A CLONAL SELECTION ALGORITHM TO INVESTIGATE

DIVERSE SOLUTIONS FOR THE RELIABLE SERVER

ASSIGNMENT PROBLEM

Abdullah Konak and Sadan Kulturel-Konak

Penn State Berks, PA, Reading, U.S.A.

Keywords: Network reliability, Reliable server assignment, Clonal selection algorithm, Artificial immune systems.

Abstract: A reliable server assignment (RSA) problem in networks is defined as determining a deployment of

identical servers on a network to maximize a measure of service availability. In this paper, a simulation

optimization approach is introduced based on a Monte Carlo simulation and embedded into a Clonal

Selection Algorithm (CSA) to find diverse solutions for the RSA problem, which is important in simulation

optimization. The experimental results show that the simulation embedded-CSA is an effective heuristic

method to discover diverse solutions to the problem.

1 INTRODUCTION

The reliable server assignment (RSA) problem on

networks is described as assigning identical servers

to network nodes to maximize a measure of service

availability under a budget constraint. Given an

undirected network G=(V, E), where V={1,…,n} is

the node set, E={(i, j)} is the edge set, the RSA

problem is expressed as

max ({ , , }, )

{0,1}

zRs sG

cs C











where s

is the binary server assignment decision

variable indicating whether a server is assigned to

node i (s

=1) or not (s

=0), c

is the cost of deploying

a server at node i, and R() is a measure of the

availability of a service provided by identical

servers.

The definition of service availability R() depends

on the nature of the service is provided by a

network. For example, a computer network will not

function properly if the servers providing the

Domain Name System (DNS) service to the network

are inaccessible. Therefore, DNS servers are

replicated over a network to ensure their availability.

In this paper, the critical service rate (CSR), which

was first defined in (Kulturel-Konak and Konak,

2009), is used as the objective function. The CSR

quantifies a network’s ability to continue providing a

service in the case of catastrophic edge or node

failures. In a catastrophic failure mode, the edges

and nodes are assumed to be in only two states,

operative or failed. Let r

(i, j)

be the reliability of edge

(i, j)E and r

be the reliability of node iV. For

computational tractability, edge and node failures

are assumed to be state-independent. When edge (i,j)

fails, the communication between nodes i and j is

interrupted if it cannot be rerouted through an

alternative path. When a node fails, the network

traffic cannot be routed through the node.

For a given server assignment S={s

,…,s

CSR(S) is defined as the probability that more than a

predetermined fraction (



) of the operational nodes

have access to at least one server in case of a

component failure as follows:

(|)

CSR( ) Pr{ }

()











(1)

where X denotes a binary state vector of the network

such that at least one node is operational,



(X|S)=1

if there exists at least one path between node i and a

server node (



(X|S)=0 otherwise), v

(X)=1 if node i

is operational in state X (v

(X)=0 otherwise), and



is the critical service level.

The RSA problem as described in this paper was

first defined in (Kulturel-Konak and Konak, 2009),

and an Ant Colony Optimization (ACO) algorithm

156

Konak A. and Kulturel-Konak S..

A CLONAL SELECTION ALGORITHM TO INVESTIGATE DIVERSE SOLUTIONS FOR THE RELIABLE SERVER ASSIGNMENT PROBLEM.

DOI: 10.5220/0003644101560161

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 156-161

ISBN: 978-989-8425-83-6

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

was developed to solve the problem. (Kulturel-

Konak and Konak, 2010) also applied a simulation

embedded-Particle Swarm Optimization (PSO)

algorithm to the problem with very promising

results. Recently, three nature-inspired heuristics,

namely, ACO, PSO, and Clonal Selection Algorithm

(CSA), were compared in (Konak and Kulturel-

Konak, 2011) in terms of their performances and

convergence properties. In this paper, a CSA is

applied to the problem with an emphasis to discover

not only good but also diverse solutions.

In general networks, exactly calculating CSR(S)

is not practical in reasonable CPU times because its

evaluation is NP-hard as it is the case in many

network reliability analysis problems (Ball, 1980).

Therefore, solutions searched by the CSA are

evaluated by a Monte Carlo (MC) simulation. The

CSA, the MC simulation, and a hashing method are

integrated in a way to minimize the computational

effort to efficiently evaluate candidate solutions and

to reduce the effect of the noise in the objective

function due to the MC simulation.

When simulation is used to evaluate alternative

solutions in a heuristic algorithm, it is important for

the algorithm to return multiple solutions because

the evaluation of solutions includes error due to

simulation. In the context of the RSA problem,

decision makers wish to have a set of alternative

server assignments with high levels of service

availability. This observation is motivated by the

fact that in most real-life reliability optimization

problems, component reliabilities are not hard facts,

but estimations based on historical observations.

Therefore, providing decision makers with a set of

alternative solutions allows them to determine the

final design by weighing other factors in addition to

system availability. The CSA in this paper returns a

set of solutions instead of a single best solution. The

CSA is compared with two previous meta-heuristics,

namely the ACO and the PSO, with respect to its

ability to discover good solutions with diverse

structures. The computational results show that the

CSA can discover more diverse set of solutions than

the ACO and PSO while its performance is par with

them.

2 A CSA ALGORITHM FOR THE

RSA PROBLEM

Artificial Immune Systems (AIS) are relatively new

bio-inspired computational algorithms. The early

theoretical immunology work (Farmer et al., 1986);

(Perelson, 1989); (Varela et al. 1988) has prepared

the origins for AIS. There has been an increasing

interest in AIS approaches to different problems,

such as scheduling, image processing, bio

informatics etc., after early applications on machine

learning (Dasgupta, 1998); (Dasgupta and Nino,

2009); (Hart and Timmis, 2009).

One of the population based AIS algorithms,

namely CSA, is applied to solve the RSA problem in

this paper. The first CSA was developed based on

the foundational clonal selection theory of Burnet

(1959). The basic immunological components are:

maintaining a specific memory set, selecting and

cloning most stimulated antibodies, removing poorly

stimulated or non-stimulated antibodies,

hypermutating (i.e., affinity maturation) activated

immune cells and generating and maintaining a

diverse set of antibodies (Dasgupta, 1998). The

principle of the theory is mainly as follows: The

antigen (i.e., the foreign molecule that the immune

system is defending against) selects the lymphocytes

(i.e., B-cells or white blood cells that detect and stop

antigens) with receptors which are capable of

reacting with a part of the antigen. The rapid

proliferation of the selected cell occurs during the

selection to combat the invasion (i.e., clonal

expansion and production of antibodies). While

duplicating the cells, copying errors occur (i.e.,

somatic hypermutation) resulting in an improved

affinity of the progeny cells receptors for the

triggering antigen. CSA has been applied to many

engineering and optimization problems. A basic

CSA has been proposed by De Castro and von

Zuben (2000) and later renamed as CLONALG by

De Castro and von Zuben (2002), and they also

provide the extensive analysis on the algorithmic

computational complexity. In the general

CLONALG first, antibodies (i.e., candidate

solutions) are selected based on affinity either by

matching against an antigen pattern or via evaluation

of a pattern by an objective function (affinity means

objective function value in optimization problems).

Then, selected antibodies are cloned proportional to

their affinity, and each clone is subject to a

hypermutation inversely proportional to its affinity.

Finally, the resulting clonal-set competes with the

antibody population to survive for membership in

the next generation, and low-affinity population

members are replaced with randomly generated

antibodies (Brownlee, 2007). The main steps of

CLONALG are as follows:

Initialization

While (stopping criteria is not) {

Antigen Selection

A CLONAL SELECTION ALGORITHM TO INVESTIGATE DIVERSE SOLUTIONS FOR THE RELIABLE SERVER

ASSIGNMENT PROBLEM

157

Exposure and Affinity Evaluation

Clonal Selection

Affinity Maturation (Hypermutation)

Metadynamics (Replacement)

}

Termination

2.1 Solution Representation and Hyper

Mutations

In the CSA in this paper, a solution j is represent by

binary server assignment vector S

={s

,…,s

} such

that s

indicates whether a server is assigned to node

i (s

=1) or not (s

=0).

In the CSA, each solution is cloned and mutated

according to its rank in the population such that a

higher-ranked solution produces a higher number of

clones than a lower-ranked solution, and it is

mutated at a lesser degree. To achieve this, solutions

in the population are sorted and ranked into three

equal sized tiers as top 1/3, middle 1/3, and bottom

1/3 based on their objective functions. Each solution

at the top, middle, and bottom tiers respectively

produces



/2,



/3, and



/4 clones which are mutated

by bitwise invert operators based on its tier. This

ensures that solutions with better objective function

values will produce a higher number of clones. On

the other hand, each clone is mutated at a rate

inversely proportional to its tier. Therefore, three

different invert mutation operators are applied to the

clones of the three tiers as follows. Top tier clones

are perturbed by inverting one or two decision

variables (Mutation1), middle tier clones by three

decision variables (Mutation2), and bottom tier

clones by four decision variables (Mutation3). In

other words, new solutions created from top tier

solutions are similar to their parents, and solutions

created from bottom tier solutions are much more

different than their parents. Top tier solutions are

promising; therefore, a local search around these

solutions is justified. On the other hand, bottom tier

solutions are probably in non-promising regions of

the search space. Therefore, moving away from

those regions by large perturbations is desired. The

mutation operators of the CSA are as follows:

Procedure Mutation1 (S

) {

Randomly select two nodes i1 and i2 such that

ji1

ji2

1

If s

ji1

ji2

=1, then flip the values of s

ji1

and s

ji2

obtain new solution S

If s

ji1

ji2

=0, then flip the value of s

ji1

to obtain

new solution S

Return S

}

Procedure Mutation2 (S

) {

Randomly select three nodes i1, i2, and i3 such

that 1 s

ji1

ji2

ji3

2

Flip the values of s

ji1

, s

ji2

, and s

ji3

to obtain new

solution S

Return S

}

Procedure Mutation3 (S

) {

Randomly select four nodes i1, i2, i3, and i4 such

that s

ji1

ji2

ji3

ji4

Flip the values of s

ji1

, s

ji2

, s

ji3

, s

ji4

to obtain new

solution S

Return S

}

2.2 Solution Evaluation

As mentioned earlier, the objective function of the

problem, CSR, is estimated using a crude MC

simulation (Konak, 2009). In the crude MC

simulation, a state vector X is sampled by

individually sampling the state of each edge (i, j) E

and node iN using Bernoulli random variables with

means r

and r

, respectively. To estimate CSR, K

state vectors are sampled in this way, and the

accessibility of servers by each operational node is

checked for each state sampled. The ratio of the

number of network states where critical service level

α is achieved to the sample size (K) yields an

unbiased estimator of CSR.

Evaluating candidate solutions using simulation

within a heuristic algorithm has two important

drawbacks. Firstly, simulation output includes a

statistical error which might affect the performance

of the algorithm. Secondly, simulation is

computationally expensive, especially if a small

margin of estimation error is required. In most

population-based heuristics, as the search converges,

same solutions will appear in the population with

increasing frequencies. Therefore, computationally

expensive simulation would be used to evaluate

same solutions again and again in the case of the

RSA problem. To address these problems, a

hierarchical solution evaluation approach with

hashing is used in the CSA.

All solutions investigated during the search are

stored in a list, called solution list (SL). A hash table

(HT) is used as a pointer to quickly access the

solutions stored in SL. A second list, called collision

list (CL) is used to store solutions with a hash

collision. After a solution S is created, the hash

value of the solution is calculated as follows:

( ) mod( ( ) , )





deHS

(2)

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

158

where H is the hash size and e

is a prime number

corresponding to decision variable s

. Hash table HT

is an integer array such that HT[d(S)]=0 if a solution

with a hash value of d(S) has not been searched yet,

or HT[d(S)]=t if the first solution with a hash value

of d(S) is the t

solution in solution list SL. In other

words, SL[t] stores the t

evaluated solution without

any hash collision. After calculating d(S) for a

solution S, there are three cases possible.

Case 1: If HT[d(S)]=0, then S has not been

investigated before.

Case 2: If HT[d(S)]=t and S=SL[t], then S has been

investigated before.

Case 3: If HT[d(S)]=t and SSL[t], then a hash

collision occurs (i.e., two different solutions have

the same hash value). In this case, S is compared

with all solutions in collision list CL. If S is not in

CL, then it has not been searched before and added

to CL.

In the CSA, instead of a single best solution-found-

so-far, the best b solutions found-so-far during the

search are maintained in a sorted list called elitist list

(EL) such that CSR(EL[1])> CSR(EL[2])>…>

CSR(EL[b]). When a new solution is created, it is

compared to previously evaluated solutions using

the hashing technique used in (Kulturel-Konak and

Konak, 2009) as briefly described above. If the

solution has not been searched before, it is first is

evaluated using a low number of simulation

replications (K

). After the first evaluation, the

solution is identified as promising if the solution has

a better estimated objective function value than the

worst elitist solution in EL, and then it is rigorously

evaluated using a higher number of replications (K

The statistical error due to simulation can be

controlled by increasing the size of EL. After the

search is terminated, all elitist solutions are

evaluated one more time using a very high number

of simulation replications (K

). The solution

evaluation procedure of the CSA is given below.

Procedure Evaluate_Solution (S) {

If solution S has not been searched before {

Evaluate S using K

simulation replications

If (CSR(S)> the worst CSR in EL) {

Evaluate S using K

simulation replications

Update EL by including S if necessary

}}}

2.3 The Overall CSA Algorithm

In this section, the overall procedure of the CSA

algorithm is presented. Initially,



solutions are

randomly generated by a simple construction

heuristics as follows:

Procedure Create_Random_Solution (S

) {

Set A=V, C



=C, s

=0 for each node iV

While (A{}) {

Randomly and uniformly select node i from A

Set s

=1 and C



-c

Set A=A\ {i} }

Return solution S

}

The replacement strategy of the CSA is to replace

the worst



% of the population with randomly

generated new solutions using Procedure

Create_Random_Solution(), and it is applied once in

every five iterations. Infeasible solutions are not

evaluated and discarded because of the high

computational cost of the MC simulation. The

overall procedure of the CSA is as follows:

Procedure CSA {

Randomly generate



solutions

While (stopping criterion is not satisfied) {

Sort and rank the population

If (replacement=TRUE) {

Replace worst



% of the population with

randomly generated solutions }

For each top tier solution S

do {

Generate



/2 clones of S

using Mutation1(S

)

Evaluate the generated clones

Replace S

if a better solution is found }

For each middle tier solution S

do {

Generate



/3 clones of S

using Mutation2(S

)

Evaluate the generated clones

Replace S

if a better solution is found }

For each bottom tier solution S

do {

Generate



/4 clones of S

using Mutation3(S

)

Evaluate the clones

Replace S

if a better solution is found }

Simulate the elitist solutions using K

replications and return them.

}

3 COMPUTATIONAL

EXPERIMENTS

In (Konak and Kulturel-Konak, 2011), the CSA was

compared with the previous PSO and ACO

approaches with promising results, and it was

reported that the performance of the CSA was par

with the PSO in some cases and between the PSO

and the ACO in most cases. Hence, there is no clear

evidence to justify the CSA to solve the RSA

problem. In (Dasgupta and Nino, 2009), CSA is

recommended as an approach to multi-modal

A CLONAL SELECTION ALGORITHM TO INVESTIGATE DIVERSE SOLUTIONS FOR THE RELIABLE SERVER

ASSIGNMENT PROBLEM

159

optimization problems where there are multiple

optimal points in the solution space. To test this

claim, we designed test problems with multiple

optimal solutions and investigated whether the three

heuristics can discover these multiple optimal

solutions in a single run. Not that in simulation

optimization approaches such as the CSA in this

paper, it is particularly important to discover a set of

diverse solutions because of the noise in solution

evaluation.

All test problems have ring topologies with

identical node reliabilities and costs (0.99 and 1,

respectively), as well as identical edge reliabilities.

The test problems were solved for C=5 and



=0.5.

In other words, the test problems were defined as

how to locate five servers on a ring network. Since

all nodes have identical reliabilities and costs, and in

addition all edges have the same reliability, there are

many alternatives with the same optimal CSR. For

example, for n=30 two solutions where five servers

are located at nodes {6, 12, 18, 24, 30} and {7, 13,

19, 25, 1}, respectively, have the same CSR. To

minimize the sampling error, simulation parameters

were set as K

=10

, K

=2x10

, K

=10

, H=99001,

and |EL|=20. Each problem was solved for ten

random replications with the CSA parameters µ=50

and



=20%. The search was terminated after 8000

solutions were searched (the stopping criterion). The

worst 20% of the population was replaced with

randomly generated solutions once in every five

iterations. The ACO and PSO were run with the

same algorithm parameters given in (Konak and

Kulturel-Konak, 2011) with the stopping criteria of

8000 maximum solutions searched.

The Hamming distances between each pair of the

elitists solutions found by the ACO, PSO, and CSA

were calculated, and the average Hamming distance

(AHD) in the final EL was used as the indicator for

the diversity of the solutions found by each heuristic

as follows:

1,...,|EL| 1,...,

AHD

|EL| (|EL| 1)/2

iki

jkjin







 





(3)

Table 1 presents the averaged CSR of the best and

worst elitist solutions found in ten replications and

the AHD among the elitist solutions. For problems

n=30, 40, 50, 60, and 70, the three heuristics

performed equally well with respect to the best and

worst elitist solutions. However, the CSA provided

the highest average Hamming distance, particularly

compared to the PSO. For problems n=80 and 100,

the ACO did not perform as well as the other

heuristics. The ACO and the CSA have the same

average Hamming distance, but the CSA performs

better. In Table 1, the column titled Normalized

Average Hamming Distance (NAHD) gives the

AHD divided by the difference between the best and

worst elitist solution. It is clear that the CSA was

able to discover the most diverse sets of elitist

solutions while it performed as well as the PSO

based on the results in Table 1. For problems n=80

and n=100, both ACO and CSA have the same level

of diversity in the elitist list, but the CSA

outperformed the ACO in these problems in terms of

the objective function. These results are consistent

with the claim in the CSA literature that CSA is a

suitable heuristic for multi-modal optimization

problems.

Figure 1 illustrates three best solutions found by

the ACO and the CSA for a multi-modal problem

with n=30. As seen the figure, the solutions found by

the CSA are very different in terms of server

assignments. For example, the best two solutions,

{20, 11, 50, 40, 29} and {19, 10, 1, 40, 30} have

only one common nodes, which is node 40. On the

other hands, the best two solutions found by the

ACO, {19, 8, 48, 38, 28} and {19, 9, 48, 38, 28},

have four common nodes. This example

demonstrates the ability of the CSA to investigate

diverse solutions.

Table 1: Results of multi-modal problems.

ACO PSO CSA

(n, m) Best Worst AHD NAHD Best Worst AHD NAHD Best Worst AHD NAHD

(30,30) 0.602 0.592 0.204 21.77 0.602 0.593 0.186 19.40 0.603 0.595 0.265 34.97

(40,40) 0.440 0.431 0.165 18.75 0.440 0.433 0.134 19.28 0.441 0.433 0.202 28.72

(50,50) 0.662 0.652 0.151 15.20 0.663 0.656 0.112 17.21 0.663 0.656 0.168 26.18

(60,60) 0.925 0.921 0.135 34.78 0.926 0.923 0.098 33.83 0.926 0.923 0.143 50.09

(70,70) 0.907 0.902 0.122 22.65 0.908 0.904 0.079 23.65 0.908 0.904 0.124 34.28

(80,80) 0.876 0.870 0.109 17.26 0.877 0.872 0.075 17.02 0.877 0.872 0.108 26.73

(100,100) 0.819 0.811 0.088 10.56 0.821 0.815 0.059 11.14 0.821 0.816 0.088 19.17

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

160

Figure 1: Best three solutions found by (a) the ACO and

(b) the CSA for multi-model problems with n=30.

4 CONCLUSIONS

The reliable server assignment problem (RSA) in

networks is studied in this paper. A simulation

optimization approach is used based on a MC

simulation and embedded into a CSA to solve the

RSA problem. During the search, hashing method is

used to rapidly detect whether a solution has been

previously investigated or not to prevent costly

evaluation process of same solutions again and

again. The experimental study shows that the CSA

algorithm is capable of searching very diverse

solutions to the problem. This is an important

concern in the RSA problem because very different

solutions may have very similar system reliability,

and decision makers can choose the best alternative

by weighing in other factors.

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A CLONAL SELECTION ALGORITHM TO INVESTIGATE DIVERSE SOLUTIONS FOR THE RELIABLE SERVER

ASSIGNMENT PROBLEM

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