The backtracking search optimization algorithm (BSA) is a population-based evolutionary algorithm for numerical optimization problems. BSA has a powerful global exploration capacity while its local exploitation capability is relatively poor. This affects the convergence speed of the algorithm. In this paper, we propose a modified BSA inspired by simulated annealing (BSAISA) to overcome the deficiency of BSA. In the BSAISA, the amplitude control factor (
Optimization is an essential research objective in the fields of applied mathematics and computer sciences. Optimization algorithms mainly aim to obtain the global optimum for optimization problems. There are many different kinds of optimization problems in real world. When an optimization problem has a simple and explicit gradient information or requires relatively small budgets of allowed function evaluations, the implementation of classical optimization techniques such as mathematical programming often could achieve efficient results [
BSA is an iterative population-based EA, which was first proposed by Civicioglu in 2013. BSA has three basic genetic operators: selection, mutation, and crossover. The main difference between BSA and other similar algorithms is that BSA possesses a memory for storing a population from a randomly chosen previous generation, which is used to generate the search-direction matrix for the next iteration. In addition, BSA has a simple structure, which makes it efficient, fast, and capable of solving multimodal problems. BSA has only one control parameter called the
However, BSA has a weak local exploitation capacity and its convergence speed is relatively slow. Thus, many studies have attempted to improve the performance of BSA and some modifications of BSA have been proposed to overcome the deficiencies. From the perspective of modified object, the modifications of BSA can be divided into the following four categories. It is noted that we consider classifying the publication into the major modification category if it has more than one modification: Modifications of the initial populations [ Modifications of the reproduction operators, including the mutation and crossover operators [ Modifications of the selection operators, including the local exploitation strategy [ Modifications of the control factor and parameter [
The research on controlling parameters of EAs is one of the most promising areas of research in evolutionary computation; even a little modification of parameters in an algorithm can make a considerable difference [
Different from the modifications of
The remainder of this paper is organized as follows. Section
BSA is a population-based iterative EA. BSA generates trial populations to take control of the amplitude of the search-direction matrix which provides a strong global exploration capability. BSA equiprobably uses two random crossover strategies to exchange the corresponding elements of individuals in populations and trial populations during the process of crossover. Moreover, BSA has two selection processes. One is used to select population from the current and historical populations; the other is used to select the optimal population. In general, BSA can be divided into five processes: initialization, selection I, mutation, crossover, and selection II [
BSA generates initial population
BSA’s selection I process is the beginning of each iteration. It aims to reselect a new
The mutation operator is used for generating the initial form of trial population
In this process, BSA generates the final form of trial population
Strategy I uses the mix-rate parameter (mix-rate) to control the numbers of elements of individuals that are manipulated by using
The two strategies equiprobably are employed to manipulate the elements of individuals through the “if-then” rule: if
At the end of crossover process, if some individuals in
In BSA’s selection II process, the fitness values in
If the best individual of
As mentioned in the introduction of this paper, the research work on the control parameters of an algorithm is very meaningful and valuable. In this paper, in order to improve BSA’s exploitation capability and convergence speed, we propose a modified version of BSA (BSAISA) where the redesign of
The modified
According to (
The design principle of the modified
The basic concept of SA derives from the process of physical annealing with solids. An annealing process occurs when a metal is heated to a molten state with a high temperature; then it is cooled slowly. If the temperature is decreased quickly, the resulting crystal will have many defects and it is just metastable; even the most stable crystalline state will be achieved at all. In other words, this may form a higher energy state than the most stable crystalline state. Therefore, in order to reach the absolute minimum energy state, the temperature needs to be decreased at a slow rate. SA simulates this process of annealing to search the global optimal solution in an optimization problem. However, accepting only the moves that lower the energy of system is like extremely rapid quenching; thus SA uses a special and effective acceptance method, that is, Metropolis criterion, which can probabilistically accept the hill-climbing moves (the higher energy moves). As a result, the energy of the system evolves into a Boltzmann distribution during the process of the simulated annealing. From this angle of view, it is no exaggeration to say that the Metropolis criterion is the core of SA.
The Metropolis criterion can be expressed by the physical significance of energy, where the new energy state will be accepted when the new energy state is lower than the previous energy state, and the new energy state will be probabilistically accepted when the new energy state is higher than the previous energy state. This feature of SA can escape from being trapped in local minima especially in the early stages of the search. It can also be described as follows.
(i)
(ii)
The Metropolis criterion states that SA has two characteristics: (1) SA can probabilistically accept the higher energy and (2) the acceptance probability of SA decreases as the temperature decreases. Therefore, SA can reject and jump out of a local minimum with a dynamic and decreasing probability to continue exploiting the other solutions in the state space. This acceptance mechanism can enrich the diversity of energy states.
As shown in (
Based on Analyses
For the two formulas of the modified
In order to verify that the convergence speed of the basic BSA is improved with the modified
Two benchmark functions and the corresponding populations, dimensions, and max iterations.
Name | Objective function | Range |
|
|
Max |
---|---|---|---|---|---|
Schwefel 1.2 |
|
|
100 | 30 | 500 |
Rastrigin |
|
|
100 | 30 | 500 |
Comparisons of modified
Modified
Original
Modified
Original
The mean values of population variances versus number of iterations using BSA and BSAISA for two functions.
Schwefel 1.2
Rastrigin
Convergence curves of BSA and BSAISA for two functions.
Schwefel 1.2
Rastrigin
(1) According to the trends of
(2) According to Figure
In general, a constrained optimization problem can be mathematically formulated as a minimization problem, as follows:
Several constraint-handling methods have been proposed previously, where the five most commonly used methods comprise penalty functions, feasibility and dominance rules (FAD), stochastic ranking,
Firstly,
The parameter setting of SA
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To illustrate the changing trend of the self-adaptive
Plot of
The pseudocode for BSAISA is showed in Pseudocode
∗
In this section, two sets of simulation experiments were executed to evaluate the effectiveness of the proposed BSAISA. The first experiment set performed on 13 well-known benchmark constrained functions taken from [
Characters of the 13 benchmark functions.
Fun. |
|
Type |
|
LI | NI | LE | NE | Active |
---|---|---|---|---|---|---|---|---|
g01 | 13 | Quadratic | 0.0003 | 9 | 0 | 0 | 0 | 6 |
g02 | 20 | Nonlinear | 99.9973 | 1 | 1 | 0 | 0 | 1 |
g03 | 10 | Nonlinear | 0.0000 | 0 | 0 | 0 | 1 | 1 |
g04 | 5 | Quadratic | 27.0079 | 0 | 6 | 0 | 0 | 2 |
g05 | 4 | Nonlinear | 0.0000 | 2 | 0 | 0 | 3 | 3 |
g06 | 2 | Nonlinear | 0.0057 | 0 | 2 | 0 | 0 | 2 |
g07 | 10 | Quadratic | 0.0003 | 3 | 5 | 0 | 0 | 6 |
g08 | 2 | Nonlinear | 0.8581 | 0 | 2 | 0 | 0 | 0 |
g09 | 7 | Nonlinear | 0.5199 | 0 | 4 | 0 | 0 | 2 |
g10 | 8 | Linear | 0.0020 | 3 | 3 | 0 | 0 | 3 |
g11 | 2 | Quadratic | 0.0973 | 0 | 0 | 0 | 1 | 1 |
g12 | 3 | Quadratic | 4.7679 | 0 |
|
0 | 0 | 0 |
g13 | 5 | Nonlinear | 0.0000 | 0 | 0 | 1 | 2 | 3 |
The recorded experimental results include the best function value (Best), the worst function value (Worst), the mean function value (Mean), the standard deviation (Std), the best solution (variables of best function value), the corresponding constraint value, and the number of function evaluations (FEs). The number of function evaluations can be considered as a convergence rate or a computational cost.
In order to evaluate the performance of BSAISA in terms of convergence speed, the FEs are considered as the best FEs corresponding to the obtained best solution in this paper. The calculation of FEs are the product of population sizes (
For the first experiment, the main parameters for 13 benchmark constrained functions are the same as the following: population size (
For the 5 real-world engineering design problems, we use slightly different parameter settings since each problem has different natures, that is, TTP (
The user parameters of all experiments are presented in Table
User parameters used for all experiments.
Problem | G01–G13 | TTP | PVP | TCSP | WBP | SRP |
---|---|---|---|---|---|---|
|
30 | 20 | 20 | 20 | 20 | 20 |
|
11665 | 1000 | 3000 | 3000 | 3000 | 2000 |
Runs | 30 | 50 | 50 | 50 | 50 | 50 |
In this section, BSAISA and BSA are performed on the 13 benchmarks simultaneously. Their statistical results obtained from 30 independent runs are listed in Tables
The statistical results of BSAISA for 13 constrained benchmarks.
Fun. | Known optimal | Best | Mean | Worst | Std | FEs |
---|---|---|---|---|---|---|
G01 |
|
|
|
|
|
84,630 |
G02 |
|
|
|
|
|
349,500 |
G03 |
|
|
|
|
|
58,560 |
G04 |
|
|
|
|
|
121,650 |
G05 |
|
|
|
|
|
238,410 |
G06 |
|
|
|
|
|
89,550 |
G07 |
|
|
|
|
|
15,060 |
G08 |
|
|
|
|
|
30,930 |
G09 |
|
|
|
|
|
347,760 |
G10 |
|
|
|
|
|
346,980 |
G11 |
|
|
|
|
|
87,870 |
G12 |
|
|
|
|
|
5430 |
G13 |
|
|
|
|
|
349,800 |
The statistical results of the basic BSA on 13 constrained benchmarks.
Fun. | Known optimal | Best | Mean | Worst | Std | FEs |
---|---|---|---|---|---|---|
G01 | −15 |
|
−15 | −15.000000 |
|
99,300 |
G02 | −0.803619 | −0.803255 | −0.792760 | −0.749326 |
|
344,580 |
G03 | −1.000500 | −1.000488 | −0.998905 | −0.990600 |
|
348,960 |
G04 | −30665.538672 |
|
−30665.538672 | −30665.538672 |
|
272,040 |
G05 | 5126.496714 |
|
5144.041363 | 5275.384724 |
|
299,220 |
G06 | −6961.813876 |
|
−6961.813876 | −6961.813876 |
|
111,450 |
G07 | 24.306209 | 24.307607 | 24.344626 | 24.399896 |
|
347,250 |
G08 | −0.0958250 |
|
−0.0958250 | −0.0958250 |
|
73,440 |
G09 | 680.630057 | 680.630058 | 680.630352 | 680.632400 |
|
348,900 |
G10 | 7049.248021 | 7049.278543 | 7053.573853 | 7080.192700 |
|
340,020 |
G11 | 0.749900 |
|
0.749900 | 0.749900 |
|
113,250 |
G12 | −1 |
|
−1 | −1 | 0 | 13,590 |
G13 | 0.0539415 | 0.0539420 | 0.1816986 | 0.5477657 |
|
347,400 |
To further compare BSAISA and BSA, the function value convergence curves of 13 functions that have significant differences have been plotted, as shown in Figures
The convergence curves of the first 9 functions by BSAISA and BSA.
Convergence curve on G01
Convergence curve on G02
Convergence curve on G03
Convergence curve on G04
Convergence curve on G05
Convergence curve on G06
Convergence curve on G07
Convergence curve on G08
Convergence curve on G09
The convergence curves of the latter 4 functions by BSAISA and BSA.
Convergence curve on G10
Convergence curve on G11
Convergence curve on G12
Convergence curve on G13
In order to further verify the competitiveness of BSAISA in aspect of convergence speed, we compared BSAISA with some classic and state-of-the-art approaches in terms of best function value and function evaluations. The best function value and the corresponding FEs of each algorithm on 13 benchmarks are presented in Table Stochastic ranking (SR) [ Filter simulated annealing (FSA) [ Cultured differential evolution (CDE) [ Agent based memetic algorithm (AMA) [ Modified artificial bee colony (MABC) algorithm [ Rough penalty genetic algorithm (RPGA) [ BSA combined self-adaptive
Comparison of the best values and FEs obtained by BSAISA and other algorithms.
Alg. | BSAISA | SR | FSA | CDE | AMA | MABC | RPGA | BSA-SA |
---|---|---|---|---|---|---|---|---|
Fun. | Best (FEs) | Best (FEs) | Best (FEs) | Best (FEs) | Best (FEs) | Best (FEs) | Best (FEs) | Best (FEs) |
G01 |
|
|
−14.993316 |
|
|
|
|
|
( |
(148,200) | (205,748) | (100,100) | (350,000) | (350,000) | (350,000) | (350,000) | |
G02 |
|
−0.8035 | −0.7549 |
|
−0.8035 |
|
|
|
(349,500) | (217,200) | (227,832) | ( |
(350,000) | (350,000) | (350,000) | (350,000) | |
G03 |
|
−1.000 | −1.0000015 | −0.995413 | −1.000 | −1.000 | −1.000 |
|
( |
(229,200) | (314,938) | (100,100) | (350,000) | (350,000) | (350,000) | (350,000) | |
G04 |
|
|
−30665.538 |
|
−30665.538 |
|
|
|
(121,650) | ( |
(86,154) | (100,100) | (350,000) | (350,000) | (350,000) | (350,000) | |
G05 | 5126.497 | 5126.497 | 5126.4981 | 5126.571 | 5126.512 |
|
5126.544 | 5126.497 |
(238,410) | (51,600) | (47,661) | (100,100) | (350,000) | ( |
(350,000) | (350,000) | |
G06 |
|
|
|
|
−6961.807 |
|
|
|
(89,550) | (118,000) | ( |
(100,100) | (350,000) | (350,000) | (350,000) | (350,000) | |
G07 | 24.307 | 24.307 | 24.311 |
|
24.315 | 24.324 | 24.333 |
|
(15,060) | (143,000) | (404,501) | ( |
(350,000) | (350,000) | (350,000) | (350,000) | |
G08 |
|
|
|
|
|
|
|
|
( |
(76,200) | (56,476) | (100,100) | (350,000) | (350,000) | (350,000) | (350,000) | |
G09 |
|
680.630 | 680.63008 |
|
680.645 | 680.631 | 680.631 | 680.6301 |
(347,760) | (111,400) | (324,569) | ( |
(350,000) | (350,000) | (350,000) | (350,000) | |
G10 | 7049.249 | 7054.316 | 7059.864 |
|
7281.957 | 7058.823 | 7049.861 | 7049.278 |
(346,980) | (128,400) | (243,520) | ( |
(350,000) | (350,000) | (350,000) | (350,000) | |
G11 |
|
0.750 | 0.749999 |
|
0.750 | 0.750 | 0.749 |
|
( |
(11,400) | (23,722) | (100,100) | (350,000) | (350,000) | (350,000) | (350,000) | |
G12 |
|
|
|
|
|
|
NA |
|
( |
(16,400) | (59,355) | (100,100) | (350,000) | (350,000) | (350,000) | ||
G13 |
|
0.053957 | 0.0539498 | 0.056180 | 0.053947 | 0.757 | NA |
|
( |
(69,800) | (120,268) | (100,100) | (350,000) | (350,000) | (350,000) | ||
Nu. |
|
|
|
|
|
|
|
|
RK | 1 | 5 | 7 | 2 | 8 | 4 | 6 | 3 |
Comparison of best solutions for the three-bar truss design problem.
Method | DEDS | HEAA | PSO-DE | DELC | MBA | BSA | BSAISA |
---|---|---|---|---|---|---|---|
|
|
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NA |
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NA |
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NA |
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NA |
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NA |
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NA |
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|
To compare these algorithms synthetically, a simple evaluation mechanism is used. It can be explained as the best function value (Best) is preferred, and the function evaluations (FEs) are secondary. More specifically, (1) if one algorithm has a better
From Table
Based on the above comparison, it can be concluded that BSAISA is effective and competitive in terms of convergence speed.
In order to assess the optimization performance of BSAISA in real-world engineering constrained optimization problems, 5 well-known engineering constrained design problems including three-bar truss design, pressure vessel design, tension/compression spring design, welded beam design, and speed reducer design are considered in the second experiment.
The three-bar truss problem is one of the engineering minimization test problems for constrained algorithms. The best feasible solution is obtained by BSAISA at
Comparison of statistical results for the three-bar truss design problem.
Method | Worst | Mean | Best | Std | FEs |
---|---|---|---|---|---|
DEDS | 263.895849 | 263.895843 |
|
|
15,000 |
HEAA | 263.896099 | 263.895865 |
|
|
15,000 |
PSO-DE | 263.895843 | 263.895843 |
|
|
17,600 |
DELC | 263.895843 | 263.895843 |
|
|
10,000 |
MBA | 263.915983 | 263.897996 | 263.895852 |
|
13,280 |
BSA | 263.895845 | 263.895843 |
|
|
13,720 |
BSAISA | 263.895843 | 263.895843 |
|
|
|
Comparison of best solutions for the pressure vessel design problem.
Method | GA-DT | MDE | CPSO | HPSO | DELC | ABC | BSA-SA |
BSA | BSAISA |
---|---|---|---|---|---|---|---|---|---|
|
0.8125 | 0.8125 | 0.8125 | 0.8125 | 0.8125 | 0.8125 | 0.8125 | 0.8125 | 0.8125 |
|
0.4375 | 0.4375 | 0.4375 | 0.4375 | 0.4375 | 0.4375 | 0.4375 | 0.4375 | 0.4375 |
|
42.0974 | 42.0984 | 42.0913 | 42.0984 | 42.0984 | 42.0984 | 42.0984 | 42.0984 | 42.0984 |
|
176.6540 | 176.6360 | 176.7465 | 176.6366 | 176.6366 | 176.6366 | 176.6366 | 176.6366 | 176.6366 |
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NA | NA | 0 |
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NA | NA |
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NA | NA |
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NA | NA |
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6059.9463 |
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6061.0777 | 6059.7143 | 6059.7143 | 6059.7143 | 6059.7143 | 6059.7150 | 6059.7143 |
Comparison of statistical results for the pressure vessel design problem.
Method | Worst | Mean | Best | Std | FEs |
---|---|---|---|---|---|
GA-DT | 6469.3220 | 6177.2533 | 6059.9463 | 130.9297 | 80,000 |
MDE | 6059.7017 | 6059.7017 |
|
|
24,000 |
CPSO | 6363.8041 | 6147.1332 | 6061.0777 |
|
30,000 |
HPSO | 6288.6770 | 6099.9323 | 6059.7143 |
|
81,000 |
DELC | 6059.7143 | 6059.7143 | 6059.7143 |
|
30,000 |
PSO-DE | 6059.7143 | 6059.7143 | 6059.7143 |
|
42,100 |
ABC | NA | 6245.3081 | 6059.7147 |
|
30,000 |
BSA-SA |
6116.7804 | 6074.3682 | 6059.7143 |
|
80,000 |
BSA | 6771.5969 | 6221.2861 | 6059.7150 |
|
60,000 |
BSAISA | 7198.0054 | 6418.1935 | 6059.7143 |
|
|
Comparison of best solutions for the tension compression spring design problem.
Method |
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|
---|---|---|---|---|---|---|---|---|
GA-DT | 0.051989 | 0.363965 | 10.890522 |
|
|
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|
0.012681 |
MDE | 0.051688 | 0.356692 | 11.290483 |
|
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|
CPSO | 0.051728 | 0.357644 | 11.244543 |
|
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|
0.012675 |
HPSO | 0.051706 | 0.357126 | 11.265083 | NA | NA | NA | NA |
|
DEDS | 0.051689 | 0.356718 | 11.288965 | NA | NA | NA | NA |
|
HEAA | 0.051690 | 0.356729 | 11.288294 | NA | NA | NA | NA |
|
DELC | 0.051689 | 0.356718 | 11.288966 | NA | NA | NA | NA |
|
ABC | 0.051749 | 0.358179 | 11.203763 |
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MBA | 0.051656 | 0.35594 | 11.344665 |
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SSOC | 0.051689 | 0.356718 | 11.288965 | NA | NA | NA | NA |
|
BSA-SA |
0.051989 | 0.356727 | 11.288425 |
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BSA | 0.051694 | 0.356845 | 11.281488 |
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BSAISA | 0.051687 | 0.356669 | 11.291824 |
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Comparison of statistical results for the tension compression spring design problem.
Method | Worst | Mean | Best | Std | FEs |
---|---|---|---|---|---|
GA-DT | 0.012973 | 0.012742 | 0.012681 |
|
80,000 |
MDE | 0.012674 | 0.012666 |
|
|
24,000 |
CPSO | 0.012924 | 0.012730 | 0.012675 |
|
23,000 |
HPSO | 0.012719 | 0.012707 |
|
|
81,000 |
DEDS | 0.012738 | 0.012669 |
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24,000 |
HEAA | 0.012665 | 0.012665 |
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|
24,000 |
DELC | 0.012666 | 0.012665 |
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|
20,000 |
PSO-DE | 0.012665 | 0.012665 |
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|
24,950 |
ABC | NA | 0.012709 |
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|
30,000 |
MBA | 0.012900 | 0.012713 |
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|
SSOC | 0.012868 | 0.012765 |
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|
25,000 |
BSA-SA |
0.012666 | 0.012665 |
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|
80,000 |
BSA | 0.012669 | 0.012666 |
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|
43,220 |
BSAISA | 0.012668 | 0.012666 |
|
|
9440 |
Comparison of best solutions for the welded beam design problem.
Method | GA-DT | MDE | CPSO | HPSO | ABC | MBA | BSA-SA |
BSA | BSAISA |
---|---|---|---|---|---|---|---|---|---|
|
0.205986 | 0.205730 | 0.202369 | 0.205730 | 0.205730 | 0.205729 | 0.205730 | 0.205730 | 0.205730 |
|
3.471328 | 3.470489 | 3.544214 | 3.470489 | 3.470489 | 3.470493 | 3.470489 | 3.470489 | 3.470489 |
|
9.020224 | 9.036624 | 9.04821 | 9.036624 | 9.036624 | 9.036626 | 9.036624 | 9.036624 | 9.036624 |
|
0.206480 | 0.205730 | 0.205723 | 0.205730 | 0.205730 | 0.205729 | 0.205730 | 0.205730 | 0.205730 |
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NA |
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NA |
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NA |
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NA |
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NA |
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NA |
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NA |
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1.728226 |
|
1.728024 |
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|
1.724853 |
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Comparison of statistical results for the welded beam design problem.
Method | Worst | Mean | Best | Std | FEs |
---|---|---|---|---|---|
GA-DT | 1.993408 | 1.792654 | 1.728226 |
|
80,000 |
MDE | 1.724854 | 1.724853 |
|
NA | 24,000 |
CPSO | 1.782143 | 1.748831 | 1.728024 |
|
24,000 |
HPSO | 1.814295 | 1.749040 |
|
|
81,000 |
DELC | 1.724852 | 1.724852 |
|
|
|
PSO-DE | 1.724852 | 1.724852 |
|
|
66,600 |
ABC | NA | 1.741913 |
|
|
30,000 |
MBA | 1.724853 | 1.724853 | 1.724853 |
|
47,340 |
SSOC | 1.799332 | 1.746462 |
|
|
25,000 |
BSA-SA |
1.724852 | 1.724852 |
|
|
80,000 |
BSA | 1.724854 | 1.724852 |
|
|
45,480 |
BSAISA | 1.724854 | 1.724852 |
|
|
29,000 |
From Tables
Figure
Convergence curves of BSAISA and BSA for the three-bar truss design problem.
The pressure vessel design problem has a nonlinear objective function with three linear and one nonlinear inequality constraints and two discrete and two continuous design variables. The values of the two discrete variables (
For this problem, BSAISA is compared with nine algorithms: BSA, BSA-SA
As shown Table
Figure
Convergence curves of BSAISA and BSA for the pressure vessel design problem.
This design optimization problem has three continuous variables and four nonlinear inequality constraints. The best feasible solution is obtained by BSAISA at
From Tables
Figure
Convergence curves of BSAISA and BSA for the tension compression spring design problem.
The welded beam problem is a minimum cost problem with four continuous design variables and subject to two linear and five nonlinear inequality constraints. The best feasible solution is obtained by BSAISA at
For this problem, BSAISA is compared with many well-known algorithms as follows: GA-DT, MDE, CPSO, HPSO, DELC, POS-DE, ABC, MBA, BSA, BSA-SA
From Tables
It is worth mentioning that from [
Figure
Convergence curves of BSAISA and BSA for the welded beam design problem.
This speed reducer design problem has eleven constraints and six continuous design variables (
Comparison of best solutions for the speed reducer design problem.
Method | MDE | DEDS | DELC | HEAA | POS-DE | MBA | BSA | BSAISA |
---|---|---|---|---|---|---|---|---|
|
3.500010 | 3.500000 | 3.500000 | 3.500023 | 3.500000 | 3.500000 | 3.500000 | 3.500000 |
|
0.700000 | 0.700000 | 0.700000 | 0.700000 | 0.700000 | 0.700000 | 0.700000 | 0.700000 |
|
17 | 17 | 17 | 17.000013 | 17.000000 | 17.000000 | 17 | 17 |
|
7.300156 | 7.300000 | 7.300000 | 7.300428 | 7.300000 | 7.300033 | 7.300000 | 7.300000 |
|
7.800027 | 7.715320 | 7.715320 | 7.715377 | 7.800000 | 7.715772 | 7.715320 | 7.715320 |
|
3.350221 | 3.350215 | 3.350215 | 3.350231 | 3.350215 | 3.350218 | 3.350215 | 3.350215 |
|
5.286685 | 5.286654 | 5.286654 | 5.286664 | 5.286683 | 5.286654 | 5.286654 | 5.286654 |
|
2996.356689 |
|
|
2994.499107 | 2996.348167 | 2994.482453 |
|
|
Comparison of statistical results for the speed reducer design problem.
Method | Worst | Mean | Best | Std | FEs |
---|---|---|---|---|---|
MDE | 2996.390137 | 2996.367220 | 2996.356689 |
|
24,000 |
DEDS | 2994.471066 | 2994.471066 |
|
|
30,000 |
HEAA | 2994.752311 | 2994.613368 | 2994.499107 |
|
40,000 |
DELC | 2994.471066 | 2994.471066 |
|
|
30,000 |
PSO-DE | 2996.348204 | 2996.348174 | 2996.348167 |
|
54,350 |
ABC | NA | 2997.058412 | 2997.058412 | 0 | 30,000 |
MBA | 2999.652444 | 2996.769019 | 2994.482453 | 1.56 | 6300 |
SSOC | 2996.113298 | 2996.113298 | 2996.113298 |
|
25,000 |
BSA | 2994.471066 | 2994.471066 |
|
|
25,640 |
BSAISA | 2994.471095 | 2994.471067 |
|
|
|
As shown in Tables
Figure
Convergence curves of BSAISA and BSA for the speed reducer design problem.
Sign Test [
Comparisons between BSAISA and other algorithms in Sign Tests.
BSAISA-methods |
|
|
|
Total |
|
---|---|---|---|---|---|
SR | 11 | 1 | 1 | 13 | 0.006 |
FSA | 12 | 0 | 1 | 13 | 0.003 |
CDE | 8 | 0 | 5 | 13 | 0.581 |
AMA | 13 | 0 | 0 | 13 | 0.000 |
MABC | 10 | 2 | 1 | 13 | 0.012 |
RPGA | 11 | 0 | 0 | 11 | 0.001 |
BSA-SA |
8 | 4 | 1 | 13 | 0.039 |
As shown in Table
On the one hand, all experimental results suggest that the proposed method improves the convergence speed of BSA. On the other hand, the overall comparative results of BSAISA and other well-known algorithms demonstrate that BSAISA is more effective and competitive for constrained and engineering optimization problems in terms of convergence speed.
In this paper, we proposed a modified version of BSA inspired by the Metropolis criterion in SA (BSAISA). The Metropolis criterion may probabilistically accept a higher energy state and the acceptance probability can decrease as the temperature decreases, which motivated us to redesign the amplitude control factor
This paper suggests that the proposed BSAISA has a superiority in terms of convergence speed or computational cost. The downside of the proposed algorithm is, of course, that its robustness does not show enough superiority. So our future work is to further research into the robustness of BSAISA on the basis of current research. Niche technique is able to effectively maintain population diversity of evolutionary algorithms [
The authors declare that they have no conflicts of interest.
This work was supported in part by the National Natural Science Foundation of China (no. 61663009), and the State Key Laboratory of Silicate Materials for Architectures (Wuhan University of Technology, SYSJJ2018-21).