RankBased Ant System with Originality Reinforcement and Pheromone Smoothing
Abstract
:1. Introduction
1.1. Motivation
1.2. Aims and Structure of This Paper
 an originality reinforcement strategy that rewards the originality (dissimilarity from already searched space) of the solutions with good fitness; and
 a pheromone smoothing mechanism that is triggered before the algorithm reaches stagnation, increasing exploration and making possible it to find better solutions.
2. Ant Colony Optimization
2.1. Ant Colony Optimization Algorithms
Algorithm 1 ACO metaheuristic 
Require: ACO parameters $\tau \leftarrow $ initialise pheromone trails while termination condition not met do for $k=1$ to m do ▹ for every ant for every step until the kth ant has completed the tour do select node j to visit next according to transition rule ▹ Equation (1) end for end for Apply local search (optional) Update pheromone trails $\tau $ end while 
2.2. Recent Trends in ACO Algorithms
3. Extension of RankBased Ant System
3.1. Originality Utility Function
3.2. Pheromone Smoothing Mechanism
3.3. Algorithm
Algorithm 2 AS${}_{Rank}$ with originality reinforcement and pheromone smoothing 

4. Experimental Results
4.1. TSP and SOP Benchmarks
 Symmetric and Asymmetric Travelling Salesman Problem (TSP and ASTP) aim to find the Hamiltonian cycle of minimum length given a graph with n cities. In case the distances between the cities are independent of the direction of traversing the edges ${d}_{i,j}={d}_{j,i}$, $\forall (i,j)$, the problem is known as symmetric TSP; otherwise—as asymmetric TSP.
 Sequential Ordering Problem (SOP) consists of finding a Hamiltonian path of the minimal length from node 1 to node n taking precedence constraints into account. The precedence constraints impose that some nodes have to be visited before some other nodes of the graph G.
4.2. Experimental Setup
4.3. Analysis of the Proposed ACO Algorithm
4.4. Comparison with ACO Algorithms
5. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Instance  Type  Optimal Solution  Dimension (n) 

brazil58  TSP  25,395  58 
kroA100  TSP  21,282  100 
ch130  TSP  6110  130 
tsp225  TSP  3916  335 
gr48  TSP  5046  48 
pr76  TSP  108,159  76 
gr202  ATSP  40,160  202 
ftv35  ATSP  1473  36 
ftv64  ATSP  1839  65 
ftv70  ATSP  1950  71 
ESC78  SOP  18,230  80 
ft70.1  SOP  39,313  71 
p43.1  SOP  28,140  44 
p43.4  SOP  83,005  44 
Instance  Percentage (%)  

10  20  30  40  50  60  70  80  90  100  
brazil58  22  7  6  5  3  2  1  1  1  0 
kroA100  17  12  9  7  4  3  3  2  2  1 
ch130  18  11  8  6  5  4  3  3  3  0 
tsp225  21  18  14  12  10  7  4  1  0  0 
gr48  14  7  4  2  2  2  2  1  0  0 
pr76  14  9  6  5  5  4  4  2  2  1 
gr202  16  14  9  7  5  4  3  2  1  0 
ftv35  18  14  10  7  4  4  4  3  3  3 
ftv64  17  13  9  8  7  7  5  3  1  0 
ftv70  21  15  5  4  2  2  1  1  0  0 
ESC78  9  6  5  5  3  2  1  0  0  0 
ft70.1  18  13  11  7  5  2  2  1  0  0 
p43.1  15  12  8  6  6  5  3  2  1  0 
p43.4  25  20  18  15  13  10  6  5  5  3 
Instance  Algorithm  Avg.  Std.  Best  PD${}_{\mathit{avg}}\phantom{\rule{3.33333pt}{0ex}}(\%)$  PD${}_{\mathit{best}}\phantom{\rule{3.33333pt}{0ex}}(\%)$  Time (s) 

brazil58  AS${}_{Rank}$  25,628  126  25,400  0.92  0.02  14 
AS${}_{Rank}^{ps}$  25,480  118  25,395  0.33  0  14  
AS${}_{ORank}$  25,677  124  25,400  1.11  0.02  13  
AS${}_{ORank}^{ps}$  25,487  121  25,395  0.36  0  15  
kroA100  AS${}_{Rank}$  21,683  252  21,306  1.89  0.11  45 
AS${}_{Rank}^{ps}$  21,357  100  21,282  0.35  0  45  
AS${}_{ORank}$  21,591  186  21,331  1.45  0.23  43  
AS${}_{ORank}^{ps}$  21,378  120  21,282  0.45  0  45  
ch130  AS${}_{Rank}$  6235  39  6169  2.04  0.97  120 
AS${}_{Rank}^{ps}$  6193  47  6141  1.36  0.51  120  
AS${}_{ORank}$  6241  48  6154  2.14  0.72  115  
AS${}_{ORank}^{ps}$  6170  33  6136  0.98  0.43  120  
tsp225  AS${}_{Rank}$  4026  31  3989  2.80  1.86  422 
AS${}_{Rank}^{ps}$  3949  23  3916  0.84  0  422  
AS${}_{ORank}$  4034  33  3978  3.02  1.58  407  
AS${}_{ORank}^{ps}$  3942  17  3916  0.65  0  424  
gr48  AS${}_{Rank}$  5117  40  5066  1.40  0.40  9 
AS${}_{Rank}^{ps}$  5104  35  5054  1.15  0.16  9  
AS${}_{ORank}$  5135  39  5074  1.76  0.55  9  
AS${}_{ORank}^{ps}$  5091  28  5049  0.88  0.06  11  
pr76  AS${}_{Rank}$  111,609  1064  109,392  3.19  1.14  60 
AS${}_{Rank}^{ps}$  110,357  1187  108,238  2.03  0.07  61  
AS${}_{ORank}$  111,507  810  110,100  3.10  1.79  60  
AS${}_{ORank}^{ps}$  109,922  900  108,159  1.63  0  64  
gr202  AS${}_{Rank}$  41,803  435  40,960  4.09  1.99  324 
AS${}_{Rank}^{ps}$  41,095  255  40,609  2.33  1.12  325  
AS${}_{ORank}$  41,602  340  41,022  3.59  2.15  317  
AS${}_{ORank}^{ps}$  41,056  228  40,554  2.23  0.98  328  
ftv35  AS${}_{Rank}$  1497  9  1473  1.65  0  5 
AS${}_{Rank}^{ps}$  1488  10  1473  0.98  0  5  
AS${}_{ORank}$  1497  11  1473  1.64  0  5  
AS${}_{ORank}^{ps}$  1483  11  1473  0.71  0  5  
ftv64  AS${}_{Rank}$  1867  21  1848  1.50  0.49  17 
AS${}_{Rank}^{ps}$  1859  8  1848  1.07  0.49  17  
AS${}_{ORank}$  1862  14  1839  1.23  0  16  
AS${}_{ORank}^{ps}$  1858  9  1839  1.05  0  17  
ftv70  AS${}_{Rank}$  2010  44  1957  3.10  0.36  21 
AS${}_{Rank}^{ps}$  1999  35  1957  2.49  0.36  21  
AS${}_{ORank}$  1989  30  1950  1.98  0  20  
AS${}_{ORank}^{ps}$  1989  37  1989  1.99  0.20  21 
Instance  Algorithm  Avg.  Std.  Best  PD${}_{\mathit{avg}}\phantom{\rule{3.33333pt}{0ex}}(\%)$  PD${}_{\mathit{best}}\phantom{\rule{3.33333pt}{0ex}}(\%)$  Time (s) 

ESC78  AS${}_{Rank}$  18,609  147  18,415  2.08  1.01  183 
AS${}_{Rank}^{ps}$  18,470  43  18,405  1.32  0.96  185  
AS${}_{ORank}$  18,584  158  18,380  1.94  0.82  185  
AS${}_{ORank}^{ps}$  18,460  69  18,300  1.26  0.38  192  
ft70.1  AS${}_{Rank}$  41,403  419  40,646  5.32  3.39  81 
AS${}_{Rank}^{ps}$  41,082  297  40,505  4.50  3.03  81  
AS${}_{ORank}$  41,053  331  40,529  4.43  3.09  80  
AS${}_{ORank}^{ps}$  40,768  341  40,092  3.70  1.98  81  
p43.1  AS${}_{Rank}$  28,333  107  28,220  0.69  0.28  21 
AS${}_{Rank}^{ps}$  28,255  65  28,220  0.41  0.28  21  
AS${}_{ORank}$  28,330  107  28,220  0.63  0.28  21  
AS${}_{ORank}^{ps}$  28,236  34  28,220  0.34  0.28  22  
p43.4  AS${}_{Rank}$  83,693  119  83,415  0.83  0.49  67 
AS${}_{Rank}^{ps}$  83,405  73  83,295  0.48  0.35  68  
AS${}_{ORank}$  83,620  120  83,405  0.74  0.48  67  
AS${}_{ORank}^{ps}$  83,416  113  83,265  0.49  0.31  68 
Instance  Algorithm  Avg.  Std.  Best  PD${}_{\mathit{avg}}\phantom{\rule{3.33333pt}{0ex}}(\%)$  PD${}_{\mathit{best}}\phantom{\rule{3.33333pt}{0ex}}(\%)$  Time (s) 

brazil58  AS  25,930  89  25,685  2.10  1.14  14 
AS${}_{Rank}$  25,628  126  25,400  0.92  0.02  14  
MMAS  25,622  51  25,480  0.89  0.33  14  
ACS  25,464  116  25,395  0.27  0  16  
AS${}_{ORank}^{ps}$  25,487  121  25,395  0.36  0  15  
kroA100  AS  22,714  198  22,221  6.73  4.41  45 
AS${}_{Rank}$  21,683  252  21,306  1.89  0.11  45  
MMAS  21,462  173  21,330  0.85  0.23  45  
ACS  21,559  279  21,282  1.30  0  53  
AS${}_{ORank}^{ps}$  21,378  120  21,282  0.45  0  45  
ch130  AS  6632  71  6482  8.54  6.09  120 
AS${}_{Rank}$  6235  39  6169  2.04  0.97  120  
MMAS  6202  34  6127  1.48  0.28  121  
ACS  6234  44  6145  2.03  0.57  145  
AS${}_{ORank}^{ps}$  6170  33  6136  0.98  0.42  120  
tsp225  AS  4374  57  4166  11.69  6.38  424 
AS${}_{Rank}$  4026  31  3989  2.80  1.86  420  
MMAS  3998  20  3962  2.10  1.17  420  
ACS  4022  42  3929  2.72  0.33  504  
AS${}_{ORank}^{ps}$  3942  17  3916  0.65  0  424  
gr48  AS  5227  36  5147  3.58  2.00  9 
AS${}_{Rank}$  5117  40  5066  1.40  0.40  9  
MMAS  5103  35  5063  1.13  0.34  9  
ACS  5095  35  5046  0.96  0  12  
AS${}_{ORank}^{ps}$  5091  28  5049  0.88  0.06  9  
pr76  AS  115,664  793  113,911  6.94  5.32  60 
AS${}_{Rank}$  111,609  1064  109,392  3.19  1.14  60  
MMAS  110,521  1027  109,271  2.18  1.03  60  
ACS  110,157  1326  108,159  1.85  0  70  
AS${}_{ORank}^{ps}$  109,922  900  108,159  1.63  0  64  
gr202  AS  45,746  482  44,368  13.90  10.48  327 
AS${}_{Rank}$  41,803  435  40,960  4.09  1.99  324  
MMAS  42,004  413  41,331  4.59  2.92  327  
ACS  41,646  340  40,720  3.70  1.39  387  
AS${}_{ORank}^{ps}$  41,056  228  40,554  2.23  0.98  328  
ftv35  AS  1504  10  1487  2.11  0.95  5 
AS${}_{Rank}$  1497  9  1473  1.65  0  5  
MMAS  1493  8  1473  1.39  0  5  
ACS  1494  19  1473  1.45  0  6  
AS${}_{ORank}^{ps}$  1483  11  1473  0.71  0  5  
ftv64  AS  1918  13  1902  4.31  3.43  17 
AS${}_{Rank}$  1867  21  1848  1.50  0.49  17  
MMAS  1857  7  1854  1.00  0.815  18  
ACS  1866  21  1842  1.85  0.16  21  
AS${}_{ORank}^{ps}$  1858  9  1839  1.05  0  17  
ftv70  AS  2149  23  2051  10.20  5.18  21 
AS${}_{Rank}$  2010  44  1957  3.10  0.36  21  
MMAS  1988  30  1950  1.95  0  21  
ACS  2044  48  1967  4.83  0.87  25  
AS${}_{ORank}^{ps}$  1989  37  1954  1.99  0.20  21 
Instance  Algorithm  Avg.  Std.  Best  PD${}_{\mathit{avg}}(\%)$  PD${}_{\mathit{best}}(\%)$  Time (s) 

ESC78  AS  20,631  227  19,950  13.17  9.43  182 
AS${}_{Rank}$  18,609  147  18,415  2.08  1.01  184  
MMAS  18,464  28  18,405  1.29  0.96  187  
ACS  18,477  97  18,290  1.35  0.33  196  
AS${}_{ORank}^{ps}$  18,460  69  18,300  1.26  0.38  192  
ft70.1  AS  44,110  469  43,221  12.20  9.94  81 
AS${}_{Rank}$  41,403  419  40,646  5.32  3.39  81  
MMAS  40,903  441  40,192  4.05  2.24  81  
ACS  42,562  715  41,014  8.27  4.33  87  
AS${}_{ORank}^{ps}$  40,768  341  40,092  3.70  1.98  81  
p43.1  AS  28,776  64  28,615  2.26  1.69  21 
AS${}_{Rank}$  28,333  107  28,220  0.69  0.28  21  
MMAS  28,258  57  28,220  0.42  0.28  21  
ACS  28,461  88  28,245  1.40  0.37  24  
AS${}_{ORank}^{ps}$  28,236  34  28,220  0.34  0.28  22  
p43.4  AS  84,218  115  83,950  1.46  1.14  68 
AS${}_{Rank}$  83,693  119  83,415  0.83  0.49  67  
MMAS  83,514  130  83,360  0.61  0.43  67  
ACS  83,601  146  83,270  0.72  0.32  69  
AS${}_{ORank}^{ps}$  83,416  113  83,265  0.49  0.31  68 
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PérezCarabaza, S.; Gálvez, A.; Iglesias, A. RankBased Ant System with Originality Reinforcement and Pheromone Smoothing. Appl. Sci. 2022, 12, 11219. https://doi.org/10.3390/app122111219
PérezCarabaza S, Gálvez A, Iglesias A. RankBased Ant System with Originality Reinforcement and Pheromone Smoothing. Applied Sciences. 2022; 12(21):11219. https://doi.org/10.3390/app122111219
Chicago/Turabian StylePérezCarabaza, Sara, Akemi Gálvez, and Andrés Iglesias. 2022. "RankBased Ant System with Originality Reinforcement and Pheromone Smoothing" Applied Sciences 12, no. 21: 11219. https://doi.org/10.3390/app122111219