Using Genetic Algorithm with Combinational Crossover to Solve

Travelling Salesman Problem

Ammar Al-Dallal

Computer Engineering Department, Ahlia University, Manama, Kingdom of Bahrain

Keywords: Genetic Algorithm, Travelling Salesman Problem (TSP), Crossover.

Abstract: This paper proposes a new solution for Traveling Salesman Problem (TSP) using genetic algorithm. A

combinational crossover technique is employed in the search for optimal or near-optimal TSP solutions. It is

based upon chromosomes that utilise the concept of heritable building blocks. Moreover, generation of a

single offspring, rather than two, per pair of parents, allows the system to generate high performance

chromosomes. This solution is compared with the well performing Ordered Crossover (OX). Experimental

results demonstrate that, due to the well structured crossover technique, has enhanced performance.

1 INTRODUCTION

Genetic algorithm is a sort of adaptive worldwide

searching probabilistic optimization algorithm,

which often simulates the procedure associated with

organisms’ inheritance and evolution. It's been

widely applied to the areas of neural network,

combinatorial optimization, economical forecasts

and pattern recognition. The principle concept of the

genetic algorithm is derived from inheritance and

evolution. It is mainly based on mechanism of

biological evolution and genetics principle, in

accordance with natural selection and survival of the

fittest principle, using simple coding technology in

order to resolve complex problems.

Travelling salesman problem (TSP) is one of the

typical NP- completeness problems in combinational

optimization (Yuan et al, 2009). It is described as

follows. A salesman has to visit a number of cities

only once and then returns home. The distances

between any pair of cities are given at first and it is

required to find the shortest path. In which order

should the cities be visited to minimize the distance

travelled?

There are two kinds of TSP, symmetric TSP and

asymmetric TSP. In symmetric TSP, the distance

travelled from city i to city j is equal to the distance

travelled from city j to city i. In asymmetric TSP, the

distance travelled from city i to city j is not equal to

the distance travelled from city j to city i. In this

paper, symmetric TSP is considered.

1.1 The Mathematical Representation

of TSP

In math, the problem may be stated as follows (Yuan

et al, 2009). Given n cities, if V denotes the set of

cities as V = {v

, v

…

, v

}, and T denotes a

permutation as T = (t

, t

…

, t

}, where t



(i=1,2,3, …,n}, and denoted t

n+1

= t

, then TSP is

given as:

 













(1)

Although this formulation is simple, TSP is hard

to solve. For a symmetric TSP with n cities the

number of possible solution is (n-1)!/2, which is a

very large number even for a relatively small n

(number of cities).

1.2 Genetic Algorithm Representation

of TSP

The search space S is a set consisting of all cities

provided in the dataset. Each path P consists of all

cities of the search space S selected in a random

order. Each chromosome in GA represents a

possible path, and each path is evaluated using a

fitness function f that accumulates the summation of

distances between these cities (more about the

fitness function is explained in Section III). A set of

possible paths (chromosomes) associated with their

Al-Dallal, A..

Using Genetic Algorithm with Combinational Crossover to Solve Travelling Salesman Problem.

In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 1: ECTA, pages 149-156

ISBN: 978-989-758-157-1

149

fitness value forms one generation. Finally, a GA

engine tries to output a set of paths that minimize the

value of f. The optimal solution is a path or set of

paths that have the minimum score (distance)

returned by the function f.

It is found that such an optimization problem can

be solved efficiently using Genetic Algorithm (GA)

(Yuan et al, 2009; Vahdati et al, 2009, Wang and

Zha, 2009; Zhao et al, 2008; Yu et al, 2011). In

addition, GA demonstrates near global optimality,

implicit parallelism, adaptability, and high

robustness (Yu et al, 2011).

In its simplest form, a GA is a probabilistic

algorithm used to simulate the mechanism of natural

selection of living organisms, and it is often used to

solve problems having expensive solutions in terms

of time and computation complexity. This is due to

the principles of selection and evolution employed to

produce several solutions for a given problem.

Generally speaking, GA’s search space is composed

of candidate solutions to the problem (represent

possible paths in TSP). Each chromosome has an

objective function value known as fitness value that

is the path length joining the cities. This measure is

used to favour selection of successful parents for

producing new offspring. Offspring solutions are

produced from parent solutions by the application of

selection, ocrossover and mutation operators

(Lianshuan, Zengyan, 2009). The Offspring forms

the next generation of possible solution. Following

generations are then created by applying these three

operators until one of the two following criteria is

satisfied. Either the maximum predefined number of

generations is created, or no further enhancement to

chromosomes (in terms of overall fitness value) is

observed, as compared to the previous generation. In

all cases, the system will return the best solution

from the last generation.

The rest of this paper is organized as follows:

Section two lists related work of TSP that is

implemented using GA. Section three describes the

proposed GA model to improve the process of

solving the symmetric TSP. Section four describes

the experiments and results. Finally, Section five

concludes this paper.

2 RELATED WORK

Many approaches and results on evolutionary

optimization for TSP have been published. Samples

of these approaches include the following:

Authors of (Yuan et al, 2009) proposed GA based

on good character breed. The searching space is

divided into segments based on the fine seed. At

first, GA runs several times and gets a perfect

individual every time to gather a fine seed set. To

strengthen the local search, every segment is treated

as a small size TSP to be optimum. This approach

demonstrates a slight improvement when applied to

a small dataset of size 150.

Vahdati et al proposed in (Vahdati et al, 2009) a

new solution for TSP using genetic algorithm. This

technique is based on applying heuristic crossover

and mutation operation to GA in order to prevent

premature convergence.

The heuristic crossover considers each

chromosome as a closed circle. Two pointers are

used for each parent. The decision whether to select

the clockwise or anticlockwise pointer is based on

the fitness. This technique is able to accelerate the

speed of convergence by reducing the number of

generations.

An improved combinational genetic algorithm for

travelling salesman problem is applied in (Wang and

Zhao, 2009). This approach adds a local search

procedure in the standard genetic algorithm. Local

search is performed on the best individual of each

generation. This method increases the stability of the

algorithm in addition to improving the precision.

The authors of (Yu et al, 2011) solved the TSP

using GA in which the algorithm employed a

roulette wheel based selection mechanism, the use of

a survival-of-the-fittest strategy, a heuristic

crossover operator, and an inversion operator. This

approach was applied in TSP with 50 cities, and it

was able to quickly obtain an optimal solution to

TSP from a huge search space. However, this study

is not provided with clear figures to demonstrate the

level of improvement over other techniques as stated

by the authors.

A multi-objective TSP that is implemented using

GA is presented in (Lianshuan and Zengyan, 2009).

It creates an initial population which satisfies the

basic qualification, then it calculates the two

objective-values: distance and cost. The aim is to

find the minimal of both the distance and the cost.

The objective value is then used to rank the

chromosomes with Pareto function. However, this

approach is not provided with clear figures to

demonstrate the level of accuracy and how quickly

the result is obtained.

Takahashi (Takahashi, 2011) suggested solving

TSP through a Combinational method called:

iterative Extended Changing Crossover Operators.

This method combines Edge Assembly Crossover

(EAX) and Ant Colony Optimization (ACO). At the

first stage, ACO tries to search for provisional

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

150

solutions by using initial circuits generated by

uniformly random numbers with a certain random

number called seed. Next EAX follow the

provisional solutions generated by ACO and

continuously searches for the better solutions and

generates individuals having higher fitness. The

provisional solutions compose the chromosomes.

The next solutions are generated by EAX and

merged with new solutions generated by ACO. With

this merged file, EAX is executed again in order to

improve the solutions to create the next optimum

solutions. This approach is applied on medium-sized

TSP dataset and the results shows that it is effective

to produce well performing offspring.

Another approach developed by (Kuroda et al,

2010) is called Z-crossover. The Z-Cross works as

follows: first is to set a zone in the travelling area

according to some rules. Secondly, the edges

connecting cities between inside and outside the

zone are cut. Thirdly, the edges inside the zone of

one parent and those of the other parent are

exchanged. Finally, the sub-tours and isolated cities

produced by the third step mentioned above are

reconnected to construct a new tour of TSP.

Razali and Geraghty in (Razali and Geraghty,

2011) examined the performance of TSP through

analysing three selection techniques. These

techniques are: Tournament Selection, Proportional

Roulette Wheel Selection and Rank-based Roulette

Wheel Selection. The experiments are performed on

datasets with sizes between 10 and 51 cities. The

results show that the tournament selection

outperforms the others where it achieved the best

solution quality with low computing times.

The authors of (Sallabi and El-Haddad, 2009)

proposed an improved genetic algorithm to solve the

TSP. They suggested two crossover techniques. The

first one is the swapped inverted crossover. It can be

applied to a one or two-point crossover. After

partitioning the chromosomes, the genes within the

each partition are flipped in their order and then

swapped between parents. The second suggested

crossover is the rearrangement crossover. It works

by finding the greatest or maximum value of a city

distance among all the adjacent cities on the tour,

and then swap city i with three other cities, one at a

time. This method may not achieve improvement

after several iterations but may take a big jump and

improve the result.

A comparative study is done by Otman and

Abouchabaka (Otman and Abouchabaka, 2012) to

investigate the performance of six crossover

techniques that are applied to solve TSP. These

crossover techniques include: Uniform Crossover

Operator, the Cycle Crossover, the Partially Mapped

Crossover, the Uniform Partially-Mapped

Crossover, the Non-Wrapping Ordered Crossover

and the Ordered Crossover (OX). The experiments

were done on one instance of TSPLIB that is

BERLIN52. The results confirm that the best known

solution for the TSP instance was obtained by using

the OX.

(Osaba et al, 2013) applied multi-crossover and

adaptive island based population algorithm to solve

routing problems such as TSP. In this technique, the

entire population is divided into subpopulations,

each with a different crossover function, which can

be switched according to the efficiency. Each

subpopulation begins with a very low value of

crossover probability and then varies with the

change of the current generation number and the

search performance of recent generations.

As mentioned above, several approaches are

applied to solve TSP. The main contribution of this

work is to develop two new crossover techniques:

Two-Point Crossover with Replacement and

Combinational Crossover as well as adopting

specific GA operators in order to investigate their

performance compared to well-performing

techniques.

3 THE PROPOSED GA FOR

SOLVING TSP

This section describes the proposed GA model by

explaining its components and operators.

3.1 Chromosome Representation

As a matter of fact, chromosomes in GA are used to

represent the possible solution for the problem under

consideration. The chromosome consists of genes

that when integrated together form the complete

solution. For TSP, the best representation is to use

integers to refer to cities numbers. Each

chromosome consists of a sequence of integers to

represent a possible path that the salesperson may

visit. The order of these cities within the

chromosome represents the order of cities to be

visited. A problem with 7 cities is represented as a

chromosome of 7 genes as shown in Figure 1. This

chromosome forms a possible solution that is a path

starts from city number 2 passes through cities 5 - 1 – 4

– 6 - 7 -3 and ends at 2 again, since the path must be

closed path.

Using Genetic Algorithm with Combinational Crossover to Solve Travelling Salesman Problem

151

Figure 1: Sample chromosome.

3.2 Initial Generation Creation

When looking at the methods of creating initial

generation, there is a trade-off between creating an

initial generation in a fast way with low quality or

slower way but with higher quality. Fast creation is

done by selecting individuals randomly without any

selection criteria. However, this method may stick at

a local optimum solution causing the results to be

less effective due to fast convergence (Aly, 1990).

Since there is no specific criterion that favours any

city, a pure random selection is applied to select the

genes (cities) of the initial generation.

3.3 Parent Selection

Once the first generation is created, the operators of

GA are applied. The first operator of GA is parent

selection. Based on the results obtained by (Razali

and Geraghty, 2011), the tournament selection is

adopted in this work.

In tournament selection (Yeh, 2007), a group of i

individuals are randomly chosen from the

population. This group takes part in a tournament

and an individual with highest fitness value wins. In

many cases i is chosen to be two, and this method is

called binary tournament selection. To further

enhance this selection, i is selected to be five. And

the best of these five is selected.

In order to speed up convergence and avoid

creating large number of generation, elitism

technique can be applied as a selection technique

(Asllani and Lari, 2007). In this technique best l

members from the current generation are selected to

form the mating pole for next generation. It is

applied to ensure gradual improvement of the

solution. In this paper, l is selected to have best two

thirds of the current generation, i.e. the best 67% of

the current population.

A validation process, before applying crossover,

is applied to insure that the two selected parents are

different from each other, i.e. P1



P2.

3.4 Fitness Function

The fitness function f is used to evaluate each

individual (chromosome) to determine the best 67%

chromosomes of the population.

It is evaluated by accumulating the sum of

distances between the adjacent cities forming a path.

Assume that a chromosome has a length of n cities,

then the fitness function is calculated as follows

(Vahdati et al., 2009):

 









(2)

And 



 is the distance between city c

and c

i+1

. If both cities have x and y coordinates, then

the distance is calculated as:

















 







 



 







(3)

3.5 Crossover Operator

One of the main operators that heavily affects the

performance of GA system is crossover. In

literature, many techniques of crossover have been

developed and analysed to study their effect on the

output from several perspectives. In all cases,

crossover is performed such that the order of the

genes, hence the developed path, is changed in a

systematic manner to produce a new visible solution.

While many approaches are developed by

researchers to enhance the performance of GA, this

work is featured by introducing two new crossover

techniques that are expected to improve the

performance of the proposed system. The first

proposed crossover technique is called Two Point

Crossover with Replacement (TPXwR). The second

proposed crossover technique is called

Combinational Crossover Technique (CXT). These

two crossover techniques will be investigated in

terms of improvement of finding the shortest path.

3.5.1 The Design of Two Point Crossover

with Replacement (TPXwR)

The technique proposed in TPXwR is derived from

the two-point crossover, and it works like this:

Two chromosomes (parents) are selected from

best 67% (two-thirds) of the current generation using

binary tournament selection. These parents are

named p1 and p2. Then, two crossover points cp1

and cp2 are selected randomly. These two points

will divide each parent into three portions. Let a, b

and c be the portions of p1 and d, e and f be the

portions of p2 as shown in Figure 2.

Two offsprings O

and O

are formed from these

parents. Offspring one is created according to the

following steps. The first and last portions a and c

are copied to the corresponding positions of O

Then portion e of p2 is copied to O

. If a gene of

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

152

portion e is already existed in O

, then it will be

replaced by a gene from portion d of p2. If genes of

d are again exist in O

, then it will be replaced by a

gene from portion f.

To create the second offspring O

, similar steps

are applied. The first and last portions d and f are

copied to the corresponding positions of O

. Then

portion b of p1 is copied to O

. If a gene of portion b

is already existed in O

, then it will be replaced by a

gene from portion a. If genes of a are again exist in

, then it will be replaced by a gene from portion c.

During this process, the uniqueness of genes

(cities) within the newly created chromosome is

preserved. Figure 2 provides an example of this

technique where the steps to create the offspring are

as follow:

1. Genes of portions a and c are copied directly

to the corresponding position of the first

offspring O

since these are the first genes

added and no need to check their uniqueness.

2. Genes of portion e are copied to the

corresponding position of the first offspring

after checking the uniqueness of genes. 9 and

7 are added from e.

3. If the added gene to the offspring is already

existed in O

(8 from e in this example) then

it is replaced by one from portion d, having

that it does not already exist in this offspring

(3 from portion d). Otherwise, it is replaced

with one from portion f.

4. Similar steps are applied when creating the

second offspring with reversing rules.

This technique is considered to be an example of

simple two-point crossover technique but with

replacement when the newly added gene is already

existed in the constructed chromosome. Apparently,

it could produce better offspring compared to their

parents. However, a good generated chromosome

(path) could be disrupted or broken if the crosspoints

happen to be somewhere within a good block

causing it to break an optimal path.

3.5.2 The Design of Combinational

Crossover Technique (CXT)

The second proposed crossover technique is the

Combinational Crossover Technique (CXT). This

technique is applied to a semi-ordered chromosome.

In the semi-ordered chromosome, the first two genes

(cities) are ordered such that these genes have the

shortest path among the paths between all genes of

this chromosome. The closest cities (shortest paths)

appear to the left of the chromosome. The process of

reordering of genes is done after creating the

chromosome.

The proposed CXT also applies the fusion

crossover (Vrajitoru, 1998), where only one offspring

is generated from two selected parents. In this

technique, the offspring inherits the genes from one

of the parents with a probability according to its

performance. The advantage of this technique is that

the good genes of both parents are inherited

simultaneously to the offspring, producing high-

quality offspring and increasing the speed of

convergence.

Figure 2: Example to illustrate the TPXwR crossover

technique.

Figure 3: Example to illustrate the CXT crossover

technique.

The CXT technique is explained through the

example depicted in Figure 3. The number of cities

in this example is 10 and the numbers inside the

chromosomes represent the length between nodes i

and i+1, and i is between 1 and 10 according to the

mathematical model explained in Section I.

Using Genetic Algorithm with Combinational Crossover to Solve Travelling Salesman Problem

153

1. Select two parents x and y randomly from the

best 67% chromosomes of the current

generation.

2. Select one crosspoint randomly. This

crosspoint will divide x into two portions a and

b. Also, y will be divided into two portions c

and d.

3. Calculate the fitness of the first gene of each

parent. That is the first gene of portions a and c.

4. The one that has the lower fitness values is

selected as parent 1. In this example, y is

selected as parent 1 since the fitness of the first

gene is 1 which is less than 2 for parent x.

5. Copy the genes from the left portion (portion c

in this example) to the left portion of the

offspring O.

6. Continue filling the offspring with the genes

from the left portion of the second parent to the

offspring (portion a). Need to maintain the

length of the offspring equal to the parent’s

length. That is the genes from the second

parent will be copied until the length of the

offspring is equal to the parent’s length.

7. If the length of the offspring is still smaller

than the parent’s after the previous step, then

more genes need to be copied from either

parent.

8. The fitness values of genes of portions b and d

are compared starting from location cp+1. The

portion with the gene that has a lower fitness

value will contribute to the offspring O. This is

done in order to generate offspring with

appropriate genes from each parent and to

guarantee that the length of O is equal to the

length of the parent.

9. Through the process of copying the remaining

genes from the parents, the uniqueness of the

copied genes must be considered, i.e., each

gene can occur only once in the new offspring.

10. Last step is to order the genes based on their

fitness value. The order is done in ascending

order as preparation for the next crossover

process.

The rationale behind selecting the portion with the

lower fitness to start with is the need to inherit the

good order of cities and maintain the good building

blocks while passing them to the resulting offspring.

4 EXPERIMENTS AND RESULTS

This section describes the data set used to test the

performance of the proposed technique. Then it

analyses the obtained results.

4.1 Dataset Description

In order to investigate the performance of the

proposed techniques, several experiments need to be

conducted on a benchmark dataset. 10 instances are

chosen from TSPLIB (Reinelt, 1991) to perform the

needed experiments. These 10 sets differ in size and

ranging from 22 cities to 318 cities.

The proposed GA system starts by creating initial

generation consisting of 50 chromosomes. Recall

that each chromosome forms a possible solution

(travelling path). Hence, the length of the

chromosome is equal to the number of cities of the

applied dataset. The stopping condition of the

proposed system is one of the following: either the

maximum predefined number of generations is

created, or no further enhancement to chromosomes

(in terms of overall fitness value) is observed, as

compared to the previous generation. The crossover

probability is chosen to be 1. These parameters are

listed in Table 1.

4.2 Analysis of the Results

In order to evaluate the proposed crossover

techniques, they are compared with the results

obtained by applying the algorithm explained in

(Otman and Abouchabaka, 2012). As mentioned in

Section II, OX has obtained best results among six

options of crossover techniques.

The considered results of the experiments are the

chromosomes of the last generation. These results

are illustrated in Table 2. The first column is the

name of the dataset. The number associated to the

dataset name reflects the size of the set. The next

three columns are the results obtained from TPXwR,

CXT and OX techniques. The last two columns

provide the improvement in the fitness value of CXT

compared to OX and TPXwR.

When comparing the two proposed crossovers,

the CXT which produces one offspring in each

crossover process, shows better performance in

general in terms of the shortest path (fitness value).

For all sets, CXT achieves shorter path than TPXwR

except for berlin52 dataset. The improvement of

CXT over TPXwR ranges between 1.3% and

17.19%. The reason behind this improvement is that

in CXT the offspring inherits the well-performing

building blocks from parents that include adjacent

cities with minimal distance. Repeating this process

in every generation provides a great chance to

achieve better results, having that both techniques

are producing the same number of generations.

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

154

By analysing the results obtained by the two

proposed techniques and the results obtained by the

well know OX technique, it can be noticed that CXT

outperforms OX by percentage starting from 2.86%

for only one dataset, i.e. lin318 dataset and increases

to reach maximum of 22.38% for the eil76.

When looking at the results of the last

generations, it ascertains that the proposed crossover

is successful in enhancing the chromosome

performance since the aim of this process is to

produce better offspring from the parents.

The main reason behind the improvement in

performance is the way of implementing CXT. The

main contribution to the high improvement is

coming from the method of adding first genes to the

offspring. The first added genes are added based on

the fitness value (path length). The cities with lower

path length form a “good” building block. One of

the objectives when developing a crossover

technique is not to destruct these well performing

building blocks. And this is what is achieved in this

technique. This “good” building block is inherited

unchanged to the offspring. Not only this but also

the best building block of the second parent is also

passed to the offspring. Hence, each crossover

process adds two “good” building blocks to the

created offspring causing an enhancement in the

fitness value.

Table 1: Parameter setting of GABIR.

Parameter Description

Value

Population size

Maximum number of generations

200

Chromosome length

No. of cities of the

running set

Crossover rate

The number of best individuals

forming the crossover mate (Elitism)

2/3 of the population

Mutation rate

0.1

5 CONCLUSIONS

One of the permanent challenges for TSP problem is

to find the optimal solution (shortest path) using a

simple solution.

This paper applied a GA-based system to solve it

by developing two crossover techniques. The first

one is the 2-point crossover with replacement

TPXwR that produces two offspringss in each

crossover operation. However, this technique is

defective, where using multiple crosspoints causes

many breaks in good behavioural building blocks.

This in turn delays the convergence process. In order

to overcome this drawback, another crossover

technique is proposed. This crossover is the

Combinational Crossover Technique CXT. It works

by applying one-point crossover on an ordered

chromosome to produce only one offspring. The

experiment was applied on 10 datasets ranging in

size from 22 and 318 cities and compared to well-

performing OX crossover technique.

The results obtained show that the CXT

outperforms both TPXwR and OX. The

improvement of CXT ranges between 1.3% to

around 17% compared to TPXwR and ranges

between 2% and 22.4% compared to OX.

To generalize the results and further demonstrate

its efficiency, the proposed techniques need to be

compared with additional crossover techniques such

as the Uniform crossover, Edge Assembly Crossover

and Ant Colony Optimization. In addition, they need

to be applied on bigger TSP datasets.

Table 2: The fitness of each date set after last generation.

Dataset

CXT

TPXwR

% Impr. of

CXT over

TPXwR

Impr.

of CXT

over

ulysses22

1310

1480

1582

11.49%

17.19

att48

1211

1461

15356

17.14%

21.12

eil51

1370

1407

1530

2.63%

10.43

berlin52

2625

2506

2951

-4.75%

11.05

st70

3129

3182

3730

1.67%

16.10

eil76

1971

2285

2539

13.74%

22.38

rat99

7617

8054

8333

5.43%

8.59

kroA100

1483

1522

1686

2.53%

11.97

gr120

9389

10182

10886

7.79%

13.76

lin318

5686

5762

5854

1.31%

2.86

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