genetic algorithm for the traveling salesman problem using minimal weight variable order selection...

8/13/2019 Genetic Algorithm for the Traveling Salesman Problem using Minimal Weight Variable Order Selection Crossover Op

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Genetic Algorithm for the Traveling Salesman Problem using Minimal Weight

Variable Order Selection Crossover Operator (MWVOSX)

Murugiah Sudharson [email protected]

Abstract

This paper introduces a new crossover operator, Minimal Weight Variable Order Selection Crossover Operator

(MWVOSX), an improved version of existing order crossover operator (OX) for genetic algorithm that generates

high quality solutions for Traveling Salesman Problem (TSP).The Minimal Weight Variable Order Selection

Crossover operator selects a specific order from multiple orders which have the least weight , and takes thebalance from the second chromosome in forward & reverse order .Now it constructs two new children using

specific order with forward order and specific order with reverse order, then it chooses the best/lease weight

new chromosome from them . The efficiency of the MWVOSX is compared against only with generally used

order crossover operator and not trying to compare with other crossover operators .Experiments show that the

new crossover operator (MWVOSX) is better the then the existing order crossover (OX) operator.

Keywords:Traveling salesman problem, Genetic algorithm,Minimal Weight Variable Order Selection Crossover,MWVOSX.

1. INTRODUCTION

The traveling salesman problem(TSP), which represents a classic optimization problem that cannot be solved using

traditional techniques and one of the benchmark and old problems in Computer Science and Operations Research.

For a given set of cities and the travel distance between each possible pairs, the TSP, is to find the best possible

way of visiting all the cities (once) that minimize the travelling distance.

For n number of cities with 'city 1' as Start Point and a travel distance (or cost, or travel time etc.,) matrix

D = [dxy] of order n associated with ordered node pairs (x,y) is given. The problem is to find a least cost complete

cycle.

Broadly, the TSP is classified as symmetric travelling salesman problem (sTSP) & asymmetric travelling salesman

problem (aTSP).For n number of cities, TSP is symmetric if dxy = dyx x, y and asymmetric otherwise.For n number of cities asymmetric TSP [1], there are (n 1)! possible solutions, one or more of which gives the

minimum Euclidean distance. For symmetric TSP, there are (n 1)! /2 possible solutions & their reverse cyclic

permutations having the same total distance.


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The TSP has many real world applications;a direct application is in the drilling problem of printed circuit boards (PCBs)

,vehicle routing,overhauling gas turbine engines, X-Ray crystallography, threading of scan cells in a testable VLSIcircuits, etc.

An implicit way of solving the TSP is simply to list all the feasible solutions, evaluate their objective function values

and pick out the good one. But it is obvious that this exhaustive search is grossly inefficient and time consuming

job because of vast number of possible solutions to the TSP even for problem of moderate size. Since practical

applications require solving larger problems, hence emphasis has shifted from the aim of finding exactly optimal

solutions to TSP, to the aim of getting good solutions in reasonable .Genetic algorithm (GA) is one of the heuristic

algorithms that have been used widely to solve the TSP instances.

In this paper, a new crossover operator named Minimal Weight Variable Order Selection Crossover

(MWVOSX) is developed and accordingly a genetic algorithm based on MWVOSX is developed for solving the TSP.

This paper is organized as follows: Section 2 develops a genetic algorithm based on MWVOSX for the TSP.

Section 3 describes computational experiments for crossover operators. Finally, Section 4 presents

comments and concluding remarks.

2. GENETIC ALGORITHM

Genetic algorithm (GA), powerful and broadly applicable derivative free stochastic optimization techniques,loosely

based on the concept of the natural selections and evolution process, are most widely known types of evolutionary

computation methods today. The benefits of using GAs are as follows;

1. GAsare parallel processing procedures, which will speed up their operations [2].

2. Applicable to both discrete and continuous optimization problems.

3. Modular, separate from application.

4. Stochastic and less likely to get trapped in local minima [2].

5. Always gives answer and answer gets better with time, etc.

GAshas many real world applications;control, design, scheduling, robotics, machine learning, signal process,

game playing, combinatorial optimization.

In general, a GA has five basic components, as summarized by Michalewicz [1]:

1. A genetic representation of solutions to the problem.

2. A way to create initial population of solutions.

3. An evaluation function rating solutions in term of their fitness.

4. Genetic operators (Reproduction, crossover, mutation) that alter the genetic composition of children

during reproduction.

5. Values for the parameter of genetic algorithms.


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And refer to the Figure 1 shows the pictorial flowchart for the genetic algorithm general structure [7].

FIGURE 1: General Structure of Genetic Algorithm.

In figure 1, first the initial population (sample chromosomes) is being built and passed to Evolution Environment. In

Evolution Environment, it evaluates the chromosomes and assigns a fitness value, then pass to GA Operators

section. In GA Operators section the chromosomes with weak fitness values will die off and the good are

reproduced and pass to for further modification to crossover operator and mutation operator.

This cycle will continue until the stopping criteria is satisfied .Once the stop criteria is satisfied, GA will return the

best/good result and GA cycle will get terminated. Now following subsections will discuss in depth about

Chromosome Encoding, Euclidean distance Calculation, The Fitness Function, Selection Operators, general Order

Crossover, newly constructed Minimal Weight Variable Order Selection Crossover (MWVOSX) and Mutation

Operator.

2.2 CHRMOSOME ENCODING

To apply GA for any optimization problem, we need to think a way for encoding the solutions as chromosomes in a

form so that a computer can process.The techniques for encoding solutions vary by problem and, involve a certainamount of art.For the TSP, solution is typically represented by chromosome of length as the number of nodes(cities) in the problem.Each gene of a chromosome takes a label of node (city) such that no node can appear twice


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in the same chromosome. There are mainly two representation methods for representing tour of the TSP

adjacency representationandpath representation. We consider thepath representationfor a tour, which simply

lists the label of nodes.

For example, let {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}be the labels of nodes/cities in a 10cities instance where 0represents

the starting point/headquarters, then a tour {1 38 2 5 74691}[4] may be represented

as [1, 3, 8, 2, 5, 7, 4, 6, 9] .But in this paper the return to starting point is not shown in the Figure 7,normally back tostart point is not taken compulsorily in practice [5].

2.3 EUCLIDEAN DISTANCE CALCULATION

The Euclidean distance Di [3], between any two cities with coordinate (X1, Y1) and (X2, Y2) [5]is calculated byequation (1) as shown below;

Di= ((|X1-X2|)2

+ (|Y1-Y2|)2

) (1)

The equation (1) will be helpful in developing a two dimensional Euclidean distance calculation matrix (EDM) which

will be a more important in calculating the total Euclidean distance for each chromosome. For example if we have

10 cities [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] and their co-ordinates (x-axis & y-axis), then the Euclidean distance calculation

matrix (EDM) will be developed as in figure 2.

FIGURE 2: Euclidean distance calculation matrix (EDM).

In figure 2, this is a two dimensional square (since its 10x10) scalar matrix, since its diagonal elements are equal (to

0).The colored x values with appropriate numbers have same value for any different city co-ordinates.

2.4 FITNESS FUNCTION

A fitness function evaluation is incorporated to assigns a value to each organism, noted as fi.This fivalue is a figure

of merit which is calculated by using any domain knowledge that applies. In principle, this is the only point in the

algorithm that domain knowledge is necessary. Organisms are chosen using the fitness value as a guide, where

those with higher fitness values are chosen more often.

The GAsare used for maximization problem. For the maximization problem the fitness function is same as the

objective function. But, for minimization problem, one way of defining a fitness function is as equation(2), where


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fiis the objective function. Since, TSP is a minimization problem; we consider this fitness function, wherefi

calculates Euclidean distance (or value) of the tour represented by a chromosome.The method employed here

was to calculate the total Euclidean distance Di for each organism first, then compute fiby using the equation (2).

fi= 1 / Di (2)

, Where Diis the Euclidean distance over organisms in the population.

2.5 SELECTION

Selection/Reproduction is usually the first operator applied to population. From the population, the chromosomes

are selected to be parents to crossover and produce offspring. Many selection/reproduction operators exist and

they all essentially do the same thing. They pick from current population, the strings of above average and insert

their multiple copies in the mating pool in a probabilistic manner.

The most commonly used methods of selecting chromosomes for parents to crossover are; Roulette wheel

selection, Rank selection, Boltzmann selection, steady state selection, Tournament selection, etc.

2.6 ORDER CROSSOVER (OX)

Order Crossover operator is one of the fastest of its kind, which makes it more attractable for practical use.

To apply OX,a swath of consecutive alleles from parent 1 drops down and remaining non-duplicative allelesare

placed in the child in the order which they appear in parent 2 [6].An Example of OX is given in figure 3.

FIGURE 3: An Example of Order Crossover.

Referring to figure 3, a swath of consecutive alleles from point 3 to point 7 (totally 5 alleles) selected and copied to

child in the same position. The remaining non-duplicativealleles which are in point 1, 4, 6 & 8 are copied to child in

the same order to fill the gaps and form a new child chromosome. Here the point 0 is selected as headquarters or

starting place and it is a constant to all chromosomes.

2.7 MINIMAL WEIGHT VARIABLE ORDER SELECTION CROSSOVER (MWVOSX)

This crossover is identical to the Order crossover (OX), except variable swaths are participants in the genetic

exchange. The following figure 4 shows the flowchart of how the Minimal Weight Variable Order Selection

Crossover (MWVOSX) works.


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(a) (b)

FIGURE 4: (a) Flowchart of OX; (b)Flowchart of MWVOSX.

DONE

Randomly select a swat

of consecutive alleles

from Parent 1 &dropdown to child.

Remaining values are

placed in the child from

parent 2 in the same

order which they

appear in parent 2.

Select a limited number

of multiple swats of

consecutive alleles from

Parent 1.

Acquire remaining from

parent 2 in forward

orderto temporary

chromosome as temp

child1.

Find a minimal weight swatof

consecutive alleles from

multiple swats in Parent 1 &

drop down to child.

Acquire remaining from

parent 2 in reverse

orderto temporary

chromosome as temp

child 2.

Make two copiesof child as

temp child 1 & temp child 2.

Calculate the total

weight(distance) of the

temporary chromosome

as temp child1.

Calculate the total

weight(distance) of the

temporary chromosome

as temp child2.

Select temporary

chromosometempchild 2 as child.

YES NO

Select temporary

chromosometempchild 1 as child.

DONE DONE

START START

Temp

child1 child2

3

2

5

4

1


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In figure 4, we have compared order crossover (figure 4(a)) and Minimal Weight Variable Order Selection

Crossover (figure 4(b)).Referring to figure 4(b) shows a flowchart of the working mechanism of Minimal Weight

Variable Order Selection Crossover (MWVOSX) and a points form brief description of how the MWVOSX works as

follows;

[1].Initially a limited number of multiple swaths with consecutive alleles in specific points are taken for next stage

from parent 1(For more details on points selections refer section 3).

[2].Now the weight calculation is performed for each selected swath using Euclidean distance calculation matrix

(EDM) and the best swath with minimal weight is selected and drop down to child chromosome.

[3].Now we make two copies of child as temp child 1 & temp child 2, and then pass them in parallel to fill their

remaining from parent 2 in forward and reverse order.

[4].Then the total weight/distance of both temp child 1 & temp child 2 calculated separately and pass for decision

maker.

[5].The child is selected from comparing the two temporary children and the smallest weighted chromosome is

selected as child.

Now consider a pictorial example of MWVOSX chromosome creation in figure 5.

FIGURE 5: An Example of MWVOSX.

Totally Four swaths with consecutive alleles are selected .Note the upcoming swaths have same specific number of

alleles (here itsfive) and upcoming swaths are taken in an incrementing point by one. Now assume the second

swath order has minimal weight comparing to other three and successfully selected and drop down to child.

Next the remaining are selected from Parent 2 in forward and reverse order as shown in figure 5, then made two

duplicate different children .

Now the weight /distance of the both two new chromosomes are compared and its assumed that reverse order

chromosome weight is smaller compared to forward order chromosome. So the reverse order chromosome is

successfully selected as the child.

2.8 MUTATION

The mutation operator randomly selects a position in the chromosome and changes the corresponding allele,

thereby modifying information. The need for mutation comes from the fact that as the less fit members of

successive generations are discarded; some aspects of genetic material could be lost forever. By performing

occasional random changes in the chromosomes, GAs ensure that new parts of the search space are reached,

Selected Order


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which reproduction and crossover alone couldnt fullyguarantee. In doing so, mutation ensures that no important

features are prematurely lost, thus maintaining the mating pool diversity.

For the TSP, the classical mutation operator does not work. We have considered the reciprocal exchangemutation

[4] that selects two nodes randomly and swapsthem.

3. COMPUTATIONAL EXPERIMENTS

For comparing the efficiency of the different crossover operators, genetic algorithms using OX & MWVOSX have

been encoded in c programming language on a Pentium 4 personal computer with speed 3 GHz and 1 GB RAM

under MS Windows XP, the program have been compiled and run by GCC (GNU C Compiler).

Initially a population of 20 sample chromosomes has been chosen for the experiment and only the new

chromosomes generated by both OX and MWVOSX are compared and no mutation is done, since this paper is only

focus on order crossover and Minimal Weight Variable Order Selection Crossover (MWVOSX) which is identical to

crossover operator.

There should be control parameters present in the experiment, where the swath size and the numbers of swaths

should be controlled. Here the swath size is fixed to five alleles and the numbers of swaths are fixed to four.

There is only one generation of new chromosomes is generated for both crossover operators and fitness function

Is evaluated by the fitness function equation (2).best chromosomes are taken from both highest fitness values.

Figure 6 shows the fitness comparison between OX and MWVOSX chromosomes, out of 20new offspring/child

chromosomes 16chromosomes which are generated by MWVOSX have high fitness values compared to

offspring/child chromosomes generated by OX. OX also produced good four chromosomes compared toMWVOSX.

Ten (5 in OX + 5 in MWVOSX) chromosomes are approximately equal to each other.

FIGURE 6: Fitness Comparison between OX & MWVOSX.

The best chromosome also being produced by the MWVOSX which is the fifth chromosome and the best for OX is

the tenth & nineteenth which have the same pattern. But the best chromosome with highest fitness for overall is

produce by MWVOSX operator which is the fifth chromosome for the first generation. The best chromosomes of

which are generated by OX and MWVOSX are shown figure 7, 7(a) belong to OX and 7(b) to MWVOSX where 7(b) is

the best offspring (minimal distance).


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(a)0197654382 (b)0723156984

FIGURE 7: (a) OX generated best offspring; (b) MWVOSX generated best offspring.

4. CONCLUSION & FUTURE WORK

I have proposed a new crossover operator named Minimal Weight Variable Order Selection Crossover (MWVOSX)

for a genetic algorithm for the Traveling Salesman Problem (TSP).In terms of quality of the solution, for the lesssized instances, MWVOSX is found to be better then OX. But as size increases the MWVOSX becomes slower

compare to OX. Note that using multiprocessors pc[8], by parallel programming the MWVOSX operator will

become more fast. Referring to figure 4(b) starting from point 3, where a parallel process is started and easily

programmed.

In-depth experiment shown in figure 8, the comparison between forward and reverse order acquiring in MWVOSX.

Note here forward order has produced 13good off springs compared to reverse order.Fouroff springs are

approximately equal to each other (2 in forward and 2 in reverse). Even though the best offspring is produced by

reverse order. So it is unpredictable to find which order(forward/reverse) will produce the best offspring, eventhe forward order produced the most best offspring, but reverse order gives the best offspring with highest fitness

value .

FIGURE 8:Comparison between Forward and Reverse order acquiring in MWVOSX.


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The swath size should not be less than the half the size of the chromosome, because a swath size with higher then

the half the size of the chromosome will make the MWVOSX operator inefficient and will limit the number of

swaths. The number of swaths also should keep minimal, excessive number of swaths will slowdown the MWVOSX

operator in finding the minimal weight swath. It is difficult to say what would the the size of the swath and how

many swaths should be presented.

In this paper I have only compared with OX, but there are other OX based crossover operators are exist in real

world. So MWVOSX operator should be compared with OX based crossover operators and other crossover

operators as well. The current studies about efficiency of MWVOSX compared with other alternatives and

development of OX based new crossover operators are under investigation.

5. REFERENCES

*[1] Donald Davendra,"Traveling Salesman Problem, Theory and Applications".

*[2] J.T.R.Jang,C.T.Sun,E.Mizutani,"Neuro-Fuzzy & Soft Computing".

*[3] S.Rajasekaran,G.A.Vijayalakshmi Pai,"Nerural Neyworks,Fuzzy Logic,and Genetic Algorithms".

*[4] RC Chakraborty,http://www.myreaders.info/09-Genetic_Algorithms.pdf

*[5]Buthainah Fahran Al-Dulaimi, and Hamza A. Ali"Enhanced Traveling Salesman Problem Solving

by Genetic Algorithm Technique (TSPGA)"

*[6]http://www.rubicite.com/Tutorials/GeneticAlgorithms/CrossoverOperators/Order1CrossoverOperator.aspx

*[7]D. E. Goldberg, Genetic Algorithm in Search, Optimization and Machine Learning

*[8]P. Borovska, T. Ivanova and H. Salem, Efficient Parallel Computation of

the Traveling Salesman Problem on Multicomputer Platform,

Proceedings of the International Scientific Conference Computer Science

2004, Sofia, Bulgaria, Dec. 2004, PP74-79.

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