In the Comparison of Genetic Operators For Solving the Traveling Salesman Problem: Selection

In comparing selection methods, for the sake of comparison it was in our best interest to leave the least to randomness except in the selection method. The mutation method was the center inverse mutation throughout all the trials and a center mutation point was chosen every time. The cutoff percentage was the same (30%) for each trial and the number of generations was a fixed 5000.

The numbers displayed below are the average of 10 trials conducted with the same input graph but a different initial population for each trial.Selection Comparison

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In the Comparison of Genetic Operators For Solving the Traveling Salesman Problem: Mutation

In attempt to statistically compare the operators, the input graph and the initial population was kept the same for each trial. The numbers displayed below are the average of 10 trials conducted with the same input graph but a different initial population. The algorithm was ran with an input graph consisting of 26 static nodes and approximately 4.03E26 possible combinations. Each trial ran 5000 generations with an input population of 5000 chromosomes. The fitness percentage was 30% throughout every trial.

Mutation Operators and Crossover Point

In this trial the method of selection was kept standard using the percentage cutoff method to avoid any influence from the selection method.

Random Crossover Point Center Crossover Point
Reverse Sequence Mutation 336 414
Center Inverse Mutation 253 310

The representation of each mutation operator over iterations was tested with a constant center crossover point.

Mutation Operator Comparison

Genetic Algorithm: Selection

In every generation, a selection agent comes to play which sifts out the fit chromosomes from the unfit chromosomes. The selection agent “kills off” a user specified percentage of organisms in the population.However, it is under the discretion of the selection agent in determining which chromosomes to kill. As mentioned earlier, fitness is defined by having the lowest weight in the circumstances put forth by the TSP. However selection may not necessarily be only off of that. This can be seen when comparing the two most prevalent types of selection operators:
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Genetic Algorithms: Crossover

The method of crossover remains fairly constant regardless of the problem and scope. Crossover is achieved by first selecting a crossover point within a pair of defined and unique organisms P1 and P2 (which are the equivalent of parents for the crossed over parent). The chromosomes are then split at the selected crossover point. The second half of P2 (P2H2)  is then appended to the first half of P`1  (P1H1) to make one child chromosome (C1). The second child (C2) is made by appending the second half of P1 (P1H2) to the first half of P2 (P2H1).Read More »

Genetic Algorithm Definitions for TSP

A genetic algorithm is a type of evolutionary algorithm and therefore TSP must be fit to fill all the constraints necessary to execute a genetic algorithm. An organism in the sense of TSP can be defined as a viable path that visits every node in the graph. Each path must start with a node, visit all the nodes present in the graph, and then return to the same node that it started with. An example of a viable path with an input graph of 10 vertices is shown below with each letter representing a node in the input graph:

{A, C, J, D, G, H, E, B, F, I, A}

The population in TSP can be defined as a set of unique paths. Fitness can be defined as the weight or distance of the path. Thus, a lower weight will result in higher fitness and vice versa. A sample population of two organisms is shown below. In front of each organism is its weight or for the cases of this exercise— its fitness:

set{ 143 : {A, C, J, D, G, H, E, B, F, I, A} , 210 : {A , B, J, D, C, E, I, F, H, G, A} }

Genetic Algorithms: Intro

In this exercise, we attempt to utilize genetic algorithms to find an optimal, but not perfect, solution to the traveling salesman problem. A genetic algorithm emulates nature in its optimization process. Nature uses several mechanisms which have led to the emergence of new species and still better adapted to their environments. The laws which react to species evolution have been known by the research of Charles Darwin in the last century: Genetic algorithms are powerful methods of optimization that utilize these rules defined by evolution in their process to find a pseudo-optimal answer. These algorithms were modeled on the evolution of species. The genetic algorithm utilizes the properties of genetics such as selection, crossover, mutation.

Introduction to the Traveling Salesman Problem

The Problem
The traveling salesman problem (TSP) is a typical example of a very hard combinatorial optimization problem. The problem is to find the shortest tour that passes through each vertex in a given graph exactly once. The TSP problem is classified as an NP-complete problem. There are some intuitive methods to find the approximate solutions, but all of these methods have exponential complexity, they take too much computing time or require too much memory. Mathematically TSP can be expressed as:
min [f(T),T = (T[1],T[2],T[3],… …,T[n])]
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