This paper proposes Parallelized Linear TimeVariant Acceleration Coefficients and Inertial Weight of Particle Swarm Optimization algorithm (PLTVACIWPSO). Its designed has introduced the benefits of Parallel computing into the combined power of TVAC (TimeVariant Acceleration Coefficients) and IW (Inertial Weight). Proposed algorithm has been tested against linear, nonlinear, traditional, and multiswarm based optimization algorithms. An experimental study is performed in two stages to assess the proposed PLTVACIWPSO. Phase I uses 12 recognized Standard Benchmarks methods to evaluate the comparative performance of the proposed PLTVACIWPSO
The Concept of swarm intelligence (SI) principle, is highly inspired by the recent advancement in the field of Neuroscience and the Behavioral Science, commonly known as intelligent paradigm in Intelligence Computational domain, to solve the optimization issues of various problems in absence of any global models [
Kennedy et al., proposed a particle swarm optimization algorithm, based on metaheuristics evolved from swarm intelligence methodologies, it can emulate cordial movement patterns of flocks of birds and flying patterns of birds, it has capabilities for intercommunication among group members, to assist them in decision making processes, in a coherent synchronized manner [
Swarms are simulated, as if flocks are out in search of their food, any particular individual in the flock, determines its relative speed and position by calculating these two attributes of its neighbors speed and positions among the flock members. During Optimization of solution space, a swarm particle in a PSO changes its positions in a multidimensional space. This phenomenon reconciles location of particle under solution space for the problem being optimized. These particles are controlled by a tradeoff of memory between the group and the individuals. As stated in [
Iteratively, the particle velocity determines its route. The particle velocity is determined by three key factors. The first component, a social component, tries to dominate the best position for all particles I during a single iteration or to keep the global best position till the next iteration, which is termed the current global best position (gbesti), where i is the particle’s index. The second component, a cognitive component, tries to dominate each particle I in the swarm individually, or the personal best for a given particle, until the present iteration, dubbed the current best for particle i. (pbesti). The third component, a momentum component, determines the influence of each particle’s past velocity and is considered a modification of the original PSO described in [
The proposed Parallelized Linear TimeVariant Acceleration Coefficients and Inertial Weight of PSO (PLTVACIWPSO) algorithm, based on a hybridization of the linearity of timevariant acceleration coefficients and a linear decrease of the inertial weight associated with the parallel processing of PSO fitness evaluation, will be used in this paper. The proposed algorithm will be used to optimize 12 known benchmark functions, varying in their properties and scales. The optimization of these functions tends to minimize the errors generated in the sequential iterations of the proposed PLTVACIWPSO
Algorithm name  Short name  Category  Referenced or implemented  Description 

Linear Inertial Weight PSO  LIWPSO  IW family  [ 
Inertial weight 
Parallelized Linear Inertial Weight PSO  PLIWPSO  Implemented  Parallelized version of LIWPSO  
NonLinear Inertial Weight PSO  NLIWPSO  [ 
Inertial weight 

Parallelized NonLinear Inertial Weight PSO  PNLIWPSO  Implemented  Parallelized version of NLIWPSO  
Linear TimeVariant PSO  LTVPSO  TVAC family  [ 
PSO acceleration variables c_{1} and c_{2} linearly vary over time. 
Parallelized Linear Time Variant PSO  PLTVPSO  [ 
Parallelized version of LTVPSO  
NonLinear TimeVariant PSO  NLTVPSO  [ 
PSO acceleration variables c_{1} and c_{2} nonlinearly vary overtime.  
Parallelized NonLinear Time Variant PSO  PNLTVPSO  [ 
Parallelized version of NLTVPSO  
Particle Swarm Optimization  PSO  Traditional algorithms  [ 
A natureinspired algorithm based on the behavior of bird swarms and fish schools. 
Differential Evolution  DE  [ 
A modified version of genetic algorithms  
Genetic Algorithms  GA  [ 
A natureinspired optimization algorithm based on human genes and their crossover and mutation  
Flower Pollination Algorithm  FPA  [ 
A natureinspired optimization algorithm based on flower pollination behavior.  
Bat  BAT  Bat algorithms  [ 
A bioinspired stochastic algorithm based on a swarm of bats 
Multiswarm BAT  MBAT  [ 
A bioinspired stochastic algorithm based on multiple swarms of bats. 
The PSO algorithm utilises metaparameters, which controls the swarm actions for efficient optimization and it effective in enhancing searchability of a particle in the swarm. The convergence characteristics parameters of PSO algorithm are dependent upon controlling parameters, hence impact on controlling parameters also impacts convergence characteristics. Therefore, the functions of these metaparameters and their influence on the conclusive results are crucially importance for designing an efficient optimization algorithm [
The random position of each particle is the initialization step in the PSO algorithm, which starts inside an iteration to keep searching for optimal solutions. The velocities and positions for each particle are determined in every iteration until the final iteration or until any stopping criteria are provided.
Nonuniformly disseminated introductory particles impact the minimal stability properties of the PSO algorithm. Furthermore, the convergence of the particle’s velocity in the swarm is dependent on the initial population [
The PSO algorithm’s control parameters are the inertial weight and acceleration coefficients. The correct setting of these parameters can have an impact on the PSO’s convergence. The PSO algorithm’s cognitive and social components are influenced by the acceleration coefficients c1 and c2 [
To increase the performance of the PSO algorithm, the TVAC was upgraded [
Another strategy for enhancing the swarm’s particle convergence rate is linear inertial weight w adaptation [
The PSO algorithm’s execution time was reduced and the fitness assessment of individual particles in the swarm was reduced by parallel processing of objective functions across multiples of the independent machine in the form of a cluster [
The capacity of particles in the swarm to cover the search space and give consistent results will be restricted if the social factor is smaller than the cognitive element utilized in classic PSO algorithms (such as those demonstrated in [
By introducing the technique that will be proven in this work, the aforementioned issues are alleviated. The Parallelized linear TVAC and inertial weight of PSO, or “PLTVACIWPSO,” is a suggested method that takes use of the benefits of linearity for each timevariant acceleration coefficient described in [
The proposed method PLTVACIWPSO takes a benchmark function as input and treats it as the goal function. The swarm iterates in PLTVACIWPSO based on changes made to the inertial weight w and TVAC parameters, as well as parallelization of the objective function evaluation and the optimized value produced for the given objective function at each iteration; finally, the smallest optimized value is chosen from all values produced by all executed iterations.
There are two stages to the experiment described in this study. Phase one compares three kinds of algorithms to the proposed method PLTVACIWPSO, which is utilized to optimize twelve distinct benchmark functions. The “IW family” is the first category, which contains algorithms that use PSO’s parallelized and nonparallelized inertial weight. The “TVAC family,” which comprises algorithms based on PSO’s parallelized and nonparallelized TVAC, is the second category. Traditional algorithms such as PSO, differential evolution (DE), the genetic algorithm (GA), and the flower pollination algorithm (FPA) are included in the third category [
The proposed PLTVACIWPSO algorithm used in phase one is a stochastic iterative PSO algorithm by nature, which indicates that an optimized result will be produced during each iteration for any given function listed in Appendix A. Therefore, the values of Worst, Best, and Mean of Mean Square Error (MSE), represented in
A series of independent runs is used to generate accurate findings from the stochastic based methods indicated in
For the proposed PLIWTVACPSO, 100 separate executions are done, and the average values of Worst, Best, and Mean, as well as the best overall execution for the proposed PLIWTVACPSO, are chosen and compared to the BAT and MBAT algorithms.
In this paper, the implementation of both PLTVACIWPSO algorithms was done using R language version 3.3.0, executed on a virtualized CentOS Linux operating system with a 2 GHz dualcore processor, 5 GB RAM, and 20 GB of nonvolatile storage.
For the IW family algorithms used in phase 1, inertial weight wrang = (0.4, 09), swarm size = 50, and iterations = 100. For the TVAC family algorithms used in phase one, inertial weight w = 0.721, c1rang = (1.28: 1.05), c2rang = (1.05: 1.28), and iterations = 100. For PSO, swarm size = 50, w = 0.721, c1 = 1.193, c2 = 1.193, and iterations = 100. For the DE algorithm used in phase I, population size = 50, crossover probability = 0.5, differential weighting factor = 0.8, and iterations = 100. For the GA used in phase I, population size = 50, crossover probability = 0.8, mutation probability = 0.1, and iterations = 100. For the FPA used in phase 1, population size = 25, probability switch = 0.8, and iterations = 100.
In phase II, the same configurations for phase I are used for PLTVACIWPSO.
Three different experiments were conducted in phase one to illustrate the capabilities of the proposed PLTVACIWPSO algorithm. The first experiment compares the proposed PLIWTVACPSO with the IW family mentioned earlier in
Functions  Values  PLTVACIWPSO  PLIWPSO  LIWPSO  PNLIWPSO  NLIWPSO 

F1  Worst  18.94  18.72  19.18  18.9  
Best  3.98  3.75  3.65  4.59  
Mean  7.28  7.58  7.55  8.43  
F2  Worst  266.8  250.5  237  265.02  
Best  2.11  1.95  2.31  2.41  
Mean  27.4  23.9  27.2  30.3  
F3  Worst  1.15E04  9.90E03  1.26E04  8.78E03  
Best  1.94E0  1.85E0  6.65E0  2.96E0  
Mean  5.74E02  5.40E02  4.69E03  4.67E02  
F4  Worst  330.80  331.5  331.5  324.7  
Best  105.4  100.2  59.7  83.1  
Mean  169  173.28  153.88  169.62  
F5  Worst  1.15E05  1.79E05  6.15E04  1.55E05  
Best  1.25E − 01  1.26E − 04  1.84E − 04  6.28E − 04  
Mean  4.37E03  4.66E03  3.88E03  4.42E03  
F6  Worst  135.6  114.7  152.9  106.5  
Best  0.09  0.04  0.04  0.03  
Mean  2.45  2.48  6.39  2.03  
F7  Worst  98.71  96.61  112.2  94.99  
Best  1.28  0.55  0.61  0.78  
Mean  10.1  6.32  7.74  7.64  
F8  Worst  43.43  44.82  61.91  46.9  
Best  0.54  0.60  2.02  0.78  
Mean  7.34  7.52  15.46  8.12  
F9  Worst  6.03E04  6.41E04  6.08E04  6.93E04  
Best  2.47E0  2.09E0  9.28E − 01  2.12E0  
Mean  3.10E03  2.45E03  2.75E03  3.76E03  
F10  Worst  1.64  1.63  1.62  1.61  
Best  0.33  0.51  0.77  0.52  
Mean  0.82  0.76  0.98  0.8  
F11  Worst  2.68E04  2.49E04  2.81E04  2.46E04  
Best  7.35E01  7.40E01  1.18E02  2.13E01  
Mean  2.41E03  2.41E03  2.55E03  2.09E03  
F12  Worst  4.78E07  4.12E07  4.47E07  4.95E07  
Best  4.10E02  4.09E02  1.60E03  2.01E03  
Mean  2.06E06  1.85E06  1.76E03  2.10E06 
The second experiment compares the proposed PLIWTVACPSO with the TVAC family, including the PLTVPSO algorithm; this family is mentioned in
Functions  Values  PLTVACIWPSO  PLTVPSO  LTVPSO  PNLTVPSO  NLTVPSO 

F1  Worst  20.67  20.73  20.73  20.87  
Best  2.98  2.71  3.91  3.82  4.31  
Mean  9.02  9.8  9.97  10.09  
F2  Worst  587.16  706.1  671.8  653.9  
Best  1.51  2.1  1.99  2.09  
Mean  65.36  65.36  62.08  72.79  
F3  Worst  6.82E04  9.27E04  7.82E04  7.28E04  
Best  2.86E00  1.74E00  3.35E00  3.62E00  
Mean  2.75E03  3.79E03  2.84E03  3.32E03  
F4  Worst  460.9  461.7  463.9  466.7  
Best  54.65  146.15  112.41  93.36  
Mean  189.59  227.32  208.47  205.96  
F5  Worst  5.58E05  6.36E05  1.30E07  1.30E06  
Best  3.64E − 01  8.36E − 05  8.96E − 04  3.82E − 04  
Mean  2.30E04  2.80E04  1.94E05  2.88E04  
F6  Worst  126.6  183.1  156.4  192.6  
Best  0.09  0.03  0.09  0.09  
Mean  6.39  7.79  7.21  7.43  
F7  Worst  210.8  219.1  242.7  240.3  
Best  0.79  1.54  1.23  1.21  
Mean  20.85  22.96  23.78  24.94  
F8  Worst  61.91  72.49  67.4  67.72  
Best  2.02  3.28  3.57  3.77  
Mean  15.46  18.12  17.17  18.94  
F9  Worst  5.57E05  5.16E05  5.59E05  1.30E06  
Best  6.05E − 01  8.13E − 01  1.65E00  4.54E00  3.82E − 04  
Mean  2.32E04  2.37E04  2.49E04  2.88E04  
F10  Worst  1.59  1.73  1.65  1.7  
Best  0.5  1  1.01  1.01  
Mean  0.97  1.18  1.24  1.24  
F11  Worst  6.39E04  8.39E04  8.12E04  7.83E04  
Best  5.17E01  8.37E01  8.33E01  7.17E01  
Mean  6.07E03  6.91E03  7.01E03  6.41E03  
F12  Worst  2.61E08  3.06E08  3.69E08  3.11E08  
Best  1.29E03  2.22E03  1.64E03  2.28E03  
Mean  1.23E07  1.32E07  1.34E07  1.32E07 
Finally, the third experiment aims at comparing the proposed PLIWTVACPSO with the traditional algorithms and the BAT algorithms as mentioned earlier in
Functions  Values  PLTVACIW 
PSO  GA  DE  FPA  BAT  MBAT 

F1  Worst  20.78  20.68  21.89  20.81  19.95  19.95  
Best  4.13  12.42  21.45  18.42  17.04  11.91  
Mean  10.46  17.02  21.69  19.87  19.43  15.79  
F2  Worst  780.7  587.16  1695.23  651.64  317.48  324.29  
Best  2.95  29.05  1269.96  104.78  120.66  30.54  
Mean  79.97  159.34  1466.61  297.13  257.51  91.67  
F3  Worst  9.99E + 04  7.38E + 04  4.85E + 05  7.86E + 04  3.40E + 05  4.20E + 04  
Best  5.37E + 00  1.87E + 02  2.74E + 05  1.78E + 03  5.20E + 03  3.00E + 03  
Mean  3.41E + 03  1.05E + 04  3.87E + 05  1.83E + 04  3.70E + 05  4.10E + 04  
F4  Worst  305.2  497.4  474.25  888.75  436.37  
Best  91.23  154.06  686.73  265.6  179.98  
Mean  127.02  211.18  245.58  779.01  347.81  252.81  
F5  Worst  1.28E + 06  7.92E + 05  1.80E + 10  6.16E + 06  3.40E + 05  
Best  7.34E + 04  1.50E − 02  1.64E + 10  2.10E + 01  843.7  77.47  
Mean  4.13E + 04  3.28E + 04  1.35E + 10  1.25E + 05  6.10E + 05  
F6  Worst  180.2  149.72  648.07  229.28  85.31  85.31  
Best  0.09  1.41  417.57  5.53  2.03  0.24  
Mean  7.77  23.55  531.11  50.23  26.52  3.46  
F7  Worst  221.2  234.51  970.31  240.11  85.31  4.20E + 04  
Best  1.76  16.42  689.64  67.67  40.31  6.86  
Mean  26.07  62.92  852.97  122.61  93.88  25.02  
F8  Worst  42.75  68.01  65.79  162.11  69.15  31.88  
Best  4.13  16.16  65.76  34.41  25.67  6.07  
Mean  17.05  31.08  125.18  46.94  36.49  14.31  
F9  Worst  5.54E + 04  7.54E + 05  3.73E + 05  2.44E + 06  5.36E + 05  3.40E + 05  3.40E + 05 
Best  3.09E + 00  2.27E + 03  1.60E + 06  1.76E + 04  1.50E + 04  1.00E + 03  
Mean  2.76E + 04  7.70E + 04  1.99E + 06  1.55E + 05  1.00E + 05  1.20E + 04  
F10  Worst  1.58  1.74  3.87  1.62  1.10E + 06  1.10E + 06  
Best  1.02  0.75  2.46  0.01  1.09  0.54  
Mean  1.29  1.02  3.47  0.53  3.69  3.09  
F11  Worst  8.89E + 04  6.97E + 04  1.75E + 05  6.73E + 04  
Best  1.69E + 02  4.03E + 04  9.79E + 04  1.03E + 04  
Mean  8.24E + 03  2.09E + 04  1.38E + 05  3.21E + 04  
F12  Worst  4.05E + 08  2.71E + 08  1.17E + 09  2.88E + 08  
Best  2.14E + 03  1.88E + 06  7.41E + 08  1.04E + 07  
Mean  1.42E + 07  4.83E + 07  9.36E + 08  8.31E + 07 
Setting the IW to the highest value, which permits particles to travel at a fast speed and participate in extensive exploration, increases the exploration behavior of particles in the proposed PLIWTVACPSO algorithm towards the nearest position of the optimal value. Following that, IW repeatedly declines linearly to lower levels, causing the particles to travel slowly and engage in more deliberate investigation. As a consequence, lowering the Worst values acquired in early iterations provides the swarm with greater Best values during its final rounds than other IW family algorithms. Furthermore, the mean value indicates that the proposed PLIWTVACPSO yields modest Worst values in early iterations and converges to optimal Best values in middle and final iterations.
The average values of the Worst, Best, and Mean are acquired from 50 different executions for the proposed PLIWTVACPSO, IW family algorithms, TVAC family algorithms, and conventional algorithms to verify that the proposed algorithm PLIWTVACPSO obtains trustworthy results. Functions F1–3 are used in this supplementary experiment. These findings are shown in
Functions  Values  PLTVACIWPSO  PLIWPSO  LIWPSO  PNLIWPSO  NLIWPSO 

F1  Worst  20.56  20.77  20.86  20.92  
Best  4.71  4.64  4.62  4.76  
Mean  8.96  8.90  8.91  9.88  
F2  Worst  232.56  238.92  232.27  232.65  
Best  1.68  1.82  1.73  1.81  
Mean  21.61  22.67  21.26  21.55  
F3  Worst  1.06E + 04  1.06E + 04  1.06E + 04  1.07E + 04  
Best  2.56E + 00  2.41E + 00  2.43E + 00  2.71E + 00  
Mean  4.66E + 02  4.51E + 02  4.43E + 02  4.46E + 02 
Functions  Values  PLTVACIWPSO  PLTVPSO  LTVPSO  PNLTVPSO  NLTVPSO 

F1  Worst  21.76  21.9  21.84  21.82  
Best  3.62  4.81  4.76  5.77  
Mean  11.46  11.77  1.85  12.23  
F2  Worst  675.81  886.54  891.97  865.28  
Best  1.64  1.63  1.62  1.55  
Mean  73.56  83.56  82.51  82.58  
F3  Worst  7.90E + 04  8.03E + 04  8.31E + 04  8.22E + 04  
Best  5.77E + 00  6.41E + 00  4.55E + 00  4.04E + 00  
Mean  4.20E + 03  4.27E + 03  4.41E + 03  4.34E + 03 
Functions  Values  PLTVACIWPSO  PSO  GA  DE  FPA 

F1  Worst  22.78  22.74  24.80  22.82  
Best  5.31  14.03  22.45  19.42  
Mean  12.52  19.98  23.69  20.39  
F2  Worst  871.85  2720.27  690.13  770.07  
Best  1.68  1750.84  40.23  80.53  
Mean  74.78  1907.92  90.71  357.11  
F3  Worst  1.09E + 04  8.89E + 05  6.80E + 06  1.51E + 05  
Best  7.98E + 00  2.35E + 02  2.11E + 05  4.77E + 03  
Mean  4.89E + 03  3.76E + 04  8.91E + 05  5.91E + 04 
We employed ten benchmark medical datasets in this phase. The datasets were taken from the University of California, Irvine’s learning machine server [
ID  Dataset  No. of instances  No. of features  Total no. of classes 

DS1  Erythematosquamous  366  34  6 
DS2  Breast cancer  569  32  2 
DS3  Hepatitis  155  19  2 
DS4  SPECTF heart data set  267  45  2 
DS5  Cervical cancer (Risk factors)  858  36  2 
DS6  Parkinson  1208  26  2 
DS7  Lung cancer  32  56  3 
DS8  Thyroid disease  7200  21  3 
DS9  Hepatitis C Virus (HCV) for Egyptian patients  1385  29  4 
DS10  Parkinson’s disease classification  197  23  2 
Three parameters are used to test PLIWTVACPSO selected features. Other requirements include accuracy of classification, number of specified features and CPU processing time have been considered [
Before applying PLIWTVACPSO  After applying PLIWTVACPSO  

Datasets  Total no. of features  Accuracy (%)  Time (sec)  Selected features  Accuracy (%)  Time (sec) 
DS1  34  91.9814  0.7199  21  94.9467  0.4129 
DS2  32  90.2838  0.0969  14  94.6010  0.0477 
DS3  19  83.7006  0.0939  5  91.0359  0.0356 
DS4  45  75.8929  0.0992  20  81.5087  0.0371 
DS5  36  63.9529  0.0995  18  74.3023  0.0523 
DS6  26  89.5345  0.2912  15  92.3475  0.1986 
DS7  56  87.2746  0.0950  20  90.7859  0.0355 
DS8  21  89.5345  0.2912  13  93.6982  0.1986 
DS9  29  88.1434  0.1230  14  91.1285  0.1023 
DS10  23  85.1255  0.5118  10  92.9413  0.4269 
The number of features and the mean accuracy of 10 folds are presented together with CPU processing time. Only a subset of characteristics may reduce classification efficiency, but computation time and storage can be greatly reduced.
The suggested PLTVACIWPSO algorithm outperforms PSO IWbased algorithms, TVAC algorithms, GA and DE algorithms, and even multiswarmbased algorithms in terms of efficiency and efficacy. This demonstrates that combining the linearity of IW and TVAC may adjust a single swarmbased PSO algorithm to get optimal outcomes. Furthermore, the suggested PLTVACIWPSO method tends to reduce the worst values from earlier iterations in order to attain the best value during the final rounds, demonstrating that the proposed algorithm is an effective and reliable global optimization technique. The suggested PLTVACIWPSO Algorithm was used to verify ten medical datasets. The findings suggest that PLTVACIWPSO can increase classification, consistency, the number of features picked, and convergence speed by a substantial amount.
Future work could include switching between the linearity of IW and the nonlinearity of TVAC and vice versa, replacing PSO parallelization across multiple cores with a dedicated board of CPUs, and finally, instead of using a single swarm, the proposed PLTVACIWPSO can use multiple swarms of particles. PLTVACIWresults PSO’s may be used to verify increasingly difficult challenges in science and contemporary engineering in the future.
The authors would like to thank Prince Sultan University, Riyadh, Saudi Arabia for paying the Article Processing Charges (APC) of this publication. Special acknowledgement to Automated Systems & Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh, Saudi Arabia. Also, the authors wish to acknowledge the editor and anonymous reviewers for their insightful comments, which have improved the quality of this publication.