Applied Mathematical Modelling 37 (2013) 888–902

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Design of computer network topologies: A Vroom InspiredPsychoclonal Algorithm

Nagesh Shukla a,b,⇑, Yogesh Dashora c, M.K. Tiwari d, Ravi Shankar e

a The Digital Lab, WMG, University of Warwick, Coventry CV4 7AL, UKb SMART Infrastructure Facility, Wollongong 2522, Australiac Sabre Travel Technologies Private Limited, Bangalore 560066, Indiad Department Industrial Engineering & Management, Indian Institute of Technology, Kharagpur, Indiae Department of Management Studies, Indian Institute of Technology, New Delhi, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 September 2011Received in revised form 16 February 2012Accepted 12 March 2012Available online 20 March 2012

Keywords:Communication and NetworkingRandom search algorithmCombinatorial optimization

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.03.027

⇑ Corresponding author at: The Digital Lab, WMGE-mail address: nagesh.nifft@gmail.com (N. Shuk

In the prevailing era of network and communication technology, the problem pertaining tothe determination of the most economic way to interconnect nodes while satisfying somereliability and quality of service constraints has been agnized as one of the most intricateand challenging problem for the modern day researchers and practitioners belonging toCommunication and Networking community. Motivated by the improved performance ofthe concepts like proliferation, affinity maturation, receptor editing, etc., over the moreprevalent generalized crossover and mutation; and by the application and effectivenessof Maslow’s need hierarchy in combinatorial optimization as well the more logical motiva-tional concepts provided by Vroom’s valence expectancy theory, authors have proposedand investigated their applications to the topological design of distributed packet switchednetworks. The extensive computations over the problems of varying complexities anddimensions prove the superiority of the proposed methodology. It has been observed thatthe proposed Vroom Inspired Psychoclonal Algorithm (VIPA) outperforms the traditionalwell established random search algorithms (i.e. Genetic Algorithm, Simulated Annealingand Artificial Immune Systems) in the context of underlying problem; the performancebeing significantly improved as the problem complexity increases.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Conceptually, a network is considered as the interconnection of various nodes with the weightless or weighted edgesdepending upon the specific applications. More precisely, the basal components of a computer network include variousswitches, computers, terminals, etc., as nodes and the communication links like telephone lines, optical fiber cables, micro-wave channels, etc., as edges [1–3]. Often, the term ‘networks’ corresponds to the active networks that aim towards custom-izing communication processing by the utilization of a sound software framework for network applications [4,5].

The concept of computer networking evolved from telecommunications and terminal-computer communication, whichconsider the connection of remote terminals to a central computing facility as the main objective. With the increase incomputer–computer interconnection, communication started utilizing the computers themselves [6–8]. The developmentof Advanced Research Projects Agency Network (ARPANet) was a pioneer work that invigorated the research and develop-ment in the computer networking sector [9–13].

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, University of Warwick, Coventry CV4 7AL, UK.la).

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The most crucial problem often encountered in the area pertaining to the network design is to obtain an economic trade-off between cost and reliability. On the basis of information exchange, three types of networks, as shown in Fig. 1, exist- (1)Centralized (star) networks; (2) Decentralized (tree) networks; and, (3) Distributed (mesh) networks. In the Centralized net-works, a central hub connects all the nodes with each other and all the information (data) is first sent to centre that routesthem to their destination. Since, any two nodes are connected by a single route only, this type of network can be regarded asa parsimonious structure; on the contrary, this structure is marked by its susceptibility due to its sole dependence on thecentral node. The demerit concerned to the single central hub is partially ameliorated in a decentralized network, whichis characterized by the existence of the several hubs. Although, the destruction of a single central hub does not abscisethe communication of the whole network, yet it affects all the nodes linked to the damaged dysfunctional hub. Again, thisproblem can be taken care in the design of distributed networks which does not have any centralized hubs. In this type ofstructure, each node is connected to many of the neighboring nodes based on various topologies. The choice among variouspossible routes to transmit data enables the distributed networks to be robust and more preferable than the other types.

Gefla and Klienrock [2] stated that the line switching techniques proves to be inefficient in the bursty modes of terminalto terminal conversations. However, the advent of packet switching techniques has increased the efficiency of data commu-nication in the bursty environment. The design of distributed packet switched networks is generally concerned to the fol-lowing concepts [3].

Topological Configuration: The set of links connecting various nodes;Traffic: Number of packets exchanged between the pair of nodes per second;Capacity Assignment: Determination of maximum bits per second that can be transmitted through each link;Routing Scheme: The selection of best path among various alternatives available;Flow Control Procedures: Limiting the overwhelming information flow;Average Packet Delay: Mean time taken by a packet to travel from a source to destination node;Reliability: It is satisfied if the k-connectivity of the network is conformed [14].

In particular, the topological design problem has been considered as one of the most intricate problems of the computernetworking; a variety of heuristic approaches exist in the literature that deals with such problems [15,16]. Earlier, Davis andCoombs [17] have applied the genetic algorithm to obtain the link capacity of the network. A system called SIDRO was uti-lized by Pierre and Hoang [18] and artificial intelligence techniques were applied for the topological design of packetswitched networks. Many of the papers have proposed Simulated Annealing based approaches in the topological design[19,20]. Pierre and Legault [3] proposed a Genetic Algorithm to generate low-cost feasible computer network topologies.In a similar attempt, this paper approaches the underlying problem by addressing it from a recently introduced PsychoclonalAlgorithm [21,22] and proposes an improvement to it utilizing the expectancy of Vroom’s valence expectancy theory [23] asthreshold to move over various need levels.

The main motivation behind the application of Artificial Immune System based Technique in the topological design ofpacket switched distribution networks has been the enticing features extended by the incorporation of clonal selection inthe evolutionary principles [24]. Clonal selection principle allows the learning of patterns during the primary phases ofthe algorithm and subsequently the retrieval of prior knowledge during the secondary phases of the algorithm. The twoalluring features of clonal selection can be defined as hypermutation and clonal expansion. Hypermutation is regarded as

Fig. 1. (a) Centralized (star) network; (b) Decentralized (tree) network; (c) Distributed (mesh) network.

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the local search that conduces to fast maturation and thus, depicts the exploitation of search space; whereas, the clonalexpansion or proliferation expedites the growth of new high affinity B cells, thus ensuring the better exploration of thesearch space (detailed in Section 3).

The last two decades have witnessed exhaustive research on the immune system dynamics and its utilization in themathematics and computing applications like differential equation based models [25,26], cellular automata models [27],classifier systems [28], genetic algorithms [29] and pattern recognition [30], etc. More recently, various immunological algo-rithms have been developed and applied in various engineering problems. Some of the worth mentioning contributions canbe summarized as-simple Immune Algorithm (IA) [31], CLONALG algorithm [32], CLIGA [31,33], Artificial Immune System(AIS) [32], aiNet [34].

Moreover, recent research has been directed towards incorporating motivational aspects to the search algorithms[21,22,35]. Motivation causes goal-directed behavior to a candidate solution. This research proposes how a candidate solu-tion viz. antibody can be motivated to acquire different levels of need with threshold motivational force. The proposedVroom Inspired Psychoclonal algorithm (VIPA) proves to be more efficient than the existing strategies in the literature whenapplied over the problems of varying dimensions; whereas the computational experiments based on the parametric config-uration of the algorithm establish its robustness, scalability and adaptability. The rest of the paper is organized in the follow-ing sequence: Section 2 mathematically formulates the topological design problem; A generic overview Psycho ClonalAlgorithm and introduction of Vroom’s valence expectancy theory is given in Section 3; Section 4 describes the applicationof the proposed VIPA to the topological design problem; Computational Results over the problems of various sizes as well thediscussion of results is presented in Section 5, and finally, Section 6 concludes the paper.

2. Problem formulation

Of the two level hierarchical structure of the distribution network- (1) communication subnetwork; (2) subnetwork ofauxiliary devices- the backbone problem adverting to the first level is the focus of study in the present paper. The end-to-end transportation of information is attributed as the prime function for such structures. The problem undertaken in thispaper is to find near-optimal design of network topologies satisfying various capacity and reliability constraints in quicktime. This section aims to mathematically formulate various aspects of the aforementioned design problem.

The network consists of a set N = {1,2 . . .n} comprising of n nodes, characterized by the set of coordinate locationsZ = {(X1,Y1), (X2,Y2) . . . (Xn,Yn)}. The attributes like flow (fij) and capacity (Cij) are defined for each link (i,j) among the nodesi and j respectively, where, fij is the effective quantity of information transported by the link and Cij is the measure of max-imum amount of information that the link can transmit. Further, both fij and Cij are measured in bits per second or bps. Also,there are certain discrete capacity options maintained by set C = {c1,c2 . . .cm}, within which the capacities of links lies. Here,m denotes number of capacity options available [36]. The average number of packets sent from source i to destination j isdefined as traffic, sij. Generally, only the average traffic values are used for the design purpose, as used in the present paper,instead of the varying traffic depending upon the time, the day and the application of use.

Given the above mentioned variables and sets, a general topological design problem can be stated as the problem of con-figuring the topology along with flow and capacity assignment subject to the delay and reliability constraints, so that thetotal link cost can be minimized [37,38]. There are many variants of the underlying problem existing in the literature onthe basis of objectives like waiting time, cost, delay, etc. In this paper the topological design problem has been investigatedon the fronts of minimizing average delay and total link cost of the network; the solution quality has been evaluated on thebasis of the comprehensive objective function formulation. First the relation between performance measures, input param-eters, design variables and the imposed constraints has been analyzed, which is followed by the formulation of overall objec-tive formulation. The objectives related to the problem formulation are described as follows:

2.1. Average time delay

The average source-to-destination packet delay T is mathematically defined as [13].

T ¼P

i;j;i–j

sij

sPij; ð1Þ

where sij average packet rate flowing from source i to destination j; Pij average packet delay from i to j; s ¼P

i;jsij, is the totaltraffic.

By the application of queuing theory on the above model, several further formulae to calculate T have been proposed [13].One of the generally accepted formulations that is sufficiently accurate for most design purposes is given as

T ¼ 1sPl

i¼1

fi

Ci � fi; ð2Þ

where Ci capacity of link i; L number of links; fi average bit rate in link i; f = (f1, f2 . . . fl) is termed as routing policy.Due to its generality and adaptability, the present paper utilizes this equation as a part of the objective function.

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2.2. Total link cost

Let d(di, Ci) represents the cost of leasing capacity Ci to the link i, and let di represents the length of link i, then the total linkcost can be formulated as:

D ¼Pl

i¼1diðdi;CiÞ: ð3Þ

2.3. The topological design problem

The underlying topological design problem aims to achieve two minimization objectives related to average time delayand link cost. The overall objective function can thus, be formulated as:

O ¼ Minimize ðw1 � Tþw2 � DÞ ð4Þ

¼ Minimize w1 �1sPl

i¼1

fi

Ci � fiþw2 �

Pl

i¼1diðdi;CiÞ

� �; ð5Þ

where w1 and w2 are the relative weights assigned to the two objectives that are introduced in order to make the addition oftime and cost objectives justified. Their values depend on the user’s preferences and thus, sometimes can be considered astuning parameters. Various requirement, capacity, delay and connectivity constraints are defined as follows:

The multi-commodity flow f must satisfy the requirement matrix [sij].The flow at any link must be less than capacity, i.e.

f 6 C ðor fij 6 Cij 8 links i; jÞ: ð6Þ

The average time delay should be less than the maximum allowable time delay, i.e.

T ¼ 1sPl

i¼1

fi

Ci � fi6 Tmax; ð7Þ

where, Tmax = Maximum Allowable Time Delay. The reliability constraint is satisfied by assuring that the topology must con-form to the k-connectivity condition.

In the past, many attempts have been made to solve the variants of aforementioned topological design problem with thehelp of various artificial intelligence techniques. Many of them provide better results but are characterized by some demeritsrelated to the inefficient trade-off among exploration and exploitation of search space. In order to tap the enhanced capa-bilities of Artificial Immune System (AIS) in getting out of local optima and ensuring more thorough search, as well the moti-vational logics to move the search further, Vroom Inspired Psychoclonal Algorithm (VIPA) has been proposed to solve theunderlying problem. The next section is dedicated to detail the basics of AIS, the two motivational theories defined and theirincorporation with the theme and search of AIS.

3. Psychoclonal algorithm and Vroom’s valence expectancy theory

The Psychoclonal algorithm is a recently proposed and established search technique that gleans its ideas from theArtificial Immune System and Maslow’s Need Hierarchy Theory, as detailed below:

3.1. Artificial Immune System

Modern day researchers and practitioners have long been fascinated by complex mechanisms of life and body systems;the development of biologically inspired evolutionary algorithms like Genetic Algorithms (GAs), Artificial Neural Networks(ANNs), Ant Colony Optimization (ACO), Particle Swarm Optimization, etc., asserts the same. The knowledge pertaining toimmune systems has governed the recent developments of clonal selection based algorithms as novel computational intel-ligence paradigm [34].

In particular, AIS can be defined as an abstract or metamorphic computational system using ideas garnered from the the-ories and components of immunology. The major attributes of immune systems are clonal selection, immune memory, affin-ity maturation and receptor editing. Clonal selection explains the response of immune systems when a non-self antigenicpattern is recognized by antibodies. From the view of optimization problems, the constraints present in the problem envi-ronment represent non-self antigens and candidate solutions at any stage of algorithm are recognized as antibodies. Inversemutation and Hypermutation guides the algorithm towards better individuals (solutions), while the algorithm comes out oflocal optima with the help of receptor editing.

The immune system mainly aims to protect the human body from the attack of foreign (harmful) organisms. It is capableof recognizing and distinguishing any malfunctioning and disease causing elements/cells, which are known as antigens. Anti-gens are heterologous proteins, when introduced causes certain body cells to elaborate antibodies. Antigens are of two types,

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one being self and other non-self. Self-antigens belong to the body and are harmless, whereas non-self antigens are disease-causing elements.

Immune systems are characterized by receptor molecules that are present on the surface of immune cells and are capableof recognizing more or less unlimited range of antigenic patterns. Whenever an individual is attacked by an antigen, someantibodies are produced from the immune cells (B-lymphocyte) derived from its bone marrow. The function of antibodies,which are present on the surface of B-cell, is to recognize and tie to an antigen. The first and the foremost requirement for animmune system to activate is the recognition of antigens. Here, it is worth noticing that the recognition is guided by theworking of cell-receptor. Recognition can only be stimulated if the antigen with the affinity greater than threshold affinityis recognized by the cell-receptor. If a B-cell encounters a non-self antigen with sufficient affinity, it follows clonal selectionprocess, through which it proliferates and differentiates into memory and effector cells. The following sub-sections explainthe clonal selection principle, immune memory and affinity maturation.

3.1.1. The principle of clonal selectionThe response of the immune system, when a non-self antigenic pattern is recognized by a B-cell is governed by the Clone

selection principle [39]. As stated earlier, once an antigen is recognized by the immune cell receptors, it stimulates the B-cellto proliferate (divide) and mature into terminal (non-dividing) antibody secreting cells called plasma cells [40]. Proliferation,that is an asexual and amitotic process, is succeeded by the cell divisions (mitosis) and the generation of clones that are cop-ies of each other. The proliferation rate of a cell is directly proportional to its recognizing degree of the antigen. During repro-duction, the B-cells off springs undergo hypermutation process that together with the strong selective pressure, results inmore powerful B-cells with enhanced affinities of antigenic receptor. This process is known as maturation of the immuneresponse. The immune cells get their learning by raising the population size and affinity (i.e. the degree of the cell recogni-tion with the antigen) of the cells. There is also an other way of learning according to which the effectiveness of the immuneresponse to secondary encounters could be considerably enhanced by storing some high affinity B-cells from the first infec-tion (i.e. memory cells), so as to form a highly effective antibody population for subsequent encounters.

These memory cells are pre-eminent in future responses to the same antigenic pattern, or a similar one. This process ofclonal selection, B-cell proliferation, and affinity maturation is diagrammatically represented in Fig. 2. The following sub-sec-tion provides a description of elaborative terms such as immune memory and affinity maturation.

3.1.2. Conception of immune memoryAs described earlier, the presence of memory cells of first encounter/infection increases the effectiveness of immune sys-

tem to work against the encountered antigen in the sub sequent encounters. The Clonal selection theory proposes to keep asmall number of best individuals, instead of working with a large data set of candidate solutions as they would multiply withthe number of generations and would create a computationally infeasible situation to handle. Hence, in accordance with theclonal selection theory, a clone will be created for the time being and progeny with low affinity will be discarded therebyensuring the quality of solution produced in the subsequent generations as well as keeping the speed of response. Thus,

Mitosis

High affinity memory cells

Selection

Dividing

Plasma Cells

Antigens

Fig. 2. The principle and process of clonal selection.

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the effectiveness of immune system increases as the number of iterations/generations increase. This phenomenon has muchimportance in the optimization field where a continuous and steady improvement towards the optimum is desired.

3.1.3. Affinity Maturation TheoryThe phenomenon which ensures that the antibodies present in the immune memory have, on average, higher affinities

than those of the early primary response, is adverted as affinity maturation in the immune response system. In the literature,two mechanisms available for the affinity maturation are hypermutation and receptor editing [34]. Random changes areintroduced into the genes of antibody molecules as per antigen-antibody interaction. These random changes are mutationalevents (hypermutation) and cause structurally different cells. Intermittently, one such event or change will lead to an incre-ment in the affinity of the antibody. The hypermutation is carried out in two phases: inverse mutation and pair wise inter-change. Inverse mutation helps in diversifying the search space in the initial stages and the pair wise interchange mutationbecomes more dominant in the later phases providing the criteria for convergence. The inferior bodies are more prone tohypermutation and strive hard to become competitive with others. This phenomenon is referred as receptor editing throughwhich B-cells delete their self reactive receptors and build entirely new receptors. This process of receptor editing may causea receptor with a better or worse affinity. Hence, hypermutation helps in exploring local regions, while receptor editing helpsin escaping out of the local minima.

3.2. Motivational theories

The basic idea behind any algorithm is to produce effective and efficient solution and thus, overall aim is to improve per-formance level of the candidate solution. This performance can be formally defined as:

Performance ¼ Ability�Motivation

Therefore, the performance level would be high if both these crucial both the parts have higher values. In order to providea robust base for the search in order to overcome local halts and other disruptions the Psychoclonal algorithm [21] usesMaslow’s need hierarchy theory to move on from one need level to another.

3.2.1. Maslow’s Need Hierarchy TheoryMaslow [41] conceptualized the priority of needs and framed a need hierarchy under the Need Hierarchy Theory.Maslow classified the needs into five levels and arranged them in a priority hierarchy. An individual (a solution) always

tries to get on to the upper levels of the hierarchy by satisfying the consecutive levels. The interpretation of five needs infunctional optimization has been as follows

3.2.1.1. Physiological needs. In the case of function optimization, the generation of feasible solutions pertaining to the prob-lem environment guarantees the fulfilment of physiological need.

3.2.1.2. Safety needs. Each candidate solution faces the threats from the constraints imposed by the problem environment.Once physiological needs are satisfied, the satisfaction of safety needs is encountered; the exposure of candidate solutionsto the constraints and their evaluation.

3.2.1.3. Social needs. The fulfilment of these needs refers to the selection of candidate solutions through interaction betweenthem, thus, exhibiting social interdependence.

3.2.1.4. Growth needs. Having satisfied the three initial needs, the individual strives to produce progenies that preserve itskind. Thus, the candidate solutions multiply to enhance the search space. This phase is important to diversify the search pro-cedure and lead it towards optima.

3.2.1.5. Self-actualization needs. Self-actualization is the highest level of needs and is concerned with the realization of one’sfull potentials and striving to enhance them. In the present context, this need refers to the attaining of best solutions throughreiterative process till the prespecified performance criterion is not met.

Although, the Psychoclonal procedures defined above possess enough motivation level and have capability to providegood solutions, they fail Maslow’s need hierarchy theory helps in identifying the needs level for desired motivation but theremust be some parameters, which assist in determining that whether a candidate solution is capable enough to enter the nextneeds level in hierarchy or not. To overcome this shortcoming, the paper incorporates the features of Vroom’s valence expec-tancy theory in the proposed algorithm and subsequently proves its robustness and efficacy.

3.3. Vroom’s valence expectancy theoryThis theory proposes a mathematical model to obtain motivational force in terms of two parameters, namely valence and

expectancy. Algebraically it can be expressed as:

Motivational Force ¼ Valence� Expectancy ð8Þ

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Valence means strength of an individual’s preference to a particular outcome. Higher the value of the valence higher themotivational force. In optimization, valence is nothing but the value of objective function. Another factor in determining themotivational force for the candidate i is expectancy, which is the probability that a particular action will lead to the needlevel, and is given by:

Expectancy ¼ ObjectiveiPiObjectivei

: ð9Þ

It is utilized with a minimum threshold motivation required for each need level, without attaining that the individual isnot allowed to strive for the next need level. More specific implementation details can be obtained in the following section.

4. VIPA: an implementation perspective

Psychoclonal Algorithm is governed by some simple guidelines that are given in the following discussion. The concept ofimmune memory of AIS motivates it to uphold a specific memory set, whereas the inception of motivational theories providemore chances to get out of the local optima. Initially, the physiological needs of the solutions are fulfilled by the generation offeasible initial solutions abiding by all the constraints. Thereafter, sequentially all the needs are satisfied provided the min-imum motivational threshold is provided. The pseudo-code of the proposed VIPA is detailed in Fig. 3.

The following subsections are devoted to the implementation aspects of the algorithm.

4.1. Representation of candidate solutions

The candidate solutions are represented in a binary array solution of size P � n(n � 1)/2, where n is the number of nodesin the network and P denotes the population size (or number of candidate solutions used for each generation). Here, it is tobe noted that the size of solution array is equal to the total number of links possible in the network of n nodes. The entity ‘1’present at a bit position represents the presence of the link, whereas, ‘0’ denotes the absence of link. A model 3-connectedtopology of 6 nodes network along with the concerned solution array is shown in Fig. 4.

Fig. 4 represents a 3-node connected topology (each node is connected with three other nodes) and corresponding solu-tion array. In Fig. 4, the number of nodes is 6, therefore, maximum number of possible links for this topology is 15 (i.e.6 � (6 � 1)/2). These possible links are ordered and numbered from 1 to 15. Therefore, first value is associated with link 1which connects nodes 1 and 2, the second value is associated with link 2 (connecting 1 and 3), and so forth.

4.2. Generation of initial feasible random solutions

This refers to the first need level to be satisfied. A small heuristic procedure [3] is adopted to generate the initial feasiblerandom solutions. The pseudo code for the procedure is presented in Fig. 5.

Fig. 3. Pseudo code of the algorithm.

Fig. 4. A representation of 3-node connected topology of 6 nodes network.

Fig. 5. Pseudo code of the initialization heuristic.

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First, a single random k connected solution is generated, which is further perturbed in order to generate the whole pop-ulation of random feasible solutions. However, it may be noted that the first solution is not completely random and utilizesthe greedy approach based on Euclidean Distance between the nodes.

4.3. Affinity calculation

In this step, the affinity of the solutions present in the solution array is calculated with the help of the objective functionand the antigen antibody interaction. The antigen antibody interaction denotes the calculation of affinity and adding penaltyto it based on the extent of constraints violated by the solution. Here, penalty is incorporated in the objective function asfollows:

PA ¼ ðpþ 1Þ � ðaffinity i:e: objective function ðSection 2:3ÞÞ; ð10Þ

where PA defines Penalized Affinity and p is the penalty factor whose value depends upon the extent of violation of con-straints. In general, p is taken to be equal to the number of constraints violated. Generally, infinite penalty is not assignedto the solutions that are infeasible because they might lead good solutions by incorporating some small changes in them.Hence, though the infeasible solutions are penalized, they have got some probability to be evaluated.

4.4. Generation of clone pool

This refers to the social need level and the solutions cannot get through this level unless they possess the threshold moti-vation. In this stage, Ps = a � P highest affinity solutions are chosen probabilistically, where a is an algorithm parameter thatdetermines number of solutions to be cloned. The chosen solutions are one by one randomly selected and cloned. The num-ber of clones to be generated for each selected solution is obtained by the following formula:

Fig. 6. Steps of hypermutation performed.

896 N. Shukla et al. / Applied Mathematical Modelling 37 (2013) 888–902

NC ¼ OiPpcj¼1Oj

& ’�X2 �X2

" #þ ðPc � NCTÞ �X1 � ð1�X2Þh i

; ð11Þ

where

X1 ¼1 if NCT < Pc;

0 otherwise;

(ð12Þ

X2 ¼1 if OjPpc

j¼1Oj

� �< ðPc � NCTÞ;

0 otherwise;

8<: ð13Þ

where NCi denotes the number of clones to be generated for ith selected solution; Pc denotes the total number clones to begenerated; NCT gives the total number of clones generated and b:c operator gives the integer P number. Now, each of theselected solution (antibody) is cloned according the above equations.

4.5. Hypermutation

Hypermutation emulate the fulfilment of growth needs; and here also the necessary motivation defined by Vroom’sexpectancy has to be satisfied. In this process, each of the clones from clone pool is hyper mutated as per the steps givenby the pseudo code of the algorithm. The two types of hypermutation are schematically shown in Fig. 6.

After hypermutation, the affinity of new clones generated is once again calculated by the antigen-antibody interaction ofthe algorithm.

4.6. Affinity maturation

This is the last need to be satisfied and thus is most difficult to fulfil. Here the threshold level determines the final solutionquality to be attained. This step is marked by the replacement of the b � P low affinity solutions, by the b � P high affinityclones from the clone pool, where b is the replacement fraction. The above steps are reiterated till the termination criteria(specified threshold motivation) is fulfilled and the best antibody/topological string from the population is given as output.

Based on the above proposed VIPA, several computational experiments were conducted and the efficacy as well as therobustness of the proposed strategy has been established.

5. Numerical results: analysis and discussions

This section has been devoted to various exhaustive computational experiments performed in order to test the applica-bility of the proposed solution strategy to solving topological design problem. Also, the experiments are performed for thepurpose of parameter tuning and a set of parameters is found to give best results. Further, in order to establish the superi-ority of the VIPA based solution strategy, a comparative study of the proposed strategy has been performed with three wellestablished peer random optimization algorithms-Genetic Algorithm (GA), Simulated Annealing (SA) and Simple ArtificialImmune System (AIS) Approach.

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5.1. Parameter tuning

In order to start the experimentations, the first task to be done is to get a better range of algorithm parameters. Fine tun-ing of the parameters and investigating their correlation with the underlying problem is a matter of great concern. Thus, it isaxiomatic that, the values of parameters can be set only after considering in detail the interactions among the problemfeatures.

In the present case, the performance of AIS is mainly affected by the variations in six user defined control parameters

� Population of Antibodies to be maintained (P).� Fraction of Antibodies (solutions) to be selected for cloning (a).� Total number of clones to be generated (Pc).� Fraction of low affinity antibodies to be replaced from the population in each iteration (b).� Rate of hypermutation (g).� Vroom valence expectancy factor (n).

In order to tune the parameters, a sample problem set of eight nodes has been used. The coordinates of these eight nodalpoints have been given in Table 1. The different capacity options and their related costs used to calculate the value of objec-tive function have been provided in Table 2.

In the experiments, the average packet size was considered to be 1000 bits, with a uniform traffic of 5 packets/sec. Thedegree of connectivity has been set as 3. The value of w1 and w2 were set to 1 and 40 respectively. These values have beenintuitively taken for the experimental purpose. However, they depend upon the user defined priority given to the relativeobjectives of time delay minimization and link cost minimization. The experiments have been performed on a 1.8 GHz,Pentium 4 processor and the algorithm has been coded in C++. In order to ease the tedious task of parameter tuning, fourlevels of each of the parameter have been considered. The L25 (56) orthogonal array [42] has been used for this purpose. Table3 provides the four levels of experimental values for each of the above mentioned parameters.

Table 4 details the average value of objective function obtained for each experiment performed as per L25 (56) array. It isto be noted that these values are the values of objective averaged over 10 algorithm runs with number of generations set to

Table 1Coordinates of the nodes of the network.

Node X coordinate Y coordinate

1 205 2302 170 4203 580 1504 90 1305 40 2106 400 3707 435 1858 530 250

Table 2Capacity options and cost.

Capacity (kbps) Variable cost ($/month/km) Fixed cost ($/month)

9.6 3 1019.2 5 1256 10 15100 15 20200 25 25560 90 60

Table 3Range of parameters.

Parameter Level 1 Level 2 Level 3 Level 4 Level 5

P 0.5 � n n 1.5� 2 � n 3 � na 0.25 0.50 0.75 0.90 1.0Pc 0.5 � n n 1.5� n 2 � 3 � nb 0.1 0.2 0.3 0.4 0.5g 0.5 0.7 0.9 0.1 0.05n 0.15 0.30 0.45 0.60 0.75

Table 4Results obtained for L’16 array.

Exp. Number P a Pc b g ⁄⁄⁄ Objective (O)

1 1 1 1 1 1 1 40,5122 1 2 2 2 2 2 48,6153 1 3 3 3 3 3 36,9934 1 4 4 4 4 4 389935 1 5 5 5 5 5 374356 2 1 2 3 4 5 388377 2 2 3 4 5 1 371978 2 3 4 5 1 2 371829 2 4 5 1 2 3 37222

10 2 5 1 2 3 4 3812311 3 1 2 3 4 5 3732512 3 2 3 4 5 1 3921813 3 3 4 5 1 2 3831714 3 4 5 1 2 3 4001815 3 5 1 2 3 4 3956116 4 1 2 3 4 5 4532117 4 2 3 4 5 1 3956218 4 3 4 5 1 2 5122319 4 4 5 1 2 3 3845620 4 5 1 2 3 4 3845621 5 1 2 3 4 5 3798622 5 2 3 4 5 1 4015923 5 3 4 5 1 2 4356724 5 4 5 1 2 3 3988725 5 5 1 2 3 4 38103

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100 in each case. From the results obtained, it is evident that the algorithm performs better for the values of parameters re-lated to experiment set 6. Without the loss of generality, for further experiments detailed in later subsections the values ofalgorithm parameters concerned with experiment number 6 have been used.

5.2. Problem variations

This subsection aims to investigate the algorithm performance with the problems of varying complexity over the wellestablished random search algorithms, GA, SA and simple AIS. In the experiments, the parameters for the GA were set as:population size = n (number of nodes); crossover probability = 0.9; mutation probability = 0.15. In order to compare the per-formance of the proposed strategy over the established strategies from the literature [3], problems of six different sizes havebeen considered. The coordinates of the nodes have been generated at random with both x and y coordinates varying in therange of (50,500). The capacity options have been kept similar to those presented in Table 2 and other values have been keptsame as detailed in the previous subsection. In the experiments related to SA, the maximum temperature, minimum tem-perature and the reduction factor have been fixed to be 5000, 1, 0.9, respectively. In the experiments related to AIS, theparameters were taken as population = n, a = 0.75, Pc = 2n, b = 0.2, g = 0.7. Number of generations in all the cased was takento be 100. However, it should be noted that the above values have been taken in accordance with a number of instances ofperformance of these algorithms reported in the literature.

Table 5Results of cost vs problem size.

P no. 1 2 3 4 5 6

n 6 10 12 15 25 30O (I.S) 38956 53357 56236 101254 256987 473254O (SA) 32735 39223 41563 81259 215984 441562% I 15.9693 26.4895 26.09183 19.74737 15.95528 6.696615O (I.S) 38451 52545 58236 99850 254749 498125O (GA) 32563 39319 41129 82564 205697 432579% I 15.313 25.17081 29.3753 17.31694 19.25503 13.15854O (I.S.) 37548 51231 57894 100056 213679 457126O (AIS) 32413 38913 41156 81006 175468 389256% I 13.67583 24.04404 28.91146 19.03934 17.88243 14.84711O (I.S.) 13.67583 24.04404 28.91146 19.03934 17.88243 14.84711O (VIPA) 37654 45987 55468 99548 224156 399875% I 32413 38654 40987 80001 175490 354987

O (I.S) – Objective function value for initial solution; O (SA) – Objective function value obtained by SA.O (GA) – Objective function value obtained by GA; O (AIS) – Objective function value obtained by AIS.O (VIPA) - Objective function value obtained by VIPA; % I Improvement shown.

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Table 5 summarizes the results obtained in terms of objective function values for the initial solution obtained by the heu-ristic defined earlier, and for the 100th iteration. In addition, the data regarding the percentage improvement in the objectivevalues for the three algorithms has also been given. From the table, it can easily be concluded that the proposed VIPA per-forms much better than the three strategies viz. GA, SA and AIS; however, the performance being significantly improved asthe problem size increased.

A set of experiments has also been conducted for another set of problems. In these the connectivity of the network wasvaried from 3 to 5. For smaller problems (with number of nodes equal to 6–10), connectivity for the nodes was kept to be 3.However, for the problems pertaining to the cases of 12 and 15 nodes, the connectivity was changed to 4. For rest of the twoproblems, the connectivity was set to be 5.

Table 6 details the results obtained with all the four algorithms under consideration. In this case, in order to visualize theeffect over the time and cost separately, the results have been presented in terms of the value of time and cost objective.Also, the last few columns of the table provide the percentage improvement of results obtained from VIPA over GA, SAand AIS respectively. From the results, it can be concluded that the performance of VIPA based search is superior irrespectiveof the increased connectivity of the network. However, it should be noted that the performance of GA, SA and AIS suffered asthe degree of connectivity of the network was increased.

5.3. Computational complexity

In general, computational complexity refers to the time and space required by the algorithm to execute [43,44]. Earlier,DeCastro and VonZuben [45] argued that the main steps contributing to the complexity of the algorithm are-(1) determiningthe affinity of antibodies, (2) selecting and re-selecting highest affinity antibodies, and (3) hypermutating the population.They proposed the time complexity of the algorithm to be of the order of O(P + PcL), where L ¼ n� n�1

2 . In order to have acomparison of run time efficiency of the algorithm, a comparative study was performed that analyze the convergence trend.The convergence trend of the three algorithms has been studied over three problems (Problems 2, 4 and 6 from Table 5). Ithas been observed that VIPA converges faster than GA, SA and AIS. Fig. 7 portrays the relative convergence of the three algo-rithms. It is quite clear from the figure that the VIPA takes very less time in order to converge to better solutions as comparedto others. This trend shown by the VIPA can be attributed to the following reasons.

� Various levels of need and hypermutation accelerate the multiple local searches.� The certainty of convergence is due to immune memory, as novel feasible antibodies are constantly introduced to make a

broader exploration of search space and prevent saturation of the population with similar antibodies.� The superior exploration of VIPA can be attributed to its ability to produce a large volume of antibodies depending upon

the needs level.� Further, motivational force and receptor editing helps in escaping local optima.

Thus, the proposed VIPA emerges out to be more logically oriented, faster and robust solution strategy for the underlyingcomputer topology design problem.

Furthermore, in order to test the performance of the proposed approach with other algorithms in literature such as GA,SA, and AIS; their application on large problem size having 1000 nodes (distributed randomly over [10,1000] rectangular

Table 6Results obtained by varying connectivity.

P No. 1 2 3 4 5 6

n 6.0000 10.0000 12.0000 15.0000 25.0000 30.0000k 3.0000 3.0000 4.0000 4.0000 5.0000 5.0000T(S) 31.6500 17.8250 47.6500 57.3500 148.1250 146.6000D(S) 31469.0000 38510.0000 39657.0000 78965.0000 210059.0000 435698.0000T(G) 28.4500 18.7500 40.8500 72.7500 27.7000 172.0250D(G) 31425.0000 38569.0000 39495.0000 79654.0000 204589.0000 425698.0000T(A) 24.9250 11.3500 41.6500 50.4250 22.4750 98.3000D(A) 31416.0000 38459.0000 39490.0000 78989.0000 174569.0000 385324.0000T(V) 23.4940 11.1300 41.7500 50.3010 21.9980 96.3240D(V) 31421 38447 39487 78945 174056 385421% T(S) 25.77 37.56 12.38 12.29 85.15 34.29% T(G) 17.42 40.64 �2.20 30.86 20.58 44.01% T (A) 5.74 1.94 -0.24 0.25 2.12 2.01% D(S) 0.15 0.16 0.43 0.03 17.14 11.54% D(G) 0.01 0.32 0.02 0.89 14.92 9.46% D (A) �0.02 0.03 0.01 0.06 0.29 �0.03

T(S), T(G), T(A), T(V) = Average Time delay for SA, GA, AIS and VIPA.D(S), D(G), D(A), D(V) = Average Link cost for SA, GA, AIS, VIPA.% T(S), % T(G), % T(V) = % Improvement in time delay of VIPA over SA and GA and AIS% D(S), % D(G), % D(V) = % Improvement in link cost of VIPA over SA and GA, AIS.

0 50 1003.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4 x 10 4

No. of Function Evaluation

Fitn

ess

Fun

ctio

nSAGAAISVIP

0 200 400 6003.8

4

4.2

4.4

4.6

4.8

5

5.2x 105

No. of Function Evaluation

Fitn

ess

Fun

ctio

n

SAGAAISVIP

0 100 200 300 400 500 6003.8

4

4.2

4.4

4.6

4.8

5

5.2x 105

No. of Function Evaluation

Fitn

ess

Fun

ctio

n

SAGAAISVIP

Fig. 7. Comparative convergence of the algorithms for various problems (fitness function vs. no of function evaluations).

Table 7Results obtained large problem size of 1000 nodes.

Method SA GA AIS VIPA

T 32065.85 30586.95 24643.54 22484.64D 398750.80 � 103 367689.55 � 103 298279.30 � 103 282188.26 � 103

%T 29.87979424 26.48943422 8.760510868 0%D 14.18493455 11.73652733 4.491200022 0

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area) is tested. All the other parameters related to problem such as capacity options have been kept similar to those pre-sented in Table 2 and other values have been kept same as detailed in the previous subsections. The results are summarizedin Table 7. It illustrates that VIPA performs better even in larger problem instances and be used for large network planningand design.

6. Conclusive remarks and future scope

This paper proposes a Vroom Inspired Psychoclonal Algorithm (VIPA) based approach for the conventional topology de-sign problem in packet switched distributed networks. This approach is inspired from the theory of immunology and adoptsthe clonal selection process in order to investigate the search space more thoroughly; also it encapsulates the motivational

N. Shukla et al. / Applied Mathematical Modelling 37 (2013) 888–902 901

aspects of need theory and defines the threshold motivational level on the basis of Vroom valence expectancy theory. In ear-lier approaches, the topological design problem was attempted with various objectives like minimization of average timedelay, minimization of link cost, etc. This paper introduces an integrated objective function that combines both objectiveswith a user defined weighted average.

Initially, the parameter tuning was performed using the standard L25(56) array and the obtained best set of parametershave been used for further experiments. The computational results over randomly generated data set of varying complexityestablish the superiority and robustness of the algorithm. In the same vein, it has been found that the algorithm performanceis least affected by the changes in the degree of connectivity. The experiments have proved the outperforming behaviour ofVIPA over GA, SA and AIS. Thus, the proposed strategy seems to be conforming to the notion of beating the challenges in theareas pertaining to distributed network design.

Future scope of the research includes the conceptualization of a more generic framework to design distributed networks;clearly, the rigorous research to establish the applicability of VIPA over a range of combinatorial optimization problems isahead. In addition, further efforts should be directed to enhancing the use of Data Mining techniques to correlate the existingknowledge base and the prediction of imprecise knowledge.

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