COOPERATING OF LOCAL SEARCHES BASED

HYPERHEURISTIC APPROACH FOR SOLVING TRAVELING

SALESMAN PROBLEM

Mitra Montazeri

, Abbas Bahrololoum

Hossein Nezamabadi-pour

Mahdieh Soleymani Baghshah

and Mahdieh Montazeri

Member of Young Researchers Society, Computer Engineering Department, Shahid Bahonar University, Kerman, Iran

Computer Engineering Department, Shahid Bahonar University, Kerman, Iran

Electronic Engineering Department, Shahid Bahonar University, Kerman, Iran

Kerman

Medical University, Kerman, Iran

Keywords: Meta-heuristic algorithm, Hyper-heuristic algorithm, Traveling Salesman Problem.

Abstract: Until now various heuristic optimization methods have been developed for solving NP-Hard problems.

These methods by trading off between exploration and exploitation attempt to find an optimum solution. In

this paper, we introduce a new optimization algorithm based on hyper-heuristic for solving TSP. A hyper-

heuristic approach has two layers. In low level, we have six local searches and in high level we use Genetic

Algorithm. Genetic Algorithm corporate local searches efficiency. The proposed method has high ability to

searching in solution space and explores and exploit appropriately. This method exploits space depended on

characteristics of the region of the solution space that is currently under exploration and also the

performance history of local searches. The proposed method is used to solve TSP and compared with well-

known methods. The experimental results confirm the efficiency of the proposed method.

1 INTRODUCTION

The traveling salesman problem, TSP, which is

known as NP-hard in the field of combinatorial

optimization has been studied by many researchers

since it appeared in the 1930. In this problem, we

have n cities and for each pair of cities a connection

cost is defined. The problem is concerned with

finding an optimum path which starts from one city

and passes all other cities exactly once and finally

returns to the starting city. The exact approaches

such as dynamic programming (Neapolitan and

Naimpour, 2004), can find exact solution but if the

number of cities is large, the problem will be

intractable. Therefore heuristic approaches been

attended in the last decades. This class of methods

has been achieved acceptable solutions within

reasonable time and is relatively efficient for dealing

NP-hard problems (Ozaglam and Cunkas, 2008).

However, since a heuristic operator always seeks to

find immediate improvement, the heuristic method

is at the risk of quickly trapping a local optimum. In

such cases, it is often desirable to guide the heuristic

search by employing a strategy known as a meta-

heuristic. This strategy encourages the discovery of

better solutions in the search space by tightening a

focus on good solutions and improving upon them

(intensification), and to encourage the exploration of

the solution space by broadening the focus of the

search into new areas (diversification) (O’Brien,

2007). Till now, many studies on TSP are performed

using the meta-heuristic

algorithms. Dorigo and

Gambardella (Dorigo and Gambardella, 2008) used

ants of the artificial colony to generate successively

shorter feasible tours by using information

accumulated in the form of a pheromone trail

deposited on the edges of the TSP graph and their

results in comparison with previous methods were

satisfying.

Some drawbacks cause that meta-heuristic

algorithms couldn’t be suitable for these problems.

One of these drawbacks is that meta-heuristic

algorithms often require domain knowledge and so

parameters are needed to be tuned expertly. For

example, in GA, which is known as a meta-heuristic

329

Mitra M., Bahrololoum A., Nezamabadi-pour H., Soleymani Baghshah M. and Montazeri M..

COOPERATING OF LOCAL SEARCHES BASED HYPERHEURISTIC APPROACH FOR SOLVING TRAVELING SALESMAN PROBLEM.

DOI: 10.5220/0003675103290332

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 329-332

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

algorithm, the chromosome of a genetic algorithm is

either the solution of the target problem or a

structure of the solution. This means that problem

specific knowledge is essential in the design of

chromosomes. No Free Lunch Theorem introduced

by Wolpert and MacReady (Wolpert and MacReady,

1997) states no individual algorithm is best and each

algorithm has its own advantages and drawbacks.

Therefore, we need algorithms which automatically

combine the strength and compensate the weakness

of the known heuristics.

In meta-heuristics, there is no further focus on

the exploitation aspect when a potential region is

identified (Ang and Tan, 2010). Thus, Memetic

Algorithms (MA) which incorporate local

improvement search into meta-heuristics, were

proposed (Ong and Keane, 2004). Experimental

studies have been shown that a hybrid of a meta-

heuristic and a local search is capable of more

efficient search capabilities (Merz and Freisleben,

1999). MA uses only one local search to explore the

whole solution space. As shown in Ref. (Ong and

Keane, 2004), the choice of the local search has an

important impact on the search performance of MA.

Therefore, inappropriate use of local search may

result in memetic algorithms performing poorer than

standard GA (Ong and Keane, 2004). Different local

search methods have different biases. These biases

may be suitable for some classes of problems but not

for others. Therefore, we need to have multiple local

searches to achieve improved search performance

and reduce the probability of utilizing an

inappropriate local method. However, we need to

have a supervisor managing the choice of local

searches that should be applied at any time.

Therefore, for solving the above mentioned

problems, new approach known as hyper-heuristic is

proposed. In this paper, we propose a novel

algorithm that is based on hyper-heuristic approach

for solving TSP. The proposed method cooperates

local searches appropriately based on hyper-

heuristic approaches.

The rest of this article is organized as follows:

Section 2 describes hyper-heuristic in details. In

Section 3, we explain the proposed method. Finally,

experimental results and conclusion are presented in

Section 4 and 5, respectively.

2 HYPER-HEURISTIC

Hyper-heuristic is an approach that was proposed in

2000 by Cowling et al (Cowling et al., 2001). It is a

heuristic approach which selects heuristics and is a

higher abstract level over meta-heuristics.

General framework of hyper-heuristic has two

levels. At the low level, there is a set of local

searches, which also known as Low Level Heuristics

(LLH)s, to modify solution locally in an attempt to

return an improved solution. At the high level, there

is a black-box choice function. It manages the

selection of LLHs. These selections depend upon the

characteristics of the region of the solution space

currently under exploration and the performance

history of the local searches.

3 THE PROPOSED METHOD

In the proposed method, GA is used as the high level

hyper-heuristic. This GA is not a direct GA, in fact

each individual in GA’s population consist of a

sequence of integer numbers. Each number is an

LLH choice which tells us which LLH must be

applied and each individual tells in which order to

apply LLHs. In fact, the key note behind this method

is its LLHs cooperation which is act of operating

together. Optimization problems in science and

engineering commonly have large search spaces,

which contain numerous local landscapes of diverse

forms. The joint operation of diverse LLHs to cope

with the large search space is facilitated via the

problem decomposition or diversity in the LLH

selection (Ong and Keane, 2004). Fig. 1 shows the

flowchart of the proposed method. In the next sub-

sections, we introduce the proposed method.

Figure 1: Flowchart of the proposed method.

3.1 Genetic Algorithm Process

3.1.1 Population of Low Level Heuristic

Hyper-heuristic concludes from a set of LLHs

created by human expert. In the proposed method,

six LLHs are used which make small change in

current solution: First LLH is 2-OPT algorithm. It

basically removes two edges from the tour, and

reconnects the two paths created. There is only one

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

330

Table 1: The best, worst and average solutions that achieve by the proposed method over 100 runs.

Database

Number

of city

)n(

Best

known

solution

Average

solutions

Minimum

solutions

Maximum

solutions

C20 20 62575 62575 62575 62575

S21 21 60000 60000 60000 60000

C30 30 62716 62716 62716 62716

F32 32 84180 85645 84180 92123

C40 40 62768 62768 62768 62768

F41 41 68168 68738 68168 91774

Table 2: Comparison average solutions of the proposed method with average solutions of well known methods on Ozcan

database on 100 runs.

Database

Number

of city

)

(

Best known

solution

The Proposed

method

Results obtained from literature

PSO (Çunka

and

Özsalam,

2009)

GA (Çunka

and

Özsalam,

2009)

TGMA-HC

(Ozcan and

Erenturk,

2004)

SSMA-HC

(Ozcan and

Erenturk,

2004)

IDGA (Lau

and Xiao,

2008)

C20 20 62575 62575 63276 63188 134497 107493 116367

S21 21 60000 60000 60786 60648 120276 93626 121439

C30 30 62716 62716 63625 63356 165795 116633 120025

F32 32 84180 85645 85535 85392 146317 108048 124640

C40 40 62768 62768 64212 63753 197829 128117 125721

F41 41 68168 68738 69995 69702 158461 115860 125461

way to reconnect the two paths so that the resulted

tour will be valid (Nilsson, 2003). Second LLH is 2-

CHANGE turns a tour into a slightly different tour.

It randomly select two edge (a, b), (c, d) where a#d,

b#c and also replaces them with (a, c), (b, d) (Keller

and Poli, 2008). Third LLH is IF 3-CHANGE that

randomly selects edges as arguments for 3-

CHANGE, and then IF 3-CHANGE makes betters

the cycle for the given arguments. It actually

executes 3-CHANGE (Keller and Poli, 2008). Forth

LLH is IF 2-CHANGE. This LLH executes 2-

CHANGE if this will generates shorter the tour

under construction (Keller and Poli, 2008). Fifth

LLH does if first LLH will be shorter (Nilsson,

2003). Sixth LLH is as a 3- CHANGE. It deletes

three mutually disjoint edges from a given tour, and

reconnects the obtained three paths so that a

different tour results (Keller and Poli, 2008).

These LLHs decode in population of GA.

3.1.2 Genetic Operators

We use the roulette wheel selection operator, one-

point crossover and a mutation operator that selects

some positions in one individual randomly and

mutates genes at these positions to other values

ranging from 1 to 6 (Davis, 1991).

3.2 Improving Diversity

Using intensive LLHs causes to miss diversity of

search space. In fact, in our GA process, we increase

exploitation and missing a few explorations due to

increasing the selection pressure. Therefore after the

LLHs apply to current solution, we add a special

mutation step. This step is used for the purpose of

maintaining the population diversity. The special

mutation which is being used for TSP is swap

mutation which selects randomly two positions and

swaps them (Banzhaf, 1990).

3.3 Change Parameters Adaptively

In this step, the mutation rate and crossover rate are

adapted in each generation according to the change

in fitness (Cowling et al., 2002).

4 EXPERIMENTAL RESULT

The proposed method has been tested on datasets

suggested by Ozcan et al. in 2004 (Ozcan and

Erenturk, 2004).

In all cases, population size is set to 80; number

of generation is 10000, crossover rate 0.7 and 0.5 for

mutation rate.

COOPERATING OF LOCAL SEARCHES BASED HYPERHEURISTIC APPROACH FOR SOLVING TRAVELING

SALESMAN PROBLEM

331

Table 1 shows the best, worst and average

solutions achieved by the proposed method over 100

runs on different datasets. As seen in these tables, in

most cases, the proposed method finds global

optimum over 100 runs. This issue is considerable.

Table 2 shows the comparison of the proposed

method with literature results. In each row the best

solution is bold. As shown in this table, in most

cases, the proposed method finds the routes better

than other methods. The results of the proposed

method have been compared with those of PSO and

GA implemented in Ref. (Çunka and Özsalam,

2009). The proposed MA in Ref. (Ozcan and

Erenturk, 2004) was introduced as Steady State

Memetic Algorithm with Hill Climbing (SSMA-HC)

and a Trans-Generational Memetic Algorithm with

Hill Climbing (TGMA-HC).

Finally, we compare our proposed method with

Iterative Deepening Genetic Annealing Algorithm

(IDGA) method to show that our method is more

efficient than both the previous methods and also a

proper hybrid of them. In Ref. (Lau and Xiao, 2008),

it was verified that IDGA is more appropriate than

SA and GA alone or hybrid for solving TSP.

5 CONCLUSIONS

In this paper, a new optimization algorithm based on

hyper-heuristic approach was introduced for solving

TSP. Proposed method searches the solution space

appropriately in which depended upon the

characteristics of the region of the solution space

currently under exploration and the performance

history of local search. Our method used GA to

select local search. In which local searches were act

of operating together, our method cooperated local

searches. The proposed method also remained robust

to increasing the number of dimension which is a

key element in the development of any evolutionary

algorithm. Our method had an excellent convergence

rate. In fact, finding the global optimum in high

speed is the salient property of our method. This

method was used to solve TSP and compared with

different well-known methods. Experimental results

confirmed the superior performance of it.

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