WAVELENGTH ROUTING AND ASSIGNMENT
IN A SURVIVABLE WDM MESH NETWORK
JEFFERY KENNINGTON and ELI OLINICK
Department of Engineering Management, Information, and Systems, School of Engineering,
Southern Methodist University, Dallas, Texas 75275-0123
[email protected] • [email protected]
AUGUSTYN ORTYNSKI and GHEORGHE SPIRIDE
Nortel Networks, Richardson, Texas 75082
[email protected] • [email protected]
All-optical networks with wavelength division multiplexing (WDM) capabilities are prime candidates for future wide-area backbone networks. The simplified processing and management of these very high bandwidth networks make them very attractive. A procedure for
designing low cost WDM networks is the subject of this investigation.
In the literature, this design problem has been referred to as the routing and wavelength assignment problem. Our proposed solution
involves a three-step process that results in a low-cost design to satisfy a set of static point-to-point demands. Our strategy simultaneously
addresses the problem of routing working traffic, determining backup paths for single node or single link failures, and assigning wavelengths
to both working and restoration paths.
An integer linear program is presented that formally defines the routing and wavelength assignment problem (RWA) being solved along
with a simple heuristic procedure. In an empirical analysis, the heuristic procedure successfully solved realistically sized test cases in under
30 seconds on a Compaq AlphaStation. CPLEX 6.6.0 using default settings required over 1,000 times longer to obtain only slightly better
solutions than those obtained by our new heuristic procedure.
Received July 2000; revisions received February 2001, November 2001; accepted November 2001.
Subject classifications: Optical networks: wavelength routing. Network design: provisioning.
Area of review: Telecommunications.
regional networks. In the literature such subnetworks are
referred to as islands of transparency (see Saleh 2000).
Because WDM transport networks are designed to carry
high volumes of traffic, network failures may have severe
consequences. Since the bandwidth lost due to link or node
failures is much larger than what would be lost in either
SONET or ATM networks, fast restoration is indispensable. Due to restoration speed considerations, most designers would prefer a simple distributed restoration algorithm
as opposed to a centralized restoration procedure.
For this investigation, we say that a network is survivable
if sufficient spare capacity exists such that operations can
be restored after either a single link or a single node failure.
We assume that a failed optical path can be detected by
measuring signal strength at the two end nodes of a path.
Our design architecture uses precomputed backup paths and
assumes that the electronic components at the end nodes of
a working path can switch to the backup path.
Designing the least-cost survivable WDM network to
satisfy a given set of static point-to-point demands can
be quite challenging compared to designing SONETbased networks as described in Grover and Stamatelakis
(1998), Herzberg and Bye (1994), Herzberg et al. (1995),
Irascko et al. (1998), Kennington and Lewis (1999), and
Kennington and Whitler (1999). Not only must the pointto-point demands be routed, but backup paths must be
determined and wavelengths must be assigned for both the
1. INTRODUCTION
All-optical networks using wavelength-division multiplexing (WDM) and optical switching are being deployed to
help meet the exponential growth of demand for bandwidth. A given node may transmit optical signals on different wavelengths that are coupled into a single fiber using
wavelength multiplexers. In an all-optical WDM network,
a logical connection between a pair of nodes, say o d,
is a path or route composed of a sequence of links from
o to d called a lightpath. The assignment of wavelengths
along a lightpath should be made such that there are no
conflicts; that is, the wavelengths assigned to lightpaths that
share a link must be distinct. Due to the accumulation of
optical impairments, optical/electrical/optical (O/E/O) conversion may be necessary at some places in the network.
When O/E/O conversion is realized, wavelength translation may be provided as well. In the absence of wavelength converters, a lightpath uses the same wavelength on
all links through which it passes. This has been referred
to as the wavelength-continuity constraint (see Banerjee
and Mukherjee 1996). The model presented in this paper
assumes the wavelength-continuity constraint and therefore
applies to networks that have O/E/O conversion, but no
wavelength translation. In addition, the model applies to
subnetworks (such as regional networks) where wavelength
translation only occurs at the boundaries connecting the
0030-364X/03/5101-0067 $05.00
1526-5463 electronic ISSN
67
Operations Research © 2003 INFORMS
Vol. 51, No. 1, January–February 2003, pp. 67–79
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/ Kennington, Olinick, Ortynski, and Spiride
working traffic and the restoration paths. There is a need
for design tools to assist network designers responsible for
solving this problem. The strategies and mathematical models used in constructing these tools must be robust, simple
to understand, and simple to use. This investigation presents
a strategy, an optimization model, a simple heuristic, and
an empirical analysis for this design problem.
1.1. Survey of the Literature
Given a set of connections (point-to-point demands), the
problem of setting up lightpaths and assigning a wavelength
for each connection is called the routing and wavelengthassignment problem (RWA) (Zang et al. 2000). These connection requests may be of three types: static, incremental, or dynamic. With static traffic, the A-Z demand matrix
is known in advance, and we seek to determine a routing
and assignment of wavelengths to minimize or maximize
some metric. Metrics for minimization models include congestion, number of wavelengths required, and cost. Some
models enforce the wavelength-continuity constraint while
others allow for wavelength translation. It is well known
that the various versions of the RWA can be modeled as an
integer linear program (ILP). In addition, it is well known
that all but trivial instances of these ILPs are computationally difficult with current state-of-the-art software such as
CPLEX (CPLEX 1997). However, ILP models are necessary to provide a formal description of the problem and
to help research groups develop effective heuristic procedures. This survey is restricted to recent work on the static
RWA. An excellent survey may also be found in Zang et al.
(2000).
Ramaswami and Sivarajan (1995) present an ILP with
an objective of maximizing total traffic in a given network.
Linear programming was applied to the continuous relaxation of their model which provided a valid lower bound. A
layered-graph model appears in Chen and Banerjee (1995).
In this approach, the entire network is duplicated for each
possible wavelength and the flows from origin to destination can appear in only one of the layers. For problems
having 80 wavelengths, this could result in a very large ILP.
Ramaswami and Sivarajan (1996) presented a node-arcbased ILP that solved the routing problem but ignored
the wavelength-assignment problem. Their objective was
to minimize the maximum load on any single link. They
only experimented with small problems having six nodes.
Banerjee and Mukherjee (1996) developed a two-phase
approach where the routing is accomplished by rounding
the continuous relaxation of an ILP followed by solving a
graph coloring problem to assign wavelengths. Mukherjee
et al. (1996) developed a nonlinear discrete optimization
model for RWA that attempts to minimize the delay. They
place no restriction on the total number of wavelengths
used. Chen and Banerjee (1996) present a node-arc-based
model in which they attempt to maximize throughput.
Banerjee and Mukherjee (1997) developed an ILP that
minimizes the average hop distance. Their model ignores
the wavelength-continuity constraint and does not account
for the cost of translators. Wuttisittikulkij and O’Mahony
(1997a, 1997b) present models and heuristic algorithms for
the RWA on rings. These models implicitly incorporate survivability considerations.
Van Caenegem et al. (1998) present an elaborate ILP
to determine a minimum cost design that also considers
restoration. They developed a simulated annealing-based
heuristic and compared it with the output from CPLEX 5.0
using a 10-minute time limit. Miyao and Saito (1998) also
developed a minimum-cost model that incorporates survivability. Their model permits wavelength translation and
they report computational results on the NSFNET T1 backbone network using CPLEX 6.0. Alanyali and Ayanoglu
(1998) (also in (1999)) present an ILP for RWA that ignores
restoration. They extend their ideas to the restoration problem by considering a fixed working network and sequentially determining restoration paths corresponding to individual failures. CPLEX was used in numerical experiments
on problems having 32 nodes and 50 links. An ILP model
to minimize congestion appears in Krishnaswamy and
Sivarajam (1998). Recovery was not considered and they
report computational experience using a rounding heuristic on a 6-node and a 14-node problem. Several heuristics for RWA appear in Park et al. (1998). Translation is
allowed, but restoration is not considered. Some computational experience with an 18-node European network is
presented. Theoretical results for RWA on rings with and
without translation may be found in Wilfong and Winkler
(1998).
Baroni et al. (1999) present a variety of ILPs for several
different versions of RWA. All their models attempt to minimize the total number of fibers needed to meet demand.
Small problem instances were solved and heuristics were
developed for larger problem instances. Ramamurthy and
Mukherjee (1999) present an arc-path-based ILP that
attempts to minimize total capacity. Their model includes
protection and they present some computational experience
on a 15-node network using CPLEX 4.0. None of their runs
obtained provably optimal solutions within their 12-hour
CPU time limit. Doshi et al. (1999) developed heuristic
methods for several problems that account for restoration.
They report computational results for a large network having 301 nodes, 449 links, and 372 demand pairs. This is the
largest problem that we have seen in the RWA literature.
1.2. Objective of the Investigation
Our objective was to develop a simple strategy for solving the WDM routing and wavelength-assignment problem
that could be used in the development of a WDM design
tool. Our work is for the static-demand case where we
assume that the demands are symmetrical and there is a
limit on the number of wavelengths available, we impose
the wavelength-continuity constraint, and we seek to minimize cost. Restoration is accomplished by precomputing
backup paths that would be stored at both end nodes of
Kennington, Olinick, Ortynski, and Spiride
every demand pair. We require 100% restoration for any
single link or node failure and use backup paths that are
link disjoint from the working paths.
We consider the routing and wavelength-assignment
problems simultaneously using an arc-path or arc-cycle
mixed integer programming (MIP) formulation. Unique
features of our work include the use of layered subnetworks that allow for ring, mesh, and/or hybrid designs.
We model wavelength assignment as a block of consecutive wavelengths rather than using individual decision variables to assign a given wavelength to a given path. That
is, we let p hp denote the smallest (largest) wavelength
assigned to path p. Then, the block of wavelengths p hp are all assigned to path p. This simplifies the models and
makes the model size independent of the maximum number
of wavelengths permitted. Demands can be split to form
smaller block sizes. In the limit, demands can be split into
blocks with a single wavelength each. However, there is a
trade-off between the size of the model and the number of
demands. So, reducing block sizes to the limit results in
an increase in model size. For the static demand case that
we address where all demands must be assigned simultaneously, we believe that assigning blocks of consecutive
wavelengths to each lightpath is reasonable. For operational problems having unknown, dynamic demands that
are assigned as they arrive, using a fixed block size, e.g.
4, 8, etc., as dictated by equipment characteristics, may be
required for implementation. Although developed for the
static case, with appropriate preprocessing our model can
be applied iteratively to solve the dynamic case.
2. THE OPTIMIZATION MODEL
To simplify the problem of wavelength routing and assignment, it has been partitioned into three components. The
layered subnetwork selection component involves determining a set of potential layered subnetworks or architectures that can be used in the final design. The idea is similar to the one used in the SONET Toolkit (see Attanasio
and Hoffman 1993, Cosares et al. 1995). The architectureselection component of the SONET Toolkit allows the
designer to select layered subnetworks from a list that
includes rings of various types, chains, trees, and point-topoint layered subnetworks. The cycle selection component
of our procedure determines a set of potential disjoint paths
associated with the o-d demand pairs. A pair of paths form
a cycle with one path used for the working traffic and the
second reserved as a backup path. The backup path is used
for restoration in the event of a failure in the working traffic path. The third component is an optimization model that
can be used to select paths and layered subnetworks to satisfy the demand and assign wavelengths to these paths.
2.1. Optical Layered Subnetworks
Because we seek to assign wavelengths along pairs of linkdisjoint paths for each o-d demand, the optical layered subnetworks must correspond to subgraphs with the property
/
69
that there are at least two distinct paths between every pair
of nodes. One way to obtain a set of such subgraphs is to
begin with a fundamental set of rings. Hence, each node
in these rings has the property required. This property is
maintained even if additional links are appended to a ring.
Let G = N E
be a graph with node set N and edge
set E. The graph G is said to be acyclic if no cycles can
be formed from N and E. A graph is said to be connected
if, for every distinct pair of nodes i j, a path can be
formed from G that links i and j. A tree is an acyclic
= N
E
is called a subgraph
connected graph. A graph G
spans G.
of G if N ⊂ N and E ⊂ E. If N = N , then G
A tree that spans G is a spanning tree. Let T = N ET be a spanning tree of G and let e ∈ E\ET . Appending e
to T results in a single cycle. The set of cycles formed
by appending each of the edges in E\ET to T is called a
fundamental set of cycles. These ideas have been used in
Hochbaum and Olinick (2001), Kennington et al. (1999),
and Olinick (1999) to obtain minimal cost ring covers.
An example of a graph, a spanning tree, and a set of
fundamental cycles is illustrated in Figure 1. Since we are
interested in mesh networks, additional links can be added
to the cycles to obtain mesh-type layered subnetworks.
By adding the link n7 n9 to Cycle 2 in Figure 1, we
obtain a set of five potential layered subnetworks as illustrated in Figure 2. We assume that each layered subnetwork can be constructed with a wavelength capacity from
W = 4 8 16 20 40 80. Since none of the layered subnetworks in Figure 2 contain both node n2 and node n6 ,
the demand with o-d pair n2 n6 would require the use
of multiple layered subnetworks. Devices such as optical
patch panels (connected to back-to-back WDM multiplexers) or photonic switches may be used to interconnect layers. Due to their increased flexibility in establishing new
connections, photonic switches are assumed to link multiple layers. A photonic switch, as described by Ramaswami
and Sivarajan (1998) and illustrated in Figure 3, allows a
given wavelength to either pass through a node on the same
fiber or be switched to another fiber. It cannot perform optical wavelength translation. Note that in Figure 3, 1 on
the upper fiber is switched to the lower fiber. This permits
lightpath routing between subnetwork layers as illustrated
j
in Figure 4, where ni represents node ni in layer j and ci
represents the photonic switch located at node ni . Note that
the photonic switch at node n9 links subnetworks in layers
1, 2, and 3.
Our strategy for selecting a set of potential layered subnetworks and potential photonic switches is a four-step procedure. First, we select a spanning tree. From this tree we
obtain a fundamental set of cycles and then append additional links to these cycles to obtain a set of potential
layered subnetworks. These additional links could be used
to help satisfy the larger demands whose origin and destination appear in a given subnetwork. Finally, we select
a set of potential photonic switches such that there are
at least two disjoint paths between every pair of original
70
/ Kennington, Olinick, Ortynski, and Spiride
Figure 1.
Examples of fundamental cycles.
n2
n2
n5
n9
n6
n3
n5
n7
n9
n6
n4
n8
n7
n1
n4
a. Example Graph
n2
n3
n1
b. Spanning Tree
Cycle 1
n9
Cycle 2
n5
n8
n9
n3
n6
n3
n7
n7
n4
n5
Cycle 3
n8
n9
n1
n9
Cycle 4
Cycle 5
n6
n6
n7
n4
c. The Five Fundamental Cycles
nodes. For the design illustrated in Figure 4, the two paths
(cycle) linking nodes n5 and n7 are illustrated in Figure 5.
The four-step procedure to obtain a potential design is a
semimanual process that should be fairly easy for an experienced network designer.
Figure 2.
2.2. Optical Cycles
The next step in the design process is to determine a
set of potential cycles corresponding to the A-Z demand
matrix. These cycles are composed of pairs of link-disjoint
paths mapped onto the network produced by the procedure
Potential layered subnetworks.
Subnetwork 2
n9
n6
n3
n7
n3
n4
n2
Subnetwork 3
n7
Subnetwork 1
n8
n5
Subnetwork 5
n9
n9
Subnetwork 4
n5
n9
n6
n7
n4
n6
n1
Kennington, Olinick, Ortynski, and Spiride
Photonic switch functionality.
……
λ1
λ1
W
D
M
W
D
M
λn
λn
λ1
λ1
W
D
M
W
D
M
λn
λn
Passthrough wavelengths
Switched wavelengths
presented in §2.1. We recommend using a successive
shortest-path strategy to obtain a set of cycles for a given
o-d pair.
Suppose we seek a cycle linking nodes n2 and n3 in the
network illustrated in Figure 4. We create two new nodes,
Figure 4.
2.3. The Optimization Model
Given sets of potential layered subnetworks, photonic
switches, and cycles, an optimization model can be constructed to select the least-cost set of layered subnetworks,
photonic switches, and cycles that can be used to satisfy
Potential design with five subnetworks in four layers via six photonic switches.
n21
Layer 1
n51
n91
n31
n 71
c5
n 81
n52
n11
n92
c9
Layer 2
n62
c3
c7
n93
Layer 3
n33
n63
c6
n 73
n43
n94
Layer 4
n 74
n44
n2
Original network
topology
Photonic
Switch
c4
n64
n5
n9
n6
n3
n7
n4
71
a source node s and a sink node t, and add links to the origin and destination nodes as illustrated in Figure 6. Using
edge lengths of 1, the shortest path from s to t is given by
the sequence of nodes s n12 n19 c9 n39 n33 t. If we remove
the edges in this path, excluding those involving s and t,
we obtain the graph illustrated in Figure 7. One shortest
path from s to t in Figure 7 is given by the sequence
s n12 n15 c5 n25 n26 c6 n36 n34 n37 n33 t. These two paths
form a cycle linking nodes n2 and n3 and we have a working path and a backup path for o-d pair n2 n3 . Using
this strategy successively yields a set of potential cycles
for each o-d pair in the A-Z demand matrix. Note that
multiple cycles for a given demand can be produced as
illustrated in Figure 8. After all cycles for an o-d pair
have been produced, all edges are restored and the procedure is repeated for the next o-d pair. While this procedure
can be automated, it can also be executed quite rapidly
using manual procedures if the network is not too large.
……
Figure 3.
/
n8
n1
72
/ Kennington, Olinick, Ortynski, and Spiride
Figure 5.
Cycle for o-d pair (n5 n7 ). Note that n36 represents node n6 in layer 3, and c9 represents the photonic switch
located at node n9 .
n21
n51
n91
n31
n 71
c5
n 81
n 52
n 92
c9
n 62
c3
c7
n 93
c6
n11
n 33
n 63
n 73
n 43
n 94
c4
n 64
n 74
n 44
Figure 6.
Shortest path from node s to node t, which corresponds to original nodes n2 and n3 , respectively. Note that
ci denotes the photonic switch installed at node ni .
s
n 21
n51
n91
n31
n71
c5
n81
n52
n92
c9
n62
n63
n33
n 74
n43
n94
n64
c3
c7
n 93
c6
n11
c4
n44
n 74
t
Kennington, Olinick, Ortynski, and Spiride
Figure 7.
73
New shortest path from node s to node t, which corresponds to original nodes n2 and n3 , respectively. Note
that ci denotes the photonic switch installed at node ni .
n 21
s
n 51
n 91
n 31
n 71
c5
n 81
n 52
c9
c3
t
c7
n 93
c6
n 11
n 92
n62
n 33
n 63
n 73
n 43
n 94
c4
n 64
n 74
n 44
Figure 8.
/
Multiple cycles for o-d pair (n4 n6 ) are illustrated in bold.
n 21
n 51
n 91
n 31
n 71
c5
n 81
n 52
n 92
c9
n 62
c7
n 93
c6
n 63
n 33
n 73
n 43
n 94
n 64
c3
c4
n 44
n 74
n 11
74
/
Kennington, Olinick, Ortynski, and Spiride
all the demands. The model can be used to size the layered
subnetworks and photonic switches selected and assign
wavelengths to the cycles so that total cost is minimized.
For our model we assume that the demands are given in
units of wavelengths and that wavelengths must be reserved
in both directions for a given o-d pair. To simplify the
model we consider only 50% of the possible wavelengths
(80 versus 160) and route demands from o to d. Hence,
all links, layered subnetworks, and photonic switches have
a capacity of 80 wavelengths. Further, we do not allow
for wavelength translation. Hence, all o-d paths used for
both working traffic and restoration use the same set of
wavelengths from origin to destination.
2.3.1. Definition of Sets. There are nine sets used in the
description of the optimization model. Let N denote the
set of nodes and W denote the set of possible wavelengths.
For our empirical testing W = 4 8 16 20 40 80. Let
D denote the set of point-to-point demand nodes o d
with o d ∈ N . Let S C, and P denote the sets of potential layered subnetworks, photonic switches, and cycles,
respectively.
Let p and q be any two cycles from P. If p and q have
at least one link in common, then we say that p and q have
a conflict. All pairs of cycles with conflicts that are used
to help satisfy the demand must use distinct wavelengths.
Let H denote the set of ordered pairs of cycles p q
p q ∈ P p < q having a conflict. Let Jod denote the set of
cycles used to satisfy demand o d ∈ D. Let Ks with s ∈ S
and Lc with c ∈ C denote sets of cycles that use layered
subnetwork s and photonic switch c, respectively.
2.3.2. Definition of Constants. There are only four types
of constants needed to define the mathematical model, three
types of costs, and a demand value. Let asw fcw denote
the construction cost for a layered subnetwork (photonic
switch) of type s ∈ S c ∈ C with size w ∈ W . Let B be a
large positive integer used to penalize unsatisfied demand
and let rod for o d ∈ D denote the demand in units of
wavelengths for the given demand pair.
2.3.3. Definition of Decision Variables. The decision
variables are of three types: continuous, integer, and binary.
Let the integer variable xp denote the number of wavelengths assigned to cycle p. Let the integer variables p
hp denote the smallest (largest) wavelength assigned to
cycle p. If xp = 10, lp = 5, and hp = 14, then the 10 wavelengths in the interval 5 14
are assigned to cycle p. Let
the continuous variable mp denote the midpoint of the interval lp hp . For the interval 5 14
, mp = 9-5.
Let p q ∈ H. Then p and q have a conflict, and therefore the intervals p hp and q hq cannot overlap. Let
the binary variable bpq = 1 if q > hp and bpq = 0 if p > hq .
+
−
Let the continuous variable dpq
= mp − mq dpq
= mq − mp if mp − mq 0 mp − mq < 0, and 0 otherwise.
Let the continuous variables uod denote the unsatisfied
demand for the pair o d ∈ D. Let the binary variables
ysw zcw be 1 if layered subnetwork s (photonic switch c)
with size w ∈ W is constructed, and 0 otherwise.
2.3.4. Definition of Constraints. There are nine types of
constraints needed in the model. The first set of constraints,
which ensure demand satisfaction, are given as follows:
p∈Jod
xp + uod = rod
∀ o d ∈ D-
(1)
The second set of constraints assign wavelength intervals
to cycles used to satisfy demands. These are given by
h p = p + xp − 1
∀ p ∈ P-
(2)
Note that when xp = 0, (2) implies that hp = p − 1. That
is, cycle p is assigned zero wavelengths from the empty
interval p p − 1
.
The third set of constraints determine the interval midpoints and are simply
mp =
p + hp 2
∀ p ∈ P-
(3)
The fourth type of constraint determines the distance
between intervals of pairs of cycles having a conflict. These
constraints are
+
−
mp − mq = dpq
− dpq
∀ p q ∈ H -
(4)
The next set of constraints force nonoverlapping intervals for cycles that have a conflict. These constraints are
+
−
dpq
+ dpq
xp + xq 2
∀ p q ∈ H -
(5)
The sixth set of constraints ensure distinct wavelength
assignment. These constraints take the form
+
dpq
801 − bpq ∀ p q ∈ H (6)
−
dpq
80bpq
(7)
∀ p q ∈ H -
The next set of constraints size the layered subnetworks
and photonic switches appropriately. That is, they ensure
that the layered subnetworks and photonic switches are
large enough (i.e., have the right size) to handle the largest
wavelengths routed across them. These constraints take the
mathematical form
w∈W
w∈W
wysw hp
∀ s ∈ S p ∈ Ks (8)
wzcw hp
∀ c ∈ C p ∈ Lc -
(9)
The eighth type of constraint ensures that layered subnetworks and photonic switches must be of a single size.
These constraints take the form
w∈W
w∈W
ysw 1
∀ s ∈ S
(10)
zcw 1
∀ c ∈ C-
(11)
Kennington, Olinick, Ortynski, and Spiride
Finally, the variables have bounds and integrality restrictions that must be enforced. These are as follows:
0 xp rod lp 1
o d ∈ D p ∈ Jod (12)
∀ p ∈ P
0 hp 80
mp 0
∀ p ∈ P
(14)
∀ p ∈ P
+
−
dpq
dpq
0
uod 0
(13)
(15)
∀ p q ∈ H (16)
∀ o d ∈ D
(17)
xp p hp are integer ∀ p ∈ P
(18)
ysw is binary ∀ s ∈ S w ∈ W (19)
zcw is binary ∀ c ∈ C w ∈ W (20)
bpq is binary ∀ p q ∈ H -
(21)
2.4. Definition of Objective Function
The objective is to minimize construction cost for satisfying demand. If the demand cannot be completely satisfied
using the cycles available, then demand can be assigned to
the unsatisfied demand variables (uod ) at a high cost. The
objective function is as follows:
minimize
asw ysw +
fcw zcw + B
uod s∈S w∈W
c∈C w∈W
(22)
od∈D
3. A HEURISTIC ALGORITHM
Our mathematical model for the wavelength routing and
assignment problem is (1)–(22) which we call Q. It has
been shown that a special case of Q where there is only
one layered subnetwork (i.e., S = 1) is NP-hard (Chlamtac
et al. 1992). Thus, Q is inherently difficult and even small
Figure 9.
bpq = 0
∀ p q ∈ R0 (23)
bpq = 1
∀ p q ∈ R1 (24)
0 bpq 1
∀ p q ∈ H -
hp
lq
lp
hq
bpq = 1
Block p to the
left of block q
hp
0 < bpq < 1
Wavelength
interference
lq
hq
lq
hq
lp
75
problem instances may require many hours of CPU time to
obtain a provably optimal solution. Consequently, practical
techniques for this problem must rely upon heuristics that
exploit the underlying problem structure.
In this section, we present a heuristic that involves solving a series of problems related to Q that are substantially
easier to solve than Q. The LP relaxation for model Q is
very weak. That is, the gap between the LP lower bound
and the MIP optimum is relatively large. The LP relaxation
of Q allows for conflicting lightpaths to share wavelengths,
whereas the MIP prohibits such conflicts. Our heuristic is
constructed to successively resolve these conflicts by carefully fixing certain subsets of the binary decision variables.
This strategy allows us to quickly find paths from the root
of the branch-and-bound tree to feasible solutions.
At each step, some of the bpq are fixed to either 0 or
1 and the remainder are permitted to assume any value in
the interval 0 1
. Since each successive problem involves
fixing additional bpq variables without changing the value
of any variable previously fixed, this strategy quickly leads
to a feasible solution for Q. By applying this simple strategy multiple times and saving the best solution found, we
hope to obtain a very good solution in a fraction of the
time required to obtain an optimum.
Recall that H is the set of pairs of cycles with a conflict and that bpq ∈ 0 1 for each p q ∈ H . If bpq is 1
(0), then the wavelengths (i.e., p hp and q hq ) will be
distinct with p hp to the left (right) of q hq as illustrated in Figure 9. Let R0 and R1 be subsets of H such that
R0 ∩ R1 = . Define a new set of constraints as follows:
Relationship between wavelength assignment for cycles having a conflict and the variable bpq .
lp
/
hp
bpq = 0
Block q to the
left of block p
(25)
76
/ Kennington, Olinick, Ortynski, and Spiride
Let IPR0 R1 be the problem defined by (1)–(20), (22),
and (23)–(25). That is, IPR0 R1 is obtained from Q
by replacing (21) with (23), (24), and (25). Let F Q
and F (IPR0 R1 ) denote the feasible regions of Q and
IPR0 R1 , respectively. Since the objective values are
identical and F Q ⊂ F (IPR0 R1 ), then IPR0 R1 is a
relaxation of Q (see Geoffrion and Marsten 1972). Let
X = x h y z u b denote the vector representation of
=
the decision variables for both Q and IPR0 R1 . Let X
¯ h̄ ȳ z̄ ū b̄ be a feasible solution for IPR0 R1 .
x̄ Then this solution is feasible for Q if and only if (21) is
satisfied. Clearly, if R0 ∪ R1 = H , then a feasible solution
to IPR0 R1 is feasible for Q.
Let BIPR0 R1 be the problem defined by (1)–(24).
Clearly, any feasible solution to BIPR0 R1 is a feasible
solution to Q. Both IPR0 R1 and BIPR0 R1 will be
used in our heuristic algorithm.
When the sets R0 and R1 are created, we seek to avoid
logical contradictions. Suppose cycles 1 2, and 3 all have
a conflict. Setting b12 to 1 implies that
1 < h1 < 2 < h2 -
(26)
Setting b23 to 1 implies that
2 < h2 < 3 < h3 -
(27)
Combining (26) and (27) implies that
1 < h 3 -
(28)
If b13 is set to 0, then h3 < 1 , which contradicts (28).
Hence, b13 must also be set to 1. To avoid contradictions,
we check for directed paths in a special directed graph
V A
. For each p ∈ P, let p be a vertex in V . For each
p q ∈ R0 , let q p be a directed arc in A and for each
p q ∈ R1 , let p q be a directed arc in A.
To determine if fixing bij to 1 will cause a contradiction,
we simply determine if V A
contains a directed path from
j to i. If so, then bij will not be fixed to 1. Likewise, to
determine if fixing bij to 0 will cause a contradiction, we
simply determine if V A
contains a directed path from i
to j. Using the above notation, the heuristic procedure can
be described as follows:
Procedure KO Heuristic
Input: An instance of Q
= x̂ ˆ ĥ ŷ ẑ û b̂, a feasible
Output: X
solution for Q
Begin
While the iteration counter is less than 100 Do
If the iteration counter is greater than 25 And
all the demand is routed Then
Terminate
While True Do
Solve IPR0 R1 If the solution is no better than the best found
so far Then Break
If no pairs can be added to Ro or R1 Then
Solve BIPR0 R1 , update the best solution
found so far if needed, Break
Else add more pairs to R0 and/or R1
End While
Increment iteration counter
End While
End
If a solution is found in which all demand is satisfied,
then the heuristic terminates after 25 iterations of the outer
loop. If no such solution is found, then the heuristic continues for the full 100 iterations. These are parameters that
can be tuned for a particular instance of Q. For a more formal and detailed description of the procedure, the reader
may consult Kennington and Olinick (2000).
4. AN EMPIRICAL ANALYSIS
The optimization model and KO heuristic procedure have
been implemented using the AMPL modeling language
(Fourer et al. 1993). The AMPL model for IPR0 R1 and code for the KO heuristic may be found in Kennington and Olinick (2000) and are available online at
http://www.engr.smu.edu/∼olinick/papers/rwa/rwa.html.
Our test suite consisted of 20 randomly generated problems of various sizes. The problem characteristics may be
found in Table 1 and the characteristics for the 20 instances
of mathematical model Q appear in Table 2. The problems
named ATT01 through ATT05 all have a topology given
in Figure 3 of Grover et al. (1991), and EUR01 through
EUR05 represent a European network described in Van
Caenegem et al. (1998). All test problems are available in
Kennington and Olinick (2000) and on the website mentioned above.
All test runs were made on a Compaq AlphaServer DS20
system with dual 6/500 Alpha processors and 4096 MB of
RAM, using the AMPL modeling language with a direct
link to the solver in CPLEX 6.6.0. Each of the 20 problems was solved twice, once using the default CPLEX
settings on Q and once using the KO heuristic (using
CPLEX to solve IPR0 R1 and BIPR0 R1 ). Each of the
20 instances of Q were given an eight-hour CPU time limit.
Table 1.
Problem characteristics.
Number of
Problem
Name
A01
A02
A03
J01
J02
J03–J07
ATT01–ATT05
EUR01–EUR05
Nodes Links Demands
6
6
6
8
10
9
11
18
9
9
9
12
15
13
23
35
9
9
9
14
24
15
16
18
Subnet- Photonic
works Switches
5
5
5
5
5
5
6
6
5
5
5
6
6
6
7
7
Kennington, Olinick, Ortynski, and Spiride
Table 2.
MIP characteristics.
Problem
Name
Total
Continuous Binary
Integer
Constraints Variables Variables Variables
A01
A02
A03
J01
J02
J03–J06
J07
ATT01–ATT03
ATT04
ATT05
EUR01
EUR02
EUR03
EUR04
EUR05
267
249
257
511
1179
444
444
651
651
651
658
658
658
658
658
131
123
127
244
582
213
213
298
298
298
296
296
296
296
296
106
102
104
162
315
148
151
197
198
197
197
201
197
198
107
30
30
30
38
60
34
31
44
43
44
40
36
40
39
40
If a run terminated prior to obtaining a provable optimum,
then the best solution found was returned. The 20 runs with
CPLEX each exited with one of the following conditions:
(i) a provable optimum was obtained, (ii) the branch-andbound tree exceeded the memory limitation, or (iii) the time
limit was exceeded.
A summary of the results comparing CPLEX to the KO
heuristic can be found in Table 3. Of the 20 problems,
Table 3.
1
2
Comparison of CPLEX with the KO heuristic (all times are in seconds).
3
4
All
KO Heuristic
Problem
Name
CPU
Time
Best
Solution
% Demand
Satisfied
Stopping
Criteria
CPU
Time
Best
Solution
A01
A02
A03
J01
J02
Average
J03
J04
J05
J06
J07
Average
ATT01
ATT02
ATT03
ATT04
ATT05
Average
EUR01
EUR02
EUR03
EUR04
EUR05
Average
Average
Scaled
1300
17000
65
29000
29000
15273
29000
29000
29000
29000
29000
29000
29000
24000
29000
29000
9800
24160
29000
29000
29000
29000
29000
29000
24358
1297
874
1903
3611
2267
100-00%
100-00%
100-00%
98-59%
68-75%
93-47%
100-00%
100-00%
100-00%
100-00%
100-00%
100-00%
94-12%
100-00%
94-44%
100-00%
100-00%
97-71%
92-86%
98-16%
81-60%
85-96%
80-49%
89-65%
95-21%
1-02
optimal
optimal
optimal
time
time
4-24
6-98
5-24
23-92
55-03
19-08
25-27
24-97
7-01
18-10
7-82
16-63
26-73
8-37
24-77
10-20
7-36
15-49
28-34
28-91
28-72
28-80
30-16
28-99
20-05
1-00
1297
982
1903
3233
2462
2885
1899
2830
4312
3585
2707
3810
5042
6930
5604
14310
16830
19422
21240
23346
77
CPLEX only obtained provably optimal solutions to the
three smallest. The KO heuristic found the optimum for two
of these problems and produced a feasible solution with a
12% higher cost on the third. Two of the problems were
terminated due to memory limitations and the other 15 ran
the full eight hours.
The 20 problems have been partitioned into four groups
with five problems in each group. Average CPU time and
average % demand satisfied is computed for each group.
For the first group, both procedures satisfied 93.5% of
the demand. On average, CPLEX required over four hours
for each problem whereas KO required only 19 seconds.
For the second group, CPLEX routed all of the demand
while KO routed 99.2%. To accomplish this, CPLEX ran
a total of 40 hours on these problems while KO never
ran more than 26 seconds on any problem. On the ATT
problems, CPLEX and KO routed the same amount of
demand (97.71%). The average run time for CPLEX was
over 1,000 times longer than that for KO (6.7 hours versus
16 seconds). For the European problems, CPLEX ran for a
total of 40 hours compared to less than three minutes for
KO (a factor of 800). The additional time for the CPLEX
runs yielded a 6% improvement in demand satisfaction.
The summary for all 20 test cases shows that CPLEX
used approximately three orders of magnitude more CPU
time than KO to achieve only a modest improvement in
CPLEX 6.6.0
Group
/
time
time
time
time
time
time
memory
time
time
memory
time
time
time
time
time
2814
2305
2816
4361
5468
2805
3810
5070
5885
6675
14310
16830
20430
21240
23550
% Demand
Satisfied
100-00%
100-00%
100-00%
98-59%
68-75%
93-47%
96-75%
99-31%
100-00%
100-00%
100-00%
99-21%
94-12%
100-00%
94-44%
100-00%
100-00%
97-71%
88-27%
91-41%
76-42%
83-04%
78-54%
83-54%
93-48%
100-00%
78
/ Kennington, Olinick, Ortynski, and Spiride
the percentage of demand satisfied. In the worst case, KO
never required more than 56 seconds. The results of this
study indicate that the KO heuristic runs very quickly and
delivers high-quality solutions.
A common practice in the telecommunications industry
is to conduct scenario analyses where one estimates the cost
of building a new network or expanding an existing one
under a variety of conditions involving different demand
patterns and equipment costs. These studies are short term
in nature and may involve solving a series of related problems. This practice was simulated in this study by the last
three sets of problems: Problems J03–J07, the ATT problems, and European problems. The European problems, for
example, differ only in the number of wavelengths required
for each o-d pair and the costs of building the layered subnetworks and photonic switches.
In practice, network planners prefer quickly identifying
reasonable solutions to searching for optimal designs. Typically, problems like the wavelength routing and assignment
problem Q are solved as subproblems of the overall network planning process. Algorithms for solving them would
be embedded as subroutines in network planning software
systems like the SONET Toolkit (Attanasio and Hoffman
1993, Cosares et al. 1995). There is a trade-off between the
quality and cost (in terms of computing resources) of solutions to these subproblems. When viewed in this light, our
results strongly indicate that the KO heuristic provides an
excellent tool for finding practical solutions to the WDM
routing and wavelength-assignment problem.
5. SUMMARY AND CONCLUSIONS
In this investigation, we present a three-step strategy
that may be used to help solve the provisioning problem for an all-optical WDM transport network. In the
literature, this is referred to as the static-demand WDM
routing and wavelength-assignment problem. Our strategy
attempts to minimize cost under the following assumptions:
(i) demands are symmetrical, (ii) 80 wavelengths are available on any link, (iii) wavelength translation is not permitted, (iv) the routing and assignment problems are considered simultaneously, and (v) 100% restoration is required
for any single-node or single-link failure. Our strategy
allows the network designer to select a ring, a mesh, or a
hybrid architecture.
The selection of layered subnetworks and location of
photonic switches determines the potential cycles that
appear in the optimization model. Hence, some knowledge
of the specific application or problem instance should be
used in construction of the layered subnetworks and photonic switch locations. Based on our strategy of using a set
of fundamental layered subnetworks, many of the demands
may have only a few candidate cycles. Thus, the optimization model may be primarily a wavelength-assignment
problem. If there is unsatisfied demand after application
of the heuristic procedure, then additional layered subnetworks containing the origin and destinations of the corresponding demands can be appended to the model. The
new resulting model is then solved and this process may
be repeated until all demand is satisfied.
A mathematical programming model that formally
defines the problem has been created and coded using
the AMPL modeling language. This model and a heuristic procedure for obtaining very good solutions appear in
Kennington and Olinick (2000) and are available for downloading from the World Wide Web htpp://www.engr.smu.
edu/∼olinick/papers/rwa/rwa.html.
In empirical experiments with test problems having up to
18 nodes, 35 links, and 18 demands, near-optimal solutions
were produced in approximately 30 seconds on a Compaq Alpha-based workstation using AMPL and CPLEX
6.6.0. The heuristic algorithm appears to be very robust. All
problem instances that we have evaluated resulted in very
good answers. Based on our computational experience, this
strategy appears to be practical for real design problems of
this type.
ACKNOWLEDGMENTS
This work was partially supported by Nortel Networks and
the Office of Naval Research under Award No. N00014-961-0315.
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