A precise estimation of isotherm model parameters and selection of isotherms from the measured data are essential for the fate and transport of toxic contaminants in the environment. Nonlinear least-square techniques are widely used for fitting the isotherm model on the experimental data. However, such conventional techniques pose several limitations in the parameter estimation and the choice of appropriate isotherm model as shown in this paper. It is demonstrated in the present work that the classical deterministic techniques are sensitive to the initial guess and thus the performance is impeded by the presence of local optima. A novel solver based on modified artificial bee-colony (MABC) algorithm is proposed in this work for the selection and configuration of appropriate sorption isotherms. The performance of the proposed solver is compared with the other three solvers based on swarm intelligence for model parameter estimation using measured data from 21 soils. Performance comparison of developed solvers on the measured data reveals that the proposed solver demonstrates excellent convergence capabilities due to the superior exploration-exploitation abilities. The estimated solutions by the proposed solver are almost identical to the mean fitness values obtained over 20 independent runs. The advantages of the proposed solver are presented.
The fate and transport of heavy metals in the soils are a serious concern due to their potential impact on the environment. These reactive substances in soils undergo complex interactions based on the soil properties, type of clay minerals, and pore-fluid parameters. The problem of identifying the fate and transport of these substances in soils requires the understanding of various sorption mechanisms of species in the soil environment [
Nature-inspired algorithms such as algorithms based on swarm intelligence (SI) are global search techniques and are widely used for many optimization problems in engineering. Particle swarm optimization (PSO) [
It will be shown that the performance of conventional optimization techniques and PSO algorithm is greatly hampered by the presence of several local optima on the functional terrain. An inverse model based on artificial bee-colony (ABC) optimization method is proposed in this work for selection and configuration of appropriate sorption isotherms using the experimental sorption data. The proposed solver is further improved by introducing several modifications to the ABC algorithm for better performance. The proposed technique is highly robust; it accurately estimates the sorption model parameters and appropriate isotherms to the experimental data.
Sorption of different chemical constituents in soils is routinely performed in the laboratory under controlled temperature, pressure, and pH. The measured sorption data is generally described by either equilibrium or a kinetic retention process [
The Freundlich isotherm is widely used for describing nonideal and reversible adsorption [
The Langmuir model is another popularly used sorption model, originally developed by Langmuir [
The Freundlich-Langmuir (FL) model is a power function based on the assumption of continuously distributed affinity coefficients [
The two-site Langmuir model is based on the assumption that sorption occurs on two types of surface sites, each with different bonding energies. One site contains a high bonding energy and reacts rapidly, while the other contains a lower bonding energy and reacts more slowly [
Accurate estimation of model parameters of (
As the aforementioned theoretical isotherms are nonlinear, optimization techniques are used to estimate the model parameters of a given model isotherm and to determine the appropriate isotherm for a given data. Nonlinear regression techniques such as Levenberg–Marquardt method are widely used in the literature for parameterization [
Gradient-based techniques are the conventional optimization tools for parameter estimation in many engineering applications. The principle behind such techniques is determining an optimum point on the search space where the derivatives of the objective function with respect to the model parameters (
It was in this work that these techniques suffered from several limitations. These conventional algorithms are highly sensitive to the fitness function (error measure) and on the initial guess. Of late, nature-inspired or swarm intelligence search techniques are used as an alternative to the conventional methods for solving several optimization problems in geoenvironmental engineering [
Swarm intelligence (SI) is a field of research which develops computational techniques by taking inspiration from the nature for solving the optimization problems. Genetic algorithms (GA), particle swarm optimization (PSO), and artificial bee colony (ABC) are some of the techniques developed by mimicking the nature for the determination of global optimum solutions to the inverse problems. This paper developed several solvers based on some of the SI techniques for predicting the model parameters of isotherm equations and for determining the appropriate isotherms for the measured data. The following SI techniques were used in developing the solvers.
PSO is a class of stochastic, derivative-free, population-based method. The working principle of PSO method is inspired by the social behaviour of animals such as flocking of birds and schooling of fishes [
PSO is widely used for solving the optimization problems in many engineering disciplines due to the simplicity. Several experiment studies on the application of PSO algorithm on benchmark functions and optimization problems revealed that the algorithm explodes quickly as the first term in the velocity term in (
The classical PSO algorithm and many of the variants suffer from few limitations during the search process. The major drawback with these algorithms is that they converge to suboptimal solutions for several problems due to the application of the same movement update equation (
Equation (
ABC algorithm is a recent addition to the class of global optimization algorithms. ABC algorithm [
The algorithm uses several steps inspired by the collective behaviour of bees for the collection and processing of the nectar. The basic ABC algorithm uses three classes of bees, that is, employed bees, onlooker bees, and scout bees. Employed bees are responsible for locating the nectar sources and sharing this information with the onlookers on the dance area of the hive. Onlooker bees select nectar source based on the quality (objective function) with some probability. Therefore, the probability of higher number of onlookers at better quality of food source is higher. The task of employed and onlooker bees is to explore the search space. Scout bees are similar to the catfishes used to exploit new food sources in the search space by random search process. Further, the employed bees with abandoned food sources become the scout bees. Each food source (a set of model parameters) is a feasible solution of the problem and the nectar amount of a food source represents quality of the solution, that is, the objective function. The quality of the solution is represented by fitness value as given by
Some modifications to the original ABC algorithm have been studied on benchmark functions. The modification of the basic ABC involves modifying the position-update equation [
The exploration-exploitation qualities of the classical ABC algorithm are improved in the present work. The following position-update equation is used for the employed bees in the modified algorithm:
The movement of the employed bees using the proposed position-update equation is based on the resultant direction of best-fit food source (position) and the randomly chosen food source location. The inertia weight and acceleration coefficient improve the exploration process. The position-update equation for onlooker bees is based on (
OBL is based on the utilization of opposition numbers of the current positions in the search space [
The performance of all the SI techniques is based on the tuning parameters. The tuning parameters of these algorithms are very sensitive to the nature of the parametric space [
Observed sorption data of metal substances, that is, Cu, Zn, Ca, and Pb, on various soils were used from the literature for selection and configuration of the described isotherms. Natural soils having various percentages of clay content, different clay minerals, and pH were chosen. The physical properties of the soils as reported in the literature were presented in Table
Description of the experimental data.
# | Soil | Reference | pH | Soil type | CEC | Grain size (%) | ||
---|---|---|---|---|---|---|---|---|
Sand | Silt | Clay | ||||||
1 | Alligator | Buchter et al., 1989 | 4.8 | Clay, Montmorillo-nitic | 30.3 | 5.9 | 39.4 | 54.7 |
2 | McLaren | Selim and Amacher, 1997 | NA | NA | NA | NA | NA | NA |
3 | Cecil | Selim and Amacher, 1997 | 5.4 | Clay, Kaolinitic | 2.4 | 30 | 18.8 | 51.2 |
4 | Kula | Selim and Amacher, 1997 | 5.9 | Sandy Loam | 27 | 66.6 | 32.9 | 0.5 |
5 | Webster | Selim and Amacher, 1997 | 7.6 | Clay Loam | 14.1 | 27.5 | 48.6 | 23.9 |
6 | Lafitte | Selim and Amacher, 1997 | 3.9 | Sandy Loam | 26.9 | 60.7 | 21.7 | 17.6 |
7 | Windsor | Buchter et al., 1989 | 5.3 | Sandy Loam | 2.0 | 76.8 | 20.5 | 2.8 |
8 | Molokai | Buchter et al., 1989 | 6.0 | Clay Loam, Kaolinitc | 11 | 25.7 | 46.2 | 28.2 |
9 | Spodosol | Buchter et al., 1989 | 4.3 | Sand | 2.7 | 90.2 | 6.0 | 3.8 |
10 | Calciorthid | Buchter et al., 1989 | 8.5 | Sandy Loam | 14.7 | 70 | 19.3 | 10.7 |
11 | Hartsells | Bolster and Hornberger, 2007 | NA | Fine Loamy | NA | NA | NA | NA |
12 | Pembroke | Bolster and Hornberger, 2007 | NA | Fine Silty | NA | NA | NA | NA |
13 | Loring | Bolster and Hornberger, 2007 | NA | Fine Silty | NA | NA | NA | NA |
Details of the input data.
# | Reactive metal substance | Soil | Number of data points | Reference |
---|---|---|---|---|
1 | Cu | Alligator | 7 | Selim and Amacher, 1997 |
2 | Cu | McLaren | 5 | Selim and Amacher, 1997 |
3 | Cu | Cecil | 5 | Selim and Amacher, 1997 |
4 | Zn | Alligator | 10 | Selim and Amacher, 1997 |
5 | Zn | Kula | 7 | Selim and Amacher, 1997 |
6 | Zn | Webster | 8 | Selim and Amacher, 1997 |
7 | Pb | Cecil | 6 | Selim and Amacher, 1997 |
8 | Pb | Spodosol | 5 | Selim and Amacher, 1997 |
9 | Pb | Lafitte | 7 | Selim and Amacher, 1997 |
10 | Pb | Alligator | 8 | Selim and Amacher, 1997 |
11 | Cr | Alligator | 15 | Buchter et al., 1989 |
12 | Cr | Windsor | 12 | Buchter et al., 1989 |
13 | Cr | Kula | 10 | Buchter et al., 1989 |
14 | Cd | Molokai | 13 | Buchter et al., 1989 |
15 | Cd | Kula | 10 | Buchter et al., 1989 |
16 | Cd | Windsor | 14 | Buchter et al., 1989 |
17 | Cd | Spodosol | 14 | Buchter et al., 1989 |
18 | Cd | Calciorthid | 7 | Buchter et al., 1989 |
19 | Ca | Hartsells | 6 | Bolster and Hornberger, 2007 |
20 | Ca | Pembroke | 6 | Bolster and Hornberger, 2007 |
21 | Ca | Loring | 6 | Bolster and Hornberger, 2007 |
A number of twenty runs were used to test the performance of the proposed solvers. The number of iterations was fixed to 150 for all the algorithms and for all the isotherms. However, the number of agents was varied based on the number of model parameters, that is, size of the dimensional (parametric) space. The algorithms used 10, 20, and 30 agents for fitting the experimental data with Freundlich isotherm (2-dimensional), Freundlich-Langmuir isotherm (3-dimensional), and two-site Langmuir isotherm (4-dimensional), respectively.
The parameter setting for different algorithms was determined empirically on synthetic test data for which the optima were known. The tuning parameters of both the PSO algorithms, such as acceleration coefficient, inertia factor, and constriction coefficient, are dependent on the given problem [
The colony size in ABC and MABC algorithms was twice the number of agents considered for other algorithms. In other words, the number of employed bees and onlooker bees was the same and equal to the number of agents for other algorithms. In these algorithms, the function was evaluated three times in each iteration. Therefore, the number of iterations was fixed to 50 to use the same number of function evaluations
The pseudocode of the proposed MABC algorithm was given in the Appendix.
The parametric space for the experimental data of Cr sorption on Alligator soil using nonlinear isotherm equations was computed. The parametric spaces were obtained using two-parameter models of Freundlich and Langmuir isotherms (obtained by substituting
Parametric space of Freundlich isotherm for Cr sorption on Alligator soil.
Parametric space of Langmuir isotherm for Cr sorption on Alligator soil.
The performance of gradient-based algorithms is dependent on the user-supplied, initial solution (guess), and the type of error measure. The sensitivity of these algorithms on the initial guess was verified qualitatively for the present problem. The “fmincon” MATLAB® routine, featuring a constrained optimization of a multivariable function using interior-point technique, was used for finding the best isotherm model parameters that provided best-fit theoretical sorption data to the measured data. The interior-point technique was found to have better convergence abilities than trust-region-reflective or any other available techniques. A finite-difference numerical solution was used to determine the first- and second-order derivatives of the objective function with respect to the model parameters. The tolerance value of the objective function was fixed to 1 × 10−8 and a very large value was used for maximum number of iterations which served as the stopping criterion. Different initial guess solutions were used to obtain the converged solutions. The performance of the gradient-based algorithm for fitting two-site Langmuir isotherm on the experimental data of Cd sorption on Molokai soil was presented in Table
Performance of the gradient-based solver on the experimental data of Cd sorption on Molokai soil using two-site Langmuir isotherm.
Initial guess | Predicted solution | Error | ||||||
---|---|---|---|---|---|---|---|---|
| | | | | | | | |
920 | 0.117 | 23.7 | 4.4 | 923.84 | 0.119 | 21.1746 | 4.884 | 0.071872 |
880 | 0.147 | 11.5 | 7 | 880.29 | 0.130 | 19.5152 | 5.054 | 0.072161 |
100 | 0.1 | 10 | 7 | 490.211 | 0.320 | 356.701 | 0.075 | 0.089178 |
100 | 0.1 | 10 | 0.5 | 515.777 | 0.308 | 324.213 | 0.075 | 0.089366 |
100 | 0.1 | 100 | 0.5 | 429.705 | 0.350 | 429.915 | 0.079 | 0.088687 |
Model parameters of Freundlich, Freundlich-Langmuir, and two-site Langmuir isotherms on the experimental data were predicted using the developed solvers. The performance of the solvers was determined based on the mean fitness value, best-fit solution, and standard deviation (STD) computed from 20 independent runs as the SI techniques are stochastic in nature and may produce different solutions in different runs. The robustness of the algorithms is evaluated by comparing the mean fitness value with the best fitness (i.e., minimum error between theoretical and experimental data) and by determining the standard deviation. The robust solver predicts best-fit solution very close to the mean fitness solution and predicts smallest standard deviation, that is, close to zero. The model parameters obtained by different algorithms for all the 21 experiments using Freundlich, Freundlich-Langmuir, and two-site Langmuir isotherms were given separately in Tables
(a) Best and worst solutions (set of model parameters) achieved by different solvers in 20 independent runs for fitting the experimental data on Freundlich isotherm model. (b) Performance of different solvers for fitting Freundlich isotherm to the experimental data.
# | Parameters | PSO | PCPSO | ABC | MABC | ||||
---|---|---|---|---|---|---|---|---|---|
Best | Worst | Best | Worst | Best | Worst | Best | Worst | ||
1 | | 252.6 | 228.5 | 252.6 | 244.2 | 252.6 | 262.2 | 252.6 | 252.6 |
| 0.533 | 0.518 | 0.533 | 0.501 | 0.533 | 0.598 | 0.533 | 0.533 | |
| | 0.151 | | 0.148 | | 0.158 | | | |
| |||||||||
2 | | 28.40 | 635.4 | 28.51 | 22.21 | 28.41 | 29.95 | 28.41 | 28.37 |
| 0.729 | 1.91 | 0.729 | 0.796 | 0.729 | 0.706 | 0.729 | 0.729 | |
| | 2.499 | | 0.126 | | 0.112 | | | |
| |||||||||
3 | | 22.25 | 4512 | 22.25 | 14.62 | 22.25 | 26.06 | 22.25 | 22.41 |
| 0.559 | 0.006 | 0.559 | 0.748 | 0.559 | 0.50 | 0.559 | 0.556 | |
| | 1.753 | | 0.114 | | 0.042 | | | |
| |||||||||
4 | | 28.50 | 4.45 | 28.50 | 42.81 | 28.50 | 28.70 | 28.50 | 28.50 |
| 0.994 | 4.19 | 0.994 | 0.973 | 0.994 | 0.985 | 0.994 | 0.994 | |
| | 2.49 | | 0.188 | | 0.080 | | | |
| |||||||||
5 | | 231.1 | 226.9 | 231.1 | 201.7 | 231.1 | 231.1 | 231.1 | 231.1 |
| 0.728 | 1.601 | 0.728 | 0.739 | 0.728 | 0.728 | 0.728 | 0.728 | |
| | 0.640 | | 0.073 | | | | | |
| |||||||||
6 | | 736.8 | 309.8 | 736.8 | 871.9 | 736.8 | 788.9 | 736.8 | 736.8 |
| 0.679 | 1.155 | 0.679 | 0.725 | 0.679 | 0.701 | 0.679 | 0.679 | |
| | 1.191 | | 0.111 | | 0.100 | | | |
| |||||||||
7 | | 265.7 | 234.9 | 265.7 | 1480 | 265.7 | 265.6 | 265.7 | 265.7 |
| 0.695 | 1.409 | 0.695 | 1.437 | 0.695 | 0.695 | 0.695 | 0.695 | |
| | 0.544 | | 0.819 | | | | | |
| |||||||||
8 | | 126.6 | 64.57 | 126.6 | 126.9 | 126.6 | 126.6 | 126.6 | 126.6 |
| 0.750 | 1.693 | 0.750 | 0.750 | 0.750 | 0.750 | 0.750 | 0.750 | |
| | 0.665 | | 0.085 | | | | | |
| |||||||||
9 | | 994.7 | 1176 | 994.7 | 977.2 | 994.7 | 1414 | 994.7 | 994.7 |
| 1.645 | 3.581 | 1.645 | 1.638 | 1.645 | 1.859 | 1.645 | 1.645 | |
| | 1.080 | | | | 0.156 | | | |
| |||||||||
10 | | 1848 | 4872 | 1848 | 6063 | 1848 | 2993 | 1848 | 1846 |
| 0.861 | 3.152 | 0.861 | 1.216 | 0.861 | 1.049 | 0.861 | 0.860 | |
| | 2.583 | | 0.283 | | 0.164 | | | |
| |||||||||
11 | | 3.493 | 37.66 | 3.493 | 3.027 | 3.493 | 3.606 | 3.493 | 3.493 |
| 0.501 | 0.515 | 0.501 | 0.541 | 0.501 | 0.460 | 0.501 | 0.501 | |
| | 1.041 | | 0.116 | | 0.104 | | | |
| |||||||||
12 | | 8.519 | 166.7 | 8.519 | 8.451 | 8.519 | 8.518 | 8.519 | 8.519 |
| 0.503 | 0.945 | 0.503 | 0.447 | 0.503 | 0.502 | 0.503 | 0.503 | |
| | 1.400 | | 0.134 | | | | | |
| |||||||||
13 | | 63.61 | 32.06 | 63.61 | 44.67 | 63.61 | 63.66 | 63.61 | 63.61 |
| 0.605 | 2.680 | 0.605 | 0.639 | 0.605 | 0.605 | 0.605 | 0.605 | |
| | 2.127 | | 0.169 | | | | | |
| |||||||||
14 | | 86.28 | 832.5 | 86.28 | 83.87 | 86.28 | 96.99 | 86.28 | 86.28 |
| 0.744 | 1.090 | 0.744 | 0.731 | 0.744 | 0.767 | 0.744 | 0.744 | |
| | 0.997 | | 0.137 | | 0.145 | | | |
| |||||||||
15 | | 194.4 | 1365 | 194.4 | 176.9 | 194.4 | 194.4 | 194.4 | 194.4 |
| 0.738 | 1.124 | 0.738 | 0.725 | 0.738 | 0.738 | 0.738 | 0.738 | |
| | 0.681 | | 0.113 | | | | | |
| |||||||||
16 | | 14.95 | 4873 | 14.95 | 18.51 | 14.95 | 31.20 | 14.95 | 14.95 |
| 0.787 | 5.339 | 0.787 | 0.783 | 0.787 | 1.348 | 0.787 | 0.787 | |
| | 5.300 | | 0.134 | | 0.661 | | | |
| |||||||||
17 | | 5.301 | 64.29 | 5.301 | 5.595 | 5.301 | 5.289 | 5.301 | 5.301 |
| 0.835 | 1.063 | 0.835 | 0.859 | 0.835 | 0.835 | 0.835 | 0.835 | |
| | 1.083 | | 0.075 | | | | | |
| |||||||||
18 | | 407.7 | 979.9 | 407.7 | 459.0 | 407.7 | 483.14 | 407.7 | 407.7 |
| 0.602 | 0.940 | 0.602 | 0.669 | 0.602 | 0.665 | 0.602 | 0.602 | |
| | 0.347 | | 0.103 | | 0.101 | | | |
| |||||||||
19 | | 81.69 | 99.29 | 81.69 | 92.05 | 81.69 | 69.36 | 81.69 | 81.69 |
| 0.264 | 1.434 | 0.264 | 0.283 | 0.264 | 0.375 | 0.264 | 0.264 | |
| | 1.052 | | 0.080 | | | | | |
| |||||||||
20 | | 43.70 | 1.428 | 43.70 | 11.57 | 43.70 | 44.22 | 43.70 | 43.73 |
| 0.419 | 2.678 | 0.419 | 1.204 | 0.419 | 0.413 | 0.419 | 0.418 | |
| | 1.121 | | 0.379 | | | | | |
| |||||||||
21 | | 83.26 | 33.30 | 83.26 | 86.52 | 83.26 | 86.20 | 83.26 | 83.26 |
| 0.312 | 1.010 | 0.312 | 0.305 | 0.312 | 0.294 | 0.312 | 0.312 | |
| | 0.468 | | 0.034 | | 0.034 | | |
# | PSO algorithm | PCPSO algorithm | ABC algorithm | MABC algorithm | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | Best | Std | Mean | Best | Std | Mean | Best | Std | Mean | Best | Std | |
1 | 0.3555 | | | 0.1452 | | | 0.1457 | | | | | |
2 | 0.4190 | | | 0.1119 | | | 0.1226 | | | | | |
3 | 0.3079 | | | 0.0311 | | | | | | | | |
4 | 0.2045 | | | 0.0819 | | | 0.0827 | | | | | |
5 | 0.0614 | | | | | | | | | | | |
6 | 0.1143 | | | | | | 0.0979 | | | | | |
7 | 0.0809 | | | | | | | | | | | |
8 | 0.1476 | | | 0.0853 | | | | | | | | |
9 | 0.1537 | | | | | | | | | | | |
10 | 0.1465 | 0.1111 | | 0.1119 | 0.1111 | | 0.1113 | 0.1111 | | | | |
11 | 0.4355 | | | 0.1335 | 0.0912 | | 0.0915 | | | | | |
12 | 0.2852 | | | 0.1450 | | | | | | | | |
13 | 0.6264 | 0.0826 | | 0.0881 | | | 0.0826 | 0.0826 | | | | |
14 | 0.4260 | | | 0.1397 | | | | | | | | |
15 | 0.1828 | | | 0.1142 | | | | | | | | |
16 | 0.1155 | | | 0.1185 | | | | | | | | |
17 | 0.6646 | | | 0.1019 | | | | 0.0677 | | | | |
18 | 0.1821 | | | 0.0891 | | | 0.0810 | | | | | |
19 | 0.2400 | | | 0.0658 | | | 0.0531 | | | | | |
20 | 0.1677 | | | 0.0437 | | | 0.0378 | | | | | |
21 | 0.4273 | 0.0310 | | 0.0444 | 0.0310 | | 0.0591 | 0.0310 | | | | |
(a) Best and worst solutions (set of model parameters) achieved by different solvers in 20 independent runs for fitting the experimental data on Freundlich-Langmuir isotherm model. (b) Performance of different solvers for fitting Langmuir-Freundlich isotherm to the experimental data.
# | Parameters | PSO | PCPSO | ABC | MABC | ||||
---|---|---|---|---|---|---|---|---|---|
Best | Worst | Best | Worst | Best | Worst | Best | Worst | ||
1 | | 1174 | 358.2 | 1166 | 574.8 | 1176 | 887.7 | 1167 | 1361 |
| 0.403 | 5.403 | 0.408 | 2.195 | 0.399 | 0.730 | 0.407 | 0.301 | |
| 0.779 | 0.998 | 0.781 | 0.987 | 0.779 | 0.905 | 0.780 | 0.725 | |
| | 0.306 | | 0.202 | | 0.132 | | 0.129 | |
| |||||||||
2 | | 711.9 | 143.3 | 668.0 | 403.2 | 744.1 | 695.6 | 674.6 | 905.3 |
| 0.034 | 4.400 | 0.036 | 0.133 | 0.032 | 0.041 | 0.036 | 0.026 | |
| 0.988 | 0.329 | 0.972 | 0.521 | 0.986 | 0.904 | 1.000 | 0.958 | |
| 0.093 | 0.406 | 0.095 | 0.243 | 0.093 | 0.102 | | 0.096 | |
| |||||||||
3 | | 645.0 | 91.33 | 791.3 | 93.47 | 1072 | 276.5 | 904.7 | 402.7 |
| 0.031 | 7.348 | 0.029 | 4.705 | 0.021 | 0.080 | 0.023 | 0.049 | |
| 0.688 | 0.418 | 0.615 | 0.520 | 0.618 | 0.790 | 0.633 | 0.758 | |
| 0.023 | 0.315 | 0.024 | 0.309 | 0.022 | 0.051 | | 0.027 | |
| |||||||||
4 | | 845.8 | 82.25 | 1261 | 79.96 | 3426 | 1586 | 5878 | 8999 |
| 0.041 | 2.705 | 0.024 | 4.236 | 0.009 | 0.019 | 0.005 | 0.003 | |
| 0.964 | 0.639 | 0.958 | 0.995 | 1.000 | 0.999 | 1.000 | 0.999 | |
| 0.152 | 0.674 | 0.131 | 0.667 | 0.078 | 0.096 | | 0.077 | |
| |||||||||
5 | | 2244 | 283.2 | 2334 | 771.3 | 2259 | 1664 | 2429 | 2280 |
| 0.132 | 2.437 | 0.125 | 0.570 | 0.131 | 0.201 | 0.119 | 0.129 | |
| 0.868 | 0.639 | 0.860 | 0.720 | 0.869 | 0.943 | 0.854 | 0.864 | |
| | 0.399 | | 0.217 | | 0.035 | | | |
| |||||||||
6 | | 1420 | 380.1 | 1460 | 3503 | 1424 | 1382 | 1420 | 1434 |
| 1.280 | 5.470 | 1.201 | 0.242 | 1.273 | 1.356 | 1.281 | 1.258 | |
| 0.950 | 0.561 | 0.940 | 0.667 | 0.949 | 0.959 | 0.950 | 0.947 | |
| | 0.381 | | 0.107 | | | | | |
| |||||||||
7 | | 1530 | 317.3 | 1639 | 3547 | 1678 | 1259 | 1690 | 1780 |
| 0.253 | 5.047 | 0.230 | 0.077 | 0.222 | 0.346 | 0.219 | 0.204 | |
| 0.871 | 0.839 | 0.856 | 0.501 | 0.853 | 0.927 | 0.851 | 0.842 | |
| | 0.358 | | 0.173 | | 0.048 | | | |
| |||||||||
8 | | 2458 | 210.7 | 1753 | 213.7 | 2234 | 1640 | 2715 | 2260 |
| 0.058 | 3.177 | 0.084 | 5.192 | 0.064 | 0.093 | 0.052 | 0.064 | |
| 0.851 | 0.686 | 0.909 | 0.993 | 0.856 | 0.940 | 0.841 | 0.865 | |
| | 0.424 | 0.084 | 0.425 | | 0.085 | | | |
| |||||||||
9 | | 2288 | 174.3 | 4857 | 237.4 | 13333 | 3801 | 13342 | 13268 |
| 0.223 | 5.257 | 0.105 | 6.490 | 0.036 | 0.136 | 0.036 | 0.036 | |
| 0.999 | 0.401 | 0.973 | 1.000 | 0.992 | 0.999 | 1.000 | 0.995 | |
| 0.266 | 0.514 | 0.263 | 0.444 | 0.249 | 0.258 | | 0.249 | |
| |||||||||
10 | | 2327 | 649.8 | 2352 | 752.4 | 2340 | 2326 | 2328 | 2390 |
| 1.402 | 1.949 | 1.389 | 6.703 | 1.399 | 1.401 | 1.402 | 1.350 | |
| 1.000 | 0.608 | 0.999 | 1.000 | 1.000 | 0.998 | 1.000 | 1.000 | |
| | 0.313 | | 0.191 | | | | | |
| |||||||||
11 | | 45.41 | 8.379 | 40.78 | 7.229 | 45.48 | 38.90 | 45.47 | 44.64 |
| 0.098 | 2.172 | 0.115 | 6.333 | 0.098 | 0.121 | 0.098 | 0.099 | |
| 0.617 | 0.639 | 0.634 | 0.999 | 0.616 | 0.650 | 0.617 | 0.615 | |
| | 0.395 | 0.071 | 0.405 | | 0.072 | | | |
| |||||||||
12 | | 45.91 | 12.61 | 45.86 | 15.79 | 45.97 | 43.91 | 45.91 | 46.06 |
| 0.342 | 1.831 | 0.343 | 4.944 | 0.341 | 0.371 | 0.342 | 0.340 | |
| 0.763 | 0.528 | 0.764 | 0.994 | 0.762 | 0.775 | 0.763 | 0.761 | |
| | 0.362 | | 0.283 | | 0.032 | | | |
| |||||||||
13 | | 748.9 | 131.3 | 758.3 | 1480 | 755.0 | 639.9 | 752.7 | 756.5 |
| 0.114 | 6.554 | 0.112 | 0.051 | 0.113 | 0.141 | 0.113 | 0.112 | |
| 0.795 | 1.000 | 0.795 | 0.467 | 0.793 | 0.833 | 0.794 | 0.792 | |
| | 0.467 | | 0.194 | | | | | |
| |||||||||
14 | | 951.2 | 96.21 | 895.3 | 125.1 | 949.0 | 924.0 | 951.3 | 954.9 |
| 0.150 | 9.307 | 0.166 | 6.732 | 0.150 | 0.157 | 0.150 | 0.149 | |
| 0.912 | 0.702 | 0.922 | 1.000 | 0.912 | 0.920 | 0.912 | 0.911 | |
| | 0.745 | 0.079 | 0.549 | | | | | |
| |||||||||
15 | | 931.6 | 155.3 | 943.3 | 157.1 | 929.1 | 959.3 | 931.4 | 933.6 |
| 0.403 | 2.035 | 0.395 | 6.723 | 0.405 | 0.377 | 0.403 | 0.402 | |
| 0.918 | 0.647 | 0.916 | 1.000 | 0.919 | 0.911 | 0.918 | 0.918 | |
| | 0.426 | | 0.357 | | | | | |
| |||||||||
16 | | 275.9 | 18.57 | 651.8 | 88.24 | 503.8 | 294.5 | 497.5 | 523.3 |
| 0.076 | 3.857 | 0.028 | 0.309 | 0.037 | 0.072 | 0.037 | 0.035 | |
| 0.916 | 0.896 | 0.869 | 0.955 | 0.870 | 0.917 | 0.869 | 0.860 | |
| 0.082 | 0.573 | 0.069 | 0.206 | | 0.079 | | | |
| |||||||||
17 | | 380.1 | 5.862 | 275.9 | 7.135 | 580.9 | 250.3 | 635.9 | 541.8 |
| 0.016 | 7.608 | 0.022 | 1.255 | 0.010 | 0.025 | 0.009 | 0.011 | |
| 0.905 | 0.701 | 0.896 | 0.979 | 0.866 | 0.936 | 0.865 | 0.869 | |
| 0.068 | 0.797 | 0.071 | 0.581 | | 0.079 | | | |
| |||||||||
18 | | 1288 | 291.0 | 1288 | 451.1 | 1296 | 1111 | 1287 | 1277 |
| 0.609 | 4.950 | 0.608 | 5.687 | 0.601 | 0.796 | 0.609 | 0.623 | |
| 0.766 | 0.746 | 0.766 | 1.000 | 0.765 | 0.797 | 0.766 | 0.770 | |
| | 0.278 | | 0.159 | | 0.035 | | | |
| |||||||||
19 | | 181.1 | 121.4 | 181.1 | 127.2 | 181.3 | 177.3 | 181.2 | 180.5 |
| 1.058 | 6.737 | 1.059 | 6.168 | 1.055 | 1.122 | 1.057 | 1.069 | |
| 0.660 | 0.241 | 0.661 | 0.997 | 0.660 | 0.684 | 0.660 | 0.664 | |
| | 0.179 | | 0.099 | | | | | |
| |||||||||
20 | | 244.8 | 100.5 | 239.3 | 101.3 | 244.5 | 201.6 | 247.8 | 267.0 |
| 0.205 | 5.056 | 0.212 | 5.464 | 0.205 | 0.255 | 0.202 | 0.185 | |
| 0.661 | 0.741 | 0.666 | 0.435 | 0.662 | 0.763 | 0.657 | 0.632 | |
| | 0.177 | 0.016 | 0.184 | | 0.019 | | 0.016 | |
| |||||||||
21 | | 266.2 | 132.2 | 273.1 | 167.2 | 267.6 | 222.6 | 265.7 | 252.1 |
| 0.501 | 5.629 | 0.480 | 1.656 | 0.499 | 0.701 | 0.502 | 0.553 | |
| 0.540 | 0.374 | 0.529 | 0.935 | 0.539 | 0.641 | 0.540 | 0.562 | |
| | 0.175 | | 0.046 | | 0.016 | | |
# | PSO algorithm | PCPSO algorithm | ABC algorithm | MABC algorithm | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | Best | Std | Mean | Best | Std | Mean | Best | Std | Mean | Best | Std | |
1 | 0.2008 | | | 0.1333 | 0.1283 | | 0.1288 | | | | | |
2 | 0.2156 | 0.0955 | | 0.1613 | 0.0940 | | 0.0975 | 0.0943 | | | | |
3 | 0.1168 | 0.0247 | | 0.0686 | 0.0239 | | 0.0329 | 0.0237 | | | | |
4 | 0.3748 | 0.1252 | | 0.2916 | 0.1386 | | 0.1214 | 0.1063 | | | | |
5 | 0.0890 | | | 0.0444 | 0.0316 | | 0.0358 | 0.0309 | | | | |
6 | 0.0852 | | | 0.0819 | | | 0.0743 | | | | | |
7 | 0.1303 | | | 0.0509 | 0.0458 | | 0.0477 | 0.0460 | | | | |
8 | 0.1234 | 0.0824 | | 0.1109 | 0.0827 | | 0.0864 | 0.0835 | | | | |
9 | 0.3188 | 0.2591 | | 0.2923 | 0.2689 | | 0.2681 | 0.2607 | | | | |
10 | 0.1800 | | | 0.1070 | | | 0.1014 | | | | | |
11 | 0.1699 | | | 0.0873 | 0.0724 | | 0.0791 | 0.0715 | | | | |
12 | 0.1354 | | | 0.0598 | 0.0310 | | 0.0331 | 0.0312 | | | | |
13 | 0.1514 | | | 0.0674 | 0.0369 | | 0.0410 | 0.0373 | | | | |
14 | 0.2191 | | | 0.1939 | 0.0788 | | 0.0811 | 0.0791 | | | | |
15 | 0.1535 | | | 0.0491 | | | 0.0405 | | | | | |
16 | 0.1775 | | | 0.1219 | 0.0829 | | 0.0756 | 0.0660 | | | | |
17 | 0.2818 | 0.0726 | | 0.1616 | 0.0834 | | 0.0971 | 0.0731 | | | | |
18 | 0.0578 | | | 0.0483 | | | 0.0344 | 0.0323 | | | | |
19 | 0.0514 | | | 0.0360 | | | 0.0245 | 0.0236 | | | | |
20 | 0.0466 | | | 0.0376 | 0.0156 | | 0.0175 | 0.0159 | | | | |
21 | 0.0428 | | | 0.0282 | 0.0113 | | 0.0140 | 0.0113 | | | | |
(a) Best and worst solutions (set of model parameters) achieved by different solvers in 20 independent runs for fitting the experimental data on two-site Langmuir isotherm model. (b) Performance of different solvers for fitting two-site Langmuir isotherm to the experimental data.
# | Parameters | PSO | PCPSO | ABC | MABC | ||||
---|---|---|---|---|---|---|---|---|---|
Best | Worst | Best | Worst | Best | Worst | Best | Worst | ||
1 | | 78.86 | 502.3 | 87.57 | 103.1 | 83.03 | 146.9 | 80.01 | 82.78 |
| 9.983 | 2.119 | 8.895 | 2.838 | 9.669 | 4.322 | 10.00 | 8.945 | |
| 944.4 | 9.705 | 956.9 | 785.6 | 942.9 | 872.8 | 969.4 | 959.4 | |
| 0.380 | 4.070 | 0.355 | 0.701 | 0.369 | 0.357 | 0.360 | 0.367 | |
| | 0.195 | 0.128 | 0.134 | | 0.130 | | 0.128 | |
| |||||||||
2 | | 737.2 | 109.5 | 484.1 | 167.9 | 353.8 | 648.5 | 685.9 | 694.3 |
| 0.031 | 6.837 | 0.056 | 0.433 | 0.031 | 0.038 | 0.035 | 0.034 | |
| 0.434 | 22.23 | 3.097 | 8.473 | 334.4 | 2.301 | 2.541 | 0.725 | |
| 3.091 | 8.363 | 3.898 | 4.842 | 0.039 | 0.370 | 0.036 | 0.121 | |
| 0.094 | 0.412 | 0.111 | 0.302 | | 0.094 | | | |
| |||||||||
3 | | 324.4 | 92.02 | 295.0 | 149.6 | 347.3 | 269.7 | 387.9 | 318.4 |
| 0.016 | 0.558 | 0.039 | 0.091 | 0.021 | 0.018 | 0.012 | 0.024 | |
| 42.26 | 17.06 | 13.12 | 12.81 | 24.72 | 59.81 | 57.98 | 24.70 | |
| 0.722 | 5.077 | 5.458 | 6.979 | 1.470 | 0.257 | 0.322 | 1.026 | |
| 0.036 | 0.245 | 0.045 | 0.114 | 0.026 | 0.034 | | 0.026 | |
| |||||||||
4 | | 953.9 | 124.5 | 3152 | 150.3 | 2934 | 860.1 | 5356 | 2199 |
| 0.035 | 0.999 | 0.009 | 0.153 | 0.010 | 0.025 | 0.005 | 0.014 | |
| 0.021 | 0.151 | 2.076 | 128.6 | 5.534 | 690.5 | 0.012 | 40.61 | |
| 1.318 | 6.773 | 1.065 | 0.144 | 0.040 | 0.020 | 1.435 | 0.026 | |
| 0.126 | 0.521 | 0.083 | 0.283 | 0.079 | 0.106 | | 0.084 | |
| |||||||||
5 | | 1701 | 344.3 | 1697 | 1257 | 1824 | 1615 | 1839 | 1991 |
| 0.161 | 0.545 | 0.161 | 0.290 | 0.138 | 0.157 | 0.140 | 0.118 | |
| 27.52 | 131.3 | 30.87 | 5.442 | 37.83 | 90.92 | 31.16 | 46.78 | |
| 9.245 | 1.002 | 7.133 | 6.675 | 7.241 | 1.663 | 9.996 | 5.910 | |
| 0.029 | 0.249 | 0.029 | 0.046 | 0.029 | 0.035 | | 0.030 | |
| |||||||||
6 | | 1327 | 938.6 | 1320 | 816.3 | 1270 | 1257 | 1352 | 1208 |
| 1.319 | 1.722 | 1.331 | 2.105 | 1.335 | 1.560 | 1.248 | 1.538 | |
| 49.07 | 164.2 | 48.82 | 565.2 | 81.96 | 33.59 | 58.03 | 94.98 | |
| 9.972 | 4.804 | 9.851 | 0.763 | 5.946 | 6.702 | 9.901 | 2.984 | |
| | 0.084 | | 0.075 | 0.075 | 0.075 | | 0.075 | |
| |||||||||
7 | | 1468 | 564.7 | 1307 | 1086 | 1218 | 1017 | 2029 | 1327 |
| 0.171 | 1.296 | 0.237 | 0.400 | 0.159 | 0.242 | 0.076 | 0.131 | |
| 122.2 | 0.0002 | 66.06 | 21.15 | 253.9 | 287.8 | 264.7 | 244.5 | |
| 2.649 | 5.624 | 4.290 | 5.282 | 1.345 | 0.939 | 1.516 | 1.673 | |
| 0.042 | 0.145 | 0.045 | 0.051 | 0.042 | 0.048 | | | |
| |||||||||
8 | | 1812 | 213.5 | 1627 | 3296 | 1495 | 1168 | 1799 | 1490 |
| 0.064 | 1.652 | 0.074 | 0.031 | 0.080 | 0.141 | 0.064 | 0.081 | |
| 22.14 | 8.328 | 20.94 | 13.98 | 28.30 | 2.741 | 22.07 | 29.66 | |
| 9.999 | 3.118 | 7.233 | 7.006 | 4.511 | 6.117 | 9.820 | 3.233 | |
| | 0.3447 | 0.078 | 0.109 | 0.080 | 0.091 | | 0.081 | |
| |||||||||
9 | | 3504 | 480.8 | 7609 | 1415 | 5915 | 1985 | 13067 | 7344 |
| 0.049 | 1.503 | 0.065 | 0.366 | 0.080 | 0.266 | 0.036 | 0.066 | |
| 2823 | 5.276 | 1.465 | 1.004 | 15.16 | 1.310 | 1114 | 4.637 | |
| 0.116 | 3.801 | 6.330 | 5.246 | 0.826 | 0.126 | 0.010 | 2.912 | |
| 0.253 | 0.347 | 0.252 | 0.279 | 0.253 | 0.266 | | 0.252 | |
| |||||||||
10 | | 2299 | 1452 | 1243 | 2318 | 2270 | 2398 | 2315 | 1717 |
| 1.399 | 2.151 | 1.399 | 1.385 | 1.442 | 1.362 | 1.404 | 1.496 | |
| 31.51 | 151.7 | 1088 | 7.399 | 9.769 | 4.237 | 12.33 | 559.0 | |
| 1.399 | 2.949 | 1.399 | 6.203 | 0.729 | 3.670 | 1.289 | 1.274 | |
| | 0.108 | | 0.101 | | | | | |
| |||||||||
11 | | 29.81 | 15.23 | 25.70 | 8.095 | 28.04 | 20.63 | 30.08 | 26.57 |
| 0.053 | 0.823 | 0.074 | 0.334 | 0.061 | 0.127 | 0.051 | 0.060 | |
| 2.836 | 2.167 | 2.609 | 1.953 | 2.696 | 2.222 | 2.908 | 2.632 | |
| 5.293 | 0.019 | 6.006 | 3.362 | 5.611 | 5.403 | 5.195 | 6.527 | |
| | 0.182 | 0.078 | 8.095 | | 0.097 | | 0.079 | |
| |||||||||
12 | | 37.94 | 21.89 | 36.58 | 19.27 | 39.82 | 40.01 | 38.18 | 36.05 |
| 0.276 | 1.282 | 0.215 | 0.607 | 0.236 | 0.150 | 0.265 | 0.199 | |
| 3.505 | 20.95 | 6.099 | 7.660 | 3.828 | 7.602 | 3.610 | 6.909 | |
| 8.607 | 0.072 | 4.588 | 3.475 | 8.296 | 3.785 | 8.471 | 4.168 | |
| | 0.058 | 0.037 | 0.102 | 0.035 | 0.041 | | 0.038 | |
| |||||||||
13 | | 664.7 | 128.3 | 608.9 | 1004 | 647.9 | 561.3 | 660.9 | 613.6 |
| 0.053 | 2.270 | 0.059 | 0.041 | 0.068 | 0.111 | 0.052 | 0.088 | |
| 66.76 | 13.99 | 79.34 | 28.35 | 53.71 | 27.61 | 71.29 | 34.55 | |
| 1.784 | 6.641 | 1.375 | 7.985 | 2.114 | 4.196 | 1.652 | 3.133 | |
| | 0.375 | 0.034 | 0.091 | 0.034 | 0.042 | | 0.037 | |
| |||||||||
14 | | 922.4 | 222.5 | 804.5 | 672.6 | 876.0 | 907.8 | 919.8 | 821.1 |
| 0.119 | 1.379 | 0.161 | 0.164 | 0.136 | 0.115 | 0.120 | 0.139 | |
| 21.01 | 8.495 | 10.72 | 59.10 | 17.56 | 39.02 | 19.75 | 14.65 | |
| 4.916 | 3.209 | 7.793 | 1.574 | 5.244 | 3.319 | 5.278 | 6.665 | |
| | 0.309 | 0.075 | 0.086 | | 0.080 | | 0.075 | |
| |||||||||
15 | | 897.9 | 265.4 | 877.5 | 220.8 | 860.1 | 993.8 | 895.7 | 797.8 |
| 0.333 | 2.947 | 0.355 | 1.430 | 0.323 | 0.216 | 0.325 | 0.375 | |
| 37.85 | 1.458 | 34.51 | 666.4 | 49.64 | 104.8 | 41.51 | 73.83 | |
| 6.720 | 4.238 | 6.991 | 0.290 | 5.345 | 3.371 | 6.266 | 3.133 | |
| | 0.232 | | 0.046 | 0.040 | 0.045 | | 0.043 | |
| |||||||||
16 | | 383.3 | 31.57 | 429.5 | 133.8 | 382.3 | 240.8 | 653.4 | 515.8 |
| 0.038 | 0.876 | 0.034 | 0.200 | 0.029 | 0.069 | 0.011 | 0.014 | |
| 4.521 | 1.361 | 2.832 | 1.973 | 11.02 | 5.644 | 34.18 | 21.70 | |
| 3.485 | 2.994 | 6.396 | 5.194 | 1.587 | 2.380 | 0.595 | 1.060 | |
| 0.070 | 0.366 | 0.072 | 0.167 | 0.071 | 0.089 | | 0.074 | |
| |||||||||
17 | | 246.9 | 4.863 | 166.9 | 8.144 | 248.7 | 106.9 | 329.6 | 100.6 |
| 0.019 | 7.005 | 0.034 | 5.133 | 0.018 | 0.045 | 0.014 | 0.069 | |
| 0.747 | 2.472 | 1.131 | 0.611 | 2.265 | 2.451 | 0.701 | 47.52 | |
| 7.408 | 8.274 | 4.310 | 6.367 | 1.709 | 1.468 | 8.615 | 0.018 | |
| 0.063 | 0.705 | 0.097 | 0.666 | 0.075 | 0.127 | | 0.119 | |
| |||||||||
18 | | 1532 | 670.5 | 1161 | 706.8 | 1117 | 990.5 | 1411 | 1397 |
| 0.171 | 2.061 | 0.277 | 1.726 | 0.259 | 0.241 | 0.200 | 0.197 | |
| 205.3 | 88.30 | 198.0 | 157.2 | 216.2 | 293.5 | 202.7 | 207.3 | |
| 9.999 | 6.937 | 9.995 | 4.266 | 9.364 | 7.183 | 9.997 | 9.775 | |
| | 0.088 | 0.043 | 0.089 | 0.046 | 0.062 | | 0.042 | |
| |||||||||
19 | | 165.1 | 77.05 | 110.5 | 135.2 | 114.4 | 114.2 | 127.5 | 98.13 |
| 0.023 | 2.275 | 3.815 | 2.047 | 3.763 | 0.687 | 0.034 | 4.354 | |
| 117.8 | 72.91 | 80.92 | 16.50 | 79.45 | 51.84 | 117.4 | 87.77 | |
| 3.519 | 2.438 | 0.111 | 5.067 | 0.088 | 9.713 | 3.506 | 0.170 | |
| | 0.039 | 0.015 | 0.038 | 0.015 | 0.027 | | 0.019 | |
| |||||||||
20 | | 167.6 | 84.16 | 165.5 | 152.5 | 153.0 | 103.2 | 166.4 | 145.1 |
| 0.135 | 2.507 | 0.135 | 0.236 | 0.125 | 0.133 | 0.131 | 0.093 | |
| 24.21 | 12.32 | 24.87 | 10.73 | 36.22 | 65.74 | 26.07 | 55.86 | |
| 7.752 | 6.932 | 8.009 | 1.760 | 2.292 | 0.750 | 5.570 | 1.244 | |
| | 0.157 | 0.015 | 0.041 | 0.015 | 0.027 | | 0.019 | |
| |||||||||
21 | | 144.2 | 105.3 | 144.4 | 117.0 | 137.0 | 140.39 | 147.4 | 139.7 |
| 0.087 | 6.227 | 0.184 | 3.267 | 0.117 | 0.269 | 0.094 | 0.283 | |
| 103.3 | 29.01 | 77.79 | 29.79 | 96.69 | 68.81 | 99.24 | 66.28 | |
| 3.563 | 4.744 | 5.675 | 4.661 | 3.978 | 6.682 | 3.827 | 7.002 | |
| | 0.122 | 0.011 | 0.089 | 0.006 | 0.014 | | 0.015 |
# | PSO algorithm | PCPSO algorithm | ABC algorithm | MABC algorithm | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | Best | Std | Mean | Best | Std | Mean | Best | Std | Mean | Best | Std | |
1 | 0.1375 | | | 0.1294 | 0.1275 | | 0.1291 | 0.1278 | | | | |
2 | 0.1870 | 0.0959 | | 0.1406 | 0.0996 | | 0.1017 | 0.0938 | | | | |
3 | 0.1562 | 0.0358 | | 0.0569 | 0.0346 | | 0.0408 | 0.0323 | | | | |
4 | 0.2318 | 0.1334 | | 0.1962 | 0.1191 | | 0.1296 | 0.1011 | | | | |
5 | 0.0670 | | | 0.0376 | 0.0294 | | 0.0357 | 0.0315 | | | | |
6 | 0.0775 | | | | 0.0744 | | 0.0747 | 0.0744 | | | | |
7 | 0.0432 | | | 0.0471 | 0.0421 | | 0.0438 | 0.0407 | | | 0.0385 | |
8 | 0.1035 | 0.0776 | | 0.0883 | 0.0783 | | 0.0843 | 0.0786 | | | | |
9 | 0.2751 | | | 0.2642 | 0.2489 | | 0.2578 | 0.2523 | | | 0.2475 | |
10 | | | | | | | | | | | | |
11 | 0.0888 | 0.0748 | | 0.1093 | 0.0831 | | 0.0802 | 0.0751 | | | | |
12 | 0.0513 | | | 0.0536 | 0.0353 | | 0.0402 | 0.0344 | | | | |
13 | 0.0776 | 0.0322 | | 0.0535 | 0.0318 | | 0.0379 | 0.0332 | | | | |
14 | 0.1240 | | | 0.0760 | 0.0721 | | 0.0746 | 0.0724 | | | | |
15 | 0.0487 | 0.0393 | | 0.0440 | 0.0396 | | 0.0495 | 0.0398 | | | | |
16 | 0.1185 | 0.0706 | | 0.1156 | 0.0695 | | 0.0838 | 0.0712 | | | | |
17 | 0.1318 | 0.0791 | | 0.1418 | 0.0686 | | 0.1091 | 0.0876 | | | | |
18 | 0.0629 | | | 0.0557 | 0.0423 | | 0.0524 | 0.0435 | | | 0.0423 | |
19 | 0.0251 | 0.0140 | | 0.0290 | 0.0216 | | 0.0190 | 0.0129 | | | | |
20 | 0.0162 | | | 0.0270 | 0.0152 | | 0.0173 | 0.0149 | | | | |
21 | 0.0220 | | | 0.0140 | 0.0100 | | 0.0136 | 0.0097 | | | | |
The mean fitness, best-fit solution, and standard deviation by different solvers for fitting the experimental data using Freundlich isotherm were presented in Table
The computed mean fit, best-fit, and standard deviation by different algorithms on the experimental data using the Freundlich-Langmuir isotherm was presented in Table
Convergence of
The performance measures by different solvers on the experimental data using the two-site Langmuir isotherm using 20 independent runs were presented in Table
Convergence of
Convergence of
Convergence of
Convergence of
Convergence of
Theoretical fits of two-site Langmuir isotherm to experimental data of Zn sorption on Alligator soil by different solvers.
Theoretical fit of Freundlich-Langmuir isotherm to the experimental data of Pb sorption on Cecil soil by MABC solver.
Theoretical fit of Freundlich-Langmuir isotherm to the experimental data of Cr sorption on Alligator soil by MABC solver.
Performance comparison of different solvers for the estimation of isotherm model parameters from the experimental data was presented earlier. The suitable isotherms are often selected in the contaminant transport studies in soils by comparing optimized fitness values obtained by fitting different isotherms to the sorption data. The best-fit isotherms for the observed data based on the computed mean fitness values by different solvers were presented in Table
Performance of different solvers for the selection of appropriate algorithm based on mean solution.
# | PSO | PCPSO | ABC | MABC |
---|---|---|---|---|
1 | Two-site Langmuir | Two-site Langmuir | | |
| ||||
2 | Two-site Langmuir | Freundlich | | |
| ||||
3 | | Freundlich | Freundlich | |
| ||||
4 | Freundlich | Freundlich | Freundlich | |
| ||||
5 | Freundlich | Two-site Langmuir | Two-site Langmuir | |
| ||||
6 | Two-site Langmuir | Two-site Langmuir | Freundlich-Langmuir | |
| ||||
7 | Two-site Langmuir | Two-site Langmuir | Two-site Langmuir | |
| ||||
8 | Two-site Langmuir | Freundlich | Two-site Langmuir | |
| ||||
9 | | | | |
| ||||
10 | | | | |
| ||||
11 | Two-site Langmuir | | | |
| ||||
12 | Two-site Langmuir | Two-site Langmuir | | |
| ||||
13 | | | | |
| ||||
14 | | | | |
| ||||
15 | | | Freundlich-Langmuir | |
| ||||
16 | Freundlich | Two-site Langmuir | | |
| ||||
17 | Two-site Langmuir | Freundlich | Freundlich | |
| ||||
18 | | | | |
| ||||
19 | | | | |
| ||||
20 | | | | |
| ||||
21 | | | | |
Performance of different solvers for the selection of appropriate algorithm based on best solution.
# | PSO | PCPSO | ABC | MABC |
---|---|---|---|---|
1 | | | | |
| ||||
2 | | | | |
| ||||
3 | Freundlich | Freundlich | Freundlich | |
| ||||
4 | Freundlich | Freundlich | Freundlich | |
| ||||
5 | | | Freundlich-Langmuir | |
| ||||
6 | | | | |
| ||||
7 | | | | |
| ||||
8 | | | | |
| ||||
9 | | | | |
| ||||
10 | | | | |
| ||||
11 | | | | |
| ||||
12 | | | | |
| ||||
13 | | | | |
| ||||
14 | | | | |
| ||||
15 | | Freundlich-Langmuir/two-site Langmuir | Freundlich-Langmuir | |
| ||||
16 | | Two-site Langmuir | | |
| ||||
17 | Freundlich | Freundlich | Freundlich | |
| ||||
18 | | | | |
| ||||
19 | | Freundlich-Langmuir | | |
| ||||
20 | | | | |
| ||||
21 | Freundlich-Langmuir | | | |
In this paper, we analyzed conventional and stochastic optimization algorithms for fitting the nonlinear isotherm models to the experimental sorption data. The disadvantage of using conventional optimization techniques is demonstrated for the present problem. These techniques were found to be highly sensitive to the initial solution and the performance is impeded by the presence of local minima. Inverse models based on swarm intelligence are developed for fitting Freundlich, Freundlich-Langmuir, and two-site Langmuir isotherm models to the observed sorption data after tuning the model parameters. However, the estimated solutions vary in different runs due to the stochastic nature of these algorithms. It was observed that all the tested optimization algorithms used from the literature are relatively slow in nature as they do not balance the exploitation and exploration components effectively. A modified bee-colony optimization algorithm is proposed in this work. Comparative analysis reveals that the proposed solver, MABC, predicts excellent results on all the experiments. The results showed that the proposed solver has superior convergence capabilities due to good exploration-exploitation components in estimating global best, mean fitness, and standard deviations. Further, the proposed solver is proved to be robust on various functional terrains. The best-fit isotherms for observed data based on the computed mean fitness values by different solvers are also obtained. The improvement in the performance of the proposed solver is significantly better and reliable for studying the fate of contaminants in soils. Therefore, a great caution should be exercised when the existing solvers are used for the selection and configuration of nonlinear isotherms.
See Pseudocode
Initialize food sources (equation ( % Evaluate nectar amount (fitness) of food sources (equation ( Produce a new food source using ( Evaluate the fitness of the new food source (equation ( Greedy selection to choose best food source; Compute probability for each food source (equation ( % Send each onlooker bees to food sources based on Produce a new food source using ( Evaluate the fitness of the new food source (equation ( Greedy selection to choose best food source; Send scout bee to a new food source based on OBL (equation ( Memorize the best solution achieved so far; Iteration = Iteration + 1;
The author declares that they have no competing interests.