digital data networks design using genetic...

Theory and Methodology

Digital data networks design using genetic algorithms

Chao-Hsien Chu 1, G. Premkumar *, Hsinghua Chou 2

Department of Logistics, Operations and Mangement Information Systems, College of Business, Iowa State University, 300 Carver Hall,

Ames, IA 50011-2063, USA

Received 14 October 1998; accepted 23 March 1999

Abstract

Communication networks have witnessed signi®cant growth in the last decade due to the dramatic growth in the use

of Internet. The reliability and service quality requirements of modern data communication networks and the large

investments in communications infrastructure have made it critical to design optimized networks that meet the per-

formance parameters. Digital Data Service (DDS) is a popular communication service that provides users with a digital

connection. The design of a DDS network is a special case of the classic Steiner-tree problem of ®nding the minimum

cost tree connecting a set of nodes, using Steiner nodes. Since it is a combinatorial optimization problem several

heuristic algorithms have been developed including Tabu search, and branch and cut algorithm. In this paper, a new

approach using genetic algorithms (GAs) is proposed to solve the problem. The results from GA are compared with the

Tabu search method. The results indicate that GA performs as well as Tabu search in terms of solution quality but has

lower computation time. However, reducing the number of iterations in Tabu search makes it faster than GA and

comparable in solution quality with GA. Ó 2000 Elsevier Science B.V. All rights reserved.

Keywords: Telecommunications; Genetic algorithms; Network design; Tabu search

1. Introduction

The use of data communication networks hasincreased signi®cantly in the last decade due to the

dramatic growth in the use of Internet for businessand personal use. As the society transforms itselfto an information society the network becomes theprimary source for information creation, storage,distribution, and retrieval. The design and devel-opment of a reliable network infrastructure tosupport the primary resource of an informationsociety becomes a very critical activity. The reli-ability and service quality requirements of moderndata communication networks and the large in-vestments in communications infrastructure havemade it critical to design optimized networks thatmeet the performance parameters. These factors

European Journal of Operational Research 127 (2000) 140±158www.elsevier.com/locate/dsw

* Corresponding author. Tel.: +1-515-294-1833; fax: +1-515-

294-2534.

E-mail address: [email protected] (G. Premkumar).1 Present address: School of Information Sciences and

Technology, Pennsylvania State University, University Park,

PA 16802, USA.2 Present address: Sprint Corporation, 3rd Floor, 10880

College Blvd., Overland Park, KS 66210, USA.

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 3 2 9 - X

have prompted researchers to develop new modelsand methodologies for network design. Recentresearch has focused on using meta-heuristics suchas Tabu search and evolutionary computationapproaches for network design problems. Thisstudy examines the use of genetic algorithms (GA),an evolutionary computation technique, for de-signing a digital data services network.

Digital Data Service (DDS) is a popular com-munication service that provides users with a dig-ital connection over leased lines. The availabilityof constant bandwidth and enhanced securitymakes this service very popular for organizationsthat need to have data networks connecting mul-tiple o�ces that are geographically dispersed. TheDDS network consists of three components ±hubs, end-o�ces, and customer locations. Thehubs are the primary nodes that form the back-bone infrastructure. The customers are connectedby leased lines to one end-o�ce and each end-of-®ce is connected to one hub or node, thereby cre-ating a star topology in the local access network. Aschematic of the network is shown in Fig. 1. Thecustomers are geographically dispersed over theentire service area, resulting in a backbone net-work infrastructure that spreads over a wide geo-graphic area. Depending on the demand indi�erent geographic areas some hubs may be ac-tive while others may be inactive. There are costsinvolved in setting up and operating the hub, thelinks connecting the hubs, the links from the end-o�ce to hub, and the links from the customer tothe end-o�ce. The network designer is primarilyinterested in designing a network infrastructurethat meets the customers' requirements at theminimum cost.

The practical network design problem discussedabove is a classic Steiner-tree problem of ®ndingthe minimum cost tree connecting a set of nodes,using Steiner nodes. This is a special case ofSteiner-tree problem, called a Steiner-tree star(STS) problem, since each target node (end-o�ces)is connected to only one active Steiner node in astar topology. The Steiner-tree problem is a classiccombinatorial optimization problem, where tradi-tional mathematical approaches, such as linearprogramming, are feasible only for small sizeproblems. The number of constraints increases

exponentially as the problem size increases, mak-ing it infeasible to use mathematical modelingtools for medium to large size problems. Hence,various heuristics are used to solve the problem.An excellent review of the research on Steiner-treeproblems is provided in Winter (1987). More re-cently, Hwang et al. (1992) provide a comprehen-sive list of algorithms used to solve variousversions of the Steiner-tree problem.

Although the traditional Steiner-tree problemhas been widely studied, the STS problem has onlybeen recently addressed by Lee et al. (1996). Intheir paper they proposed a heuristic procedure togenerate an upper bound and then use local im-provement procedures to improve the initial solu-tion. Xu et al. (1996) proposed a Tabu searchmethod to solve this design problem. Recentlythere has been an increased interest in using GA tosolve network design problems. Table 1 shows theuse of GA in various network design applications.Discussion of these applications is provided inSection 5. A cursory analysis of Table 1 indicatesthat studies have not examined the use of GA forsolving the STS problem.

The primary objective of this paper is to eval-uate the use of GA for designing STS networksand compare its performance with an alternativedesign approach, namely, the Tabu search heu-ristic. The paper is organized as follows ± Section 2provides a background on the network design;Section 3 describes the mathematical formulationof the problem; Section 4 discusses the Tabusearch heuristic; Section 5 describes the proposedgenetic algorithm; Section 6 describes the researchhypotheses and the experimental design; Section 7presents the results, and its implications; andSection 8 discusses the conclusions.

2. Network design

Network design problems can be classi®ed un-der four broad categories ± network topology de-sign, network routing and ¯ow control, networkperformance, and network reliability. Althoughthey are listed as separate categories for simplicityand similarity in design models, the problemsin these categories are highly interrelated. The

C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 141

network topology is clearly in¯uenced by the ¯owparameters as well as the requirements of networkreliability and levels of performance. The designproblem for this study falls under the networktopology category. Network topology design isprimarily concerned with design of backbone net-works that will satisfy certain performance, cost,and reliability parameters. Researchers in opera-tions research have examined this problem underthe broad category of `minimum cost ¯ow'problem.

It is typical to consider communication net-works as trees to design optimal solutions (Ahujaet al., 1993). Trees are particular types of graphswith speci®c properties. A tree (or subtree) of ageneral undirected graph G � �V ;E� with a node(or vertex) set V and edge set E is a connectedsubgraph T � �V 0;E0� containing no cycles. A

spanning tree is a tree that spans all the nodes ofan undirected network. T is a spanning tree (of G)if T spans all the nodes V of G, that is, V 0 � V . Agraph of n nodes is a spanning tree only if it isconnected and contains nÿ 1 edges. The mostcommon form of tree optimization is as follows:Given a graph G � �V ;E� and a weight de®ned foreach edge e 2 E, ®nd a tree T in G that optimizes(maximizes or minimizes) the total weight of theedges in T. There may be other constraints im-posed such as number of nodes in a sub tree, de-gree constraints on nodes, ¯ow and capacityconstraints on any edge or node, type of servicesavailable on nodes or edges, etc. The commoncategories of tree optimization problems are:· Minimal spanning tree problem ± Find a tree that

spans all the nodes of a graph G, and minimizesthe overall weight of the edges in the tree.

Fig. 1. Illustration of the DDS network.

142 C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158

· Rooted subtree problem ± Given a tree and a rootnode r, ®nd a subtree rooted at this node andthat minimizes the overall weight of the nodes(and/or arcs) in the subtree.

· Steiner-tree problem ± Find a tree that containsa set of terminal nodes that need to be connect-ed to each other whose total edge weight is min-imal. The optimal tree might contain nodes,called Steiner nodes, other than the terminalnodes T. When T � V it is the minimum span-ning tree problem.

· K-median problem ± Find K or fewer node dis-joint subtrees of a network, each with a root,that minimizes the total edge weight of the edgesin the subtrees.

· C-capacitated tree problem ± Find a rooted sub-tree with an additional constraint of each sub-tree being limited to C nodes.Minimum spanning tree (MST) and Steiner-tree

are two common models that are extensively usedin designing communication networks. In MST thedesign attempts to ®nd a minimum cost tree thatconnects all the nodes of the network. The links oredges have cost associated with them, which could

be based on their distance, capacity, quality of lineetc. Since the complexity of the problem increasessigni®cantly for mid-to-large size problems, manyheuristic algorithms, such as Kruskal (1956) andPrim (1957) algorithms are used to solve the MSTproblem. The Steiner Problem is one of the oldestoptimization problems in mathematics, tracing itsorigins to Jacob Steiner in early nineteenth cen-tury, who proposed to connect three villages A, B,and C using the shortest distance of roads by in-troducing a fourth point P. There are many net-work design problems, which are closely related tothe Steiner-tree problem. They are generalizedSteiner-tree problem (GSP) (Zelikovsky, 1993),Steiner-tree problem in Euclidean metrics (Smithet al., 1981; Winter and Zachariasen, 1997),Steiner-tree problem in probabilistic networks(Chopra and Rao, 1989), Steiner minimal treeproblem (Chang and Wu, 1997), and Steiner-treestar problem (Lee et al., 1996). A good summaryof work in this area appears in Winter (1987) andHwang et al. (1992).

Steiner-tree problem is a subset of MST. InMST we design a tree with ®xed vertices or nodes,

Table 1

GA applications in telecommunication networks design

Topics References

Topology and graph

General communication network Coombs and Davis (1987); Davis and Coombs (1989); Abuali et al. (1993);

Palmer and Kershenbaum (1995) and Charddaire et al. (1995)

Steiner tree problem Hesser et al. (1989); Kapsalis et al. (1993) and Esbensen (1995)

Probabilistic MST Abuali et al. (1995)

Concentrator location problem Oyman and Solvinf (1994)

Degree constrained MST Zhou and Gen (1996)

Network performance

Broadcasting Hoelting et al. (1996)

Routing and ¯ow control

Telecom. tra�c routing Liu et al. (1994) and Murgu and Lahdelma (1995)

Dynamic routing control Shimamoto et al. (1993) and Marin et al. (1994)

Routing table optimization Sinclair (1993)

Networks reliability

Fault-tolerant network Kumar and Babu (1994)

Others (Emerging technologies)

Frequency assignment problem UEA CALMA (1994)

Channel assignment problem Cuppinim (1994)

Optical WDM networks Tan and Pollard (1995)

Optical ®ber (Ring-loading) Karunanithi and Carpenter (1993)


A1;A2; . . . ;An, connected without loops at thelowest cost. In Steiner-tree problem we add extravertices besides the existing vertices Ai�1 to n toconstruct a lower cost tree connectingA1;A2; . . . ;An. The extra-introduced vertices arecalled Steiner points. For example, if we have threenodes that need to be connected we can ®nd afourth node P to create a minimum distance con-necting all three that is better than the MST for thethree vertices (sum of the two shortest edges). TheSteiner-tree problem is stated as follows: IfG � �V ;E� is an undirected graph with a ®nite setof vertices, V; and an edge set, E; and c : E ! R bea cost function assigned a positive real value foreach edge in G. Assume we have a set, S � V , ofspecial vertices, the Steiner minimal tree probleminvolves ®nding a subgraph G0 � �V 0;E0� of G suchthat: (1) V 0 contains all the vertices in S, (2) G0 isconnected, and (3) c: e 2 E0 is minimal.

The STS problem can be considered as a vari-ation of Steiner-tree problem. Lee et al. (1994)developed an integer programming formulationand used branch and bound approach to solvesmall size problems. They showed that the prob-lem is complex for medium to large size problems.Later, Lee et al. (1996) introduced a branch andcut heuristic to solve medium size problems. Theyused a heuristic procedure to generate an upperbound and then used local improvement procedureto improve the initial solution. However, this ap-proach is not promising for large size problems.

3. Mathematical formulation

The STS network consists of m target nodesthat are interconnected through n Steiner nodes;each Steiner node j that is used to connect to atleast one target node or other Steiner nodes iscalled an active Steiner node and will incur a ®xedcost bj; each target node i must be connected toexactly one active Steiner node, incurring a con-nection cost Cij; two distinct Steiner nodes j and kthat are directly connected incur a connection costdjk. The design problem is to ®nd the minimumcost tree that spans all target nodes through se-lected active Steiner nodes. The STS problem canbe formulated as an integer-programming model

(Xu et al., 1996). The following notation is used inthe formulation:

Indices:i: index of target nodes; i � 1; 2; . . . ;M ;j, k: index of Steiner nodes; j; k � 1; 2; . . . ;N ;M: set of target nodes;N: set of Steiner nodes.

Parameters:Cij: cost of connecting target node i to Steinernode j;Djk: cost of connecting Steiner nodes j and k;Bj: cost of activating Steiner node j;S: subset of N;W: node of set S.

Decision variables:Xij :� 1 if and only if target node i is linked toSteiner node j; otherwise Xij � 0;Yjk :� 1 if and only if Steiner node j is linkedto Steiner node k; otherwise Yjk � 0;Zj :� 1 if and only if Steiner node j is selectedto be active; otherwise Zj � 0.

The formulation is presented as follows:

MinimizeXi2M

Xj2N

CijXij �Xj2N

Xk>jk2N

DjkYjk �Xj2N

BjZj

�1�subject toXj2N

Xij � 1; i 2 M ; �2�

Xij6 Zj; i 2 M ; j 2 N ; �3�

Yjk 6 �Zj � Zk�=2; j < k; j; k 2 N ; �4�Xj2N

Xk>j

k2N

Yjk �Xj2N

Zj ÿ 1; �5�

Xj2S

Xk2Sk>j

Yjk 6X

j2fSÿwgZj; w 2 S; S � N ; jSjP 3;

�6�

Xij 2 f0; 1g; i 2 M ; j 2 N ; �7�


Yjk 2 f0; 1g; k > j; j; k 2 N ; �8�

Zj 2 f0; 1g; j 2 N : �9�In this formulation, the objective function (1) is tominimize the total cost, which includes the con-nection cost between a target node and a Steinernode, the connection cost between Steiner nodes,and the ®xed cost to activate Steiner nodes. Con-straint set (2) indicates that each target node isconnected to exactly one Steiner node. Constraintset (3) indicates that the Steiner node must be ac-tivated before target node can be connected.Constraint set (4) indicates that two Steiner nodesmust be activated before they can be connected toeach other. Constraint set (5) expresses the con-straints of a spanning tree that the total connec-tions among Steiner nodes must be equal to thenumber of activated Steiner nodes less one. Con-straint set (6) is an anti-cycle constraint that en-sures that the backbone network solution is aspanning tree without any cycles. Constraints (7)±(9) indicate that the three decision variables arebinary.

In the formulation, constraint set (2) containsM constraints. Constraint set (3) generates M � Nconstraints. Constraint set (4) provides�N 2 ÿ N�=2 constraints. Constraint sets (5) and (6)sum up to N�2Nÿ1 ÿ N� � 1 constraints. The totalconstraints are N�2Nÿ1 �M ÿ �N ÿ 1�=2�� M � 1.Eqs. (7)±(9) contain 1=2�N 2 � N� �M � N � Ndecision variables. Although, a number of studieshave attempted to solve the problem, the compu-tational complexity makes it di�cult to solve largesize problems. Therefore, researchers have beendesigning e�cient heuristics to solve large-scaleproblems.

4. Tabu search

Tabu search is a meta-heuristic approach thatderives its name from the use of a `Tabu' list, a listof solutions that is avoided in subsequent itera-tions to overcome `local optimality entrapment' inoptimization problems (Glover and Laguna,1997). It has been successfully used in manycombinatorial optimization problems (Hansen,

1986; Glover and Laguna, 1993; Hertz et al., 1996;Skorin-Kapov, 1990). Glover (1989) provides agood summary of the research in the Tabu searcharea. Recently, it has been used in a wide variety ofnetwork design problems such as capacitatedminimum spanning tree (Sharaiha et al., 1997), callrouting (Anderson et al., 1993), hub location(Skorin-Kapov and Skorin-Kapov, 1994), networktopology design (Glover et al., 1991), and band-width packing in telecommunication networks(Laguna and Glover, 1993).

Tabu search is a meta-level search heuristic thatguides search methods to overcome local opti-mality. The search moves in each iteration from asolution to its best admissible neighbor, even if thiscauses the objective function to deteriorate, toavoid the trap of local optima. To avoid cycling,solutions that are recently explored are consideredTabu (or forbidden) for a certain number of iter-ations. The Tabu list is stored in short-termmemory. The solutions are in the Tabu list for acertain period of time (Tabu tenure) and are thenremoved from the list so that they may be con-sidered for future iterations. The algorithm startswith an initial feasible solution. In a backbonenetwork design problem, a `greedy' algorithm suchas Prim's algorithm is used to develop an initialminimal spanning tree solution. Then the neigh-borhood space is searched for a series of ex-change. Each exchange is checked with the Tabulist and it is allowed if the node is not on the list.Intensi®cation and diversi®cation strategies areused to improve the search. In intensi®cation,the search is increased in promising regions of thefeasible solution. In diversi®cation, the search isbroadened to consider the broad area of the so-lution space. Long-term memory functions areused along with short-term memory to facilitatethese strategies.

The Tabu search heuristic used in this paper isbased on the paper by Xu et al. (1996). In thisheuristic the Steiner nodes are divided into twodisjoint subsets of active node (A) and inactivenode ( ). There are two elementary moves in thesearch space ± constructive and destructive moves.Constructive move transfers a node from to Aand adds a new node to the initial spanning tree.Destructive move moves a node from A to and


removes an active node from the spanning tree.The moves can be individual or pair-wise ex-change. After the moves the new solution is eval-uated for the total network cost. The nodes in Aare used to create the backbone minimum span-ning tree network using Prim's algorithm. Thenevery target node (end-o�ces) is connected to anactive Steiner node that has the least connectioncost. Then the total network cost is computed asthe sum of the backbone network cost and theconnection cost of every target node to the back-bone node. The list in short term memory keepstrack of recent moves and ensures that for a cer-tain number of iterations (Tabu tenure) a move isnot reversed; i.e. if a node is moved from A to itis not reversed from to A. Similarly reverseswap moves are also restricted. This restriction canbe overridden by an aspiration criterion when thecandidate solution is considerably better than theexisting solution. The long-term memory uses afrequency based memory structure to achieve di-versi®cation by encouraging exploration in regionsless frequently visited.

5. Genetic algorithms

Genetic algorithm, introduced by Holland(1975), refers to a class of adaptive search proce-dures based on principles derived from naturalevolution and genetics. In recent years there hasbeen increasing interest in the use of GA forsolving telecommunication network design prob-lems. Table 1 provides a list of network designapplications where GA has been employed. Palmerand Kershenbaum (1995) compared GA to tradi-tional heuristics to solve the optimal communica-tion spanning tree problem and found that thesolutions generated by GA were equal or betterthan the heuristics. Zhou and Gen (1996) used GAto solve degree constrained spanning tree prob-lems. Kapsalis et al. (1993) developed a GA ap-proach to solve Steiner minimal tree problem ingraphs, and compared the results of GA with fourother heuristics to solve a set of problem in the ORlibrary (Beasley, 1990). GA found the optimalsolution in all the 18 problems. Esbensen (1995)used GA to solve the Steiner-tree problem in a

graph and found that GA-based solutions were 1%from a global optimum in 93% of the time andcompared favorably with other approaches interms of computation time. The ®ndings fromthese studies indicate that GA is an attractive ap-proach to solve network design problems and canbe expected to provide near-optimal solutionswithin reasonable computation time.

Fig. 2 illustrates the GA implementation for theSTS problem. The key components of GA areproblem representation, population initialization,selection, evaluation using a ®tness function, andreproduction using crossover and mutation.

5.1. Problem representation

The problem is represented using a chromo-some, which is a ®nite length string that representsimportant characteristics of the problem. Encod-ing the problem in a string is a very critical issue inGA implementation. Most GA studies use binary

Fig. 2. Procedure of genetic algorithms for Steiner-tree-star

problem.


strings for encoding due to its simplicity in im-plementation of crossover and mutation opera-tors. However, integer and other coding schemeshave been used to solve more complicated prob-lems (Goldberg, 1989). Binary strings have beenpredominantly used for problem representation inSteiner-tree problems. For example, researchershave used it for Steiner minimum tree (Hesseret al., 1989) and Steiner problem in graph(Esbensen, 1995).

A careful analysis of the traits of the STS prob-lem reveals that a key determinant to solving theSTS problem lies in determining the hub locations.Since the end-o�ces have ®xed locations they neednot be included in the encoding. Fig. 3 shows ourbinary scheme, where each gene represents a Steinernode (hub). Each gene takes a value of either 0 or 1,where 1 indicates that it is an active hub and 0 in-dicates that it is inactive. The length of the encodingstring is equal to the number of hubs. This is a node-based direct encoding strategy.

5.2. Population initialization

Random initialization strategy is often adoptedto generate the initial population. In our case, a

random number generator is used to generate a 0or 1 and then assigned to each gene until all thehubs have a number.

5.3. Evaluation ± ®tness function

The objective of the STS problem is to connectthe hubs to the target nodes at the minimum cost.Since the coding strings contain the active/inactivestatus of the hub, the problem can be transformedinto a minimal spanning tree problem (MST),where all the active nodes in the encoding need tobe connected at the minimum cost. Hence, ``Prim''algorithm is used to obtain an MST solution.

The Prim's MST algorithm can be illustrated asfollows:

The backbone network cost is calculated by ag-gregating the connection, bridging and ®xed nodecost. The local access network design ®nds theleast cost connection from each target node (end-o�ce) to a backbone node. The total cost of thebackbone and local access network is computedfor each chromosome. Then the ®tness functioncompares the cost of each chromosome with thehighest value in the population and assigns thedi�erence as the ®tness value for the chromosome.Chromosomes with higher ®tness values are re-tained for the next generation.

5.4. Selection

There are various methods to select the intialpopulation for evolution in each generation(Goldberg, 1989; Michalawecz, 1992). Two possible

S0 G is the given non-trivial n-vertexweighted connected graph

S1 Set i � 1 and E � ;. Select any vertex, sayv, of G and set V1 � fvg

S2 Select an edge ei � �p; q� of minimumweight such that ei has exactly one endvertex, say pi. De®ne Vi�1 � Vi [ fqg,Ei � Eiÿ1 [ ei, and Ti � Eiÿ1 [ ei

S3 If i < nÿ 1, set i � i� 1 and return to S2.Otherwise, let Tmin � Tnÿ1 and halt

Fig. 3. GA representation scheme for Steiner-tree-star prob-

lem.


approaches to selecting the mating pool are to useonly the o�springs, or a mixture of parent ando�springs and choose the ``®ttest'' from the en-larged pool. Researchers prefer the latter since itreduces possibility of duplicate chromosomes(Back and Ho�meister, 1991; Back et al., 1997;Gen and Cheng, 1997). There are two methods inthe enlarged pool strategy. We could mix them andchoose the required number for the new popula-tion �l� k�, or choose a ®xed number from o�-springs and the remaining from the parentpopulation �l; k� to create the new population. Inour study, we chose the �l� k� method with sto-chastic selection. In this method, the child andparent populations are mixed, the chromosomesare sorted based on their ®tness values and pickedfrom the top till the mating pool is full. After-wards, a stochastic algorithm rearranges theranking of selected chromosomes. Generally, en-larged sampling space selection methods do notduplicate chromosomes based on their ®tnessprobability but rather gather both parent and childchromosomes thereby preventing premature clo-sure. For a higher exploration probability, a rela-tively high crossover and mutation rate is used inevolution.

5.5. Stop criterion

The completion decision can be based on threecriteria ± number of total generations, executiontime, and ®tness convergence. In this study we use®tness convergence as our criterion. It stops theevolution process when all the chromosomes in thepopulation have the same ®tness value.

5.6. Reproduction operator ± crossover

Two reproduction operators, mutation andcrossover, are used to generate o�springs from theparent population. In crossover, the genes of theparent are interchanged in some fashion to createtwo new o�springs with di�erent characteristics.The popular variations of the general crossoveroperator are one-point, two-point, and uniformcrossover. In one-point crossover, a random point

is identi®ed in the two parent chromosomes, thestring is cut at that point, and the ends are inter-changed between the two chromosomes to gener-ate two new o�springs. In two-point crossover, tworandom positions, head and tail are generated, andthe genes of the ®rst chromosomes from the headposition to the tail are exchanged with the secondchromosomes in the same range. Uniform cross-over is a dynamic and less deterministic methodsince the algorithm does not decide how many orwhat positions to replace. It starts from randomlygenerating a set of positions called mask within thelength of the chromosome. Then a pair of chro-mosomes exchange their genes between each otherbased on the generated positions. There are tworandom decisions in the algorithm, the positions toreplace and the number of genes to replace. Sinceall the random positions may not be neighbors, thealgorithm may replace genes in non-continuouspositions. Since the binary values in the genesrepresent active or inactive nodes, crossover mayincrease or decrease the number of active nodes.One-point and two-point crossovers replace genesin a set thereby resulting in a continuous range ofhubs becoming activated or deactivated. Uniformcrossover will cause hubs in a given set of randompositions to be activated or deactivated. Based oninitial experiments, uniform crossover was foundto provide better performance compared to theother two in this context. Hence, uniform cross-over was used in our study.

5.7. Reproduction operator ± mutation

Mutation, unlike crossover, occurs within thechromosome rather than across a pair of chro-mosomes. The simplest form of mutation is ran-dom mutation, which ¯ips one of the bits in arandom fashion with a certain probability. In re-cent years many new mutation methods have beendeveloped for solving complex problems. Twopopular mutation methods suitable for binarycoded chromosomes are insert and exchange mu-tation. Insert mutation randomly generates twopositions in a given chromosome. The algorithminserts the gene from the ®rst position in the sec-ond position and shifts all the genes on the right by


one position. This algorithm changes a set of hubsfrom active to inactive and vice versa, but does notchange the number of active and inactive nodes.Exchange mutation randomly selects two positionsin a given chromosome and exchanges both genes.The remaining genes are kept intact. The totalnumber of active and inactive hubs remains thesame. Unlike insert mutation, exchange mutationwill impact only the randomly selected hubs andall the remaining hubs are kept intact. Based oninitial experiments, exchange mutation was foundto provide better performance. Hence, exchangemutation was used in this study.

5.8. Problem illustration in GA

In this section, we use a 5-hub and 5 end-o�ceexample to illustrate our proposed GA approach.Table 2(a) and (b) are distance tables that containthe distance information of hub to hub and hub toend-o�ces. Steiner nodes are represented as S andend-o�ces are represented as T. Table 2(c) is acustomer allocation table that shows the numberof customers assigned to each end-o�ce. Table2(d) is a cost table based on the latest FederalCommunications Commission (FCC) regulationaccording to which cost is sensitive to the distance

Table 2

Numerical illustration

(a) Distance between hubs S1 S2 S3 S4 S5

S1 0 28 65 28 8

S2 28 0 73 56 31

S3 65 73 0 70 56

S4 28 56 70 0 28

S5 8 31 56 28 0

(b) Distance from targets to hubs T1 T2 T3 T4 T5

S1 35 64 29 40 33

S2 60 87 12 49 31

S3 48 116 61 24 42

S4 23 45 56 51 54

S5 29 68 28 32 27

(c) Customer allocation of each target

End-o�ce Allocated customer

T1 2

T2 1

T3 2

T4 1

T5 3

(d) Fixed and variable cost level

Cost category Cost/dollars

Fixed bridging cost 82.0

Bridging cost per line 41.0

Line connecting cost Mileage Fixed cost Variable cost

<1 30.0 0.0

<16 125.0 1.2

<46 130.0 1.5

<100 150.0 2.0

<150 180.0 2.5


and the variable cost is scaled down with increas-ing distance (Xu et al., 1996).

For each active hub, ®xed cost is interpreted asbridging cost. The line connecting cost is chargedfor each incoming and out-coming line. Fig. 4shows the initial solution when the nodes and end-o�ces are not connected.

We use two randomly generated chromosomesC(I) and C(II) to illustrate GA procedure.

C�I� �1 0 0 1 0� C�II� �0 1 1 1 0�

Two randomly selected positions 1 and 4 composethe crossover mask for uniform crossover. The

Fig. 4. Illustration of the layout among end o�ces and hubs.

Fig. 5. Illustration of uniform crossover.

Fig. 6. Illustration of exchange mutation.

Fig. 7. Illustration of DDS layout decoded from C00(I).


masked genes with the same ®xed position in a pairof chromosomes would exchange with each other.Fig. 5 (after uniform crossover) shows that in thenew chromosomes C0(I) the number of active hubsis reduced from 2 to 1 and in C0(II) the number ofactive hubs is increased from 3 to 4.

Mutation is performed on the chromosomesafter crossover. Exchange points for C0(I) are 1and 3 and for C0(II) are 2 and 5. Fig. 6 shows theexchange mutation. The move for C0(I) is to ex-change the same two genes and become C00(I),which, in this case, is same as C0(I). For C0(II)mutation activates the last hub and deactivates the

second one. The physical networks created by thenew chromosomes are shown in Figs. 7 and 8 forC00(I) and C00(II), respectively.

The new chromosomes formed by crossoverand mutation are evaluated for total network cost.To evaluate the cost of C00(I) and C00(II), the costtable in Table 2(d) is used. Table 3 shows the costcalculations. Although C00(II) uses four hubs toreduce the connection costs between hubs and end-o�ces, the higher hub cost o�sets the savings inconnection cost.

6. Experimental design

Since the objective of the study is to comparethe performance of Tabu search and GA we usetwo performance measures ± value of the objectivefunction and computation time. These two mea-sures have been extensively used in many studies(Gen and Cheng, 1997). We formulate two hy-potheses to statistically test the di�erence betweenthe two methods on these two performance mea-sures.

Hypothesis Ha: There is no di�erence in solutionquality performance between Tabu and GA.

Hypothesis Hb: There is no di�erence in compu-tation time between Tabu and GA.

The e�ectiveness of GA and Tabu search insolving the STS problem was evaluated using anexperiment. Both the methods generated solutionsFig. 8. Illustration of DDS layout decoded from C00(II).

Table 3

Numerical illustration: cost evaluation

(a) Cost evaluation of C"(I)

Bridging cost ± ®xed 82:0� 1 � 82:0Bridging cost ± variable 41:0� �2� 1� 2� 1� 3� � 369:0

Connecting cost ± ®xed 130:0� 3� 150:0� 6 � 1290:0

Connecting cost ± variable 1:5� �23� 2� 45� � 2:0� �56� 2� 51� 54� 3� � 786:5

Total cost 82:0� 368:0� 1290:0� 786:5 � 2526:5

(b) Cost evaluation of C"(II)

Bridging cost ± ®xed 82:0� 4 � 328:0

Bridging cost ± variable 41� �6� 2� 1� 2� 1� 3� � 615:0

Connecting cost ± ®xed 125� 10� 130� 150 � 1575

Connecting cost ± variable 1:2� 8� 2� 56� 1:5� �28� 23� 2� 45� 28� 2� 24� 27� 3� � 541:6

Total cost 328:0� 615:0� 1575� 541:6 � 3059:6


for a series of experimental network problems andthe results in terms of cost and computation timewere compared to test the two hypotheses. Thedata set for the experiment was generated byrandomly picking the location of target andSteiner nodes from a Euclidean space �0; 500�. Theset-up cost for each Steiner node is randomlygenerated from the interval �10; 1000�.

The experiments were performed on a DigitalAlpha Station (255/300 EV4.5) 300 MHz serverwith 64 Mbytes memory. The data set used forthe experiment are shown in the ®rst column ofTable 4. The data sets ranged from small sizeproblems (10 ´ 10 nodes) to large size problems(300 ´ 300 nodes). The same data set was used for

GA and Tabu methods. A iteration limit of30,000 was set for Tabu search, as suggested byXu et al. (1996).

7. Results

The results of the experiment are shown inTables 4 and 5. The solution quality (cost) andcomputation time (CPU-seconds) are shown forboth the approaches for each data set.

The values in Table 4 indicate that in terms ofsolution quality both the methods seem to obtainvery similar values most of the time. While Tabusearch has better solution in three data sets, GA

Table 4

The comparison of GA and Tabu search (iteration� 30,000)

Data set (M ´ N)a Genetic algorithms Tabu search

Cost CPU (s) Cost CPU (s)

10 ´ 10 1752 0 1752 3

20 ´ 20 4300 0 4300 8

30 ´ 30 4899 1 4899 12

40 ´ 40 5943 3 5943 32

50 ´ 50 7391 3 7391 35

60 ´ 60 7840 7 7840 38

70 ´ 70 7940 13 7940 42

80 ´ 80 10,422 19 10,422 51

90 ´ 90 11,354 23 11,354 70

100 ´ 100 16,166 40 16,166 180

150 ´ 100 19,359 134 19,359 360

200 ´ 100 22,948b 243 25,102 600

125 ´ 125 16,307 180 16,307 540

175 ´ 125 21,046 300 21,046 900

225 ´ 125 26,223 361 26,213b 1380

150 ´ 150 19,329 360 19,329 1320

200 ´ 150 24,358 377 24,358 1920

250 ´ 150 28,248 772 28,248 2880

175 ´ 175 20,907 760 20,907 2700

225 ´ 175 25,003 903 25,003 4020

275 ´ 175 27,672 835 27,672 4440

200 ´ 200 22,892 1189 22,876b 3900

250 ´ 200 26,122 1179 26,122 5520

300 ´ 200 29,879 1557 29,879 7440

250 ´ 250 25,566b 2330 25,573 6940

300 ´ 250 29,310 3120 29,310 8520

350 ´ 250 32,290 2580 32,290 6900

100 ´ 300 13,120 1654 13,120 2510

200 ´ 300 21,238 3060 21,238 3600

300 ´ 300 28,732 4500 28,728b 6120

a N: Steiner nodes (hubs), M: Target nodes (end o�ces).b The method performs better than the other.


has better solution in two data sets. However, thedi�erences in values are minimal. The results ofpaired t-tests, shown in Table 5(a), indicate thatthe di�erence in solution quality between Tabusearch and GA is not statistically signi®cant.Hence, hypothesis Ha is supported.

In terms of computation time GA performsbetter than Tabu search in almost all the data sets.The results of paired t-tests, shown in Table 5(b),indicate that the di�erences are signi®cant at 0.01level. Hence, hypothesis Hb is not supported.

The results indicate that the two methods gen-erate solutions that are comparable. Prior researchindicates that Tabu and GA often generate solu-tions that are optimal or very near optimal (Xuet al., 1996; Kapsalis et al., 1993; Esbensen, 1995).Hence, signi®cant improvement in solution qualityis not feasible. Future research could examinethe e�ect of much larger node sizes on solutionquality.

The results indicate that there is signi®cantdi�erence in computation time between the twomethods. The GA approach consistently providedfaster solutions than the Tabu approach. Theseresults are consistent with prior research thatfound basic Tabu search to be not very e�cient incomputation time. For instance, Xu et al. (1996)found that computation time for basic Tabusearch was signi®cantly higher than other heuristic

methods. However, they found that computationtime could be reduced by integrating additionalheuristics in the Tabu search that improves itssolution search capability. They used techniquessuch as short-term memory with probabilisticmove selection to reduce the computation time.

There could be various reasons for the di�er-ences in computation time between the two algo-rithms. The ®rst reason could be due to the stopcriterion used for Tabu search. It was based onmaximum number of iterations, which causes thealgorithm to continue even if there is no appre-ciable improvement in the objective function val-ue. In GA we use ®tness convergence strategy forstop criterion which results in stopping the algo-rithm once it reaches a convergence value. Thesecond reason could be due to the basic charac-teristic of the algorithm. Tabu search starts from asingle solution and uses a Tabu list to preventsolutions from cycling in local optima. This could,however, lead to multiple wasted iteration beforemoving to a better solution. For example, if amove that could result in a better solution is in theTabu list, it could not be used until the Tabutenure is completed. Hence, it may cycle throughother iterations without appreciable improvementin the objective function. In GA the algorithmstarts from a set of solutions. The entire solutionspace is examined in the evaluation phase therebyreducing/eliminating iterations that do not resultin improvement of the objective function value.The third reason could be the computational logic,which is simpler in GA. Tabu search starts with alower bound solution obtained from a heuristicand if the results obtained from Tabu is not betterthan the lower bound then the search is continuedwhich results in signi®cant search time. In GA weuse general crossover and mutation operators andonly use the Prim algorithm in the evaluationfunction to generate the MST and calculate cost.The fourth reason is the overhead in computationdue to maintenance of short term and long termmemory in Tabu search. The Tabu list and longterm memory list can get quite huge for large sizenetwork problems and have to be constantlysorted and maintained for every iteration. This isevident from Table 4 where computation timesincreases rapidly for large size problems.

Table 5

Paired t-test ± Tabu search (iteration� 30,000) and GA

GA Tabu

(a) Solution quality

Mean 18,618.53 18689.56

Variance 81,149,963.09 81,938,501.84

Observations 30.00 30.00

Df 29.00

t-Statistics )0.98

Signi®cant 0.33

(b) Computation

time

Mean 883.46 2429.33

Variance 1,354,209.22 7,223,611.13


Df 29.00

t-Statistics )4.62

Signi®cant 0.000


To examine the impact of stop criterion (# ofiterations) on the performance of Tabu search al-gorithm, we reduced the number of iterations from30,000 to 5000. We expected the reduction innumber of iterations to improve computationtime, but at the cost of reduced solution quality.The results of Tabu search with the new stop cri-terion are shown in Table 6.

The results indicate that the solution qualityhas only marginally decreased with ®ve solutionsthat are worse than GA. Paired t-test, shown inTable 7(a), indicate that the di�erences are notsigni®cant. The results on computation time indi-cate there is a signi®cant improvement in perfor-

mance and Tabu search is consistently faster thanGA in all but two instances. Paired t-tests, shownin Table 7(b), indicate that the di�erences are sig-ni®cant at p < 0.001. The computation time for GAdeteriorates for larger size problems. Hence, wenote that Tabu search is sensitive to the stop cri-terion, and for mid-size problems, as in our study,Tabu search performs as well as GA. Future re-search could explore the impact of stop criterionon solution quality in Tabu search for large sizeproblems. We would expect the number of itera-tions to be increased to generate good quality re-sults. Prior studies have also found GA to be lesse�cient in computation time. Esbensen (1995)

Table 6

Comparison of GA and Tabu search (iteration� 5000)

Data set (M ´ N)a Genetic algorithms Tabu search

Cost CPU (s) Cost CPU (s)

10 ´ 10 1752 0 1752 0

20 ´ 20 4300 0 4300 0

30 ´ 30 4899 1 4899 0

40 ´ 40 5943 2 5943 1

50 ´ 50 7391 4 7391 2

60 ´ 60 7840 8 7840 4

70 ´ 70 7940 13 7940 9

80 ´ 80 10,422 17 10,422 14

90 ´ 90 11,354 25 11,354 20

100 ´ 100 16,166 40 16,166 55

150 ´ 100 19,359 134 19,359 180

200 ´ 100 22,948b 243 25,102 300

125 ´ 125 16,307 180 16,307 120

175 ´ 125 21,046b 300 21,051 242

225 ´ 125 26,223 361 26,223 350

150 ´ 150 19,329 360 19,329 184

200 ´ 150 24,358 377 24,358 369

250 ´ 150 28,248 772 28,248 524

175 ´ 175 20,907b 760 20,918 312

225 ´ 175 25,003 903 25,003 540

275 ´ 175 27,672 835 27,672 765

200 ´ 200 22,892 1189 22,876b 540

250 ´ 200 26,122 1179 26,122 733

300 ´ 200 29,879 1557 29,879 1380

250 ´ 250 25,566b 2330 25,573 1402

300 ´ 250 29,310b 3120 29,324 1542

350 ´ 250 32,290 2580 32,290 1980

100 ´ 300 13,120 1654 13,132 660

200 ´ 300 21,238 3060 21,238 1740

300 ´ 300 28,732 4500 28,728b 3890

a N: Steiner nodes (hubs), M: Target nodes (end o�ces).b The method performs better than the other.


found that while GA provided better solutionquality compared to other heuristic approaches, itdid not compare well on computation time forsome problems. They found that the branch andcut heuristic found very fast solutions for smallproblems but were not capable of solving large sizeproblems. GA provided solutions within reason-able time for all the problem sizes. Further, GA ismore amenable for parallel processing and perhapsthat may facilitate in obtaining faster solutions.Another possibility would be to change the stopcriterion to number of generations rather than®tness convergence to reduce the computationtime, but it could be at the cost of solution quality.Very often, the ®rst few generations provide thebiggest improvement in value, and it takes manygenerations to reach a convergence value withoutsigni®cant improvements in value.

8. Conclusion

Communication networks have witnessed dra-matic growth in the last decade due to widespreaduse of Internet. The increased reliability and ser-vice quality requirements of modern networkscombined with signi®cant increase in investments

in communications infrastructure have made itcritical to design optimized networks that meet theperformance parameters. In this paper, we evalu-ated the use of genetic algorithms in designing aSTS network that is extensively used in digital datanetwork design. The results of GA were comparedwith the results from Tabu search for the samenetwork. Initially, the STS problem was solvedusing a genetic algorithm based on binary encod-ing system, enlarged sampling space with sto-chastic �l� k� selection, uniform crossover andexchange mutation for reproduction, and Prim'sMST in the evaluation function. Subsequently, aTabu search heuristic was used to solve the sameproblem. The two algorithms were evaluated on awide variety of problem sizes ranging from 10 ´ 10to 300 ´ 300 nodes. The results indicate that GAgenerates solutions that are better than Tabusearch in two instances and worse than Tabusearch in three instances. In terms of computationtime GA performs better than Tabu search (using30,000 iterations) in almost all instances. However,after reducing the number of iterations to 5000 forthe Tabu search, the computation time for Tabusearch was better than GA in all but two instances,but with a marginal decrease in solution quality in®ve instances.

8.1. Research implications

This study demonstrates the versatility of ge-netic algorithms in network design problems. Thebinary encoding scheme for network representa-tion proved to be e�cient. The performance of thealgorithm did not deteriorate for large size prob-lems. This study can be considered, in an indirectway, a validation of the Tabu search heuristic.Kershenbaum (1997) suggested that GA can beused for validating existing heuristic algorithmsthat may have been developed in one context witha set of parameters, but may or may not performthe same in a new context. Since GA approaches aproblem with no preexisting biases it will provideinformation to validate or invalidate the heuristicin the new context.

Future research could examine how the twoalgorithms work for very large size problems. We

Table 7

Paired t-test ± Tabu search (iteration� 5000) and GA

GA Tabu

(a) Solution quality

Mean 18618.53 18691.30

Variance 81,149,963.09 81,951,847.18


Df 29.00

t-Statistics )1.01

Signi®cant 0.31

(b) Computation time

Mean 883.46 595.26

Variance 1,354,209.22 713,877.02


Df 29.00

t-Statistics 3.65

P(T<� t) two-

tail

0.000


noticed that reducing the number of iterations inTabu search did a�ect the solution quality, al-though only to a limited extent. We need to ex-amine how it behaves for larger networks. Also,we need to examine if the di�erences between GAand Tabu, which were not signi®cant for mid-sizenetworks, will become signi®cant for largernetworks.

Another potential area for future research is theuse of hybrid algorithms that blends the best ofheuristic and GA. Recently, Ahuja and Orlin(1997) have highlighted potential opportunities inhybrid algorithms that improve performance.Aggarwal et al. (1997) combine GA with optimi-zation techniques to generate optimal children thatlead to better performance. Glover et al. (1995)provide interesting ideas for combining GA withTabu search to create hybrid algorithms that are®ne-tuned to speci®c problem domains. WhileTabu search excels in systematic exploration ofmemory functions in the search process, GA ex-ploits the idea of combining solutions and allow-ing the ®ttest to survive. Some features of Tabusearch can be incorporated in GA to create di-versity in o�spring population. The crossover op-erator in GA does not use any knowledge of the®tness of the parent or o�spring. If we can incor-porate selective reproduction using prior knowl-edge of parents' traits, we can generate childrenthat produce better solutions faster. Another ideawould be to make some changes to the crossoveroperator. Typical general crossover implementa-tion generates two children from two parents.Promising ideas include generating more than twochildren from a single pair of parents and choosingthe best for further evolution (Ahuja and Orlin,1997). For example, Glover (1994) developed theidea of `path relinking' that incorporates some ofthese ideas. Researchers could also examine usingheuristics, such as Tabu search, in generating theoperators for mutation.

A random strategy was used to initialize thepopulation. It has been observed in heuristics thatstarting with a good initial solution improves itsperformance. Research could explore heuristicsthat will generate good initial solutions for GA butalso have su�cient diversity to cover the entiresolution space.

Acknowledgements

The authors would like to thank Dr. Xu andGlover for providing the Tabu search algorithmand data sets for our experiment.

References

Abuali, F.N., Schoenefeld, D.A., Wainwright, R.L., 1993. The

design of a multipoint line topology for a communication

network using genetic algorithms. Proceedings of the

Seventh Oklahoma Conference on Arti®cial BL Intelligence,

101±110.

Abuali, F.N., Wainwright, R.L., Schoenefeld, D.A., 1995.

Determinant factorization: A new encoding scheme for

spanning trees applied to the probabilistic minimum span-

ning tree problem. Proceedings of the Sixth International

Conference on Genetic Algorithms, 470±477.

Aggarwal, C.C., Orlin, J.B., Tai, R.P., 1997. Optimized

crossover for the independent set problem. Operations

Research 45, 226±234.

Ahuja, R.K., Orlin, J.B., 1997. Developing ®tter genetic

algorithms. INFORMS Journal of Computing 9 (5), 251±

253.

Ahuja, R.K., Magnauti, T.L., Orlin, J.B., 1993. Network

Flows: Theory Algorithms Applications. Prentice-Hall,

Englewood Cli�s, NJ.

Anderson, C., Fraughnaugh, K., Parker, M., Ryan, J., 1993.

Path assignment for call routing: An application of Tabu

search. Annals of Operations Research 43, 301±312.

Back, T., Ho�meister, F., 1991. Extended selection mechanism

in genetic algorithms. Proceedings of the Fourth Interna-

tional Conference on Genetic Algorithms, 92±99.

Back, T., Fogel, D., Michalawecz, Z. (Eds.), 1997. Handbook

of Evolutionary Computation. Oxford University Press,

Oxford.

Beasley, J.E., 1990. OR Library, Imperial College, UK. URL:

mscmga.ms.ic.ac.uk/info.html.

Chang, C.-C., Wu, W.-B., 1997. E�cient path and vertex

exchange in Steiner tree algorithms. Networks 29 (2), 81±92.

Charddaire, P., Kapsalis, A., Mann, J.W., Rayward-Smith,

V.J., Smith, G.D., 1995. Applications of genetic algorithms

in telecommunications. Proceedings of the Second Interna-

tional Workshop on Applications of Neural Networks to

Telecommunications, 290±299.

Chopra, S., Rao, M.R., 1989. On the Steiner tree problem I &

II. Working Paper, New York University.

Coombs, S., Davis, L., 1987. Genetic algorithms and commu-

nication link speed design: Constraints and operators. In:

Grefenstette, J.J. (Ed.), Proceedings of the Second Interna-

tional Conference on Genetic Algorithms and their Appli-

cations, pp. 257±260.

Cuppinim, M., 1994. A genetic algorithm for channel assign-

ment problems. European Transactions on Telecommuni-

cations Related Technologies, 285±294.


Davis, L., Coombs, S., 1989. Optimizing network link sizes with

genetic algorithms. Modeling and Simulation Methodology,

317±331.

Esbensen, H., 1995. Computing near-optimal solutions to the

Steiner problem in a graph using genetic algorithm.

Networks 26 (4), 173±185.

Gen, M., Cheng, R., 1997. Genetic Algorithms and Engineering

Design. Wiley, New York, 1997.

Glover, F., 1989. Tabu search ± part I. ORSA Journal of

Computing 3, 190±206.

Glover, F., 1994. Genetic algorithms and scatter search: Unsus-

pected potentials. Statistics and Computing 4, 131±140.

Glover, F., Lee, M., Ryan, J., 1991. Least cost network

topology design for a new service: An application of Tabu

search. Annals of Operations Research 33, 351±362.

Glover, F., Kelly, P.J., Laguna, M., 1995. Genetic algorithms

and Tabu search. Computers and Operations Research 22

(1), 111±134.

Glover, F., Laguna, M., 1993. Tabu search. In: Reeves, C.

(Ed.), Modern Heuristic Techniques for Combinatorial

Problems, Basil Blackwell, Oxford.

Glover, F., Laguna, M., 1997. Tabu Search. Kluwer Academic

Publishers, Boston, MA.

Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimi-

zation, and Machine Learning. Addison-Wesley, Reading,

MA.

Hansen, P., 1986. The steepest ascent mildest descent heuristic

for combinatorial programming. Numerical Methods in

Combinatorial Optimization, Capri, Italy.

Hertz, A., Taillard, E., de Werra, D., 1996. Tabu Search, Local

Search in Combinatorial Optimization. Wiley, Chichester.

Hesser, J., Manner, R., Stucky, O., 1989. Optimization of

Steiner trees using genetic algorithms. Proceedings of the

Third International Conference on Genetic Algorithms,

231±236.

Holland, J., 1975. Adaptation in Natural and Arti®cial

Systems. University of Michigan Press, Ann Arbor, MI.

Hoelting, C., Schoenfeld, D., Wainwright, R., 1996. A genetic

algorithm for the minimum broadcast time problem using a

global precedence vector. Proceedings of the ACM Sympo-

sium on Applied Computing, 258±262.

Hwang, F.K., Richard, D.S., Winter, P., 1992. The Steiner tree

problem. Annals of Discrete Mathematics 53, 59±85.

Kapsalis, A., Rayward-Smith, V.J., Smith, G.D., 1993. Solving

the graphical Steiner tree problem using genetic algorithms.

Journal of the Operations Research Society 44 (4), 397±406.

Karunanithi, N., Carpenter, T., 1993. A ring loading applica-

tion of genetic algorithms. Technical Report TM-ARH-

023337, Bellcore, Bell Communication Research, Red Bank,

NJ.

Kershenbaum, A., 1997. When genetic algorithms work best.

INFORMS Journal of Computing 9 (3), 254±255.

Kruskal, J.B., 1956. On the shortest spanning subtree of a

graph and the traveling salesman problem. Proceedings of

the American Mathematical Society 7, 48±50.

Kumar, G.P., Babu, G.P., 1994. Optimal network partition-

ing for fault-tolerant network management using evolu-

tionary programming. Information Processing Letters 25,

145±149.

Laguna, M.J., Glover, F., 1993. Bandwidth packing: A Tabu

search approach. Management Science 39 (4), 492±500.

Lee, Y., Chiu, S.Y., Ryan, J., 1996. A branch and cut algorithm

for Steiner tree-star problem. INFORMS Journal on

Computing 8 (3), 100±120.

Lee, Y., Lu, L., Qiu, Y., Glover, F., 1994. Strong formulations

and cutting planes for designing digital data networks.

Telecommunication Systems 2, 261±274.

Liu, X., Sakamoto, A., Shimamoto, T., 1994. Genetic channel

router. IEICE Transactions on Fundamentals of Electronics

Communications and Computer Sciences 3, 492±501.

Marin, F.J., Gonzalez, F.J., Sandoval, F., 1994. The routing

problem in tra�c control using genetic algorithms. Pro-

ceedings of the Second IFAC Symposium on Intelligent

Components and Instruments for Control Application, 8±

10.

Michalawecz, Z., 1992. Genetic Algorithms + Data Struc-

tures�Evolution Programs. Springer, Berlin.

Murgu, A., Lahdelma, R., 1995. Tracking statistics in commu-

nication tra�c control using a genetic algorithm. Proceed-

ings of the First Nordic Workshop on Genetic Algorithms

and Their Applications, 241±258.

Oyman, A.I., Solvinf, C.E., 1994. Concentrator location-

problems using genetic algorithms. Proceedings of the

Seventh Mediterranean Electrotechnical Conference 3,

1341±1344.

Palmer, C.C., Kershenbaum, A., 1995. An approach to a

problem in network design using genetic algorithms. Net-

works 26 (3), 151±163.

Prim, R.C., 1957. Shortest connection networks and some

generalizations. Bell System Technical Journal 36, 1389±

1401.

Sharaiha, Y.M., Gendreau, M., Laporte, G., Osman, I.H.,

1997. A Tabu search algorithm for the capacitated shortest

spanning tree problem. Networks 29, 161±171.

Shimamoto, N., Hiramatsu, A., Yamasaki, K., 1993. Dynamic

routing control based on a genetic algorithm. Proceedings of

IEEE International Conference on Neural Networks, 1123±

1128.

Sinclair, M.C., 1993. The application of a genetic algorithm to

trunk network routing table optimization. Proceedings of

the Tenth UK Teletra�c Symposium, 2±6.

Skorin-Kapov, J., 1990. Tabu search applied to the quadratic

assignment problem. ORSA Journal on Computing 2, 33±

45.

Skorin-Kapov, D., Skorin-Kapov, J., 1994. On Tabu search for

the location of interacting hub facilities. European Journal

of Operational Research 73, 502±509.

Smith, J.M., Lee, D.T., Liebman, J.S., 1981. An O(n log n)

Heuristic for Steiner minimal tree problems on the Euclid-

ean metric. Networks 11 (1), 23±39.

Tan, T.K., Pollard, J.K., 1995. Determination of minimum

number of wavelength required for all-optical WDM

networks using graph coloring. Electronic Letters, 1895±

1897.


UEA CALMA Group, 1994. Genetic algorithms approaches to

solving the radio link frequency assignment problem.

Technical Report 56, University of East Anglia, Norwich.

Winter, P., 1987. Steiner problem in networks: A survey.

Networks 17 (2), 129±167.

Winter, P., Zachariasen, M., 1997. Euclidean Steiner minimum

trees: an improved exact algorithm. Networks 30 (3), 149±

166.

Xu, J., Chiu, S.Y., Glover, F., 1996. Using Tabu search to

solve the Steiner tree-star problem in telecommunication

networks design. Telecommunication Systems 6 (2), 117±

127.

Zelikovsky, A.Z., 1993. A faster approximation algorithm for

the Steiner tree problem in graph. Information Processing

Letters 46, 79±83.

Zhou, G., Gen, M., 1996. A note in genetic algorithms for

degree constrained spanning tree problems. Networks 30

(2), 91±97.


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