Using the Expanded IWO Algorithm to Solve the Traveling Salesman

Problem

Daniel Kostrzewa and Henryk Josiński

Institute of Informatics, Silesian University of Technology, Akademicka 16, Gliwice, Poland

Keywords: Traveling Salesman Problem, Expanded IWO Algorithm, Inver-over Operator.

Abstract: The Invasive Weed Optimization algorithm (IWO) is an optimization metaheuristic inspired by dynamic

growth of weeds colony. The authors of the present paper have expanded the strategy of the search space

exploration of the IWO algorithm introducing a hybrid method along with a concept of the family selection

applied in the phase of creating individuals. The goal of the project was to evaluate the expanded IWO

version (exIWO) as well as the original IWO by testing their usefulness for solving some test instances of

the traveling salesman problem (TSP) taken from the TSPLIB collection which allows to compare the

experimental results with outcomes reported in the literature. The results produced by other heuristic

algorithms as well as the methods based on the self-organizing maps served as the reference points.

1 INTRODUCTION

The Invasive Weed Optimization (IWO) algorithm

is an optimization metaheuristic, in which the

exploration strategy of the search space (similarly to

the evolutionary algorithm) is based on the

transformation of a complete solution into another

one. The authors of the original version of the

algorithm from University of Tehran were inspired

by observation of dynamic spreading of weeds and

their quick adaptation to environmental conditions

(Mehrabian and Lucas, 2006).

Usefulness of the IWO was confirmed for both

continuous and discrete optimization tasks. The

research was focused inter alia on minimization of

the multimodal functions and tuning of a second

order compensator (Mehrabian and Lucas, 2006),

antenna configurations (Mallahzadeh et al., 2008),

electricity market dynamics (Sahraei-Ardakani et al.,

2008), a recommender system (Sepehri Rad and

Lucas, 2007), and the join ordering problem for

database queries (Kostrzewa and Josiński, 2011).

The goal of the present paper is to introduce an

expanded version of the IWO (exIWO)

distinguished by the hybrid strategy of the search

space exploration proposed by the authors.

Evaluation of the suggested modification is based on

the solution of some test instances of the traveling

salesman problem (TSP) taken from the TSPLIB

collection (Reinelt, 1991) of the Research Group

Discrete and Combinatorial Optimization at the

Ruprecht-Karls-Universität Heidelberg (available at

www.iwr.uni-heidelberg.de/groups/comopt/software/

TSPLIB95/tsp/).

The overview of bibliography describing the

methods for solving the TSP would be unusually

spacious. Numerous studies related to the usage of

exhaustive, greedy, and evolutionary methods were

mentioned in (Michalewicz and Fogel, 2004),

whereas the IWO algorithm, according to the

authors’ knowledge, has never been used to this

purpose by other researchers.

The organization of this paper is as follows –

Section 2 contains a brief description of the modi-

fied IWO algorithm taking into serious consideration

the proposed hybrid method of the search space

exploration. Discussion of the transformations used

for creation of a new individual in case of the TSP is

presented in Section 3. Section 4 deals with proce-

dure of the experimental research along with its

results. The conclusions are formulated in Section 5.

2 DESCRIPTION OF THE

EXPANDED IWO ALGORITHM

The simplified pseudocode mentioned below

describes the exIWO algorithm by means of

451

Kostrzewa D. and Josi

nski H..

Using the Expanded IWO Algorithm to Solve the Traveling Salesman Problem.

DOI: 10.5220/0004224204510456

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 451-456

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

terminological convention consistent with the

“natural” inspiration of its idea. Consequently, the

words “individual”, “plant”, and “weed” are treated

as synonyms.

Create the first population.

For each individual:

Compute the value of the fitness

function.

While the stop criterion is not

satisfied:

For each individual from the

population:

Compute the number of seeds.

For each seed:

Choose the dissemination

method.

Create a new individual.

Compute the value of its

fitness function.

Create a new population.

The population of initial solutions of the given

optimization task is randomly or greedily generated

over the search space. Next, the degree of

individuals’ adaptation to environmental conditions

is estimated by the value of their fitness function,

which at the same time determines the number of

seeds produced by each plant according to the

following formula:



















minmax

minmin

ffSS

indind

(1)

where S

max

, S

min

denote maximum and minimum

admissible number of seeds generated, respectively,

by the best population member (characterized by

fitness f

max

) and by the worst one (fitness f

min

The seeds are scattered over the search space.

The hybrid strategy of the search space exploration,

proposed by the authors of the present paper, makes

use of the following component methods: dispersing,

spreading and rolling down. Probability values of

selection assigned to the particular methods form

parameters of the algorithm.

Construction of a new individual according to

the dispersing method is based on transformations

(see Section 3) performed on the copy of the parent

individual. The number of transformations equals to

the conventional distance between the parent

individual and the descendant in the search space.

The distance is described by normal distribution

with a mean equal to 0 and a standard deviation

truncated to nonnegative values. The standard

deviation is decreased in each algorithm iteration

(i.e. for each population) and computed for the

iteration iter,





max

,1 iteriter



according to the

following formula:



finfininit

iter

iteriter





















max

(2)

The total number of iterations (iter

max

), equivalent to

the total number of populations, can be either used

with the purpose of determination of the stop

criterion or can be estimated based on the stop

criterion defined as the execution time limit. The

symbols σ

init

, σ

fin

represent, respectively, initial and

final values of the standard deviation, whereas m is a

nonlinear modulation factor. A tendency to gradual

reduction of the distance for subsequent populations

resulting from the formula (2) accords with intention

of the authors of the original IWO algorithm version.

The spreading is a random disseminating seeds

over the whole of the search space. Therefore, this

operation is equivalent to the random construction of

new individuals.

The rolling down is based on the examination of

the neighborhood of the parent individual. In case of

discrete optimization task the neighborhood

comprises individuals that differ from the parent by

exactly one transformation, whereas for the

continuous optimization the term “neighbors” stands

for individuals located at the same randomly

generated distance from the considered one. The

best adapted individual is chosen from among the

determined number of neighbors, whereupon its

neighborhood is analyzed in search of the next best

adapted individual. This procedure is repeated k

times (k is a parameter of the method) giving the

opportunity to select the best adapted individual

found in the k-th iteration as a new one.

Creation of the next population is based on the

concept of the family selection. Each plant from the

first population is a protoplast of a separate family.

A family consists of a parent weed and its direct

descendants. According to the family selection rules

only the best individual of each family survives and

becomes member of the next population.

3 ADAPTATION OF THE

EXPANDED IWO TO THE TSP

The expanded IWO algorithm is a metaheuristic.

Hence, its application for solving a given

optimization task requires a formulation of a single

solution representation as well as a definition of

transformations used for creation of a new solution.

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From among significant concepts related to the

form of a single solution it is worthwhile to mention

three vector representations proposed in the

literature: path, ordinal, and adjacency as well as a

matrix representation (Michalewicz and Fogel,

2004). A plant used by the exIWO was designed

according to the simple and natural rule of the path

representation – a tour is an ordered list of cities (i.e.

expressed as a vector [2 3 9 4 1 5 8 6 7]) and the

order of visitation is determined by the order of

vector elements (2–3–9–4–1–5–8–6–7–2).

Specific character of weeds reproduction

mechanism functioning in the IWO implies rather

application of transformations based on a single

individual used as a sole parent. In case of the TSP

transformations are realized by means of operators.

Basic unary operators – inversion, insertion,

displacement, and reciprocal exchange – are

discussed in (Herdy, 1991). The inver-over operator

is admittedly based on the inversion of a segment of

cities, but the inverted segment is most often

determined by means of another individual which is

randomly selected from the same population. Due to

specific character of this operation a fragment of the

“second parent” is guaranteed to be a part of the

offspring. In this way the inver-over operator

combines features of unary as well as binary

techniques. However, very promising results

reported by its inventors (Tao and Michalewicz,

1998) decided on implementation of both –

inversion and inver-over – operators in the exIWO.

In case of the TSP the individuals from the first

population are greedily generated – the nearest city

will be visited as the next one.

4 EXPERIMENTAL RESEARCH

The goal of the experiments was to compare the

quality of solutions found by the exIWO with the

outcomes generated by other methods using 17 test

instances of the TSP taken from the TSPLIB

collection. As reference points served the results

presented in (DePuy et al., 2005), (DePuy et al.,

2002) and (Bai et al., 2006). The former study

presents application of the metaheuristic for

randomized priority search (Meta-RaPS), whereas

the latter discusses results achieved by means of the

self-organizing maps (SOM). Both papers include

comparison of the outcomes reported by other

researchers, who investigated methods from

different areas of artificial intelligence. Therefore

the exIWO algorithm operated on the same test

instances as those used in the aforementioned works.

It is worthwhile to mention that a number of cities is

included in the name of each test set (e.g., the route

described as “bier127” consists of 127 cities).

The workstation used for experiments is

described by the following parameters: 2×Intel Xeon

E5620 2.40 GHz, RAM 16 GB, MS Windows

Server 2008 R2 Datacenter 64-bit SP1.

Evaluation of solutions generated by particular

methods was based on 2 criteria – minimum and

average difference from optimal or best known

solution. Average values for the exIWO were

computed based on results of 100 experiments for

each test set.

Table 1 shows the minimum difference between

the solutions produced by the IWO variants (exIWO

and the original version) using inversion or the

inver-over operator and optimal or best known

solution for particular test instances expressed as a

percentage.

Table 1: Minimum difference between the solution

produced by one of the tested variants of the IWO

algorithm (exIWO and the original version) and optimal or

best known solution [%].

Test set exIWO IWO

Inversion Inver-over Inversion Inver-over

kroA100 0 0 0 0

kroB100 0 0 0 0

kroC100 0 0.096 0.096 0

kroD100 0.169 0 0.169 0.169

kroE100 0 0 0 0

bier127 0 0 0 0

eil51 0 0 0 0

eil76 0 0 0 0

kroA200 0.200 0.449 0.446 0.446

lin105 0 0 0 0

pcb442 1.554 1.829 1.544 1.916

pr107 0 0 0 0

pr136 1.588 2.265 1.130 2.585

pr152 0.294 0.455 0.277 0.563

rat195 0.689 0.861 0.732 0.646

rd100 0 0 0 0

st70 0 0 0.148 0.148

The optimal or best known solution was found

by the exIWO in case of 12 from among 17 test

instances.

Average run times of the exIWO implementation

in Java for 10000 iterations (equivalent to the

number of populations) are included in Table 2. The

number of individuals in a single population depends

on the number of cities in a test set – for routes with

less than 150 cities a single population contains 200

individuals, in other cases – 50. The best exIWO

results reported in all tables were achieved for

UsingtheExpandedIWOAlgorithmtoSolvetheTravelingSalesmanProblem

453

particular test instances using different values of

other algorithm parameters (a nonlinear modulation

factor m, probabilities of selection of dispersing,

spreading and rolling down, maximum and

minimum admissible number of seeds S

max

, S

min

initial and final values of the standard deviation σ

init

fin

). Only the number k of neighborhoods examined

during the rolling down remained equal to 2 in the

most successful exIWO experiments.

Table 2: Average run times of the exIWO variants [s].

Test set Inversion Inver-over

Number of

individuals

kroA100 58.8 59.9 200

kroB100 62.2 62.7 200

kroC100 67.1 67.6 200

kroD100 70.9 68.8 200

kroE100 74.1 73.0 200

bier127 89.4 89.5 200

eil51 24.4 28.9 200

eil76 38.3 39.6 200

kroA200 35.5 25.6 50

lin105 62.8 61.6 200

pcb442 69.6 51.9 50

pr107 70.2 68.8 200

pr136 79.6 90.6 200

pr152 27.1 20.4 50

rat195 37.1 23.2 50

rd100 58.1 59.6 200

st70 47.8 43.1 200

The average run times of both exIWO variants

are similar in most cases.

Tables 3-6 present average difference from

optimal or best known solution for the chosen

methods expressed as a percentage. Test instances

considered in Table 3 belong to the Kro?100

collection (the question mark substitutes for one of

the following symbols {A, B, C, D, E}), whereas

results of the optimization methods other than

exIWO were taken from (DePuy et al., 2005) and

(DePuy et al., 2002). Particular names represent the

following approaches: Meta-RaPS TSP – a Meta-

heuristic for Randomized Priority Search, Priority

rule based on the simple TSP heuristics – cheapest

insertion and node insertion, GRASP – Greedy Ran-

domized Adaptive Search Procedure – all algorithms

were described in (DePuy et al., 2005), Christofides

& 2opt 3opt, Convex hull & 3opt, NN & 2opt 3opt

(“NN” denotes “Nearest Neighbor”), 2opt 3opt –

algorithms using local search methods (Lawler et al.

1985), Lin-Kernighan algorithm (Padberg and

Rinaldi, 1991), Modified Lin-Kernighan algorithm

(Mak and Morton, 1993), CCAO – Convex hull,

Cheapest insertion, Angle selection and Or-opt – a

heuristic which exploits geometrical properties of

symmetric Euclidean TSP (Golden and Stewart,

1985), Triangul. – a Delaunay triangulation-based

heuristic (Krasnogor et al., 1995), I^3 – a composite

heuristic consisting of 3 phases: construction of an

Initial envelope, Insertion of the remaining vertices,

and Improvement procedure (Renaud et al., 1996),

P-SEC, F-SEC – a Preliminary and a Full Subpath

Ejection Chain method, respectively (Rego, 1998).

Abbreviations related to the IWO variants have the

following meanings – “inv” denotes inversion, whe-

reas “i-o” represents the inver-over operator. The

underlined values outperform results produced by

the exIWO.

Table 3: Average difference from optimal or best known

solution for the chosen methods mentioned in (DePuy et

al., 2005), (DePuy et al., 2002) and variants of the IWO

based on the Kro?100 test sets [%].

Method A B C D E

Meta-RaPS

TSP

0 0.25 0 0 0.17

Priority rule 0.5 2.46 0.82 1.43 1.1

GRASP 0 0.55 0.31 0.42 0.37

Christofides

& 2opt 3opt

2.51 1.4 1.53 0.17 3.03

Convex hull

& 3opt

0.37 1.46 1.06 0.04

2.46

NN & 2opt

3opt

0.14 1.46 1.06 0.73 2.46

2opt 3opt 0.81 1.44 0.53 1.74 0.18

Lin-

Kernighan

0.26 0

0.7 0.17 0.16

Modif. Lin-

Kernighan

0 0.17 0 0 0.21

CCAO 0 0.97 0.5 0.97 2.54

Triangul. 0.51 2.13 2.79 3.81 2

I^3 0 0.9 0.5 2 2.6

P-SEC 0 0.32 0.02 0.75 0.33

F-SEC 0 0 0 0 0

exIWO inv 0 0.01 0.24 0.49 0.19

exIWO i-o 0 0.01 0.25 0.51 0.20

IWO inv 0 0.01 0.25 0.54 0.20

IWO i-o 0 0.01 0.24 0.54 0.21

Results presented in Tables 4-6 are related to the

SOM-based methods and were taken from (Bai et

al., 2006) and compared with outcomes produced by

the IWO variants. Particular acronyms represent the

following approaches: PKN – Pure Kohonen

Network (Hueter, 1988) and (Fort, 1988), GN –

Guilty Net (Burke and Damany, 1992), AVL – the

procedure of Angéniol, de la Croix Vaubois and Le

Texier (Angéniol et al., 1988), KL, KG – 2 variants

of the Kohonen Network Incorporating Explicit

Statistics (KNIES) – Local and Global, respectively

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(Aras et al., 1999), SETSP – SOM Efficiently

applied in the TSP (Vieira et al., 2003), MGSOM –

the Modified Growing ring SOM approach for TSP

(Bai et al., 2006).

Table 4: Average difference from optimal or best known

solution for the SOM-based methods and variants of the

IWO – part I [%].

Method bier127 eil51 eil76 kroA200

PKN 3.322 4.202 6.171 5.311

GN 31.181 10.493 14.182 34.058

AVL 3.713 4.108 6.190 5.540

KL 2.762 2.864 4.981 2.836

KG 3.079 2.864 5.483 3.667

SETSP 1.850 2.221 4.234 3.119

MG SOM 1.097 1.398 3.384 1.972

exIWO

inv

0.061 0.099 0.338 0.757

exIWO i-o 0.069 0.127 0.292 0.716

IWO inv 0.090 0.131 0.346 0.749

IWO i-o 0.097 0.129 0.323 0.747

Table 5: Average difference from optimal or best known

solution for the SOM-based methods and variants of the

IWO – part II [%].

Method lin105 pcb442 pr107 pr136

PKN 6.921 – 0.454 7.343

GN 7.584 – 81.661 –

AVL 6.487 17.472 1.791 6.893

KL 1.985 11.072 0.734 4.531

KG 1.291 10.447 0.425 5.147

SETSP 1.301 10.160 0.409 4.400

MG SOM 0.028 8.577 0.172 2.154

exIWO

inv

0.070 2.580 0 3.768

exIWO i-o 0.033 3.011 0 3.834

IWO inv 0.078 2.713 0.001 3.993

IWO i-o 0.106 3.199 0.001 3.895

The column “avg” in Table 6 includes average

values computed on all 12 test sets taken into

account in Tables 4-6.

The exIWO variants produce results which

outperform the most of the outcomes of the SOM-

based methods with the exception of the MGSOM in

case of 3 test instances.

In the majority of cases solutions obtained by the

exIWO are slightly better than those of the original

version.

Similarity of both exIWO variants concerns not

only the run times but the minimum and average

differences from optimal or best known solution as

well.

Table 6: Average difference from optimal or best known

solution for the SOM-based methods and variants of the

IWO – part III [%].

Method pr152 rat195 rd100 st70 avg

PKN 1.523 – – 2.637 –

GN 42.817 – 10.382 11.956 –

AVL 1.302 15.420 4.498 2.711 6.344

KL 0.968 12.238 2.095 1.511 4.048

KG 1.285 11.916 2.622 2.326 4.213

SETSP 1.169 11.192 2.601 1.600 3.688

SOM

0.741

5.984 1.172 1.183 2.322

exIWO

inv

0.863 1.721 0.127 0.533 0.905

exIWO

i-o

0.924 1.868 0.105 0.594 0.969

IWO inv 0.943 1.857 0.123 0.624 0.971

IWO i-o 0.944 2.055 0.153 0.636 1.024

The average difference from optimal or best

known solution for both exIWO variants computed

on all 17 test instances mentioned in this paper

amounts to 0.741 % in case of the inver-over

operator and 0.693 % for the inversion.

5 CONCLUSIONS

The authors of the present paper have modified the

IWO metaheuristic introducing a hybrid strategy of

the search space exploration as well as a concept of

the family selection. Analysis of the exIWO results

presented in this paper enables to expect solutions of

good quality in a reasonable amount of time. It is

also worth mentioning that each iteration of the

exIWO generates a population of individuals

representing feasible tours.

The experiments revealed the usefulness of the

exIWO algorithm for solving discrete optimization

tasks and confirmed the concept of using

randomness as a mechanism to avoid local optima.

The method can compete with other heuristics,

although the influence of the hybrid strategy

components (dispersing, spreading, rolling down) on

the solution found by the exIWO requires further

research – at present the algorithm takes part in the

World TSP Challenge (www.tsp.gatech.edu/world/

index.html) and in the Mona Lisa TSP Challenge

(www.tsp.gatech.edu/data/ml/monalisa.html).

ACKNOWLEDGEMENTS

All tests were performed using the computer server

UsingtheExpandedIWOAlgorithmtoSolvetheTravelingSalesmanProblem

455

funded by the Ministry of Science and Higher

Education, Poland, Grant No. N N516 265835.

Authors would like to thank Dr Bożena Małysiak-

Mrozek and Dr Dariusz Mrozek from the Institute of

Informatics, Silesian University of Technology,

Gliwice, Poland for the possibility to perform

calculations on this computer.

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