A SelfAdaptive Discrete PSO Algorithm with Heterogeneous Parameter Values for Dynamic TSP
Abstract
:1. Introduction
1.1. SelfAdaptivity
1.2. Contributions
 We propose a method for automatically setting the values of four crucial DPSO parameters. This method is based on discrete probability distributions defined to diversify the behaviors of the particles in the heterogeneous DPSO. The aim of this diversification is to improve the convergence of the algorithm.
 We perform an analysis of the convergence of the proposed algorithm based on computational experiments conducted on a set of DTSP instances of varying sizes. We discuss the relationships between the values of the DPSO parameters and their effect on particle movement through the problem’s solution search space.
 We study the diversity of the population of particles in the proposed heterogeneous DPSO and the original approach based on the information entropy calculated in two ways. The former method considers the edges, which are building blocks of the solutions to the TSP and DTSP. The latter focuses only on the quality of the solutions
 We compare the efficiency of the proposed heterogeneous DPSO with that of the base DPSO and two algorithms based on ant colony optimization (ACO). The results show that the proposed algorithm outperforms the base DPSO and is competitive with the ACObased algorithms.
2. Dynamic Traveling Salesman Problem
3. Heterogeneity
 Neighborhood heterogeneity: This concerns cases in which the size of the neighborhood is different for every particle, and hence, the virtual topology of connections between particles is not regular. Some particles can have a wider influence than others on the movement of the swarm.
 Bestparticle heterogeneity: Here, there can be variations in the method of selecting the best particle, i.e., the particle whose position is used when updating the current velocity and position. For instance, one particle might update its position following the best particle in its (small) neighborhood, while the second particle might be fully informed and follow the global best particle.
 Heterogeneity of the position update strategy: Here, the particles differ in their patterns of movement (searching) through the solution space. For example, one group of particles might explore the solution space, while the other group might conduct a local search by restricting their velocities or even positions to a certain range. This type of heterogeneity diversifies the population to the greatest extent, since it provides the greatest flexibility in diversifying particle movement.
 Heterogeneity of parameter values: Here, each particle or group of particles in the swarm can have different values of the parameters. For example, some particles might have a large inertia $\omega $ and explore the solution space, whereas other particles might have a small value of $\omega $ and perform the search locally (around the best position found). Although this type of heterogeneity is not as flexible as the heterogeneity of the position update strategy, it requires relatively few changes to the PSO, since only the values of the particle parameters need be set individually. It is this strategy that we apply in the proposed heterogeneous DPSO algorithm.
4. DPSO with Pheromone
 It alters the probability of edge selection during the solution construction process; i.e., the higher the value of the pheromone, the greater is the probability of selecting the corresponding edge. In other words, the pheromone serves as an additional memory of the swarm, allowing it to learn the structure of highquality solutions and, potentially, improve the convergence of the algorithm.
 The pheromone matrix created while solving the current DTSP subproblem is retained and used when solving the next subproblem. This allows knowledge about the previous solution search space to be transferred with the aim of helping the construction of highquality solutions to the current subproblem. This implicitly assumes that the changes between consecutive subproblems are not very great, so that the highquality solutions to the current subproblem share most of their structure with the highquality solutions to the previous one.
5. Heterogeneous Swarm
 ${c}_{1}$: $P\left(0.1\right)$ = 0.4, $P\left(0.75\right)$ = 0.15, $P\left(1.5\right)$ = 0.3, $P\left(1.75\right)$ = 0.15;
 ${c}_{2}$ and ${c}_{3}$: $P\left(0.1\right)$ = 0.4, $P\left(1\right)$ = 0.15, $P\left(1.5\right)$ = 0.15, $P\left(2\right)$ = 0.3;
 $\omega $: $P\left(0.1\right)$ = 0.4, $P\left(0.25\right)$ = 0.2, $P\left(0.5\right)$ = 0.4.
6. Experimental Results
6.1. Convergence Analysis for Various Sets of Parameters
6.2. Comparative Study
Algorithm 1 Outline of the procedure for solving the DTSP. 

6.3. Entropy Study
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
ACO  ant colony optimization 
DPSO  discrete particle swarm optimization 
DTSP  dynamic traveling salesman problem 
PACO  population ant colony optimization 
PSO  particle swarm optimization 
TSP  traveling salesman problem 
Appendix A. The Entropy Study for the Other TSP Instances
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Year  Authors  Algorithm  DTSP Variant 

2001  Guntsch and Middendorf [21]  ACO with local and global reset of the pheromone  Addition/removal of vertices 
2002  Eyckelhof and Snoek [22]  ACO with various variants of pheromone matrix update to maintain diversity  Changes in edge lengths with time (simulated traffic jam on a road) 
2006  Li et al. [19]  GSInverover and gene pool with the $\alpha $measure [23]  CHN145 + 1: 145 cities and one satellite 
2010  Mavrovouniotis and Yang [24]  ACO with immigrants scheme to increase population diversity  Coefficients: frequency and size of changes 
2011  Simões and Costa [25]  CHCalgorithm  A test involving the addition of changes and their subsequent withdrawal [26]. In that way, the optima at the beginning and the end are the same. 
2014  Tinós et al. [27]  EA algorithm  Random changes in the problem 
2014  Zhang and Zhao [28]  Hopfield neural network  Simulation of various types of real random events in a street 
2016  Eaton et al. [29]  ACO with immigrants scheme  Changes in edge lengths. Simulated delays of trains. 
2016  Mavrovouniotis and Yang [30]  MMAS  Encoding of the problem is changed, but the optimal solution remains the same 
2017  Mavrovouniotis et al. [31]  ACO  Distances between cities are changed. The problem can be transformed to an asymmetric one. 
2018  Chowdhury et al. [32]  ACO  Random DTSP, dynamic changes occur randomly. Cyclic DTSP, dynamic changes occur with a cyclic pattern. 
2018  Schmitt et al. [33]  MMAS  Acyclic DTSP with changes in edge lengths with time 
2018  Yirui Wang et al. [34]  ACO  
2018  YanWei Huang et al. [35]  MCTS  Addition/removal of vertices 
No.  ${\mathit{c}}_{1}$  ${\mathit{c}}_{2}$  ${\mathit{c}}_{3}$  $\mathit{\omega}$  Description 

1  0.1  0.1  0.1  0.1  Favors quick changes of position 
2  2.0  0.1  0.1  0.1  Emphasis on the information from $pBest$ 
3  0.1  2.0  0.1  0.1  Emphasis on the information from $gBest$ 
4  0.1  0.1  2.0  0.5  Very slow changes of position 
5  0.75  1.0  1.0  0.25  Weak $pBest$, $gBest$ influence 
6  1.25  1.5  1.5  0.5  Stronger $pBest$, $gBest$ influence 
7  1.5  2.0  2.0  0.5  Strong $pBest$, $gBest$ influence 
8  1.75  2.0  2.0  0.75  Very strong $pBest$, $gBest$ influence 
Problem  ${\mathit{c}}_{1}$  ${\mathit{c}}_{2}$  ${\mathit{c}}_{3}$  $\mathit{\omega}$  $\mathit{SwarmSize}$  Neighborhood 

berlin52  0.5  0.5  0.5  0.2  32  7 
kroA100  0.5  0.5  0.5  0.5  64  7 
kroA200  0.5  0.5  0.5  0.5  80  7 
gr202  0.5  0.5  0.5  0.5  101  10 
pcb442  0.5  1.5  0.5  0.5  104  15 
gr666  0.5  1.0  1.5  0.6  112  30 
Rank  Parameters  Number of $\mathit{gBest}$ Improvements  

${\mathit{c}}_{\mathbf{1}}$  ${\mathit{c}}_{\mathbf{2}}$  ${\mathit{c}}_{\mathbf{3}}$  $\mathit{\omega}$  
1  0.1  0.1  0.1  0.5  113 
2  0.1  0.1  0.1  0.1  102 
3  0.1  2  0.1  0.1  49 
4  0.1  2  2  0.5  46 
5  0.1  2  2  0.1  42 
6  0.1  1.5  0.1  0.5  39 
7  0.1  1  0.1  0.1  38 
8  0.1  1  0.1  0.25  34 
9  0.75  2  2  0.25  27 
10  0.1  1  2  0.1  26 
11  0.1  2  0.1  0.25  24 
12  0.75  2  2  0.1  22 
13  1.5  1.5  2  0.5  21 
14  1.5  2  0.1  0.1  21 
15  1.5  2  0.1  0.25  20 
Iterations  Parameters  Number of $\mathit{gBest}$ Improvements  

${\mathit{c}}_{\mathbf{1}}$  ${\mathit{c}}_{\mathbf{2}}$  ${\mathit{c}}_{\mathbf{3}}$  $\mathit{\omega}$  
0–1250  0.1  0.1  0.1  0.1  94 
0.1  0.1  0.1  0.5  93  
0.1  2  2  0.5  38  
0.1  2  0.1  0.1  38  
0.1  2  2  0.1  32  
1250–2500  0.75  2  2  0.25  12 
1.5  2  0.1  0.1  10  
0.1  2  0.1  0.1  10  
0.1  1.5  0.1  0.5  9  
0.1  1  2  0.1  8  
2500–3750  0.1  0.1  0.1  0.5  10 
1.5  1.5  2  0.5  4  
1.5  2  0.1  0.25  4  
0.75  2  2  0.25  3  
0.1  1  2  0.1  3  
3750–5000  0.1  1  0.1  0.1  2 
0.1  1  2  0.1  2  
0.75  0.1  2  0.5  2  
1.5  0.1  2  0.1  2  
1.5  2  1.5  0.1  2  
5000–6144  1.75  2  1  0.5  3 
1.5  2  1.5  0.1  2  
1.75  0.1  2  0.5  2  
0.1  1.5  0.1  0.5  2  
1.5  1  1  0.1  1 
Homogeneous DPSO  Heterogeneous DPSO  Common Parameters  

Problem  ${\mathit{c}}_{\mathbf{1}}$  ${\mathit{c}}_{\mathbf{2}}$  ${\mathit{c}}_{\mathbf{3}}$  $\mathit{\omega}$  Problem  ${\mathit{c}}_{\mathbf{1}}$  ${\mathit{c}}_{\mathbf{2}}$  ${\mathit{c}}_{\mathbf{3}}$  $\mathit{\omega}$  $\mathit{SwarmSize}$  Neighborhood 
berlin52  0.5  0.5  0.5  0.2  berlin52  Chosen randomly as described in Section 5  32  7  
kroA100  0.5  0.5  0.5  0.5  kroA100  64  7  
kroA200  0.5  0.5  0.5  0.5  kroA200  80  7  
gr202  0.5  0.5  0.5  0.5  gr202  101  10  
pcb442  0.5  1.5  0.5  0.5  pcb442  104  15  
gr666  0.5  1.0  1.5  0.6  gr666  112  30 
Problem  Iterations  DPSO Algorithms  Counterparts  

Homogeneous  Heterogeneous  ACS  PACO  
T (s)  G (%)  D (%)  T (s)  G (%)  D (%)  G (%)  G (%)  
berlin52  104  0.13  0.15  0.32  0.13  0.13  0.15  0.96  0.96 
berlin52  416  0.3  0.01  0.04  0.28  0.01  0.05  0.5  0.5 
berlin52  1664  0.98  0  0  0.89  0.01  0.05  0.46  0.46 
kroA100  100  1.03  5.44  2.47  0.86  2.68  1.4  1.8  2.97 
kroA100  400  1.63  1.28  1.02  1.27  1.05  0.81  1.31  2.13 
kroA100  1600  4.11  0.64  0.69  3.38  0.78  0.77  0.82  1.36 
kroA200  160  2.49  15.63  2.77  2.18  5.14  1.84  2.41  3.33 
kroA200  640  5.13  4.45  1.62  4.46  2.89  1.09  1.62  2.71 
kroA200  2560  15.6  1.62  0.81  13.18  2.02  0.8  1.47  2.28 
gr202  128  8.82  13.75  2.06  8.17  4.19  1.2  6.26  4.91 
gr202  512  11.54  6.81  2.11  10.88  1.97  0.66  4.88  3.9 
gr202  2048  23.01  1.52  0.6  21.98  1.53  0.55  3.93  3.34 
pcb442  272  11.22  29.31  5.33  11.16  6.73  1.68  6.18  4.44 
pcb442  1088  28.52  13.41  5  30.69  2.87  0.89  4.87  3.56 
pcb442  4352  102.78  3.13  1.52  108.25  1.92  0.79  3.91  3.3 
gr666  384  85.19  10.84  1.52  91.83  9.58  0.86  9.18  5.89 
gr666  768  98.36  7.37  1.0  115.19  6.88  0.78  7.46  4.77 
gr666  1536  124.84  5.62  0.84  163.48  5.33  0.57  6.09  4.51 
gr666  3072  180.66  4.88  0.63  259  4.52  0.88  5.67  4.14 
gr666  6144  296.83  3.99  0.77  453.83  3.8  0.78  4.92  4.21 
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Strąk, Ł.; Skinderowicz, R.; Boryczka, U.; Nowakowski, A. A SelfAdaptive Discrete PSO Algorithm with Heterogeneous Parameter Values for Dynamic TSP. Entropy 2019, 21, 738. https://doi.org/10.3390/e21080738
Strąk Ł, Skinderowicz R, Boryczka U, Nowakowski A. A SelfAdaptive Discrete PSO Algorithm with Heterogeneous Parameter Values for Dynamic TSP. Entropy. 2019; 21(8):738. https://doi.org/10.3390/e21080738
Chicago/Turabian StyleStrąk, Łukasz, Rafał Skinderowicz, Urszula Boryczka, and Arkadiusz Nowakowski. 2019. "A SelfAdaptive Discrete PSO Algorithm with Heterogeneous Parameter Values for Dynamic TSP" Entropy 21, no. 8: 738. https://doi.org/10.3390/e21080738