | Optimal cost | |||
---|---|---|---|---|---|
|
|
|
| ||
|
|
|
|
|
|
MFO [ |
0.2057 | 3.4703 | 9.0364 | 0.2057 | 1.72452 |
GGSA [ |
0.215917 | 3.314955 | 8.896195 | 0.215917 | 1.770829 |
GA (Coello Coello) [ |
N/A | N/A | N/A | N/A | 1.8245 |
GA (Deb) [ |
N/A | N/A | N/A | N/A | 2.3800 |
GA (Deb) [ |
0.2489 | 6.1730 | 8.1789 | 0.2533 | 2.4331 |
HS (Lee and Geem) [ |
0.2442 | 6.2231 | 8.2915 | 0.2443 | 2.3807 |
Random [ |
0.4575 | 4.7313 | 5.0853 | 0.6600 | 4.1185 |
Simplex [ |
0.2792 | 5.6256 | 7.7512 | 0.2796 | 2.5307 |
David [ |
0.2434 | 6.2552 | 8.2915 | 0.2444 | 2.3841 |
APPROX [ |
0.2444 | 6.2189 | 8.2915 | 0.2444 | 2.3815 |
The results of Table
The objective function of this problem is to minimize the total weight of the speed reducer as illustrated in Figure
Structure of speed reducer design.
This problem has also been popular among researchers and optimized in many studies. The heuristic algorithms that have been employed to optimize this problem are Akhtar et al. [
Comparison results of the speed reducer design problem.
Algorithm | Optimal values for variables | Optimum weight | ||||||
---|---|---|---|---|---|---|---|---|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
Akhtar et al. [ |
3.506122 | 0.700006 | 17 | 7.549126 | 7.85933 | 3.365576 | 5.289773 | 3008.08 |
Mezura-Montes et al. [ |
3.506163 | 0.700831 | 17 | 7.460181 | 7.962143 | 3.3629 | 5.3090 | 3025.005 |
CS [ |
3.5015 | 0.7 | 17 | 7.6050 | 7.8181 | 3.3520 | 5.2875 | 3000.981 |
HCPS [ |
3.5 | 0.7 | 17 | 7.3 | 7.71532 | 3.350215 | 5.286654 | 2994.47107 |
SCA [ |
3 |
0 |
17 | 7 |
7 |
3 |
5 |
2994 |
( |
3 |
0 |
17 | 7 |
7 |
3 |
5 |
2996 |
ABC [ |
3 |
0 |
17 | 7 |
7 |
3 |
5 |
2997 |
In this paper, an improved version of MFO algorithm based on Lévy-flight strategy, which is named as LMFO, is proposed. In order to benchmark the performance of LMFO, nineteen unconstrained benchmark functions and two constrained engineering design problems were conducted.
According to the values of Best, Worst, Mean, and Std and
As we can see in Section
In Section
In our study, nineteen benchmark functions have been applied to evaluate the performance of LMFO. We also test our proposed method on the real-world engineering problems. Moreover, we will compare LMFO with other optimization algorithms.
Due to the limited performance of MFO, Lévy-flight strategy has been introduced into the standard MFO to develop a novel Lévy-flight moth-flame optimization algorithm for optimization problems. As shown in Section
Furthermore, this paper also considers solving two classical engineering problems by using the LMFO algorithm. The high level of exploration and exploitation of this algorithm were the motivations for this study. The comparative results in Section
For future works, two research directions can be recommended. Firstly, we are going to apply the LMFO to solve more real-world engineering problems. Secondly, it is recommended to develop binary and multiobjective versions of the MFO algorithm.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by National Science Foundation of China under Grants no. 61463007 and 6153008.