On stacking bi-directional self-healing-rings on a … stacking bi-directional self-healing-rings on...

On stacking bi-directional self-healing-rings on a conduit ringq

Byung-Suk Min*, Dong-Wan Tcha

Graduate School of Management, KAIST (Korea Advanced Institute of Science and Technology), 207-43 Cheongryangri-Dong,

Dongdaemun-Gu, Seoul 130-012, South Korea

Accepted 20 February 2003

Abstract

In this paper, we consider a network design problem of stacking multiple bi-directional self-healing-rings

(BSHRs) on a given physical conduit ring in order to meet given demand requirements. The objective is then to

find a cost-minimizing solution which not only specifies how many and what kinds of BSHR rings to install, but

also indicates how to split the demand requirements among the installed BSHRs. The problem has been formulated

as an IP model and then reformulated to a set-partitioning model via the column generation approach. An iterative

solution method of alternating between two phases of generating and fixing columns was developed and tested

with three different demand patterns. Computational experience gained from a total of more than 100 test runs for

each demand pattern strongly supports the real-world applicability of the proposed model and the solution method.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Self-healing-ring; Integer programming; Column generation

1. Introduction

Explosive growth of the Internet has dramatically increased traffic volume between central offices

(COs). Despite the advent of new transmission technologies, the synchronous technology (synchronous

digital hierarchy, SDH, or synchronous optical network, SONET) is expected to play a major role for the

time being in transporting Internet traffic as well as the plain old voice traffic (Struyve et al., 2000).

Particularly the self-healing-ring (SHR) is spotlighted due to its resilient capability of providing

survivability and very high-speed service restoration in the event of a single link or node failure. Most

0360-8352/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0360-8352(03)00022-6

Computers & Industrial Engineering 45 (2003) 127–139www.elsevier.com/locate/dsw

q Processed by Area Editor Tom Cavalier.

E-mail address: [email protected] (D.-W. Tcha), [email protected] (B.-S. Min)1

* Corresponding author. Tel.: þ82-2-958-3377; fax: þ82-2-958-3376.

http://www.elsevier.com/locate/dsw

telecommunication companies have been operating SHR networks, the ring architecture of which is also

adopted by many Internet service providers (ISP).

In SHRs, nodes having Add-Drop Multiplexer (ADM) equipment are linked in a cycle. Each link has

an amount of working capacity that consists of a given number of channels of unit capacity. SHR has two

generic architectures: unidirectional self-healing-ring (USHR, also known as unidirectional path

switched ring (UPSR)) and bi-directional self-healing-ring (BSHR, also known as bi-directional line

switched ring (BLSR)). Generally, BSHR to be addressed in this paper is more resource-efficient than

USHR, since each pair demand is transported bifurcatedly on both ring directions.

In a USHR, traffic between pairs of nodes on the ring travels in a single direction. Thus, a single unit

of reciprocal demand between a pair of nodes (that is, demand both from a node i to a node j; as well as

from j to i) would utilize a single channel on each link, all the way around the ring. In a BSHR, half the

working capacity is oriented in one direction around the ring and half in the other direction. A reciprocal

unit of demand between a pair of nodes is then routed on working capacity on both directions, using only

the links on the corresponding ring side. Therefore for a demand of multiple units between a pair of

nodes i and j; with i , j; there could be two different sets of links, one on each ring side (i.e. the

‘clockwise’ set, passing through nodes {i; i þ 1;…; j 2 1; j} and the ‘counter-clockwise’ set, passing

through nodes {j; j þ 1;…; n; 1;…; i 2 1; i}). The problem of bifurcating each integral demand into such

two sets of links on the different ring sides is called the ring-loading problem in the BSHR. Exemplar

operations in both architectures are described in Fig. 1. Note the significant difference in the loads

imposed on both USHR and BSHR. For details of the demand transporting operation of each

architecture, see Klincewicz, Luss, and Yan (1998) and Wu (1992).

Consider a region with a single isolated conduit ring, wherein the traffic demand to cover far exceeds a

single BSHR’s capacity. Then multiple BSHRs should be deployed in a stacked manner on the conduit

ring. Fig. 2 shows a real instance of three stacked BSHRs operated by an ISP in Korea. The decision

Fig. 1. Illustrative operations of USHR and BSHR.

B.-S. Min, D.-W. Tcha / Computers & Industrial Engineering 45 (2003) 127–139128

problems naturally arising from the situation are then how many BSHRs to place and how to allocate

node pair demands to each established BSHR. Assumed from the real-world practice is that BSHRs have

equal capacity which is large enough to cover any single pair demand.

In the literature, there are only a few related studies, but most of them have addressed the decision

problem for the USHR, which is easier to deal with than BSHR. The virtually same problem with ours

but on the USHR was studied by Lee, Sherali, Han, and Kim (2000) and Sutter, Vanderbeck, and Wolsey

(1998), who, respectively, presented a branch-and-cut algorithm and a branch-and-price algorithm for

the solution. Similar problems on the USHR were also dealt with by Armony, Klincewicz, Luss, and

Rosenwein (2000) and Sherali, Smith, and Lee (2000). Besides this setting with a given single conduit

ring, there are studies dealing with placing multiple USHRs in a given region, but on differing multiple

conduit rings (Kang, Lee, Park, Park, & Kim, 2000; Klincewicz et al., 1998; Luss, Rosenwein, & Wong,

1998).

Stacking BSHRs, however, has not been seriously studied as yet, to our knowledge, except by

Gawande and Klincewicz (1999). This scarcity, despite the practical importance of resource-efficient

BSHRs, may be attributed to the problem complexity that the subproblem of intra-ring demand loading

on a BSHR with given capacity should be solved repetitiously.

In this paper, we consider a problem of stacking BSHRs on a given conduit ring, which covers both

intra-ring demand loading and demand partitioning into multiple BSHRs. Note that for the BSHR, the

issue of intra-ring demand loading itself is complicated enough to have been dealt with as the ring

loading problem in the literature (Cosares & Saniee, 1994; Myung, 2003; Myung, Kim, & Tcha, 1997;

Schrijver, Seymour, & Winkler, 1998), contrary to the simple USHR case. Between each node pair of the

Fig. 2. Three stacked BSHRs (17 nodes).

* Corresponding author. Tel.: þ82-2-958-3377; fax: þ82-2-958-3376.

B.-S. Min, D.-W. Tcha / Computers & Industrial Engineering 45 (2003) 127–139 129

conduit ring, given is a demand which is small enough to be covered by a single installed ring. We

assume a cost structure in which placing optical fibers along the conduit incurs negligibly small cost

compared to installing ADMs. The objective of our design problem is then to minimize the required

number of ADMs when stacking BSHRs under the following constraints: each node pair demand be

assigned to a single BSHR, the demand loaded on a BSHR should not exceed its capacity, common to all

BSHRs, when the demand between node pair, u and v; is loaded on a BSHR, then an ADM should be

installed on that BSHR at each of the two nodes, u and v:

Now the following is given to highlight the difference of stacking BSHRs from stacking USHRs.

Consider SHRs with the capacity of STM-4 (622 Mbps) which can transport 12 E3 (45 Mbps) demand

units. Fig. 3 shows an exemplar conduit ring. Also given is a demand graph where an edge corresponds

to a demand between its two end nodes, with the number representing its demand size measured in E3

units. In Fig. 4, the optimal solution with two stacked BSHRs is illustrated, the configuration of which

provides reasoning why an elaborate partitioning and loading scheme is needed for the BSHR case. For

each of the two constituent BSHRs, given, respectively, in Fig. 4(a) and (b) are ADM locations marked

with dark nodes, the node pair demands covered, and the associated intra-ring demand load-balancing

(routing).

Fig. 5 is given to compare the USHR mechanism with the BSHR. With demand routing allowed only

in the clockwise direction, the solution shows four stacked USHRs requiring a total of eleven ADMs.

Note the significant reduction in the number of ADMs from eleven for USHR to eight for BSHR. This is

large enough to offset a small increase in per ADM equipment cost, rendering the BSHR system more

cost-effective. Though digressed, the reader is reminded of the other merits of the BSHR in providing

better traffic transport services than USHR, due to the shortened length of transmission paths.

The remainder of this paper is organized as follows. In Section 2, an integer programming formulation

is given for the problem. Section 3 summarizes our solution approach, and Section 4 presents

computational results. Several remarks and further study issues are given in Section 5.

2. Problem formulation

A conduit ring infrastructure is given with node set N; sequentially ordered and indexed by i [ N ¼

{1;…; n} ðn $ 2Þ: Let L ¼ {ð1; 2Þ; ð2; 3Þ;…; ðn 2 1; nÞ; ðn; 1Þ} be the set of links on the ring. For each

node pair e ¼ ði; jÞ; dij or de (assumed to be less than or equal to the capacity of a BSHR), denotes the

number of required channels to carry the traffic demand between nodes i and j; and oðeÞ and dðeÞ denote

Fig. 3. An example of the problem.


the origin and the destination of demand e; respectively. Under the assumption of the symmetric demand

pattern (that is, dij ¼ dji), we assume, without loss of generality, oðeÞ , dðeÞ for each demand pair. Note

that this simple indexing comes from the ring structure of the conduit system, which would otherwise be

quite involved.

An input instance of the BSHR design problem can then be represented as a demand graph G ¼

ðN;EÞ; where E is the set of edges, each edge corresponding to a positive demand pair, that is, E ¼

Fig. 4. BSHR optimal solution.

Fig. 5. USHR optimal solution.


{e ¼ ði; jÞli , j; de . 0}: Let m ¼ lEl: Define b as the capacity of a BSHR that is measured by

the number of channels available in a constituent link l [ L: Let R ¼ {1;…; r} be the index set of

BSHRs to install (evidently, r # m). Also, for A # E; denote GðAÞ and NðAÞ as the subgraph induced by

edge subset A and the node set of GðAÞ; respectively.

As noted in Section 1, the demand e ¼ ðoðeÞ; dðeÞÞ can be transported in two directions. We say that a

flow is routed in the clockwise [counter-clockwise] direction if a flow passes through the node sequence

{oðeÞ; oðeÞ þ 1;…; dðeÞ2 1; dðeÞ} ½{oðeÞ; oðeÞ2 1;…; 1; n;…; dðeÞ þ 1; dðeÞ}�: Further, for each e [ E;

let Lþe ¼ {ði; i þ 1Þ [ LloðeÞ # i , dðeÞ} and L2

e ¼ L\Lþe ; that is, Lþ

e and L2e are the set of links

contained in the clockwise and counter-clockwise direction paths from oðeÞ to dðeÞ; respectively. Finally,

for each l [ L; let Eþl ¼ {e [ Ell [ Lþ

e } and E2l ¼ {e [ Ell [ L2

e } ¼ E\Eþl ; that is, Eþ

l and E2l are the

sets of edges whose clockwise and counter-clockwise direction paths pass l; respectively.

Then, our design problem of stacking BSHRs can be conceptualized as an edge-capacitated graph

partition problem. That is, its objective is to partition the edge set E into a collection of edge subsets

{E1;…;Er} such thatS

1#r#r Er ¼ E; Ep > Eq ¼ B for all p – q [ R; and each subset Er satisfies the

BSHR capacity constraint, so that the sum of subset costs might be minimized. Note that the cost of an

edge subset Er is lNðErÞl; the number of nodes in the subgraph induced thereby, which is the number of

the constituent ADMs of the corresponding BSHR.

For model formulation, we further define the following. Let f ri ¼ 1 if an ADM is installed at node i for

BSHR r; and f ri ¼ 0 otherwise. For each e ¼ ði; jÞ [ E and each r [ R; the 0–1 variable gr

e denotes

whether or not demand e is assigned to BSHR r: Also, hrþe and hr2

e denote the integral parts of de routed

in the clockwise and counter-clockwise direction on BSHR r; respectively. We now present an integer

program (P) for our problem

ðPÞ minXi[N

Xr[R

f ri ð1Þ

s:t:Xr[R

gre ¼ 1; e [ E ð2Þ

gre # f r

i and gre # f r

j ; e ¼ {i; j} [ E and r [ R ð3Þ

hrþe þ hr2

e ¼ de·gre; e [ E and r [ R ð4ÞX

e[Eþl

hrþe þ

Xe[E2

l

hr2e # b; l [ L and r [ R ð5Þ

f ri ; g

re [ {0; 1}; i [ N; e [ E; r [ R ð6Þ

hrþe ; hr2

e [ {0; 1;…; de}; e [ E and r [ R ð7Þ

Constraint (2) ensures that each pair demand be covered by a single BSHR. Constraint (3) enforces two

ADMs to be installed at both nodes i and j on BSHR r; if demand e [ E is covered by BSHR r:

Constraints (4) shows that each demand loaded on a BSHR can be transported bifurcatedly in both

directions, and Eq. (5) indicates the capacity limit b on link l [ L:

The formulation (P) is very weak, rendering lower bounds very loose (see the study on USHR

by Sutter et al., 1998). Furthermore, (P) has too many 0–1 decision variables, which makes

the direct application of the branch-and-bound process, seemingly the only viable approach,


not desirable at all. This motivates us to develop an alternative formulation as in Sutter et al.

(1998).

Consider an admissible BSHR k covering the set of demand edges Ek with the cost ck ¼ lNk ¼

NðEkÞl: This leads to the following set partitioning master problem (MP).

Master ProblemðMPÞ minXk[K

ckzk ð8Þ

s:t:Xk[K

ykezk ¼ 1; e [ E ð9Þ

0 # zk # 1; k [ K ð10Þ

zk integer; ; k [ K ð11Þ

where K is the set of such admissible BSHRs, the 0–1 coefficient yke is 1 if demand edge e is covered by

BSHR k; and the 0–1 variable zk is 1 if BSHR k is selected to be stacked. Each column vector yk ¼ ðykeÞ is

feasible to

ðx; y;wþ;w2Þ xi; ye [ {0; 1};i [ N and ;e [ E;wþ

e ;w2e [ {0;…; de};e [ E

G ¼X

e[Eþl

wþe þ

Xe[E2

l

w2e # b; ;l [ L

ye # xi and ye # xj; ;e ¼ ði; jÞ [ E

wþe þ w2

e ¼ de·ye; ;e ¼ ði; jÞ [ E

ð12Þ

or by substituting w2e ¼ deye 2 w2

e and redefining variables,

ðx; y;wÞ xi; ye [ {0; 1};i [ N and ;e [ E; we [ {0;…de};e [ E

G ¼X

e[Eþl

we þX

e[E2l

ðdeye 2 weÞ # b; ;l [ L

ye # xi and ye # xj; ;e ¼ ði; jÞ [ E

we # de·ye; ;e ¼ ði; jÞ [ E

ð13Þ

Note that the reformulation is enabled by the column generation approach, which has successfully been

applied for solving many complicated combinatorial problems arising in such areas as routing and

scheduling (see Nemhauser & Wolsey, 1998).

3. Solution method with generating and fixing columns

A solution approach is now described for finding a near-optimal solution to our problem. A column

generation scheme is used to obtain an admissible BSHR, combined with a column fixing operation to

improve the incumbent solution with small computational burden. A flowchart in Fig. 6 is provided to

overview the solution process.

Consider the LP relaxation of (MP), denoted by (MPL). Since the number of admissible BSHRs is

exponentially increasing in problem size, it is impractical to fully enumerate all the BSHRs for solving


Fig. 6. Flowchart of our algorithm.


(MPL). Thus, we start with only a few columns, and more columns are generated to be included as the

solution process for (MPL) proceeds. This column-generating process is formalized as the admissible

BSHR generation problem (ABGP). Let ue be the dual variable associated with constraint (2), then the

pricing subproblem, or (ABGP), takes the following form:

ðABGPÞ z ¼ minXi[N

xi 2Xe[E

ueye : ðx; y;wþ;w2Þ [ G

( )ð14Þ

If no column (BSHR) is found admissible any further, i.e. if z $ 0; a feasible solution of (MPL) at hand

is optimal. Otherwise a column corresponding to the obtained z ð, 0Þ is added to the MPL for the next

pivot operation. If the LP optimal solution is integral, then it is optimal to (MP) as well.

When the optimal solution of (MPL) is not integral, we define a restricted version of (MP), denoted by

(RMP), the constituent columns of which are confined to those (BSHRs) generated during the (MPL)

solution process. A branch-and-bound (B&B) procedure is applied to generate an integer optimal

solution of (RMP), and thus a near optimal solution of (MP). Due to the availability of a lower bound, the

optimal objective value of (MPL), the quality of an integer solution generated by the B&B procedure can

be evaluated.

To improve the quality of the integral solution of (MP) at hand, acting as the incumbent upper bound,

we consider a constrained (MP), denoted by (CMP), which is obtained from the (MP) by fixing one 0–1

variable zk at one, i.e., by fixing one column (BSHR) to open. Since reduced costs are readily available at

the final simplex tableau when solving the LP relaxation of the (CMP), that is (CMPL), the column

having the smallest reduced cost is naturally chosen as a fixing one. Though one and only one zk is

further fixed at the (CMP), it brings forth the effect of fixing a large number of related y�es, thereby

reducing the resultant size of (ABGP) by that much. In other words, the column generation solution

process for the (CMPL) requires much less computation than for its predecessor (MPL).

To facilitate understanding, consider a grand iteration as consisting of a number of column-fixing

iterations, which begins with defining a (CMP) by fixing some (BSHR) to open, solves the associated

(CMPL), and ends with obtaining an integral solution of (CMP), and thus of (MP), by applying the B&B

procedure for solving the associated version of (RMP). The number of BSHRs installed on a single

conduit ring is at most half a dozen for almost all practical input data. This implies that only after several

such fixing iterations, a grand iteration ends at which no further fixing is possible, i.e., at which an

integer solution of (MP) is found. Note that the computational load required for later fixing iterations is

far less than for an earlier one, as explained in the previous paragraph. If an integer solution of (MP)

obtained after a grand iteration is not satisfactory, one may repeatedly apply a number of grand iterations

until a desirable level of integer solution is obtained.

4. Computational results

The proposed algorithm has been tested on conduit rings of small, medium, and large size (lNl ¼ 15;

20, 25). We considered demand graphs from sparse to dense with the numbers of demand-edges, i.e. the

numbers of node-pairs with demand, (lEl ¼ 20; 25, 30, 45 60). From among 3 £ 5 ¼ 15 possible

combinations of graph scale (lNl; lEl), selected are six scales. For each selected scale, we considered

three different graph types, star, ring and mesh, by placing the given number of demand-edges roughly in


accordance with the type, which is to check how sensitive our algorithm is to the demand pattern. For

each of 6 £ 3 ¼ 18 resulting demand graphs, five input instances have been generated, by arbitrarily

assigning a demand size on each demand edge from among four values, 1, 2, 3 and 4 units. In the

assignments, minor adjustment was made to fit reality, i.e. the occurrences of large-sized edge demands

are given less frequently.

The solution procedures were coded in C, and run on a Pentium PC (350 MHz), using the CPLEX

callable mixed integer library as an eventual subproblem solver. The computational results are

summarized in Tables 1–3 with one table for each graph type. In the tables, NAME indicates one of five

input instances for the corresponding graph scale and type; CPU the total CPU time measured in

seconds; COL the number of columns (SHRs) generated; LB the lower bound (the optimal objective

value of MPL); UB1 the upper bound (the objective value of MP obtained prior to column-fixing

iterations); UB the tightened upper bound (the objective value of MP at termination), and GAP

represents the normalized gap between LB and UB.

Throughout the experiments, only a single grand iteration is applied to solve each input instance. Still,

near optimal solutions within the normalized gap of 10% were obtained within reasonable time with a

few exceptions. Note that the reported gaps are slightly larger for dense graphs, and that they are in

general smallest for the star-type, then for the mesh-type, and the largest for the ring-type.

It is noteworthy to find the definitive role of the column-fixing scheme in improving the solution

quality. The scheme reduced on the average the relative gaps by 10.8% for 78 of the 90 instances. Such

occurrences of gap reduction tend to be higher as the scale of the demand graphs, i.e., the numbers of

Table 1

Results for the star-type demand pattern

NAME CPU COL LB UB1 UB GAP (%) NAME CPU COL LB UB1 UB GAP (%)

(a) (lNl; lEl) ¼ (15, 20) (b) (lNl; lEl) ¼ (15, 30)

1 6 81 19 21 20 0.00 1 36 200 21.9 27 23 4.30

2 6 108 18.3 19 19 0.00 2 37 188 22.3 24 24 4.20

3 5 70 20 22 20 0.00 3 24 192 21.2 25 24 8.30

4 10 103 17.7 21 19 5.30 4 30 234 22.6 29 25 8.00

5 8 103 18.6 22 20 5.00 5 31 200 23.1 24 24 0.00

(c) (lNl;lEl) ¼ (20, 25) (d) (lNl; lEl) ¼ (20, 45)

18 153 23.7 25 25 4.00 1 245 403 34 38 37 8.10

2 21 122 24.1 27 25 0.00 2 239 391 34.7 41 38 7.90

3 17 168 23.5 28 24 0.00 3 479 366 31.5 38 34 5.90

4 18 164 22.7 27 23 0.00 4 151 399 35.5 43 38 5.30

5 25 149 23.9 28 24 0.00 5 130 349 33.5 39 36 5.60

(e) (lNl; lEl) ¼ (25, 30) (f) (lNl; lEl) ¼ (25, 60)

1 30 174 29.2 35 30 0.00 1 709 683 42.9 50 48 10.40

2 27 179 28.4 34 30 3.30 2 1,496 569 44.7 56 49 8.20

3 37 230 28.2 31 29 0.00 3 2,145 637 44.7 55 47 4.30

4 28 170 29.8 34 30 0.00 4 1,989 592 44.4 54 48 6.30

5 33 194 29.5 33 30 0.00 5 1,574 685 40.4 54 43 4.70

CPU: total CPU time measured in seconds. COL: number of generated columns. LB: lower bound (the optimal objective

value of MPL). UB1: upper bound (the objective value of MP without column fixing iterations). UB: tightened upper bound (the

objective value of MP at termination). GAP: normalized gap between LB and UB. GAP ¼ ((UB-dLBe)/UB).


Table 2

Results for the mesh-type demand pattern


(a) (lNl; lEl) ¼ (15, 20) (b) (lNl; lEl) ¼ (15, 30)

1 11 89 18.4 20 19 0.00 1 53 194 20.1 26 23 8.70

2 10 108 14.4 18 16 6.30 2 52 179 22.7 27 25 8.00

3 11 84 18 18 18 0.00 3 83 213 19.5 27 22 9.10

4 11 77 14.2 16 15 0.00 4 41 145 23.7 29 26 7.70

5 12 97 18.4 19 19 0.00 5 44 189 23.4 28 26 7.70

(c) (lNl, lEl) ¼ (20, 25) (d) (lNl; lEl) ¼ (20, 45)

1 34 136 19.3 22 20 0.00 1 1,782 368 32.2 40 36 8.30

2 60 123 21.9 25 23 4.30 2 1,619 356 36.7 43 39 5.10

3 37 141 20.1 22 22 4.50 3 1,344 367 32.6 41 35 5.70

4 49 135 22.1 26 25 8.00 4 648 330 35.4 41 38 5.30

5 55 149 18.3 21 21 9.50 5 557 334 33.5 40 36 5.60

(e) (lNl; lEl) ¼ (25, 30) (f) (lNl; lEl) ¼ (25, 60)

1 121 218 21.8 22 22 0.00 1 3,352 555 46.1 55 50 6.00

2 84 215 22.5 29 25 8.00 2 3,153 634 43.9 54 50 12.00

3 156 194 24.1 29 27 7.40 3 4,530 505 45.9 55 50 8.00

4 153 182 22.9 27 25 8.00 4 3,475 580 45.5 56 49 6.10

5 134 262 22.5 26 24 4.20 5 6,252 558 39.7 52 44 9.10

Table 3

Results for the ring-type demand pattern


(a) (lNl; lEl) ¼ (15, 20) (b) ðlNl; lElÞ ¼ ð15; 30Þ

1 11 91 14.5 18 16 6.30 1 55 180 19.1 26 22 9.10

2 8 101 14.4 15 15 0.00 2 59 184 21 29 23 8.70

3 17 80 17.3 19 19 5.30 3 58 199 18.1 25 21 9.50

4 13 95 14.1 17 15 0.00 4 79 192 19.2 25 22 9.10

5 10 85 15.8 18 17 5.90 5 63 229 20.8 27 23 8.70

(c) ðlNl; lElÞ ¼ ð20; 25Þ (d) ðlNl; lElÞ ¼ ð20; 45Þ

1 74 154 16.9 18 17 0.00 1 2,362 388 29.4 41 33 9.10

2 117 161 19.4 24 20 0.00 2 3,144 424 29.8 41 33 9.10

3 117 180 18.5 21 21 9.50 3 4,404 423 27.1 42 31 9.70

4 42 143 18.7 22 21 9.50 4 1,205 433 31.2 40 35 8.60

5 71 137 16.9 20 18 5.60 5 849 340 31 37 35 8.60

(e) ðlNl; lElÞ ¼ ð25; 30Þ (f) ðlNl; lElÞ ¼ ð25; 60Þ

1 224 267 19.5 25 22 9.10 1 23,790 750 37 53 41 9.80

2 82 219 21.3 25 22 0.00 2 16,913 672 38.6 57 43 9.30

3 251 198 20.8 26 22 4.50 3 17,384 669 39.3 55 46 13.00

4 200 279 20.9 22 21 0.00 4 21,072 687 38.5 55 44 11.40

5 134 245 19.9 25 22 9.10 5 22,662 781 35.1 56 42 14.30


nodes and edges, increases. This observation indirectly hints that the column generating process has

selected profitable columns well enough. Also observed as expected is that CPU time increases as the

scale and the density of demand graphs increase.

Most of the computation time (89–99%) was spent on solving the subproblem ABGP. Close

inspection reveals that solving a single subproblem itself is a determining factor, since the total number

of generated columns seems generally dependent on the number of demand edges of the given demand

graph. If one wants to improve the overall efficiency of the solution method, more discretion is thus

required on upgrading the efficiency of the subproblem ABGP.

5. Conclusion

This paper has dealt with a new network design problem of stacking multiple BSHRs on a conduit

ring. This problem is of great concern to Telcos and ISPs. However, it has not been extensively

addressed as yet. This scarcity, despite the practical importance of resource-efficient BSHRs, may be

attributed to the problem complexity.

The problem has been formulated as a natural IP model and reformulated as a set-partitioning model

via the column generation approach. An iterative solution method of alternating between two phases of

generating and fixing columns was developed and tested with demand graphs of six different and

practical scales. Computational experience gained from solving a total of 3 £ 6 £ 5 ¼ 90 test problems

strongly supports the real-world applicability of the proposed model and the solution method.

Furthermore, the proposed solution framework is flexible enough to easily accommodate the real-

world features like various ADM capacities or the other architecture (USHR) by simply modifying the

subproblem structure. An immediate extension of practical importance would be to stack both types of

SHRs on a given conduit ring infrastructure.

Acknowledgements

The authors are indebted to an anonymous referee for considerably improving the presentation.

References

Armony, M., Klincewicz, J. G., Luss, H., & Rosenwein, M. B. (2000). Design of stacked self-healing rings using a genetic

algorithm. Journal of Heuristics, 6, 85–105.

Cosares, S., & Saniee, I. (1994). An optimization problem related to balancing loads on SONET rings. Telecommunications

Systems, 3, 165–181.

Gawande, M., & Klincewicz, J. G. (1999). Designing a family of bi-directional self-healing-rings. SPIE Conference on

Performance and Control of Network Systems III, 3841, 210–218.

Kang, D., Lee, K., Park, S., Park, K., & Kim, S. (2000). Design of local networks using USHRs. Telecommunications Systems,

14, 197–217.

Klincewicz, J. G., Luss, H., & Yan, D. C. Y. (1998). Designing tributary networks with multiple ring families. Computers and

Operations Research, 25(12), 1145–1157.

Lee, Y., Sherali, H. D., Han, J., & Kim, S. (2000). A branch-and-cut algorithm for solving an intra-ring synchronous optical

network design problem. Networks, 35(3), 223–232.


Luss, H., Rosenwein, M. B., & Wong, R. (1998). Topological network design for SONET ring architecture. IEEE transactions

on System, Man, and Cybernetics, 28(6), 780–790.

Myung, Y. S. (2003). An efficient algorithm for the ring loading problem with integer demand splitting. SIAM Journal on

Discrete Mathematics, in press.

Myung, Y. S., Kim, H. G., & Tcha, D. W. (1997). Optimal load balancing on SONET bi-directional rings. Operations Research,

45(1), 148–152.

Nemhauser, G. L., & Wolsey, L. A. (1998). Integer and combinatorial optimization. New York: Wiley.

Schrijver, A., Seymour, P. D., & Winkler, P. (1998). The ring loading problem. SIAM Journal on Discrete Mathematics, 11(1),

1–14.

Sherali, H. D., Smith, J. C., & Lee, Y. (2000). Enhanced model representations for an intra-ring synchronous optical network

design problem allowing demand splitting. INFORMS Journal on Computing, 12(4), 284–298.

Struyve, K., Wauters, N., Falcao, P., Arijs, P., Colle, D., Demeester, P., & Lagasse, P. (2000). Application, design, and

evolution of WDM in GTS’s pan-European transport network. IEEE Communication Magazine, 38(3), 114–121.

Sutter, A., Vanderbeck, F., & Wolsey, L. A. (1998). Optimal placement of add/drop multiplexers: Heuristic and exact

algorithms. Operations Research, 46(5), 719–728.

Wu, T. H. (1992). Fiber network service survivability. Artech House.


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