digital data networks design using genetic...
TRANSCRIPT
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)
C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 143
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�
144 C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158
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
C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 145
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.
146 C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158
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.
C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 147
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
148 C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158
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
C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 149
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).
150 C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158
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
C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 151
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.
152 C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158
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
Observations 30.00 30.00
Df 29.00
t-Statistics )4.62
Signi®cant 0.000
C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 153
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.
154 C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158
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
Observations 30.00 30.00
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
Observations 30.00 30.00
Df 29.00
t-Statistics 3.65
P(T<� t) two-
tail
0.000
C.-H. Chu et al. / European Journal of Operational Research 127 (2000) 140±158 155
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.
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