SOLVING THE RING ARC-LOADING PROBLEM USING

A HYBRID SCATTER SEARCH ALGORITHM

Anabela Moreira Bernardino, Eugénia Moreira Bernardino

Research Center for Informatics and Communications, Dep. of Computer Science, School of Technology and Management

Polytechnic Institute of Leiria, Leiria, Portugal

Juan Manuel Sánchez-Pérez, Juan Antonio Gómez-Pulido and Miguel Angel Vega-Rodríguez

Dep. of Technologies of Computers and Communications, Polytechnic School, University of Extremadura, Cáceres, Spain

Keywords: Communication Networks, Weighted Ring Arc-Loading Problem, Scatter Search Algorithm, Bio-inspired

Algorithms.

Abstract: Resilient Packet Ring (RPR) is a standard that uses Ethernet switching and a dual counter-rotating ring

topology to provide SONET-like network resiliency and optimised bandwidth usage, while it delivers

multipoint Ethernet/IP services. An important optimisation problem arising in this context is the Weighted

Ring Arc Loading Problem (WRALP). That is the design of a direct path for each request in a

communication network, in such a way that high load on the arcs will be avoided, where an arc is an edge

endowed with a direction. The load of an arc is defined as the total weight of those requests routed through

the arc in its direction. WRALP ask for a routing scheme such that the maximum load on the arcs will be

minimum. In this paper we study the loading problem without demand splitting and for solving it we

propose a Hybrid Scatter Search (HSS) algorithm. Coupled with the Scatter Search algorithm we use a Tabu

Search algorithm to locate the global minimum. We show that HSS is able to achieve feasible solutions to

WRALP instances, improving the results obtained by previous approaches.

1 INTRODUCTION

The past two decades have witnessed tremendous

research activities in optimisation methods for

communication networks. Resilient Packet Ring

(RPR), also known as IEEE 802.17, is a standard,

designed to optimise the transport of data traffic

through optical fiber ring networks (Davik et al.,

2004; RPR Alliance, 2004; Yuan et al., 2004). The

RPR aims to combine the appealing functionalities

of Synchronous Optical Network/Synchronous

Digital Hierarchy (SONET/SDH) networks with the

advantages of Ethernet networks. The load balancing

model for RPR differs from the SONET/SDH ring

loading. Namely, SONET/SDH demands are bi-

directional and the demands assigned to go

clockwise compete for common span capacity with

the demands assigned to go counter-clockwise. In

RPR there are two distinct rings (clockwise and

counter-clockwise) and the demands do not compete

for common capacity. In this paper we consider the

Weighted Ring Arc-Loading Problem (WRALP)

which arises in engineering and planning of RPR

systems. Specifically, for a given set of non-splitable

and unidirectional point-to-point demands, the

purpose is to find the routing for each demand so

that the maximum link segment load will be

minimised (Karunanithi and Carpenter, 1994; Cho et

al., 2005; Kim et al., 2008; Bernardino et al., 2010a).

There are three variants to solve this problem: (i)

demands can be split in two parts, and then each one

is sent in a different direction; (ii) demands are

allowed to be split in two parts, but restricted to be

integrally split; (iii) each demand must be entirely

routed in either one of the two directions, clockwise

or counter-clockwise. In this paper we study the

third variant, where NP-hardness can be drawn from

the results in literature (Cosares and Saniee, 1994;

Kubat and Smith, 2005).

Cosares and Saniee (1994) and Dell’Amico et al.

(1998) studied a similar no-split loading problem on

Moreira Bernardino A., Moreira Bernardino E., Manuel Sánchez-Pérez J., Antonio Gómez-Pulido J. and Angel Vega-Rodríguez M..

SOLVING THE RING ARC-LOADING PROBLEM USING A HYBRID SCATTER SEARCH ALGORITHM.

DOI: 10.5220/0003076200600069

In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 60-69

ISBN: 978-989-8425-31-7

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

SONET/SDH rings. For the split problem, several

approaches have been summarised by Schrijver et al.

(1998) and their algorithms are compared in Myung

and Kim (2004) and Wang (2005). Recently Kim et

al. (2008) presented an Ant Colony Optimisation

(ACO) algorithm using different strategies to solve

the loading problem on SONET/SDH rings.

The non-split WRALP considered in the present

paper is identical to the one described by Kubat and

Smith (2005) (non-split WRALP), Cho et al. (2005)

(non-split WRALP and split WRALP) and Yuan and

Zhou (2004) (split WRALP), that studied the

loading problem on RPR systems .

We verify that the main purpose of previous

works was to build feasible solutions for the loading

problems in a reduced amount of time. Our purpose

is different - we want to compare the performance of

our algorithm with others in the achievement of the

best-known solution. Using the same principle

Bernardino et al. (2008, 2009a, 2009b, 2010a,

2010b) presented several Evolutionary Algorithms

(EAs) and a Tabu Search (TS) algorithm to solve the

non-split loading problem on SONET/SDH rings

and several EAs and Swarm Optimisation

algorithms to solve the non-split WRALP.

The WRALP problem is a NP-complete

combinatorial optimisation problem (Cosares and

Saniee, 1994; Kubat and Smith, 2005). It means that

we cannot guarantee to find the best solution in a

reasonable amount of time. In practice, approximate

methods are used to find a good solution to complex

combinatorial optimisation problems where classical

heuristics have failed to be efficient. The existing,

successful methods in approximate optimisation fall

into two classes: Local Search (LS) and population-

based search. There are many LS and population-

based optimisation algorithms.

This paper presents an application of a

population-based optimisation algorithm called the

Scatter Search (SS) algorithm combined with a LS

technique called the Tabu Search (TS).

The SS is an EA that has recently been found to

be promising to solve combinatorial optimisation

problems. The SS was first introduced in 1977 by

Fred Glover and extensive contributions have been

made by Manuel Laguna (2002). The SS operates on

a small set of solutions and makes only limited use

of randomisation as a proxy for diversification when

searching for an optimal solution.

Embedded in the SS algorithm we use a TS

algorithm, which is used to improve the solutions’

quality. The TS algorithm is a mathematical

optimisation method, which belongs to the class of

LS techniques.

We compare the performance of Hybrid SS

(HSS) algorithm with five algorithms: Probability

Binary Particle Swarm Optimisation (PBPSO),

Genetic Algorithm (GA), Hybrid Differential

Evolution (HDE) algorithm, Hybrid ACO (HACO)

algorithm and Discrete Differential Evolution

(DDE), used in literature.

The paper is structured as follows: in Section 2

we present the problem definition; in Section 3 we

describe the implemented HSS algorithm; in Section

4 we discuss the computational results obtained and

in Section 5 we report about the conclusions.

2 PROBLEM DEFINITION

An optimal loading balance in RPR systems is of

paramount importance as it increases system

capacity and improves the overall ring performance.

Considering a given set of non-split and

unidirectional point-to-point requests (weights), the

purpose is to find the routing for each request in

such a way that the maximum arc load will be

minimised (Schrijver et al., 1998).

Let R

be a n-node bidirectional RPR ring with

nodes {n

,…,n

} labelled clockwise. Each

edge {e

k+1

} of R

, 1≤ k ≤ n, is taken as two

arcs with opposite directions, in which the data

streams can be transmitted in either direction:

)

1kkk

e,(ea

or )e,(ea

k1kk +

−

= .

A communication request on R

is an ordered pair

(s,d) of distinct nodes, where s is the source and

d is the destination. We assume that data can be

transmitted clockwise or counter-clockwise on the

ring, without splitting. We use P

(s,d) to denote

the directed (s,d) path clockwise around R

and

(s,d) the directed (s,d) path counter-

clockwise around R

Often a request (s,d) is associated with an

integer weight w>=0; we denote this weighted

request by (s,d;w). Let Z={(s

),...,(s

)} be a set of

integrally weighted requests on R

. For each request

) we need to design a directed path P

of R

from s

to d

. A collection P={P

: i=1,2,…,m}

of such directed paths is called a routing for Z.

In this work, the solutions are represented using

binary vectors (Table 1). If a position has the value

1 the demand flows in the clockwise direction, if it

has the value 0, it flows in the other way.

We assume that weights cannot be split, that is,

for some integer L

=1, 1≤ i ≤ m, the total amount

of data is transmitted along P

); L

=0, the

SOLVING THE RING ARC-LOADING PROBLEM USING A HYBRID SCATTER SEARCH ALGORITHM

total amount of data is transmitted along P

). The vector L= (L

,…,L

)determines

a routing scheme for Z.

Table 1: Representation of the solution.

Pair(s, t) Demand

1: (1, 2) Æ 15

2: (1, 3) Æ 3

3: (1, 4) Æ 6

4: (2, 3) Æ 15

5: (2, 4) Æ 6

6: (3, 4) Æ 14

n=numberNodes=4

m=numberPairs=6

15 C

3 CC

6 CC

15 C

6 CC

14 C

C - clockwise

CC - counter clockwise

Representation (x)

Pair

1 0 0 1 0 1

3 SCATTER SEARCH

ALGORITHM

This metaheuristic technique derives from strategies

proposed by Glover (1977) to combine decision

rules and constraints, and was successfully applied

to a large set of problems (Glover et al., 2003). The

basic idea is to create a set of solutions (the

reference set), that guarantees a certain level of

quality and diversity. The iterative process consists

in selecting a subset of the reference set, combining

the corresponding solutions through a strategy, in

order to create new solutions and to improve them

through a LS optimisation technique. The process is

repeated with the use of diversification techniques,

until certain stopping criteria are met.

In SS algorithm it is built an initial set of

solutions (reference set) and then the elements of

specific subsets of that set are systematically

combined to produce new solutions, which hopefully

will improve the best-known solution (see Glover et

al., 2003 for a comprehensive description of the

algorithm).

The basic algorithmic scheme is composed of

five steps:

1. Generation and improvement of solutions;

2. Construction of the reference set;

3. Subset selection;

4. Combination;

5. Reference set update.

The standard SS algorithm stops when the reference

set cannot be updated. However, the scheme can be

enhanced by adding new steps in which the

reference set is regenerated. Our algorithm uses a

diversification mechanism after a pre-defined

number of nid iterations without improving the best

solution found so far. The reinitialisation can be very

useful to refocus the search on a different search

space region and to avoid the early convergence of

the algorithm.

The main steps of the HSS algorithm applied to

the WRALP are detailed below:

Initialise Parameters

Generate initial set of Solutions

Evaluate Solutions

Apply Improvement Method

Generate Reference Set

WHILE TerminationCriterion()

Select subsets

Apply Combination Method

Apply Improvement Method

Update Reference Set

IF (no new solutions) THEN

Regenerate Reference Set

IF (nid iterations without improve

best solution) THEN

Apply Diversification Mechanism

The next subsections describe each step of the

algorithm in detail.

3.1 Initialisation Parameters

The following parameters, must be defined by the

user: (1) mi– number of iterations; (2) ni– number

of initial solutions; (3) b1– number of best solutions

in the reference set; (4) b2– number of most

different feasible solutions in the reference set and

(5) nid- number of iterations without improvement

(used for diversification).

3.2 Generation of Solutions

The initial solutions can be randomly created or in a

deterministic form based in a Shortest-Path

Algorithm (SPA). The SPA is a simple traffic

demand assignment rule in which the demand will

traverse the smallest number of segments.

3.3 Evaluation of Solutions

To evaluate how good a potential solution is relative

to other potential solutions we use a fitness function.

The fitness function returns a positive value (fitness

value) that reflects how optimal the solution is.

The fitness function is based on the fitness

function used in (Bernardino et al., 2008, 2009a,

2009b, 2010a, 2010b ):

,…,w

between(s

),…,(s

) (1a)

, …, L

= 0 Æ P

)

ICEC 2010 - International Conference on Evolutionary Computation

1 Æ P

) (1b)

Load on arcs:

Load(L,

∑

∈ )d,(sPa:i

iik

(2a)

Load(L,

−

∑

−−

∈ )d,(sPa:i

iik

(2b)

∀k=1,…,n; ∀i=1,…,m

Fitness function:

max {max Load(L,

max Load(L,

−

)} (3)

For a given ring, between each node pair (s

)

there is a demand value >=0. Constraint sets (1)

state that each positive demand value is routed in

either clockwise (C) or counter-clockwise (CC)

direction.

For an arc, the load is the sum of w

for

clockwise or counter-clockwise between nodes e

and e

k+1

(2). The purpose is to minimise the

maximum load on the arcs of a ring (3).

3.4 Generation of Reference Set

The best b1 solutions in the initial set of solutions

are selected to be in the reference set. The b2

feasible solutions in the initial set of solutions that

are the most different when compared to the

solutions already in the reference set, are also

selected to be in the reference set.

As a measure of the difference between two

solutions, we compute the total number of different

assignments between the two solutions.

3.5 Subset Selection

In literature, several methods can be applied to

generate the subsets. In our implementation, the

subsets are formed by combining two solutions from

the reference set:

(1,2), (1,3), (1,4),…, (1,b1+b2),

(2,3),…, (b1+b2-1,b1+b2).

We adopt Type-1 (Glover et al., 2003). This

method consists of ((b1+b2)

- (b1+b2))/2

pair wise combinations of the solutions.

All pairs of solutions in the reference set are

selected for the combination procedure (see

subsection 3.6).

3.6 Combination Method

This method combines the solutions in each subset

to form new solutions.

First a random node is chosen and then the pairs

with that node are exchanged (see Fig. 1) between

the two solutions.

Figure 1: .Combination Method – produces two combined

solutions – example with n=4 (number of nodes) and m=6

(number of pairs). The node chosen was “1”.

The combination method consists of the

following steps:

node= random(n)

FOR i=1 TO m DO

IF Solution1(i) has node OR

Solution2(i) has node THEN

CombinedSolution1(i)= Solution2(i)

CombinedSolution2(i)= Solution1(i)

ELSE

CombinedSolution1(i)= Solution1(i)

CombinedSolution2(i)= Solution2(i)

The combination method produces two combined

solutions.

The combined solutions go through the

improvement phase (see subsection 3.7).

3.7 Improvement Method

A TS algorithm is applied to each solution in the

initial set of solutions in order to reduce its cost, if

possible. After the combination, the TS algorithm is

also applied to improve the quality of the combined

solutions.

The basic concept of TS was described by

Glover (1986). TS allows the search to explore

solutions that decrease the objective function value

only in those cases where these solutions are not

forbidden. This is usually obtained by keeping track

of the action used to transform one solution into the

next. When an action is performed it is considered

tabu for the next T iterations, where T is the tabu

status length. A solution is forbidden if it is obtained

SOLVING THE RING ARC-LOADING PROBLEM USING A HYBRID SCATTER SEARCH ALGORITHM

by applying a tabu action to the current solution.

In our implementation, the TS only exploits a

part of the neighbourhood. The most common and

simplest way to generate a neighbour is to exchange

the direction of the traffic of one request. In our

implementation, some positions of the solution are

selected and their directions are exchanged (partial

search). This method can be summarised in the

following pseudo-code steps:

p1 = random (m)

p2 = random (m)

N = neighbourhoods of ACTUAL-

SOLUTION (one neighbourhood results of

interchange the direction of p1 and/or

p2)

SOLUTION = FindBest (N)

If ACTUAL-SOLUTION is worst than

SOLUTION

ACTUAL-SOLUTION = SOLUTION

The positions which directions are exchanged are

classified as tabu attributes. A candidate can be

chosen as a new current solution, if the positions

which directions are exchanged are not the same as

those in the tabu list. Normally in TS algorithm, if a

neighbour is the best solution found so far it could

be selected as a move, even when it is tabu. In our

implementation, we don’t explore neighbours when

the two pairs chosen are in the tabu list. In

aspiration, just the best neighbour not tabu with a

fitness value lower than the best is selected.

The TS ends when a maximum number of

iterations is reached. Based on preliminary

observations, we consider a maximum number of 10

iterations. With a higher value of iterations, the

algorithm slows down. We also observed that a high

number of iterations does not produce significant

better results.

For the tabu list, we consider m/20 elements. In

the tests carried out with TS, it was verified that the

number of elements in the tabu list does not have a

significant influence on the efficiency and quality of

the search. However, if the number of elements is

high, the search space will be small, which may lead

to a premature convergence of the algorithm. On the

other hand, if the number of elements is small, the

search space will be large, which may take a long

time to obtain a good solution.

The improved solutions are considered for

inclusion in the reference set (see subsection 3.8).

3.8 Reference Set Update

The purpose is to maintain a good level of quality

and diversity.

We adopted the dynamic reference set update

(Glover et al., 2003).

A new feasible solution immediately enters in

the reference set, if its quality is better than the

quality of the worst solution, or if its diversity is

greater than the diversity of the less different

solution. Solutions that are equal to others already in

the reference set are not allowed to enter under any

condition.

If the reference set is not updated, then the

algorithm restarts the reference set (see subsection

3.9).

3.9 Regeneration of Reference Set

The algorithm creates another set of solutions - Ps

(with the same size of the initial set of solutions).

The new solutions go through the improvement

phase (see subsection 3.7).

A new feasible solution immediately enters in

the reference set, if its quality is better than the

quality of the worst solution.

The b2 solutions with greater diversity are

erased from the reference set and the b2 feasible

solutions in Ps that are the most different when

compared to the solutions already in the reference

set are selected to be in the reference set.

3.10 Diversification Mechanism

This mechanism restarts the best b1 solutions in the

reference set.

The algorithm creates another set of solutions -

Pd (with the same size of the initial set of solutions).

The new solutions go through the improvement

phase (see subsection 3.7).

The best (b1-1) solutions in Pd are selected to

be in the reference set. For the following iteration,

we kept the best solution.

3.11 Termination Criterion

The algorithm stops when a maximum number of

iterations (mi) is reached.

4 RESULTS

We evaluate the utility of the algorithms using the

ICEC 2010 - International Conference on Evolutionary Computation

same instances produced by Bernardino et al.

(2009a, 2009b, 2010a, 2010b). The studied

examples arise by considering six different ring sizes

– 5, 10, 15, 20, 25 or 30 nodes. A ring in a

telecommunication network will typically contain

between 5 and 20 nodes. The instances consider the

5, 10 and 15 node rings to be ordinary-sized rings

and the 20, 25 and 30 node rings to be extremely

large rings. The demand cases are:

 Case 1: complete set of demands between

5 and 100 with uniform distribution;

 Case 2: half of the demands in Case 1 set

to zero;

 Case 3: 75% of the demands in Case 1 set

to zero.

 Case 4: complete set of demand between 1

and 500 with uniform distribution. This case was

only used for the 30 nodes ring.

It was generated 1 different problem instance for

each case. This yields 3 instances for each ring size

(4 instances for the 30 nodes ring). For

convenience, they are labelled C

, where 1<i<6

represents the ring size and 1<j<4 represents the

demand case.



Figure 2: Number of initial solutions – Average

Fitness/Execution Time/Number of Best-known Solutions

- b1 =[4,8], b2=[4,8] and nid= [m/10, m/2] .



Figure 3: Number of best solutions in the reference set

(parameter b1) – Average Fitness/Execution Time/

Number of Best-known Solutions – ni=10.

We perform comparisons between all parameters

(using all instances) in order to establish the correct

parameter setting for the HSS algorithm.

We consider the same instance – C41 (a problem

with average difficulty) to show the comparisons

between parameters. To compute the results we use

50 iterations.

The best results obtained with the HSS algorithm

use ni between 40 and 100, b1 between 4 and 10,

b2 between 4 and 10 and nid between m/10 and

m/2. These parameters were experimentally

considered good and robust for the problems tested.

The number of initial solutions was set to {10,

20, 30, 40, 50, 60, 70, 80, …, 200}. We studied

the impact on the execution time, the average fitness

and the number of best-known solutions found. The

number of solutions has a significant impact on the

execution time (see Figure 2).

The best results obtained with HSS use ni

between 40 and 100. With these values, the

algorithm can reach, in a reasonable amount of time,

a reasonable number of best-known solutions (see

Figure 2). With a higher number of solutions, the

algorithm is more time consuming.

The number of solutions in the reference set is

typically small - 20 solutions or less (Glover et al.,

2003). In our experiments the number of solutions

b1 and the number of solutions b2 were set to {1,

SOLVING THE RING ARC-LOADING PROBLEM USING A HYBRID SCATTER SEARCH ALGORITHM

2, 3, 4, 5, 6, 7, 8, 9, 10}. We studied

the impact on the execution time, the average fitness

and the number of best-known solutions found. The

number of solutions in the reference set has a

significant impact on the execution time (see Figure

3 and Figure 4).



Figure 4: Number of most different feasible solutions in

the Reference Set (parameter b2) – Average

Fitness/Execution Time/Number of Best-known Solutions

– ni=10.

The results show that the best results obtained use

b1>=4 and b2>=4 (see Figure 3 and Figure 4).

These parameters were experimentally found to be

good and robust for the problems tested. With

b1+b2>20 the algorithm can reach a better average

fitness but it is more time consuming.

We observe that a small number of solutions in

the reference set allows an initial faster convergence,

but a worse final result, following to an increased

amount of suboptimal values. This can be explained,

because the quality of the initial best-located

solution previous to the first restart highly depends

on the reference set size: they need more diversity to

avoid premature stagnation.

For parameter nid, the number of iterations

used for diversification, the values between m/10

and m/2 have been shown to be experimentally

more efficient.

Phenomena of stagnation and insufficient

intensification have been observed for values of nid

lesser than m/10 and greater than m/2.

In general, the experiments have shown that the

proposed parameter setting is very robust to small

modifications.

In this paper, we only compare our algorithm

with: PBPSO (Bernardino et al., 2009a), GA

(Bernardino et al., 2008), HDE (Bernardino et al.,

2009b), HACO (Bernardino et al., 2010a) and DDE

(Bernardino et al., 2010b) because the authors: (1)

use the same test instances; (2) adopt the same

fitness function; (3) implement the algorithms using

the same language (C++) and; (4) adopt the same

representation (binary).

Suggestions from literature helped us to guide

our choice of parameter values for PBPSO, GA

HDE, HACO and DDE (Bernardino et al., 2008,

2009a, 2009b, 2010a, 2010b).

PBPSO was applied to populations of 40

particles and we consider the value 1.49 for the

parameters C1 and C2, and for the inertia velocity

(W) values in the range [0.6,0.8].

GA was applied to populations of 200

individuals; it uses “Uniform” as recombination

method, “Multiple” as mutation method and

“Tournament” as selection method. For GA, we

consider crossover probability in the range

[0.6,0.9] and mutation probability in the range

[0.5,0.7].

HDE was applied to populations of 50

individuals, it uses the “Best1Bin” strategy, CR in

the range [0.3,0.5] and factor F in the range

[0.5,0.7].

For the HACO, we consider populations of 40

individuals, 30 modifications, Q=100, x1 in the

range

[0.6,0.8], x2 in the range [0.7,0.8]

and q in the range [0.7,0.8].

For the DDE, we consider populations of 50

individuals, 5 perturbations, pc in the range

[0.1,0.2], pp in the range [0.6,0.8] and the

LS method “Exchange Direction”.

Finally, the parameters of the HSS algorithm

were set to ni=50, b1 between 4 and 8, b2

between 4 and 8, number of iterations of the TS =3

and nid between m/10 and m/2. The six

algorithms were executed using a processor Intel

Quad Core Q9450. The initial solutions of the six

algorithms were created using random solutions. For

the instance C64 the SPA was used to create the

ICEC 2010 - International Conference on Evolutionary Computation

Table 2: Best obtained results.

Instance Nodes Pairs Best Fitness Iterations

C11

5 10

161 25

C12

5 8

116 10

C13

5 6

116 10

C21

10 45

525 50

C22

10 23

243 25

C23

10 12

141 10

C31

15 105

1574 100

C32 15 50 941 50

C33 15 25 563 25

C41 20 190 2581 300

C42 20 93 1482 100

C43 20 40 612 50

C51 25 300 4265 500

C52 25 150 2323 400

C53 25 61 912 250

C61 30 435 5762 1500

C62 30 201 2696 1000

C63 30 92 1453 500

C64 30 435 27779 500

Table 3: WRALP results – run times and number of iterations.

Inst.

PBPSO GA HDE HACO DDE HSS

Time IT Time IT Time IT Time IT Time IT Time IT

C11 <0.001 2 <0.001 2

<0.001 2 <0.001

<0.001 2

C12 <0.001 2 <0.001 2

<0.001 2 <0.001

<0.001 2

C13 <0.001 1 <0.001 1

<0.001 1 <0.001

<0.001 1

C21 <0.001 15 <0.001 15

<0.001

<0.001 10

C22 <0.001 3 <0.001 5

<0.001 3 <0.001

<0.001 3

C23 <0.001 3 <0.001 3

<0.001 3 <0.001

<0.001 3

C31 0. 1 20 0. 1 30

0. 1

0. 1 30

0. 1

0. 1 10

C32 <0.001 8 <0.001 15

<0.001

<0.001 5

C33 <0.001 5 <0.001 5

<0.001

<0.001 3

C41 0.2 50 0.1 50

0.1 30

0.15 50

0.1 25

0.1 20

C42 0.075 20 0.075 40

0.05 10

0.06 25

0.05 8

0.05 10

C43 <0.001 5 <0.001 10

<0.001

<0.001 5

<0.001

<0.001 5

C51 0.75 80 0.75 80

0.75 40

0.6 100

0.5 30

C52 0.1 25 0.1 40

0.1 15

0.1 30

0.1 15

C53 0.01 15 0.01 25

0.01 10

0.01 20

0.01 8

0.01 10

C61 2 130 1.75 130

1.75 40

1.75 150

1.5 50

1.3 40

C62 0.4 50 0.2 60

0.25 20

0.4 60

0.25 25

0.25 20

C63 0.075 15 0.075 30

0.075 10

0.075 20

0.06 10

0.05 10

C64 0.5 40 0.3 30

0.25 5

0.5 5

0.1 3

0.1 5

Table 4: WRALP results – Average Fitness / Average Time / Standard Deviation.

Inst It PBPSO GA HDE HACO DDE HSS

AF AT SD AF AT SD AF AT SD AF AT SD AF AT SD AF AT SD

C41 50 2594,36 0,26 7,70 2587,62 0,17 3,46 2584,31 0,27 1,15 2591,23 0,16 7,73 2582,06 0,16 1,18 2581,50 0,22 0,71

C51 75 4291,52 0,86 16,85 4273,18 0,43 2,97 4271,27 0,7 5,10 4279,49 0,76 10,10 4268,96 0,53 5,71 4265,68 0,65 1,16

C61 100 5837,58 3,10 23,19 5784,62 1,34 10,05 5783,18 1,87 7,45 5793,68 2,23 14,17 5781,52 1,39 9,78 5763,72 1,44 2,25

SOLVING THE RING ARC-LOADING PROBLEM USING A HYBRID SCATTER SEARCH ALGORITHM



PBPSO GA HDE HA CO DDE HSS

Algorithms

Percentage of best solutions

Figure 5: Percentage of best-known solutions obtained by the six algorithms – instance C41 (50 iterations).

initial populations. Table 2 presents the best

obtained results. The first column represents the

instance number (Instance), the second and third

columns show the nodes’ number (Nodes) and the

pairs’ number (Pairs), the fourth column

demonstrates the minimum fitness values obtained

and the fifth column demonstrates the number of

iterations used to test each instance. The number of

iterations was selected based upon preliminary

observations.

Table 3 presents the best WRALP results

obtained with the six implemented algorithms. The

first column represents the instance number (Inst.)

and the remaining columns demonstrate the obtained

results (Time – Run Times, IT – Iterations) by the

six algorithms. The presented values have been

computed based on 100 different executions for

each test instance, using the best combination of

parameters found and different seeds. Table 4 only

considers the 30 best executions. The six algorithms

reach feasible solutions for all test instances and all

the algorithms reach the best-known solutions before

the run times and the number of iterations presented.

In comparison, the HSS algorithm produces a

higher number of best-known solutions using the

same number of iterations (Figure 5). The DDE

algorithm obtains a reasonable number of best-

known solutions and a good average fitness in a

better running time (Figure 5, Table 4). The PBPSO

is the slowest algorithm and it obtains a smaller

number of best-known solutions comparing with the

other algorithms (Figure 5).

When using the SPA to create the initial

solutions, the times and number of iterations

decrease – instance C64. This instance is

computationally harder than the C61 however the

best-known solution is obtained faster. Based on

preliminary observations we consider more efficient

to initially apply a SPA and after, a metaheuristic to

improve the solutions.

Table 4 presents the WRALP average fitness and

the WRALP average time obtained with PBPSO,

GA, HDE, HACO and DDE using a limited number

of iterations for the instances C41, C51 and C61

(harder instances). The first column represents the

instance number (Instance), the second column

demonstrates the number of iterations used to test

each instance and the remaining columns show the

obtained results (AF – Average Fitness, AT –

Average Time, ST – Standard Deviation) by the six

algorithms. The results have been computed based

on 100 different executions for each test instance

using the best combination of parameters found and

different seeds.

As it can be seen, the average fitness and

standard deviations for the HSS are smaller. It

means that the HSS is more robust than the other

algorithms. DDE also presents a good average

fitness and a good standard deviation.

5 CONCLUSIONS

In this paper we present a Hybrid Scatter Search

algorithm to solve the WRALP. The Hybrid Scatter

Search Algorithm is an evolutionary optimisation

technique, able to perform simultaneous local and

global search.

The performance of' Hybrid Scatter Search

algorithm is compared with five algorithms from

literature, namely: PBPSO, GA, HDE, HACO and

DDE.

Relatively to the problem studied, the Hybrid

Scatter Search algorithm presents better results. The

computational results show that it had a stronger

performance, improving the results obtained by

previous approaches. Moreover, in terms of standard

ICEC 2010 - International Conference on Evolutionary Computation

deviation, the algorithm also proved to be more

stable and robust than the other algorithms.

Experimental results demonstrate that the

proposed algorithm is an effective and competitive

approach in composing satisfactory results with

respect to solution quality and execution time for the

WRALP.

In literature the application of Scatter Search

algorithm for this problem is nonexistent. For that

reason, this article shows its enforceability in the

resolution of this problem.

The continuation of this work will be the search

and implementation of new methods to speed up the

optimisation process.

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