Ring–mesh topology design in a SONET–WDM network

Photon Netw Commun (2010) 20:43–53DOI 10.1007/s11107-010-0244-4

Ring–mesh topology design in a SONET–WDM network

Youngjin Kim · Youngho Lee · Junghee Han

Received: 18 July 2009 / Accepted: 25 January 2010 / Published online: 19 February 2010© Springer Science+Business Media, LLC 2010

Abstract This article deals with a ring–mesh networkdesign problem arising from the deployment of an opticaltransport network. The problem seeks to partition the set ofdemand pairs to a number of rings and a mesh cluster, andto determine the location of the optical cross-connect system(OXC), while minimizing the total cost of optical add-dropmultiplexers (OADMs), OXCs, and fiber links. We formu-late this problem as a zero-one integer programming prob-lem. In strengthening the formulation, we develop some validinequalities for the zero-one quadratic (knapsack) polytopeand a column generation formulation that eliminates the sym-metry of ring configurations. Also, we prescribe an effectivetabu search procedure for finding a good quality feasible solu-tion, which is also used as a starting column for the columngeneration procedure. Computational results show that theproposed solution procedure provides tight lower and upperbounds within a reasonable time bound.

Keywords Optical transport network · Ring–mesh topol-ogy · Integer programming · Tabu search ·Column generation · Valid inequality

1 Introduction

In this article, we consider a ring–mesh network design prob-lem that arises from the deployment of an optical transportnetwork. Optical transport networks such as the synchronous

Y. Kim · Y. Lee (B)Division of Information Management Engineering, KoreaUniversity, Sung-Buk Ku Anam Dong 5-1, Seoul 136-701, Koreae-mail: [email protected]

J. HanCollege of Business Administration, Kangwon National University,Hyoja-dong, Chunchon, Gangwon, 200-701, Korea

optical network (SONET) have been widely deployed in tele-communication networks. SONET supports several networkarchitectures with enhanced network survivability, for exam-ple, self-healing ring (SHR), dual homing, and point-to-pointdiverse protection schemes. In particular, SHR-based opticalnetworks are most widely deployed due to the fast restora-tion speed and simple network architecture. However, due tothe difficulty of reconfiguring the SHR network, mesh typeoverlay networks along with the existing SHR networks areconsidered [5,8,9]. Accordingly, we consider a ring–mesharchitecture for deploying optical transport networks.

In recent years, SONET-based optical networks are beingupgraded to SONET–WDM (Wavelength Division Mul-tiplexing) based optical networks to meet the increaseddemand of telecommunication services. WDM technologyenables us to stack multiple wavelengths on a single opti-cal fiber such that we can economically upgrade the existingSONET-based networks to SONET–WDM based networks.Thus, the combination of SONET and WDM technologiesbased on a ring–mesh architecture can be an attractive choicefor network planners since the capacity of survivable opticalnetworks can be increased economically.

To route the traffic between nodes on a ring, an optical add-drop multiplexer (OADM) should be placed at each node ofthe ring. An OADM adds and drops traffic to and from a ring,and routes the traffic on the ring. Also, we need an opticalcross-connect system (OXC) to connect two rings at a com-mon node or to route traffic on a mesh network. If we cannothandle all the demand pairs in the network on a ring, we parti-tion the set of demand pairs into a number of rings and a meshcluster within capacity and cardinality constraints. Also, wedetermine the route of each demand pair assigned to a meshcluster considering the capacity of an OXC. Among all thefeasible configurations, we seek to find one that minimizesthe total cost of OADMs, OXCs, and links in a mesh cluster.

123

44 Photon Netw Commun (2010) 20:43–53

(a) (b)

(c)

Fig. 1 Proposed ring–mesh architecture

Figure 1 shows the nature of the problem. Part (a) depictsa given demand pattern, where the numbers on links repre-sent traffic demands (the number of wavelengths requiredbetween nodes). Part (b) shows that every demand pair isassigned either to a ring cluster (logical ring 1 or logicalring 2) or to a mesh cluster. As shown in part (b), we needseven OADMs to cover the demand pairs allocated to ring 1and ring 2 since nodes 1 and 3 appear in both rings. Here, weassume that the capacity of a ring is 10. Hence, demand pair(4, 5) is treated as an inter-ring traffic due to the capacity limitof ring 2 even though nodes 4 and 5 are on ring 2. Demandpairs in a mesh cluster are routed over OXCs, and part (c)shows an example of demand pair routing in a mesh cluster,where two OXCs are installed. Nodes 2 and 4 are connectedto OXC 1, and node 5 is connected to OXC 2. Note that thetotal traffic demands covered by OXC 1 and OXC 2 are 9and 6, respectively. If the capacity of an OXC is 8, we mayroute the demand pair (4, 5) via OXC 2, in which case node4 needs another connection to OXC 2 for routing demand(4, 5) as an inter-ring traffic.

For the last decade, several studies have been conductedfor designing an optical network with a ring topology (seeLee et al. [10,11], Smith et al. [15], Goldschmidt et al. [4],Sutter et al. [16]) and with a mesh topology (see Kenningtonet al. [7]). In particular, Arijs et al. [1] provides an overviewof solution approaches and design issues of ring and meshnetworks. Grover and Martens [5] consider a hybrid approachthat interconnects regional ring networks by a mesh net-

work. Lee and Koh [9] deal with a similar ring–mesh networkdesign problem, where demand pairs are assigned either torings or to a mesh cluster. However, they assume that OXCsare uncapacitated. This assumption entails additional loadbalancing problems in a mesh cluster for assuring the robust-ness. Also, they do not consider any link cost among OXCsand between OXCs and end nodes of demand pairs assignedto a mesh cluster. This may result in a solution installingOXCs at nodes that are widely separated, which may not becost-effective. Recently, Kim et al. [8] address a ring–meshnetwork design problem considering the OXC capacity andlink cost in a mesh cluster. They consider link cost for eachpath i −s − t − j associated with demand pair (i, j) assignedto a mesh cluster, where s and t denote a pair of OXC nodesthat the traffic of demand pair (i, j) is routed through. Sincethe link cost is defined for path i − s − t − j , we cannot con-sider the economies-of-scale at links i − s, s − t , and t − j .In this article, we break down the aggregated link cost forpath i − s − t − j into two parts; link cost among OXCsand link cost between OXCs and end nodes of demand pairsassigned to a mesh cluster. This cost structure is realisticwhen a network operator builds up new fiber lines insteadof leasing capacities of links from back-bone network opera-tors for routing the traffic of demand pairs assigned to a meshcluster.

The remainder of this article is organized as follows. InSect. 2, we present an integer programming formulationfor the ring–mesh network design problem. In Sect. 3, we

123

Photon Netw Commun (2010) 20:43–53 45

develop a column generation formulation to obtain enhancedlower bounds and tight upper bounds. In Sect. 4, we devisean effective tabu search (TS) procedure to find good qualityfeasible solutions within a reasonable time bound. Computa-tional results are presented in Sect. 5, and Sect. 6 concludesthe article.

2 Problem formulation

In this section, we develop a zero-one integer programmingmodel for the ring–mesh network design problem. Considera set N of (demand) nodes, a set M of candidate OXC nodesand a set K of rings. Let di j be the traffic demand betweennodes i and j (> i) ∈ N . We define an edge set E = {(i, j) :di j > 0, ∀i, j (> i) ∈ N }. We assume that edge set Ecovers all the nodes in N . That is, N (E) = N , where N (S)

is the set of nodes induced by an edge set S ⊆ E . Let ba

be the capacity of an OADM and let bc be the capacity ofan OXC. Also, let R be the maximum number of OADMsallowed on a ring. Let αi be the OADM installation cost atnode i ∈ N and let βs be the OXC installation cost at nodes ∈ M . If OXCs are installed at nodes s and t ∈ M , link costγst arises. Also, if demand node i ∈ N is connected to OXCnode s ∈ M , link cost θis arises.

We define decision variables xik, where xik = 1 if nodei ∈ N is assigned to ring k ∈ K , and 0 otherwise. Also,define fi jk = 1 if demand pair (i, j) ∈ E is assigned to ringk ∈ K , and 0 otherwise. Let yi jst = 1 if (inter-ring) trafficdemand (i, j) ∈ E is routed via OXCs s and t ∈ M, and 0otherwise. Let zs = 1 if an OXC is installed at node s ∈ M ,and 0 otherwise. Let ust = 1 if a link between OXC nodes sand t ∈ M is used, and 0 otherwise. In order to guarantee thesurvivability in the mesh cluster, we assume that OXCs arefully connected. Thus, we assume that ust = zs zt . Finally, wedefine variable vis , where vis = 1 if node i ∈ N is connectedto OXC node s ∈ M , and 0 otherwise. Then, the ring–meshnetwork design problem, denoted by RM, can be formulatedas follows:

RM : Minimize∑

i∈N

∑

k∈K

αi xik +∑

s∈M

βs zs

+∑

s,t (>s)∈M

γst ust +∑

i∈N

∑

s∈M

θisvis

subject to∑

i∈N

xik ≤ R, k ∈ K , (1)

∑

(i, j)∈E

di j fi jk ≤ ba, k ∈ K , (2)

∑

k∈K

fi jk+∑

s,t∈M

yi jst=1, (i, j) ∈ E, (3)

∑

(i, j)∈E

di j

⎛

⎝yi jss +∑

t∈M :t �=s

(yi jst + yi j ts)

⎞

⎠

≤ bczs, s ∈ M, (4)

fi jk ≤ xik, (i, j) ∈ E, k ∈ K , (5)

fi jk ≤ x jk, (i, j) ∈ E, k ∈ K , (6)

yi jst ≤ vis, (i, j) ∈ E, s, t ∈ M, (7)

yi jst ≤ v j t , (i, j) ∈ E, s, t ∈ M, (8)

ust ≤ zs, s, t (> s) ∈ M, (9)

ust ≤ zt , s, t (> s) ∈ M, (10)

ust ≥ zs + zt − 1, s, t (> s) ∈ M, (11)

fi jk ∈ {0, 1}, (i, j) ∈ E, k ∈ K , (12)

xik ∈ {0, 1}, i ∈ N , k ∈ K , (13)

yi jst ∈ {0, 1}, (i, j) ∈ E, s, t ∈ M, (14)

zs ∈ {0, 1}, s ∈ M, (15)

ust ∈ {0, 1}, s, t (> s) ∈ M, (16)

vis ∈ {0, 1}, i ∈ N , s ∈ M. (17)

The first term of the objective function represents the OADMcost. The second term represents the OXC cost and the thirdterm represents the link cost among OXCs. The fourth termrepresents the link cost for connecting demand nodes toOXCs. Constraint (1) forces the number of OADMs on a ringto be no more than R. Constraint (2) forces the sum of trafficdemands assigned to a ring to be no more than the capacityof an OADM given by ba . Constraint (3) ensures that eachdemand pair is assigned either to a ring or to a mesh clus-ter. Constraint (4) forces the sum of demand pairs routed byan OXC to be no more than the capacity of an OXC givenby bc. Constraints (5) and (6) ensure that if a demand pairis assigned to a ring, two OADMs associated with the endnodes of the demand pair should be installed on the ring. Con-straints (7) and (8) ensure that if a demand pair is connectedby OXCs, links between demand nodes and OXCs should beplaced. Constraints (9–11) are the linearized expression ofthe non-linear constraints ust = zs zt . Constraints (12–17)represent that all the variables are binary.

Remark 1 By surrogating the constraints (7) and (8) over tand s( �= t) ∈ M, respectively, and noting that∑

s,t∈M yi jst ≤ 1 for (i, j) ∈ E, we obtain∑

t∈M

yi jst ≤ vis, (i, j) ∈ E, s ∈ M, (18)

and∑

s∈M

yi jst ≤ v j t , (i, j) ∈ E, t ∈ M. (19)

Note that inequalities (18) and (19) are valid and dominatethe constraints (7) and (8), respectively. Now, let us considersome inequalities defining an optimal solution. Hereafter,

123


we denote yi jss by yi js for (i, j) ∈ E and s ∈ M. Observethat any optimal solution satisfies the inequality

∑

(i, j)∈E

yi js +∑

(i, j)∈E

∑

t ( �=s)∈M

(yi jst + yi j ts) ≥ zs, s ∈ M.

(20)

Also, note that constraint (11) can be weak for a small frac-tional value of z. Thus, we consider

zs ≥ vis, s ∈ M, i ∈ N , (21)

which is satisfied by an optimal solution. �Proposition 1 If

∑(i, j)∈E di j>bc|M |, the following inequal-

ity is valid:

∑

i∈N

∑

k∈K

xik ≥ 2

⌈∑(i, j)∈E di j − bc|M |

ba

⌉. (22)

Proof We need at least τ =⌈∑

(i, j)∈E di j −bc|M|ba

⌉rings to

cover all the demand pairs. Also, note that at least two nodesshould be assigned to ring k = 1, . . . , τ. Thus, we have that∑

i∈N

xik ≥ 2, k = 1, . . . , τ.

By surrogating the above inequality over k, we obtain (22).This completes the proof. �

Let us strengthen inequality (22).

Corollary 1 If∑

(i, j)∈E di j>bc|M |, the following inequal-ity is valid:

∑

i∈N

∑

k∈K

xik ≥ 2∑

(i, j)∈E di j − ν(K P)

ba�, (23)

where ν(K P) is the optimal objective value of problem KP:

KP : Maximize∑

(i, j)∈E

∑

s∈M

di j yi js

subject to∑

s∈M

yi js ≤ 1, (i, j) ∈ E,

∑

(i, j)∈E

di j yi js ≤ bc, s ∈ M,

yi js ∈ {0, 1}, (i, j) ∈ E, s ∈ M.

Proof Observe that the optimal objective value of KP repre-sents the maximum traffic demand that can be allocated to amesh cluster. Thus, the minimum traffic demand covered byrings is

∑(i, j)∈E di j − ν(K P). The rest of the proof is the

same as that of Proposition 1. �

Proposition 2 Define ζ =⌈∑

(i, j)∈E di j − ba |K |bc

⌉. Then,

the following inequality is valid:∑

s∈M

zs ≥ ζ. (24)

If ζ ≥ 2, the following inequality is also valid:∑

s,t (>s)∈M

ust ≥ ζ × (ζ − 1)/2. (25)

Proof Note that from (2),∑

k∈K

∑

(i, j)∈E

di j fi jk ≤ ba |K |. Thus,

from (3) and (4), we have∑

(i, j)∈E

di j − ba |K | ≤∑

(i, j)∈E

di j −∑

k∈K

∑

(i, j)∈E

di j fi jk

=∑

(i, j)∈E

∑

s,t∈M

di j yi jst

≤∑

(i, j)∈E

di j

∑

s∈M

(yi jss

+∑

t∈M :t �=s

(yi jst + yi j ts)

⎞

⎠

≤∑

s∈M

bczs .

By dividing the above inequality by bc and rounding up, weobtain (24). Since ζ is a lower bound on the number of OXCs,we obtain (25) by (11). This completes the proof. �

Now, let us strengthen inequality (24).

Corollary 2 Define ζ ′ =⌈

max{∑(i, j)∈E di j −τ1,τ2}bc

⌉, where τ1

is obtained from

Maximize τ1 =∑

(i, j)∈E

∑

k∈K

di j fi jk

subject to∑

i∈N

xik ≤ R, k ∈ K ,

∑

(i, j)∈E

di j fi jk ≤ ba, k ∈ K ,

∑

k∈K

fi jk ≤ 1, (i, j) ∈ E,

fi jk ≤ xik, (i, j) ∈ E, k ∈ K ,

fi jk ≤ x jk, (i, j) ∈ E, k ∈ K ,

fi jk ∈ {0, 1}, (i, j) ∈ E, k ∈ K ,

xik ∈ {0, 1}, i ∈ N , k ∈ K ,

and where τ2 is obtained from

Minimize τ2 =∑

(i, j)∈E

∑

s∈M

di j yi js

subject to∑

k∈K

fi jk +∑

s∈M

yi js = 1, (i, j) ∈ E,

∑

(i, j)∈E

di j yi js ≤ bc, s ∈ M,

123


∑

(i, j)∈E

di j fi jk ≤ ba, k ∈ K ,

∑

i∈N

xik ≤ R, k ∈ K ,

fi jk ≤ xik, (i, j) ∈ E, k ∈ K ,

fi jk ≤ x jk, (i, j) ∈ E, k ∈ K ,

0 ≤ fi jk ≤ 1, (i, j) ∈ E, k ∈ K ,

0 ≤ xik ≤ 1, i ∈ N , k ∈ K ,

yi js ∈ {0, 1}, (i, j) ∈ E, s ∈ M.

Then, the following inequality is valid:∑

s∈M

zs≥ζ ′. (26)

Proof Note that τ1 represents the maximum traffic demandthat can be covered by rings and as a result,

∑(i, j)∈E di j −τ1

denotes the minimum traffic demand that should be coveredby OXCs. Also note that τ2 is the minimum traffic demandthat should be covered by OXCs. Hence, ζ ′ is a lower boundon the number of OXCs. This completes the proof. �Proposition 3 Define δ(i) = { j ∈ N : di j > 0 or d ji > 0}for i ∈ N . Then, the following inequality is valid:∑

k∈K

xik +∑

j∈δ(i)

∑

s,t∈M

y[i j]st/|δ(i)| ≥ 1, i ∈ N , (27)

where [i j] = (i, j) if i < j , and [i j] = ( j, i) if j < i.

Proof We assume that∑

k∈K xik = 0 for some i ∈ Nsince, otherwise, (27) becomes trivial. Then, (27) becomes∑

j∈δ(i)∑

s,t∈M y[i j]st = |δ(i)|. Thus, (27) holds true. Thiscompletes the proof. �

For V ⊆ N , we say that V is a dependent set if |V | > R;otherwise, V is an independent set. A dependent set V isminimal if all of its subsets are independent. Let δ(i, A) ={ j ∈ N : (i, j) ∈ A} for i ∈ N and A ⊆ E, and letδ(i) = δ(i, E). Below, we consider two valid inequalitiesbased on a dependent set V ⊆ N (see Johnson et al. [6]).

Remark 2 Let V ⊆ N be a dependent set and the subgraphinduced by V is connected, and let T be a set of edges induc-ing a spanning tree subgraph on the nodes in V . Then, thefollowing inequality is valid:

∑

(i, j)∈T

fi jk ≤∑

i∈V

(|δ(i, T )| − 1)xik, k ∈ K . (28)

Inequality (28) can be generalized to forests. Let V ⊆ N be aminimal dependent set and let F be a set of edges that inducea forest on the nodes in V . Also, let F have p components.Then, the following inequality is valid:

∑

(i, j)∈F

fi jk ≤ p − 1 +∑

i∈V

(|δ(i, F)| − 1)xik, k ∈ K . (29)

The inequalities (28) and (29) are identified by finding a(minimal) dependent set, which is NP-hard (see, Nemhauserand Wolsey [12]). Thus, we resort to heuristic proceduresfor finding the violated inequalities. Let ( f ∗, x∗) be a frac-tional (partial) solution of the LP-relaxation of RM. For agiven k ∈ K , in finding a violated tree inequality (28), westart with an edge (i, j) having the biggest fractional valueof f ∗

i jk . That is, let V = {i, j} and T = {(i, j)}. Then, weadd an edge (i, j), where i ∈ V and j �∈ V, having thesmallest value of x∗

ik − f ∗i jk such that f ∗

i jk is fractional, andlet V = V + { j}. We repeat this process until the node setin T is dependent or until there is no edge of fractional f ∗such that one end node is in V and the other end node isnot. If such an edge is not found, the tree obtained so farrepresents a component of a forest. To find a violated forestinequality (29), we seek to find the other components of theforest. Toward this end, we find another component of a tree.We start with an edge (i, j) �∈ F, where i, j �∈ V, having thebiggest fractional value of f ∗

i jk . We terminate this processwhen we obtain a forest of which node set is dependent orwhen there is no fractional edge that is not in F.

Proposition 4 If (|δ(i)| > R or∑

j∈δ(i) di j > ba) and∑j∈δ(i) di j > bc for some i ∈ N , the following inequal-

ity is valid:∑

k∈K

xik +∑

s∈M

vis ≥ 2. (30)

Proof If all the demand pairs in δ(i) are covered by rings forsome i ∈ N , we need at least two rings, which implies that∑

k∈K xik ≥ 2. Also, if all the demand pairs in δ(i) are cov-ered by OXCs for some i ∈ N , we need at least two OXCs.Thus, we find that

∑s∈M vis ≥ 2. While, if some demand

pairs in δ(i) are covered by rings and others are covered byOXCs, we see that

∑k∈K xik ≥ 1 and

∑s∈M vis ≥ 1. This

completes the proof. �Remark 3 Note that constraints (9) and (10) become redun-dant in an optimal solution since the objective function ofminimizing the link cost forces the constraint (11) binding.Also, constraints (11) can be replaced by the following validinequality:∑

s∈S

zs ≤∑

s∈S

∑

t (>s)∈S

ust + 1, (31)

where S ⊆ M. As proved by Padberg [13], inequality (31)is a facet of the integer polytope described by (9), (10), (11),(15), and (16).

�

3 Column generation formulation

Now, we develop a column generation formulation CGRMfor problem RM. For this purpose, let us define some

123


notations. Let S ⊆ E be a subset of demand pairs satisfying∑(i, j)∈S di j ≤ ba and |N (S)| ≤ R. Let Q be the set of all

the ring configurations. We denote the OADM cost of a ringconfiguration q ∈ Q by cq . Let ξq = 1 if a ring configurationq ∈ Q is selected, and 0 otherwise. Let ηi jq be an indicatorbeing equal to 1 if demand pair (i, j) is covered by a ringconfiguration q ∈ Q, and 0 otherwise. Then, an alternativeformulation CGRM can be described as follows.

CGRM : Minimize∑

q∈Q

cqξq +∑

s∈M

βs zs

+∑

s,t (>s)∈M

γst ust +∑

i∈N

∑

s∈M

θisvis

subject to∑

q∈Q

ηi jqξq +∑

s,t∈M

yi jst = 1, (i, j) ∈ E, (32)

∑

(i, j)∈E

di j

⎛

⎝yi jss +∑

t∈M,t �=s

(yi jst + yi j ts)

⎞

⎠

≤ bczs, s ∈ M, (33)

yi jst ≤ vis, (i, j) ∈ E, s, t ∈ M, (34)

yi jst ≤ v j t , (i, j) ∈ E, s, t ∈ M, (35)

ust ≤ zs, s, t (> s) ∈ M, (36)

ust ≤ zt , s, t (> s) ∈ M, (37)

ust ≥ zs + zt − 1, s, t (> s) ∈ M, (38)∑

q∈Q

ξq ≤ |K |, (39)

ξq ∈ {0, 1}, q ∈ Q, (40)

yi jst ∈ {0, 1}, (i, j) ∈ E, s, t ∈ M, (41)

zs ∈ {0, 1}, s ∈ M, (42)

ust ∈ {0, 1}, s, t (> s) ∈ M, (43)

vis ∈ {0, 1}, i ∈ N , s ∈ M. (44)

Note that we can enhance the LP-relaxation of CGRM byreplacing (34) and (35) with (18) and (19), and by adding (20)and (21). Also, note that solving CGRM with all the possiblering configurations is intractable since Q may have expo-nentially many ring configurations. Thus, we generate ringconfigurations as needed starting from a restricted CGRMhaving a few ring configurations. Below, we describe a pro-cedure for generating a ring configuration (column). Let πi j

be the dual variable of CGRM indicating the LP-relaxation ofCGRM associated with constraint (32), and let π0 be the dualvariable associated with constraint (39). Then, the columngeneration subproblem can be defined as follows:

SPRMring(π) : Minimize∑

i∈N

αi xi −∑

(i, j)∈E

πi j fi j − π0

subject to

∑

i∈N

xi ≤ R,

∑

(i, j)∈E

di j fi j ≤ ba,

fi j ≤ xi , (i, j) ∈ E,

fi j ≤ x j , (i, j) ∈ E,

fi j ∈ {0, 1}, (i, j) ∈ E,

xi ∈ {0, 1}, i ∈ N .

If the optimal objective value of SPRMring(π) is less thanzero, we add the generated ring configuration to the currentrestricted CGRM.

4 Tabu search procedure

Dealing with the inherent computational complexity of theproblem RM, we devise an effective TS heuristic procedure.For details of TS, see Glover and Laguna [3] and Glover [2].In order to exploit the combinatorial structure of the ring andmesh architecture, we propose a two-level hierarchical searchscheme with long-term and short-term memory processes. Ina long-term memory process, we partition E into τ(≤ |K |)rings Ering(k) for k = 1, . . . , τ and a mesh Emesh . Note thatany partition of E, for example, Ering(k) for k = 1, . . . , τ

and Emesh itself is not a feasible solution since the alloca-tion of demand pairs in Emesh to OXCs are not determined.In a short-term memory process, we allocate demand pairsin Emesh to OXCs with the TS-Mesh procedure, and con-sider an alternative partition of E − Emesh to τ rings withthe TS-Ring procedure. Since Ering

⋂Emesh = φ, where

Ering = ⋃τk=1 Ering(k), the TS-Ring and TS-Mesh proce-

dures can be implemented independently. The outline of theproposed TS procedure is as follows. Given an initial parti-tion of E to τ rings and a mesh, we perform the short-termmemory process with the TS-Ring and TS-Mesh proceduresfor a given number of iterations. Then, we perform the long-term memory process to find a new partition of E to Ering(k)

for k = 1, . . . , τ and Emesh . With this new partition, weresume the short-term memory process. We terminate theproposed heuristic procedure after generating new partitionsT S_LT M_M AX times. Below, we describe the short-termand long-term memory processes in detail.

4.1 Short-term memory process

4.1.1 TS-Ring procedure

For a given set of rings, Ering = ⋃τk=1 Ering(k), the

TS-Ring procedure seeks to find a (near) optimal partitionof Ering to τ(≤ |K |) rings, Ering(k) for k = 1, . . . , τ. Forthis purpose, we use two moves: transfer and swap. A transfer

123


move relocates a demand pair of a ring to another ring, whilea swap move exchanges two demand pairs of different rings.We define the neighborhood of a current solution by the unionof the neighborhood of a transfer move and the neighborhoodof a swap move. At the iteration of the TS-Ring procedure, wemove to the neighborhood of a current solution that improvesthe current solution most. If there is no neighborhood thatimproves the current solution, we move to the neighborhoodthat aggravates the current solution least. In the TS-Ringprocedure, we perform a T S_ST M_Ring number of iter-ations for neighborhood search (or move). Also, we employa recency-based tabu memory defined for each demand pair.That is, we prohibit the demand pair(s) associated with thecurrently selected move from being considered for the nextT abu_T enure_Ring number of iterations.

4.1.2 TS-Mesh procedure

For a given Emesh, the TS-Mesh procedure determines boththe location of OXCs (denoted by M ⊆ M) and the allo-cation of demand pairs in Emesh to OXC nodes in M . Inthe TS-Mesh procedure, we generate a number of differ-ent subset M by using a neighborhood search, in which thesearch space is defined by adding (and deleting) an OXCnode to (and from) the current M . That is, the neighbor-hood of the current M is the collection of subset M

′, where

M′ = M + {s} for s ∈ M − M or M

′ = M − {s} for s ∈ M .Evaluation of a M

′in the neighborhood of the current M

is done by a greedy algorithm that allocates demand pairsin Emesh to OXC nodes in M

′. Among all the subsets M

′in

the neighborhood of the current M, we choose one that mini-mizes the sum of the OXC related costs β, γ, and θ. Also, weemploy a recency-based tabu memory for M

′with tabu tenure

T abu_T enure_Mesh. That is, we prohibit subsets M′being

investigated within the previous T abu_T enure_Mesh iter-ations from being selected. The greedy algorithm allocatingdemand pairs in Emesh to OXC nodes in M

′is as follows.

Greedy algorithm for demand allocation:

Step 0. Let N = N (Emesh). Let Di = { j ∈ N : (i, j) or( j, i) ∈ Emesh}. For each node i ∈ N , sort s ∈ M in non-decreasing order of θis . Let si (hi ) ∈ M be the hth OXCnode for demand node i ∈ N , where h = 1, . . . , |M |. Sethi = 1 for each node i ∈ N .Step 1. For i ∈ N and s ∈ M , compute θ̃is =θis/

∑j∈Di

d[i j], where d[i j] = di j if i ≥ j and d[i j] = d ji

otherwise. Select node i ′ = argmaxi∈N �i , where �i =|θ̃i,si (hi +1) − θ̃i,si (hi )| if hi < |M | and θ̃i,si (hi ) otherwise.Step 2. Assign demand pair (i ′, j) for j ∈ Di ′ to the OXCat node si ′(hi ′) considering the capacity limit of an OXC,where a demand pair (i ′, j) with larger d[i ′ j] is assignedfirst. Remove from Di ′ all the nodes j associated with the

demand pair (i ′, j) assigned to the OXC at node si ′(hi ′).If hi ′ = |M| and Di ′ �= φ, stop with infeasibility.Step 3. If Di ′ = φ, let N = N − {i ′}, otherwise, let hi ′ =hi ′ + 1. If N = φ, stop, otherwise go to Step 1.

4.2 Long-term memory process

The long-term memory process enables us to explore adiversified solution space by providing a new partition ofE; Ering(k) for k = 1, . . . , τ and Emesh . Also, we manage alist L consisting of up to Max_L non-empty subsets of E .

Thus, Ering(k) for k = 1, . . . , τ and Emesh can be elements(referred to as demand clusters) of L. Each demand cluster inL is labeled with the objective function value (not the cost ofthe demand cluster) of the solution to which the demand clus-ter belongs, and is sorted in nondecreasing order of label. Thelist L is updated with Ering(k) for k = 1, . . . , τ and Emesh ofthe short-term memory process. If |L| > Max_L , we delete|L|−Max_L demand clusters from L, where a demand clus-ter with the largest label is deleted first. The procedure forgenerating a new partition from L is as follows.

Step 0. Set L′ = L and τ = 0.Step 1. Repeat this step until L′ becomes empty.

Step 1-1. Choose demand cluster l from L′ probabilisti-cally, where the probability of choosing a demand clusterof the lth highest (or worst) label is set by 2l/|L′|(|L′|+1). From L′, remove all the demand clusters containingat least one demand pair belonging to the demand clus-ter l. If the selected l is a ring cluster, go to Step 1-2.Otherwise go to Step 1-3.Step 1-2. Set τ = τ + 1, and let Ering(τ ) denote thering cluster l. If τ = |K |, remove all the ring clustersfrom L′.Step 1-3. Let Emesh denote the edge set of mesh clusterl, and remove all the mesh clusters from L′.

Step 2. Let S = E−{⋃τk=1 Ering(k)

⋃Emesh}. If τ = |K |,

go to Step 3. Otherwise repeat Steps 2-1 and 2-2 whileτ < |K |.Step 2-1. Make a new ring Ering(τ+1) and assign as manydemand pairs in S as possible to the ring, where the demandpair of the largest demand is assigned first, while satisfyingthe node cardinality and capacity constraints.Step 2-2. Remove demand pairs in Ering(τ+1) from S. Setτ = τ + 1.Step 3. If S �= φ, set Emesh = Emesh

⋃ S.

The proposed long-term memory process requires an ini-tial list of demand clusters. We find I local solutions to adddemand clusters of them to the list. To generate a local solu-tion, we first find an initial solution which has |K | ring clus-ters. Each ring cluster is generated by choosing a demand

123


Table 1 Computation results (|N | = 15, |M | = 7, |E | = 21, |K | = 2)

Problems Lower bounds Upper bounds Elapsed time Gaps (%)

r ZL P1 ZL P2 ZCG Z I P ZT S ZCG I P TI P TCG Gap1 Gap2 Gap3

1.0 188.3 320.2 368.0 413 420 414 195.9 118.1 41.2 22.5 0.2

1.0 267.0 390.2 455.3 550 596 545 3600.0 389.0 31.6 29.1 0.9

1.0 168.4 306.3 352.4 400 457 400 591.4 144.7 45.0 23.4 0.0

1.0 195.8 331.1 380.3 462 530 474 3600.0 1634.4 40.9 28.3 2.6

1.0 150.8 295.6 334.0 400 452 403 3600.0 311.0 49.0 26.1 0.8

1.5 207.7 330.4 375.1 463 493 457 3600.0 375.5 37.1 28.6 1.3

1.5 233.1 361.5 416.3 523 544 516 3600.0 2038.1 35.5 30.9 1.3

1.5 215.2 338.1 376.8 472 511 463 3600.0 299.4 36.4 28.4 1.9

1.5 215.9 353.5 399.4 450 541 474 318.7 162.0 38.9 21.4 5.3

1.5 234.2 357.6 420.0 496 518 496 3424.1 396.9 34.5 27.9 0.0

2.0 226.8 343.0 384.3 438 470 447 1255.5 141.6 33.9 21.7 2.1

2.0 265.2 382.8 440.3 517 594 521 2766.2 209.7 30.7 26.0 0.8

2.0 214.2 336.4 377.5 484 531 488 3600.0 3038.0 36.3 30.5 0.8

2.0 252.1 368.6 427.5 492 505 492 2233.5 164.6 31.6 25.1 0.0

2.0 226.9 345.3 386.1 483 531 490 3600.0 2639.8 34.3 28.5 1.4

Table 2 Computation results. (|N | = 15, |M | = 7, |E | = 21, |K | = 3)



1.0 240.3 324.0 426.1 484 511 490 3600.0 98.1 25.8 33.1 1.2

1.0 175.5 260.3 328.1 381 403 390 3600.0 98.1 32.6 31.7 2.4

1.0 201.4 277.9 354.8 395 402 397 1234.1 27.6 27.6 29.6 0.5

1.0 175.1 253.5 324.6 366 398 370 511.4 58.2 30.9 30.7 1.1

1.0 176.9 242.7 306.1 335 352 335 283.0 12.4 27.1 27.6 0.0

1.5 160.9 202.9 252.3 275 278 278 340.6 16.6 20.7 26.2 1.1

1.5 246.5 351.4 436.8 514 535 515 3600.0 166.6 29.8 31.6 0.2

1.5 278.6 364.3 463.4 510 540 510 3047.6 59.8 23.5 28.6 0.0

1.5 246.1 308.6 401.4 474 496 483 3600.0 699.9 20.3 34.9 1.9

1.5 253.9 303.9 398.9 431 462 457 385.1 49.1 16.5 29.5 6.0

2.0 229.1 248.2 315.0 361 361 361 1932.3 19.9 7.7 31.2 0.0

2.0 194.8 249.6 313.3 344 361 353 347.4 23.8 22.0 27.4 2.6

2.0 233.4 279.7 357.0 401 421 401 710.7 16.7 16.6 30.3 0.0

2.0 262.1 317.9 411.0 447 448 448 337.7 28.5 17.6 28.9 0.2

2.0 252.0 344.3 435.7 512 505 497 3600.0 82.3 26.8 32.8 2.9

pair at random and adding an unselected demand pair repeat-edly, which incurs the least additional number of OADM, aslong as the OADM capacity constraint and node cardinalityconstraint are satisfied. Remaining demand pairs are used tomake a mesh cluster that provides an initial solution of TS-Mesh. Then, the initial solution is improved by the TS-Ringand TS-Mesh procedures of the short-term memory process.Demand clusters of the local optimal solution obtained fromTS-Ring and TS-Mesh are added to the list, labeled with the

objective value of the solution to which it belongs and sortedin increasing order of label.

As noted in [14], the proposed long-term memory processincorporates both diversification and intensification implic-itly. The long-term memory process allows us to extendthe diversity of solutions. The intensification effect is ratherimplicit. Observe that the same demand cluster can be foundfrom more than one local solution and thereby a demand clus-ter is included multiple times, one for each solution in the list

123


Table 3 Computation results (|N | = 15, |M | = 7, |E | = 21, |K | = 4)



1.0 183.0 214.4 312.5 347 352 347 3600.0 22.9 14.7 38.2 0.0

1.0 192.4 232.0 335.8 367 367 367 3600.0 15.4 17.0 36.8 0.0

1.0 154.6 180.0 254.7 264 264 264 144.4 0.6 14.1 31.8 0.0

1.0 140.3 160.7 230.9 259 261 261 3600.0 5.7 12.7 38.0 0.8

1.0 125.7 150.0 200.5 210 210 210 90.7 0.3 16.2 28.6 0.0

1.5 264.9 285.0 401.5 418 418 418 720.0 0.6 7.0 31.8 0.0

1.5 175.7 180.0 252.0 252 264 252 41.4 0.1 2.4 28.6 0.0

1.5 245.2 262.3 377.1 421 421 421 3600.0 27.3 6.5 37.7 0.0

1.5 180.0 187.0 278.6 308 322 308 3600.0 14.3 3.7 39.3 0.0

1.5 268.0 286.4 423.3 484 484 484 3600.0 19.7 6.4 40.8 0.0

2.0 177.7 180.0 242.9 252 252 252 34.4 0.3 1.3 28.6 0.0

2.0 238.4 247.0 368.4 411 411 411 3600.0 9.1 3.5 39.9 0.0

2.0 195.0 195.0 294.5 299 299 299 354.0 0.4 0.0 34.8 0.0

2.0 220.5 235.7 346.7 405 403 399 3600.0 30.0 6.5 41.8 1.5

2.0 265.0 270.0 384.0 396 396 396 434.6 0.8 1.9 31.8 0.0

L. In addition, when we choose demand clusters to generate anew solution, we assign the probability of being chosen suchthat a better demand cluster is selected more often. Thus, byrepeating the procedure, good clusters are more frequentlyused in constructing a new solution. As a result, the searchprogressively changes from a diversification process to anintensification process.

5 Computational results

We have tested the proposed solution procedure for severaltest problems. For |N | nodes, we generated |N | − 1 links atrandom such that the resulting graph became a tree spannedby the node set N and added (|E |− |N |−1) links at randomon the spanning tree. Traffic demand di j for (i, j) ∈ E is setfrom a uniform distribution in the range of [3, 10]. We setR = 8 and ba = bc = 32. Also, we set αi = α for all i ∈ N ,

where α is an arbitrary integer value between 10 and 20, andset βs = r × α. Link costs γst for s and t (> s) ∈ M and θis

for i ∈ N and s ∈ M are set from a uniform distribution inthe range of [20, 30]. The coding is done in C and all runsare made on a Pentium IV 1.5 GHz PC with CPLEX version9.0 as a LP/MIP solver. Computation times are measured inseconds, and a CPLEX run is limited to 1 h (3,600 s).

To set the parameters of our TS procedure, we conductsome preliminary experiments and find that T abu_T enure_Ring = 5, T abu_T enure_Mesh = 3, T S_ST M_Ring =5 × T abu_T enure_Ring, T S_ST M_Mesh = 5 × T abu_T enure_Mesh, and T S_LT M_M AX = 20 is a good

combination in terms of computation time and solution qual-ity. We set the number of initial local solutions I = 10, andlimit the size of list L, Max_L , to 20 × (|K | + 1).

Computational results are shown in Tables 1–3. ZL P1

denotes the lower bound obtained by solving the LP-relax-ation of initial formulation. ZL P2 denotes the LP bound ofthe enhanced formulation obtained by replacing (7) and (8)by (18) and (19) and by adding (20), (21), and (31). ZCG

denotes the lower bound obtained by solving the LP-relax-ation of the column generation model CGRM. Z I P denotesthe objective function value of the optimal solution or ofthe best integer feasible solution that CPLEX finds within3,600 s by implementing the enhanced formulation used toobtain ZL P2. ZT S denotes the objective function value ofthe TS procedure and ZCG I P denotes the objective functionvalue of the (integer) restricted master problem of modelCGRM after adding all the rings generated by solving theLP-relaxation of CGRM as well as obtained by the TS. TI P

and TCG denote the elapsed time to obtain Z I P and ZCG I P ,respectively. Gap1 = (ZL P2−ZL P1)

ZL P2× 100% represents the

improvement of LP lower bound obtained by adding validinequalities. Gap2 = (Z I P−ZL P2)

Z I P× 100% denotes the gap

between the improved LP lower bound and Z I P , and Gap3= |ZCG I P−Z I P |

Z I P× 100% denotes the gap between Z I P and

ZCG I P .From ZL P2 (or Gap1), we observe that adding valid

inequalities is quite effective for improving the LP-relaxa-tion bound. Also, we see that the column generation lowerbound ZCG is much better than the LP-relaxation boundZL P2. By comparing the columns of “Upper bounds,” we

123


see that column generation upper bound starting from an ini-tial solution obtained from the TS is quite good in finding afeasible solution with (average, maximum) of Gap3 beingequal to (1.2%, 5.3%), (1.15%, 6.0%), and (0.0%, 0.0%)for the test problems in Tables 1–3, respectively, when anoptimal solution is identified. Moreover, when an optimalobjective value is not identified, observe that column gen-eration provides better or equal upper bounds ZCG I P com-pared with Z I P , for four out of eight problems, one out offive problems, and seven out of eight problems in Tables1–3, respectively, while consuming far less computationtimes. From this observation, we see that the proposed col-umn generation heuristic combined with TS is a promis-ing alternative solution procedure for finding a good qualitysolution for the ring–mesh problem.

6 Conclusions

In this research, we addressed a ring–mesh optical networkdesign problem and developed a zero-one integer program-ming formulation. By exploiting the polyhedral structure ofthe problem, we developed effective valid inequalities forimproving the LP-bounds. Also, we devised a promising heu-ristic procedure by combining TS with a column generationalgorithm. Computational results demonstrate that the pro-posed solution approach finds tight upper bounds and thatthe lower bounds derived from the column generation formu-lation are tighter than those obtained from the formulationenhanced by the proposed valid inequalities.

Motivated by the performance of the column genera-tion approach, we consider finding an optimal solution by abranch and price procedure. Toward this end, we are examin-ing elaborate branching rules for effective search on a branchand bound tree.

Acknowledgements This research is supported by the Korea Univer-sity Grant, 2008.

References

[1] Arijs, P., Van Caenegem, B., Demeester, P., Lagasse, P., VanParys,W., Achten, P.: Design of ring and mesh based WDM transportnetworks. Opt. Netw. Mag. 1(3), 25–40 (2000)

[2] Glover, F.: Tabu search and adaptive memory programming—advances, applications and challenges. In: Barr, R., Helgason, R.,Kennington, J. (eds.) Advances in Metaheuristics, Optimizationand Stochastic Modeling Technologies, pp. 1–75. Kluwer, Dordr-echt (1997)

[3] Glover, F., Laguna, M.: Tabu Search. Kluwer, Boston (1998)[4] Goldschmidt, O., Laugier, A., Olinick, E.: SONET/SDH

ring assignment with capacity constraints. Discret. Appl.Math. 129(1), 99–128 (2003)

[5] Grover, W., Martens, R.: Combined ring–mesh optical transportnetworks. Cluster Comput. 7(3), 245–258 (2004)

[6] Johnson, E., Mehrotra, A., Nemhauser, G.: Min-cut cluster-ing. Math. Program. 62(1), 133–151 (1993)

[7] Kennington, J., Olinick, E., Ortynski, A., Spiride, G.: Wavelengthrouting and assignment in a survivable WDM mesh network. Oper.Res. 51(1), 67–79 (2003)

[8] Kim, Y., Lee, Y., Han, J.: A ring–mesh topology design problemfor optical transport network. J. Heuristics 14(2), 83–202 (2008)

[9] Lee, C., Koh, S.: Assignment of add-drop multiplexer (ADM)rings and digital cross-connect system (DCS) mesh in telecom-munication networks. J. Oper. Res. Soc. 52(4), 440–448 (2001)

[10] Lee, Y., Sherali, H., Han, J., Kim, S.: A branch-and-cut algorithmfor solving and intraring synchronous optical network designproblem. Networks 35(3), 223–232 (2000)

[11] Lee, Y., Han, J., Kang, K.: A fiber routing problem in designingoptical transport networks with wavelength division multiplexedsystems. Photonic Netw. Commun. 5(3), 247–257 (2003)

[12] Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimiza-tion. Wiley, New York (1988)

[13] Padberg, M.: The boolean quadric polytope: some characteristics,facets and relatives. Math. Program. 45(1), 139–172 (1989)

[14] Rochat, Y., Taillard, E.: Probabilistic diversification and intensifi-cation in local search for vehicle routing. J. Heuristics 1(1), 147–167 (1995)

[15] Smith, J., Schaefer, A., Yen, J.: A stochastic integer programmingapproach to solving a synchronous optical network ring designproblem. Networks 44(1), 12–26 (2004)

[16] Sutter, A., Vanderbeck, F., Wolsey, L.: Optimal placement ofadd/drop multiplexers: heuristic and exact algorithms. Oper.Res. 46(5), 719–728 (1998)

Author Biographies

Youngjin Kim received Ph.D. degree atthe Department of Industrial Engineer-ing of Korea University in Seoul, SouthKorea, in 2007. From 2007, he works as amanager with Business Strategy ResearchTeam at SK Telecom. His research inter-est includes telecommunication managementstrategy, telecommunication network design,and combinatorial optimization.

Youngho Lee completed Ph.D. degree inIndustrial Systems Engineering at VirginiaTech, and received his bachelor and masterdegrees in industrial engineering from SeoulNational University in Korea. His researchinterest is in combinatorial optimization withapplications to the design of emerging tele-communications networks. He is now a pro-fessor at Korea University and the director ofthe management systems lab. Before coming

to the Korea University, he was a distinguished-member technical-staffat US WEST Advanced Technologies, Boulder, CO. He has been activewith various journals and societies. He has authored and coauthoredover 20 journal publications, and over 50 conference publications. Hehas won awards for his scholarly work.

123


Junghee Han received Ph.D. degree at theDepartment of Industrial Engineering ofKorea University in Seoul, South Korea, in1999. From 2000 to 2004, he worked as asenior engineer with WCDMA R&D groupat LG Electronics. From 2004, as an associateprofessor he is with the faculty of Col-lege of Business Administration of Kang-won National University, South Korea. Hisresearch interest includes telecommunication

network design, frequency assignment, and scheduling problems withemphasis on combinatorial optimization and heuristic algorithm.

123

Documents

Ring–mesh topology design in a SONET–WDM network