asm*
HD28
M414
•
no-
5233
V
,
JAN
1991
/
"Using a Hop-constrained Model
Alternative
Generate
Communication Network Designs
to
A. Balakrishnan, K. Altinkemer
MIT
Sloan
School Working Paper #3233-91 -MSA
Revised:
December 1990
\
.
Using
Hop-constrained Model
Generate
Alternative Communication Network Designs
a
to
Anantaram Balakrishnan
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Ketnal Altinketner
Krannert Graduate School of Management
Purdue University
West Lafayette Indiana 47907
Revised:
Supported
and
in part
a grant
December 1990
by grant DDM-8996120 from the National Science Foundation
from the
AT&T
Foundation
Abstract
Designing the link topology and selecting capacities in a backbone network of
a
communication system involves complex tradeoffs between investment and
operating costs and service considerations such as network reliability and
vulnerability, delays,
and blocking. Incorporating
all
these design criteria
simultaneously in a comprehensive model results in a large-scale, non-linear,
discrete optimization
problem
alternate optimization-based
In this paper,
that is intractable.
methodology
the
method
propose an
that generates several cost-effective
backbone network designs with varying cost and performance
Network planners can use
we
in conjunction
characteristics.
with detailed performance
evaluation techniques to assess the cost impact of different service requirements, and
select a design that achieves the
proper balance between conflicting objectives. To
generate different configurations, the method parametrically varies a set of hop
constraints that restrict the
Reducing the
maximum number
but incurs higher
constraints,
number
network design
of
total cost for the
we develop
a
of links over
which messages can be transmitted.
hops increases the number of alternate routes
communication system. For a given
set of
hop
Lagrangian-based algorithm to identify a cost-minimizing
that satisfies all internode traffic requirements.
We
report results for
extensive computational tests of the algorithm using several randomly generated
test
problems.
method
Our
identifies
results
good
demonstrate
that,
heuristic solutions
even for relatively large problems, the
and
tight
lower bounds that confirm the
near-optimality of the selected designs. Using a 25-node example,
the
model can be used
to evaluate the cost versus
Keywords: Communication network
Lagrangian relaxation
we
illustrate
performance tradeoff.
design, cost-performance tradeoff,
how
1.
Introduction
This paper presents a methodology to support topological design and link
capacity planning decisions for the backbone network interconnecting geographically
dispersed gateway nodes and switching centers in a telecommunication system.
Topological design decisions are very important in backbone network planning
because of the enormous capital investments required for transmission and
switching
and the
facilities,
service levels.
significant impact of
In selecting the configuration
and
network configuration choices on
backbone network,
capacities for a
planners face conflicting objectives of reducing the total investment and operating
costs while ensuring adequate service levels, expressed in terms of
performance measures such as
reliability
capabilities, blocking probabilities,
and
network
vulnerability, alternate routing
and queueing and transmission delays. With
increasing competition and service diversity in the communications sector, these
service considerations are
becoming increasingly important.
Because of the multiple, conflicting objectives, non-linearities in service
criteria,
and
discrete design choices, formulating
and solving
a
comprehensive
optimization model to support backbone network planning decisions
is
impractical.
Indeed, even with a single objective such as cost minimization, the design task
very complex and challenging because the decision model must
(i)
is
incorporate
binary network design decisions to capture the significant fixed costs, and
(ii)
simultaneously account for the close linkages between topological design, link
capacity,
and routing
In this paper
based method
performance
to
decisions.
we propose an
alternate
approach that uses an optimization-
generate several alternative network designs with varying cost-
characteristics.
Our approach
regarding network design practice.
First,
is
motivated by several observations
network planners must often accomodate
various implicit and qualitative design considerations that cannot be adequately
represented in optimization models; hence, they prefer a decision support system
that identifies a limited set of effective designs rather than a single 'optimal'
solution for an imprecise model.
Second, in most practical contexts
completely characterize network performance
(reliability, delays,
we
cannot
blocking) in a
convenient form for mathematical modeling, except under very restrictive
assumptions about
traffic
patterns
assess the network's performance,
and network behaviour. Therefore,
we
to accurately
require detailed evaluative techniques such as
simulation and queueing network analysis. Third, the total
number
of possible
network designs grows exponentially with the dimensions of the network. Since
detailed performance evaluation of each design
1-
is
a
time-consuming process,
planners require a principled method to prune the
list
of
all
possible network
configurations to a few cost-effective designs with varying performance
Finally, to
characteristics.
keep the model
minimization) model must necessarily
tractable,
even a single objective
make some approximations.
(cost-
In terms of
computational complexity, most (deterministic) discrete choice network design
problems in telecommunication planning are NP-hard. Therefore,
we must
on
rely
optimization-based heuristic procedures that provide provably good solutions.
Thus, to effectively use decision support models,
framework
for
we propose
a hierarchical
telecommunication network planning. The approach consists of the
following four steps:
(1)
Generate, using an optimization-based model, a small set of good 'base'
designs that span a wide range of cost and service levels.
(2)
Refine each base design
meet
(3)
(e.g.,
selectively
add buffer
capacities)
and modify
it
to
implicit or qualitative design requirements.
Evaluate the true cost and performance of each design using detailed
analytical or simulation models.
(4) Select
an appropriate design that achieves the desired tradeoff between cost
and performance.
The designs generated
enhancements
in Step 2.
in
Step
1
serve as the basis for marginal user-directed
Each base design specifies an underlying network topology,
the preferred routes between various origin-destination pairs,
'rough-cut' capacity plan for switching
and a corresponding
and transmission resources. The base designs
should account for tradeoffs and interrelationships between topological design and
routing issues, between fixed and variable
costs
2,
and connectivity
(e.g.,
the
number
(traffic
dependent) costs, and between
of alternate routes in the network).
total
In Step
the planner might refine each base design (perhaps, using a decision support
model) by selectively adding buffer capacities to improve network performance, and
making minor changes
to the
topology and routes to accomodate context-specific
Step 3 consists of detailed evaluation of each candidate design, and Step
constraints.
4 involves choosing a design that balances the planners' preferences for cost versus
The four
service.
performed
steps in this approach are closely interrelated,
iteratively with, say, Step 2
In this paper,
Step
1.
We also
refinements
we
describe
providing feedback to Step
1
and so
on.
focus on a model to generate alternative network designs in
model extensions
to incorporate
some
of the design
— allocating buffer capacities, and creating alternate routes — contained
in Step 2 of the hierarchical
all
and might be
planning approach. Since the base designs incorporate
major topological decisions, they largely determine the possible levels of service.
-2-
In particular, a base design that
alternative routes
provides routing
and
is,
is
sparse
(i.e.,
contains few links) cannot accomodate
therefore, inherently unreliable, while a
Thus,
flexibility.
order to generate a spectrum of base designs
in
we must identify topologies that differ
we propose a deterministic model that
with varying cost-performance characteristics,
For this purpose,
in their connectivity levels.
seeks cost-effective designs to meet internode
routing restrictions which
the
hop
We
refer to the
we
call
dense design
traffic
hop constraints.
requirements subject to a class of
For each origin-destination pair,
number of links (or hops) on the interconnecting
on the number of hops between each origin and
constraint limits the
upper
limit
destination as the route's
Maximum Hop
Parameter
hop parameter in
(or,
route.
brief).
By
hop parameters, we can generate topologies with
varying arc densities and costs. In particular, reducing the hop parameter values
increases the number of alternate paths, and impacts delays while increasing total
systematically changing these
Thus, hop constraints serve as a convenient surrogate for service restrictions
generate alternate base designs. In addition, limiting the number of hops for each
to
cost.
message also simplifies the task of managing
traffic
example, Ash, Murray and Cardwell (1981) and
over the network
Monma
and Sheng
(see, for
Recently,
(1986)).
LeBlanc and Reddoch (1990) independently developed a hop-constrained model with
delay costs for reliable topology/capacity design and routing in backbone
explicit
networks.
For a given
set of
hop
constraints,
mixed integer program, and propose
we
formulate the design problem as a
a Lagrangian relaxation
approximately, but with performance guarantees.
method
to vary the
hop parameters
We
method
it
then propose a systematic
in order to generate a variety of
The network designer can then apply
to solve
models
detailed evaluative
design solutions.
to these
candidate
designs in order to accurately assess the cost impact of different service requirements.
The
rest of this
paper
is
organized as follows. In Section
2,
we
formally
describe the design problem, state our modeling assumptions, and present the
mathematical programming formulation. Section 3 describes our Lagrangian-based
solution approach to generate upper
solution, for a given set of
hop
and lower bounds on the
constraints.
In Section 4
computational results for several randomly generated
computational
tests focuses
we
test
cost of the optimal
present extensive
problems.
Our
first set
of
on evaluating the effectiveness of the proposed
algorithm for various problem sizes, cost structures, and hop restrictions. These
results
demonstrate that the method constructs near-optimal solutions even for
relatively large
second
set of
problems with upto 30 nodes, 420 edges, and 420 commodities. Our
computational experiments
gain insights about the tradeoffs between cost
-3-
how
model can be used to
and performance by varying the hop
illustrates
the
some variants and enhancements of the basic hopconstrained network design model to allocate buffer capacities, and provide for
Section 5 describes
restrictions.
alternate routes.
Section 6 summarizes the paper and outlines directions for further
research.
2.
Problem Definition and Formulation
The backbone network design problem involves selecting a subset of links to
include in the topology, determining the capacity on each selected link, and routing
all
We
internode
emphasize
routing
requirements
traffic
at
minimum
that the three decisions
total
investment and operating
cost.
— topological design, capacity planning, and
— are closely interrelated, and we seek an integrated model that addresses
all
three decisions simultaneously.
Many
authors have proposed methods to separately solve two components of
problem
the backbone design
problem
— the capacity assignment problem and the routing
— assuming that the network topology
assignment problem
(see, for
Maruyama and Tang
(1976))
is
prespecified.
example, Bonucelli (1981),
assumes
Chou
The
capacity
et al. (1978),
that traffic routes (and, hence, the
network
topology) are given, and selects the best capacity for each link from a discrete set of
available link capacities in order to minimize total cost subject to delay restrictions.
On
the other hand, the routing or flow assignment problem starts with a given
choice of link topology and capacities, and seeks to identify primary routes between
various origin-destination pairs in order to minimize the average or
message delay
example, Ahuja (1979), Bertsekas (1980, 1984), Cantor and
(see, for
Gerla (1974), Courtois and Semal (1981), Frank and
(1981),
and
Yum
Chou
(1971), Gerla (1973),
Tymes
(1981)).
Combined approaches
that iterate
decisions have been discussed by Gerla
Maruyama, and
maximum
Fratta
and Tang
between capacity and flow assignment
et al. (1974),
(1977).
Gerla and Kleinrock (1977), and
Gavish and
Neuman
(1986,1987),
Gavish
and Altinkemer (1987) and Pirkul and Narasimhan (1987,1988) describe integrated
optimization methods for simultaneous capacity assignment and routing.
papers also account for
objective function),
traffic
node and
delay (either as a constraint or as a
link failures,
and/or
practical connectivity constraints in fixed-charge
Eswaran and Tarjan
(1976)
show
that the
enhancement of existing networks
restrictions
is
alternate routing.
NP-hard.
-4
the
Incorporating
network design models
problem of finding the
to satisfy
'cost' in
These
is difficult.
least cost
even some special types of connectivity
LeBlanc and Reddoch (1990) introduce two combined topology/ capacity
planning and routing models for reliable backbone network design. These models
two
are similar to our approach, but differ in
respects: they
(non-linear) delay cost in the objective function,
and
capacities,
The
model considers
first
(ii)
assume continuous
model includes an additional
linear capacity costs (our
installing links).
and
include an explicit
(i)
traffic
with different
fixed cost for
priorities,
and finds
the least (delay + capacity) cost design containing alternate routes for single link
The second model uses hop
failures.
Both
constraints as a surrogate for reliability.
models permit bifurcated routing. The authors exploit the linear capacity cost
structure to solve problems with
up
to 100
nodes using the flow deviation
algorithm.
present a
we
some notation, state our modeling assumptions, and
mixed-integer programming formulation for the backbone network design
Next,
introduce
problem with hop
2.1
constraints.
Notation and Modeling Assumptions
The hop-constrained network design problem
network G:(N,E) whose
vertices
N
is
defined over an undirected
represent the backbone nodes and intermediate
switches; the edges in E correspond to possible direct connections. For every pair of
nodes
p,
q
in the
demand) from p
network,
to q.
We
let
d
represent the projected peak internode traffic (or
between each pair of nodes as a separate
treat the traffic
commodity, with <p,q> denoting the commodity that flows from node p to node
Let
K
denote the
set of all
commodities,
i.e.,
K
= {<p,q> p,q e
:
each commodity <p,q>, the designer specifies an upper
number
the
of
hops
for
messages originating
(maximum) hop parameter
for selecting these
To
two types
for
at
limit,
p and destined
q.
N with dpq > 0}.
For
denoted as h
on the
We refer to
for q.
commodity <p,q>. Section
,
2.3 describes a
h
as
method
hop parameters.
establish transmission capacity
of (investment
and operating)
on each
link
(i,j)
costs: a fixed cost,
from
variable cost c^ that represents the cost per unit of capacity
might consist of investments
of the network, we incur
denoted as F-, and a
i
to
j.
The
fixed cost
for acquiring land, building the infrastructure,
installing/operating the communication link, while the variable cost
and
component
approximates expenses for acquiring and maintaining transmission and switching
equipment. In Section 5
we
indicate
how
these fixed
and variable
account for fixed or proportional buffer capacities on each
accommodates economies of
scale
and volume discounts
link.
in the
costs can also
Our model
also
form of piecewise-
linear,
concave costs (see Figure
we assume
1);
the simpler fixed plus variable cost
As LeBlanc and Reddoch
structure for expositional ease.
(1990) note, even
though
transmission cables and switches can be purchased and leased only in discrete sizes,
telecommunication companies often have the option of using public
telecommunication
traffic;
facilities
(and even leasing fractional capacities) for overflow
hence, piecewise linear, concave functions adequately represent capacity costs.
Our model
includes an explicit design constraint specifying that the selected
network topology must be connected. This
though not
restriction,
essential for our
algorithm, strengthens the problem formulation and improves the lower bound.
when
For certain problem contexts,
the
commodity flow pattern does not
necessitate
network transformation ensures connectedness.
a connected solution, the following
Consider a "commodity graph" containing the original nodes, and edges
{p,q} for
> 0. Let v be the number of connected components
every commodity <p,q> with d
pq
in this graph. If v = 1, we say the demand pattern is 'complete'; in this case, every
feasible design
valid.
must necessarily be connected, and our connectedness
When demand
network as follows
say,
in
node
0,
not complete
is
to create a
every component
r
=
1,2,...,v
connecting node
commodity graph. Each
demand and
cost.
r
}
Add
restriction of
1.
v
dummy
Observe
is
can augment the original
of the
with unit
the
we
{0,i
and a very high variable
hop
1),
edges
fixed cost
a
v >
dummy
complete demand pattern: introduce a
dummy
and add v
(i.e., if
restriction
to
one selected node
dummy
commodities
that the
node,
ij.
edge has zero
<0,i >, each
r
commodity graph
augmented network has a single component; and, the optimal design
problem together with the dummy edges {0,^} is optimal for the
for
for the
original
transformed problem.
assumption
wishes
Thus, augmenting the network makes the connectedness
Finally, the
valid.
to explicitly
network transformation
impose connectedness
routing) even with incomplete
demand;
is
unnecessary
if
the planner
(for instance, to facilitate alternate
our algorithm generates a
in this case,
higher lower bound relative to the 'unrestricted' lower bound obtained using the
augmented network. For our computational
origins
and destinations resulted
with sparse
2.2
demand
patterns);
in
random choice
tests, a
complete demand for
all test
of
commodity
networks (even
we, therefore, did not augment the networks.
Mathematical Programming Formulation
Our mixed-integer programming formulation
for the
backbone network
design problem distinguishes the direction of flow on each edge of the original
undirected network G.
corresponding
to
We consider
two
directed arcs,
each original undirected edge
-6-
{i,j}.
denoted as
The
(i,j)
and
(j,i),
original set of undirected
edges
is
denoted as
E; let
A
represent the corresponding set of directed arcs.
formulation uses two sets of binary decision variables, y^ and
y-
=
if
1
edge
{i,j}
is
zPjl,
Our
defined as follows:
included in the design,
otherwise, and
zPjl
=
if
1
commodity <p,q>
is
routed on arc
(i,j),
otherwise.
The y variables model the edge
selection decisions, while the z variables represent
the (primary) routing decisions for each commodity.
The backbone design problem can now be represented mathematically
following integer formulation called
£
minimize
[HCDP]
F ijyjj
+
(i,j)
subject
hop-constrained
(for
£ £ ^pq
<p,q>
2
design
as the
problem):
(Z1)
??
0,j)
to:
£
je
N
zW
-
]T
je
zP[l
=
1
if
i
= p
-1
if
i
=q
if
i
* p ,q
for all
<p,q> e K,
N
Y
2,
(i,j)
zW
z
lj
<
-
h
n
(2.2)
for all
pq
<p,q> e K,
Constraints
ensure
(2.2)
that, for
each commodity <p,q>, the arcs
form a route from p
to q.
commodity <p,q>;
specifies that
it
Constraint
(2.3)
represents the
hop
<p,q> must be routed over
is
commodity <p,q> can flow
included in the topological design. Finally, constraint
selected design
solution
must be
a connected
subnetwork of the
method does not require an
explicit
most h
at
zW = 1
arcs.
and edge
on edge
in either direction
with
restriction for
Inequalities (2.4) are forcing constraints that relate the routing
variables:
(i,j)
{i,j}
selection
only
if
this
edge
the
(2.5) specifies that
network G. Our
original
mathematical representation of
this
constraint.
To incorporate piecewise linear, concave capacity cost functions, we replace
and j, where
{i,j} of the original network with rj: parallel edges connecting
each edge
Tjj
i
denotes the number of linear segments in the expansion cost function for edge
(see Figure
r
F[-
Let
1).
segment; the
F[:
r
and a variable
and
c[:
parallel
cost of
c[:.
{i,j}
denote, respectively, the y intercept and the slope of the
edge
in
our augmented network carries a fixed charge of
Because the cost function
is
concave, the model does not
require explicit capacity constraints to enforce the upper limit of flow for each cost
range.
23
Setting the
Maximum Hop
Parameters
The backbone network design model's primary
minimizing solutions as the hop
role
is
to generate various cost
restrictions are parametrically
We
changed.
propose one possible scheme, called the Demand-based Hop Restriction Method, to
hop
hop parameter h
systematically vary the
selects the
hop parameter
values.
largest desired
hop
restrictions.
For each commodity <p,q>, this method
from a user-specified window or range of permissible
Let hj and h 2 (> hj) denote, respectively, the smallest and
parameter. Within the range [h-j, h 2 ], our method selects lower
h
values for high-demand commodities, and vice versa. In particular,
= hj
for the
and h
we
set
h
commodity <p,q> with the largest demand (among all commodities),
= h 2 for the commodity with the lowest demand. For commodities with
intermediate
demand
values, the
interpolation (rounded
down
hop parameters are calculated by
to the next
By imposing more stringent hop
the hop restriction method gives
restrictions for
to select
[hj,
h 2 ].
commodities with higher demand,
greater importance to these commodities in
determining the network configuration. In practice, designers
and methods
linear
lower integer values) in the range
may
and systematically vary the hop parameters
commodity.
8-
use other
for each
criteria
Observe
the resulting
say, the
that as hj
and h 2 decrease, the hop constraints become
and
tighter,
network designs should become more dense. Also, when hj = h 2 =
method
selects the
same hop parameter value hg
Hop
refer to this special case as the Uniform
for all
Restriction Method.
commodities.
hg,
We
Section 4.3,
presents computational results to illustrate the effect of various hop parameter
settings using the general
methods. Next,
we
(commodity-dependent) and uniform hop
discuss a Lagrangian-based algorithm for generating good upper
and lower bounds on the optimal value
3.
restriction
of
[HCDP].
Solving the Hop-constrained Network Design Problem
The hop-constrained design problem generalizes the fixed-charge network
design problem which
Kan
(1978)).
itself is
known
to
be NP-hard (Johnson, Lenstra and Rinnooy
Finding the optimal hop-constrained solution
computational
In this section,
task.
that simultaneously generates
good
we develop
is,
therefore, a difficult
method
a Lagrangian-based solution
feasible solutions as well as
lower bounds on the
optimal cost, for any given set of hop parameters. These bounds provide
performance guarantees so that the user can assess the quality of the heuristic
solutions.
3.1
Lagrangian Relaxation Scheme
The Lagrangian
relaxation
difficult discrete optimization
The method
(1981)).
method has been
problems
(see, for
successfully applied to several
example, Geoffrion (1974), Fisher
exploits special structure in the
problem formulation by
dualizing the complicating constraints, and solving the resulting subproblems using
For a given
efficient, specialized algorithms.
the optimal Lagrangian solution serves as a lower
original problem.
Lagrange multipliers, the
set of
bound on the optimal
The Lagrangian subproblem solutions
c
st
of
cost of the
also provide useful
information to construct feasible designs for the original problem. The gap between
the Lagrangian lower
maximum
bound and
the cost of the best feasible solution measures the
possible cost differential between the optimal
and
heuristic solutions.
To
generate good lower bounds, the Lagrange multipliers are changed iteratively using
techniques such as subgradient optimization or dual ascent (see, for example, Fisher
(1981), Balakrishnan et
al.
(1989)).
For the hop-constrained design problem,
scheme
we
consider a Lagrangian relaxation
that dualizes the forcing constraints (2.4) using nonnegative Lagrangian
multipliers \iV% for
all
edges
(i,j)
and
all
commodities <p,q>. Observe that when
9-
,
removed from formulation [HCDP], the problem decomposes
constraints (2.4) are
into
two main subproblems: a Routing subproblem denoted as [RSPQi)], and an Edge
Selection subproblem denoted as [ESP(p.
The routing subproblem determines the
)].
optimal values of the route selection variables
z^, while
subproblem calculates the values of the design variables
further
Vj:.
The routing subproblem
decomposes by commodity; we denote the subproblem corresponding
commodity <p,q> as
greater detail,
3.2
the edge selection
to
RSPp^fi)]. The next two sections describe these subproblems in
[
and discuss methods
to solve
them.
Routing Subproblem
For any given set of Lagrange multipliers
commodity <p,q> and every
variable cost for each
c«
\i
A
Cjj
+
Then, the routing subproblem
[
H^/dpq
RSP p
= {^8}, we define an
arc
for all
(^)]
(i,j)
(i,j)
adjusted
as follows:
€ A,
all
<p,q> e K.
(3.1)
corresponding to commodity <p,q> has the
following formulation:
[RSPpqM
^(u)
c^
min
=
c^ z^
£
(i,j)e
(3.2)
A
subject to
ifi
1
IN
z
??
I^M
-
je
je
=
=
p,
-lifi = q ,and
(3.3)
N
otherwise
IE
>
2
node p
that
to
subproblem
(34)
1
j
zW
Observe
Vr 3™
s
??
[
RSP
node q containing
a
(u)
=
]
or
1
for all
(i,j)
e A.
essentially seeks the shortest directed path
maximum
of
h
arcs; the adjusted variable costs
serve as arc lengths for this hop-constrained shortest path problem.
value Lpq(n) of the routing subproblem equals d
hop-constrained path.
-10
(3.5)
from
c^
The optimal
times the length of the shortest
We
solve the hop-constrained shortest path subproblem using a truncated
Let U: denote
version of the method of successive approximations (Lawler (1976)).
node p
the length of the shortest path from
h
arcs.
u] =
Initially,
known
of U: are
c^
for all
if (p,j)
nodes
e A; otherwise u ] =
The
first
term
(u
h
,
min
j
i,
for
containing
j
At the end of stage
u
h
cW
+
j
we
h,
at
values
use the
e N:
:
i
equation
e N,
(i,j)
(3.6) is
e
A
]
(3.6)
}.
the length of the shortest
p
to
j,
which must consist of a h-arc shortest path
some intermediate node
plus the arc from
i,
i
to
j.
To solve subproblem [RSP pq (|i)L we terminate the algorithm after h pq
The value u obtained
).
the method's computational complexity is 0(n h
final stage
from p
(i.e.,
to q.
most
containing at most h arcs. The second term represents the length of
the (h+l)-arc shortest path from
to
for all
in the right-hand side of
path from p to
from p
[
°°.
node
network. During stage (h+1),
in the
j
following expression to compute u |
h 1
u j = min
(the origin) to
with h = h
We can
pq )
stages;
at the
gives the length of the shortest hop-constrained path
trace the actual path for
commodity <p,q> by performing
the
usual backtracking procedure.
33 Edge
Selection Subproblem
Let us
now
consider the second Lagrangian subproblem involving the edge
selection variables Vy.
Fjj
Given a
A
Fjj
set
-
n of Lagrangian multipliers,
^
for all
represent the adjusted fixed cost for edge
{i,j).
The edge
involves selecting a subset of edges that connects
minimum
select all
total (adjusted fixed) cost.
We
all
{i,j}
let
e E,
selection
(3.7)
subproblem
nodes of the network
at
solve this subproblem as follows: First,
edges with negative adjusted fixed
costs,
in order of nondecreasing adjusted fixed cost.
and arrange the remaining edges
At each stage
we
refer to the
subnetwork defined by the currently selected edges as the current network. Consider
each edge
{i,j}
in
sequence from the sorted
list
of unselected edges.
connects two different components of the current network, select
If
it
edge
and update the
current network by merging the two components; otherwise, discard edge
Repeat the procedure for
all
edges in the
must be connected. The sum
list.
At termination, the
of the adjusted fixed costs in this
(i,j)
final
{i,j}.
network
network gives the
optimal value of the edge selection subproblem. The computational effort required
to solve this
subproblem
is
dominated by the edge sorting procedure which requires
-11
0( E log
I
I
I
E
I
)
operations.
connectedness condition
follows: set
Vj:
=
1 if Fj:
Note
(2.5),
<
0,
formulation
that, if
the edge selection
and
otherwise, for
[HCDP] does not contain
subproblem
all
edges
(i,j)
is
the
easily solved as
e E; this solution
may
have a lower value than the connected subproblem solution.
The optimal value
(i)
Z(|i) of the
complete Lagrangian subproblem
the edge selection subproblem value, and
route selection subproblems for
nonnegative multipliers,
sum
of
the optimal values L_>q(n) of the
(i
of
lower bound on the optimal value of the original
subproblem. To obtain good lower bounds,
instance, Held, Wolfe,
the
commodities <p,q>- For any given vector
all
Z(p.) is a
(ii)
is
and Crowder
we
use a subgradient method
(see, for
(1974) or Fisher (1981)) to iteratively adjust the
Lagrange multipliers.
We
note that our Lagrangian relaxation scheme does not satisfy the
integrality property (Geoffrion (1974)); for
programming
relaxation of the routing
Therefore, the best lower
the linear
3.4
example, the optimal solution to the linear
subproblem might possibly be
bound obtained using our
programming lower bound
relaxation
for formulation
scheme may exceed
[HCDP].
Lagrangian-based Heuristic Procedure
At each subgradient
iteration, the solution to the routing
subproblem
provides a set of feasible routes (that satisfy the hop restrictions) for
Note
to
that the
edge selection subproblem
may
include
some edges
therefore, ignore the solution to the
the routing solution to construct an
initial
design consists of
all
all
that
any of the selected routes and/or may omit some edges belonging
We,
fractional.
commodities.
do not belong
to these routes.
edge selection subproblem, and use only
initial feasible
design.
Our Lagrangian-based
edges belonging to the routes chosen in the routing
subproblem. For this design, we solve the hop-constrained shortest path problem for
each commodity using the original variable costs Cj: as arc lengths. The sum of the
fixed costs for the selected arcs
and the
(true) variable costs for all
commodities gives
the cost of the heuristic solution.
We
then attempt to improve this starting solution by applying a Drop
heuristic (Billheimer
method
and Gray
(1973)).
The drop procedure
that iteratively eliminates existing
reduce the
total cost.
When we drop an
variable costs might increase since
is
a local
improvement
edges from the current design
in
order to
edge, the total fixed cost decreases while the
some commodities must now flow on more
expensive alternate routes. At every stage, the procedure evaluates the net savings
12
Aj:
obtained by dropping each edge
zero or negative for
all
from the current design.
{i,j}
If
the net savings
edges, the procedure terminates. Otherwise,
it
is
drops an edge
with positive net savings, updates the current design and commodity routings, and
decreases the upper bound.
Evaluating the revised routing costs
a hop-constrained shortest path
savings for every edge
a candidate
list
savings in the
at
each
of edges to evaluate.
to this
list.
The
local
entails solving
that previously
we do
used
not evaluate the
maintain and iteratively update
contains
all
edges that yield net
we only evaluate the savings
maximum savings is dropped
The edge with the
savings are also deleted from the
by evaluating the savings
we
Initially, this list
from the current design and removed from the
it
dropped
In subsequent iterations,
first iteration.
edges that belong
Instead,
iteration.
is
commodity
for each
Since this computation can be time-consuming,
this edge.
for
problem
when an edge
for all
improvement
When
list.
edges
list;
the
edges that do not give any net
list
becomes empty, we
reinitialize
in the current design.
heuristic can
be applied
at the
subgradient iteration. However, to reduce computation time,
procedure intermittently, say, once every 100 iterations or
if
end of each
we
only apply the drop
the current starting
design has a lower cost than the previous best starting design. The next section
describes our computer implementation of the Lagrangian-based solution procedure,
and presents extensive computational
results for several
randomly generated
test
problems.
4.
Computational Results
We
implemented the Lagrangian-based algorithm
design problem in standard
FORTRAN
tested the algorithm using several
describes
random
some
test
on an IBM 3083 (model BX) computer, and
randomly generated problems. This section
features of our implementation,
problems. In
Our computational
all,
we
first set
of tests focuses
heuristic solutions, as a function of the
We
first
to generate
applied the algorithm to over 65 problem instances.
the al gorithm, in terms of computation time
restrictions.
and the method we used
tests (discussed in Sections 4.3
performance aspects. The
for the hop-constrained
and
4.4)
on evaluating the performance of
and quality
problem
address two different
of the Lagrangian-based
size, cost structure,
and hop
then focus on a single problem instance in order to assess the
impact on cost and performance of systematically varying the hop restrictions
.
exercise demonstrates the effectiveness of our approach for generating several
alternative base designs.
13
This
4.1
Implementation Details
In addition to the features described in Section 3, our implementation of the
Lagrangian relaxation algorithm incorporates:
upper bound before
(i)
method
a
initiating the subgradient procedure,
generate an
to
and
initial
a non-standard
(ii)
multiplier initialization method.
Initial Heuristic Solution
Generating an
To generate an
initial
upper bound, we
first
construct a feasible design using a
method which we call the BUILD heuristic, and subsequently apply the drop
procedure to improve this design. The Build heuristic first sorts all commodities
decreasing order of demand;
commodity
it
in
builds a feasible design by iteratively considering each
At the beginning of stage
in the sorted order.
E
k, let
(k-1)
denote the
set
of edges in the current design. This design contains feasible paths (satisfying the
respective
hop
the algorithm
constraints) for each of the
augments
commodity <p,q> has
this
first (k-1)
commodities. During stage
commodity,
design to ensure that the k
good feasible path. For
constrained shortest path problem from node p
a
this
purpose,
we
node q with
to
k,
say,
solve a hop-
arc lengths l^ defined
as follows:
=
=
Essentially,
edge
if
{i,j}
if (i,j)
Cjj
/jj
Cjj
+
Fydpq
Ek_1 and
e
,
k_1
E
if(i,j)«
already belongs to the current design,
commodity <p,q>. Otherwise, we add
Cj:, effectively forcing commodity <p,q>
(4.1)
.
we
assign only the
variable cost q: to
the per unit cost Fj:/d
the variable cost
to
cost of
edge
{i,j}.
Let
P denote
to
Thus, at the end of stage
commodities
demand,
E
(k_1)
all
the
list.
of
Pk
,
i.e.,
algorithm reroutes
paths in E
I
K
I
all
set of
edges E
I
K
I
is
first
.
k
in decreasing order of
high-demand commodities
developing the design.
a feasible design.
As
When
the
a final step, the
commodities over the respective shortest hop-constrained
using the original variable costs
drop procedure (described
Build method.
then
we set E k = E (k_1) u Pk
By considering commodities
to the variable cost) in
procedure terminates, the
is
design contains feasible paths for the
the Build heuristic gives greater importance to
(which contribute more
absorb the entire fixed
The current design
l^.
new edges
k, the current
in the sorted
to
the set of edges belonging to the shortest hop-
constrained path from p to q using arc lengths
augmented by adding
pq
Cj:
as arc lengths.
We then apply the
in Section 3.4) to the starting solution constructed
The improved solution provides the
-14
initial
upper bound.
by the
,
Initializing
the Lagrange Multipliers
we
Instead of initializing the Lagrange multipliers to zero,
use the
heuristic solution to determine the starting values for the multipliers.
the total flow on each edge
then set equal to Fjj/x-
uP^
otherwise,
is
method completely
Lagrange multipliers
all
for all
commodities
computational
size factor (see, for instance,
tests,
Held
et al. (1974)) to 2;
Random Problem
4.2
We
test
to a
if
the gap
for the
halves the step size factor
(ii)
restrictions.
number
The program
of
problem
To generate
and commodity
nodes
w
first
must
a test problem, the user
n,
edges m, and commodities
for the
random component
locates the required
feasibility).
It
random spanning
number
in
I
K
edge
later); (iv)
tree.
number
I
;
first
(ii)
cost
(iii)
and fixed-
costs,
average
network
demand
d;
nodes randomly on a 1000 x
nodes
(to
ensure
of additional edges
(i.e.,
m-(n-
Having constructed the underlying network G:(N,E),
and variable
i and
denote the Euclidean distance between nodes
then computed
of
tree over these
then adds the required
edges) randomly to the
j.
costs for each
The
edge
in G.
fixed cost F^ of
Let
edge
(i,j)
as:
Fjj
where
densities, cost
seed for random number generator;
(i)
the problem generator calculates the fixed
is
of
hop parameters.
1000 grid, and generates a
ejj
wide range
to construct a
hop-restriction information: the lower and upper limits, h| and h^, defining
the range of desired
1)
terminates
(iii)
between the current best upper
to-variable cost ratio r ''these parameters are explained
(v)
and
if
Lagrangian relaxation algorithm. The problem generator can
structure information: weight
and
initializes the step
(i)
Generation
specify the following parameters:
size information:
to the
very small value).
create test problems with different sizes, edge
and hop
Fj:
this edge.
implemented a random problem generator
problems
structures
on
for 15 consecutive iterations;
after 250 subgradient iterations (or earlier
and lower bounds reduces
in the initial design, the
our subgradient procedure:
bound does not improve
the lower
{i,j}
allocates the fixed cost
that flow
|j.P9 is
in the initial solution;
{i,j}
Thus, for each edge
initialized to zero.
multiplier initializatio:
For
commodity <p,q> uses edge
if
Let x- denote
The multiplier
of the initial heuristic solution.
(i,j)
initial
=
w is the user-specified
(1 -
w)
weight
ej
(0
from a uniform distribution with range
can generate cost values that are either
j
<
+
wC
(4.2)
w < 1), and
£ is a
Observe
random number derived
by varying w, the user
completely random (when w = 1) or directly
[0,1000].
15-
that,
w = 0); intermediate values of w give a
We refer to w as the cost randomness factor.
proportional to Euclidean distance (when
mixture of random and Euclidean
Note
that
when
w>
The variable
costs.
the fixed costs
0,
cost
for
Cj:
edge
may
{i,j}
is
not satisfy the triangle inequality.
obtained by dividing the fixed cost
the user-specified fixed-to-variable cost ratio
are negligible
compared
r.
When
r is
very small, the fixed costs
to the variable costs; in this case, the
union of the hop-
constrained shortest paths (using the variable costs as arc lengths) for
commodities gives the best design. At the other extreme, when
variable costs are negligible,
and the best solution
is
F-by
r is
all
very large, the
network with the smallest
a
total
fixed cost that contains at least one hop-constrained path for every origin-destination
pair.
In particular,
when h
for large values of r
= (n-1) for
all
commodities <p,q>, the optimal solution
the minimal spanning tree.
is
After calculating the edge costs, the problem generator
the required
number
(
I
K
I
)
of origin-destination pairs
commodities. For each commodity, the
distribution ranging from
problems (even with IK
commodities), the
demand
I
random
to 2d;
less
demand d
we set d =
than
50%
which define the
selected from a uniform
is
5 units in all cases. For
of the
identifies
maximum
all
our
number
possible
test
of
choice of origin-destination pairs resulted in complete
patterns (as defined in Section
2.1).
program generates the hop parameters
Finally, the
randomly
for each
commodity using
demand-based method described in Section 2.3, i.e., by interpolation in the range
with high-demand commodities having lower hop parameter values and
the
[hi, h^]/
vice versa.
problem
Given
a set of
feasibility.
If
either increase the
hop parameter(s) or
some
try a different
sections describe the results of our computational experiments
to assess the effectiveness of the
Lagrangian relaxation algorithm, and generate
different cost minimizing designs
by varying the hop
restrictions.
Testing the Performance of the Lagrangian Relaxation Algorithm
We
use two performance measures to evaluate the effectiveness of the
Lagrangian relaxation algorithm:
best
checks for
seed.
The next two
4.3
first
the network does not contain any feasible path for
commodity, the user must
random number
hop parameters, our implementation
(i)
the
%
upper and lower bounds expressed as
gap, defined as the difference between the
a percentage of the best
-16-
lower bound,
which measures the quality of the
(in
CPU
heuristic solutions,
and
(ii)
the computation time
seconds) required for the entire procedure.
Our computational
characteristics
-
problem
tests
attempt to evaluate the effect of three problem
and hop
size, cost structure,
restrictions
performance measures (% gap and computation time). Problem
number
of
nodes
of commodities.
on the algorithmic
size
depends on the
network, the number (or density) of edges, and the number
in the
The
-
cost structure is
determined by two
randomness factor w, and the fixed-to-variable
cost ratio
characteristic affecting algorithmic effectiveness
is
factors: the cost
The
r.
third
problem
the tightness of the hop
measured, for instance, by (hj + h 2 )/2 (the mid-point of the userspecified window). For convenience, in this section we consider only the uniform
hop restriction method, with h^ = h 2 = hQ results for the more general hop
restrictions
;
restriction
4.3.2
method (with
h^
< h 2 ) are reported
Effect of Cost Structure
To
in Section 4.4.
and Hop Restrictions
isolate the effect of each of the three
performance,
randomness
we
first
consider different cost structures
and
factors
problem
characteristics
(i.e.,
different cost
and hop
fixed-to-variable cost ratios)
on algorithmic
restrictions for a 20-
network
node network with 180 edges and 180 commodities. For
this
generated 3 different problem instances (using different
random number
varying cost randomness factors, namely,
w=
0, 0.5,
and
1.
size,
we
seeds) with
For each network,
we
applied the Lagrangian relaxation algorithm with three fixed-to-variable cost ratios,
=
10, 20,
and
50,
and three
different (uniform)
hop parameter
values, hg =
3, 4,
and
5
r
hops. This set of parameters represents a wide spectrum of possible problem
from Euclidean costs
structures, ranging
to
to
high fixed costs (relative to the variable
Table
and
summarizes the
I
hg, this table
shows:
(i)
(iii)
commodities, which
where h
'
h
(<
)
is
is
%
r,
FC);
the
(ii)
number
of edges included in the final
number
of
hops h
(ave. hop) over all
defined as
the actual
<pq>
of hops
number
gap between the
as a percentage of the best lower
IBM
our analysis. For each combination of w,
results of
the weighted average
heuristic solution; (iv) the
seconds on an
costs).
the fixed cost as a percentage of total cost in the final
heuristic solution (denoted as
design (# of edges);
completely random costs, and moderate
bound (%
final
gap);
for
<pq>
commodity <p,q>
in the final
upper and lower bounds expressed
and
(v) the total
CPU
time
(in
3083) for generating heuristic solutions and subgradient
-17-
optimization (CPU). The
three statistics describe the structure of the final
first
two measure the algorithm's
heuristic solution, while the last
As
expected, the results in Table
while the number of edges
from 10
ratio increases
For ratios of 10 and 20,
to 50.
and proved
its
in this category, the
all
gaps are
gap increase
less
gaps are larger (ranging from 4
than or equal to
less
algorithm generated the optimal
bound equal
optimality (by generating a lower
bound); for 8 other instances, the gap was
ratio of 50, the
% FC and %
that the
design decreases as the fixed-to-variable cost
in the final
5%. For one problem instance
solution
show
I
effectiveness.
to the
upper
than 1%. With a fixed-to-variable cost
These
to 23%).
latter
problems
represent extreme values of fixed costs, and might have inherently higher duality
gaps. For consistency,
all
the
%
gaps reported in Table
between the best upper and lower bounds
at the
I
correspond to the difference
end of 250 subgradient
iterations.
(The gaps might possibly be lower if we increased the limit on the number of
subgradient iterations. For example, for the test problem with \\q = 3 and r = 50,
were able
to reduce the
20% gap
to
100 more iterations; however, the
As
the
hop
the final design
restrictions
17% by applying
CPU
become
becomes more dense
difficult to solve (the
%
gap
good
of
initial
CPU
1 .0
tighter
all
when h^
Interestingly, the
randomness
random
upper bounds with very
time in almost
(i.e.,
costs).
%
subgradient procedure requires around
initial
In Section 4.3.1,
and hop
we
fixed the cost
We considered
respectively.
Hop
fixed the
restrictions
than 2 seconds
effort (less
at intermediate
by 4.5% on average. The
of the total computational time,
and
randomness
Restrictions
problem
size
and considered the
We now
on algorithmic performance.
of problem size on the quality of
we
0.0
lower bound by an average of 30.6%.
43.2 Effect of Problem Size and
structure
70%
any
heuristic generates
Applying the drop heuristic
instances).
3),
exhibit
changes from
The Build
stages of the subgradient procedure reduces the initial cost
improves the
gap does not
factor
computational
little
decreases from 5 to
problems also become more
as expected, but the
increases).
(completely
the subgradient procedure for
time increased by 75%.)
consistent pattern of variation as the cost
(Euclidean costs) to
we
bounds and computation
factor at 0.5,
and the
time.
size,
we
(with correspondingly sparse and dense
For this
effect
set of tests,
and 30 nodes,
generated sparse as well as dense networks
demand
patterns).
performance of the Lagrangian relaxation algorithm for 8
different sizes, for various (uniform)
study the
fixed-to-variable cost ratio at 20.
four basic network sizes containing 10, 20, 25
For each network
effect of cost
hop
restrictions.
18-
Table
test
II
reports the
networks with
%
Again, the
gaps decrease as the hop parameter increases. In
instances, the gaps reduce to
consistent variation as the network size
show any
node problem the
for the
below 10% within 250
%
gap
is
iterations.
and density
The
%
all
but three
gaps do not
increase; for the 10-
higher for the denser network, while the converse
true
is
20-node problem. As expected, the computation times increase with problem
size.
summary, our computational
In
algorithm
of
up
is
quite effective.
that are generally
We
emphasize
For almost
method generates
to 20, the
results suggest that the
tight
all
problems with fixed-to-variable cost
many
less
from the optimal solutions.
problems are very
of our test
ratios
lower bounds and good heuristic solutions
guaranteed to be within 10% or
that
Lagrangian-based
difficult to solve optimally.
For instance, the integer programming formulation corresponding to the 30-node,
420-edge
problem
test
problem contains 353,220 integer
size is
general integer
4.4
and 542,640
constraints; this
state-of-the-art,
programming methods.
The Network Cost-Performance Tradeoff
The
is
variables,
well beyond the realm of capabilities of current
results of the previous section
effective in generating
show
that the Lagrangian-based algorithm
good cost-minimizing solutions
for the hop-constrained
design model for a wide range of problem structures. This section attempts
demonstrate
that,
to
by systematically varying the hop parameters, the hop-constrained
model achieves our
original objective of generating a variety of base
network designs
with different cost-performance characteristics. In general, evaluating the
performance
(i.e.,
reliability, blocking, delay, etc.) of
detailed context-specific analyses
Instead, to illustrate the effect of
performance metrics
-
and simulations which we did not implement.
hop
constraints on service level,
we use two sample
average number of hops, and average connectivity.
The (weighted) average number of hops
as a
each design would require
h, defined in Section 4.3,
might serve
measure of operational (routing) complexity, with lower values corresponding
to easier routing.
Furthermore, under certain assumptions, the average delay per
message T
in a packet
number
hops
of
h.
switched network
(For example,
if
is
directly proportional to the average
the traffic arrivals
Kleinrock's assumptions, say, Possion arrivals
distributed message lengths with
buffer capacity B, then
mean
\/\l
-19
rates satisfy
and independent, exponentially
on every
link,
T = h/uB.) Thus, h might serve
changes in delay performance.
and service
and
if
each link has a fixed
as a rough indicator to assess
The (weighted) average connectivity
network
number
For a given commodity <p,q>,
vulnerability.
from p
of alternate arc-disjoint paths
bound on
number
the
statistic is a
of arcs that
blocked. Using the relative
must
demands
to
q
before
fail
surrogate measure for
we
define connectivity as the
in the final design;
all
demand commodities
commodities as the weights
for all
maximum
implementation uses a
1
unit, the
maximum
for their
Our
flow algorithm to compute the connectivity of
When
each commodity in a given design.
of
is
Thus, providing multiple
considered more important.
is
gives a lower
p-to-q communication
respective connectivity values gives average connectivity.
paths for high
it
flow from p
each edge of the final design has a capacity
q gives the number of alternate arc-disjoint
to
paths for commodity <p,q>.
To study the
effect of various
hops, and average connectivity,
we
hop
on
restrictions
cost,
average number of
consider a single problem instance with 25 nodes,
300 edges, and 300 commodities. Observe that this network has a complete topology
(i.e., it
contains one edge connecting every pair of nodes) and complete
K
the set of commodities
randomness
factor
is
contains one
commodity
fixed at 0.5, the average
for each
demand
node
at 5 units,
pair).
demand
The
and the
(i.e.,
cost
fixed-to-
variable cost ratio at 20.
Table
The
first
(=h-[=
III
shows
the computational results for 22 different
hop
7 solutions correspond to uniform
h 2 ) ranging from
1
restrictions,
to 7; the other 15 solutions
hop
correspond to commodity-
means and widths. For example,
dependent hop
restrictions with varying
restriction [2,2]
corresponds to a uniform restriction of 2 hops for
On
the other hand, the restriction
[2,5]
all
the
hop
commodities.
generates commodity-dependent hop
demand
and low demand commodities may use
parameters with a range width of 3 and
mean
commodities have hop parameters close
to 2
up
restrictions.
with parameter hg
of 3.5; in this case, high
to 5 edges.
Since the data
is
randomly generated, the actual
solutions are not very meaningful.
Instead,
we
cost values for different final
consider a normalized cost index
that expresses the actual cost as a percentage of the lowest total cost over all 22
solutions (obtained with the least stringent
cost index of 140
is
40% more expensive than
normalized cost criterion
hop
hop
facilitates
restrictions).
Thus, a design with a
the lowest cost design. Using this
our assessment of the cost-benefit tradeoff as the
restrictions vary.
-20-
show that total cost and average
connectivity increase, while the average number of hops decreases as the hop
constraints become more restrictive. In particular, with a uniform hop parameter h Q
As
of
expected, the results of Table
must contain
(solution #1), the design
1
commodity must have
III
a single-hop route,
all
edges of the original network, every
and each commodity has 24 alternate
disjoint paths (including the single-hop route).
design
the
also the
is
hop
most expensive because of the
restriction with hg =
Observe
same
solution
final
that, as
we
increase the
Finally,
[3,7].
we
hop
To
a certain threshold,
For example, in the uniform hop case, the
7. Similarly, with commodity-
in
all
final
design satisfies the hop restrictions
but one instance the
Table
III
closely
%
gap
is less
[3,6]
than 6.5%,
approximate the optimal costs for
restrictions.
better assess the cost versus performance tradeoff, Figure 2
number
shown in
graphical display of the average
(Solution #
solutions.
hop
increases from 6 to
note that for
implying that the costs shown
different
relax
expensive while average connectivity
hop parameters beyond
design remains optimal.
dependent hop parameters, the same
and
As we
significant fixed costs.
7.
unchanged when h Q
is
less
this
cost design corresponds to the least restrictive (uniform)
The lowest
declines.
the
become
constraints, the designs
For our problem instance,
arc-
1
is
not
of
hops and normalized cost
this figure since
it
shows a
for different
has a normalized cost of
over 400, while the remaining solutions have normalized costs ranging from 100
150.)
As
this figure illustrates, the
method generates
For instance, solution # 2 (with hop restriction
hop
[2,2])
several 'dominated' solutions.
dominates solution
restriction [1,4]) since the latter design has higher cost
number
of
#10
(with
and higher average
hops (and lower connectivity). Similarly, solution # 3 dominates
solution #18.
The
set of
undominated solutions define an
characterizes the cost-benefit tradeoff
The piecewise
number
to
when
the average
"efficient" frontier that
number
of hops changes.
linear curve in Figure 2 shows, for instance, that reducing the average
of hops from 1.74 (solution # 2) to 1.71 (solution # 9) increases the total cost
by over 16%
(relative to the lowest cost design).
To summarize, our hop-constrained network design model helps
selected subset of
good solutions with varying performance
pictorial representation
such as Figure 2 can greatly
to identify a
characteristics.
assist the
A
network designer
in
assessing the tradeoffs between conflicting criteria before selecting an appropriate
backbone network topology.
and average number
of
We
emphasize
that
we have used
average connectivity
hops as measures of service performance only for
-21
illustrative
purposes. In general, similar tradeoff analyses can be performed using any desired
performance measure.
5.
Model Variants and Extensions
In Section 2
that selects
we
presented a basic version of the hop-constrained design model
primary routes
commodity and provides adequate capacity on
describes how to adapt and extend the model to
for each
these routes. This section briefly
plan buffer capacities and select alternate routes.
5.1
Planning buffer capacities
Buffer capacities improve service levels by decreasing delays,
accomodating contingency
allocate these capacities
5.1.1
-
traffic.
We
briefly outline
two methods
and
to judiciously
a two-step process and an integrated model.
Two-step process
Planning buffer capacities might be viewed as the second step of the
hierarchical process described in Section
model
first
In this scheme, the hop-constrained
1.
generates several network topologies and corresponding base capacities.
For each topology, a subsequent step augments capacities on selected links
improve performance. This second step might,
the approach proposed
by Kleinrock
(1976).
for instance, use a
Kleinrock's
model
model
to
similar to
selectively allocates
buffer capacities to links in the base design in order to minimize average delay per
message subject
about
to a
traffic arrivals,
budget
restriction.
message
sizes,
and
In particular,
under certain assumptions
link expansion costs, Kleinrock
De
proposes a
"square-root" formula for optimally allocating a budgeted
amount
average delay. Bitran and Tirupati (1988) propose
manufacturing context)
enhancements and variants of
this
model
to
(in the
to
reduce
improve the performance of queueing
networks. For instance, they describe models to minimize the total additional
investment required
to
achieve a prespecified delay target, and to reallocate capacities
from one part of the network
5.1.2
to
another for reducing delays.
Integrated models
Instead of applying a separate buffer allocation
hop-constrained model can
itself
(Fjj
enhancements
step, the
be easily modified to account for certain types of
The modifications involve changing the fixed and variable
and cj and the demand values d
We describe the model
buffer capacities.
parameters
model as the second
cost
.
for three different types of buffers
-22-
-
fixed buffers, proportional buffers,
and commodity-dependent
Designers might use any combination of these
buffers.
three buffer types.
of the
Consider first a fixed buffer policy which specifies that each selected link (i,j}
network must have a (fixed) spare capacity of bj: units. The hop-constrained
model can incorporate
this
requirement by using a higher fixed
cost; in particular,
we
A
use an inflated fixed cost F^ which
(the
second terms
A
is
is
equal to the original fixed cost
Fj:
plus bj^Cj:
the incremental cost of the required buffer capacity).
proportional buffer policy specifies the required buffer level on each link as
volume on
a proportion of the traffic
for each link
{i,j}, i.e., {i,j}
that link.
Suppose we specify
must have spare capacity of
flow of X units. To represent
this
a proportion
at least Pj:*X units
if it
(3jj
carries a
requirement in the hop-constrained design model,
A
we use
inflated variable costs c ^ = (1+Pjj)*Cjj,
capacity on link
(i,j).
we
X
increases.
model can
also handle
can represent capacity-dependent
where the incremental buffer proportion,
buffers,
the original cost per unit of
is
In fact, since the hop-constrained
piecewise-linear concave costs,
of flow
where c^
proportional
say, Yj;(X) declines as the
This type of buffering strategy
is
volume
consistent with Kleinrock's
(1976) observation that, as throughput increases, a smaller buffer proportion suffices
to
maintain the same delay performance. In general,
concave buffer capacity requirement
root
law
we
can approximate any
(as reflected, for instance, in Kleinrock's
square
hop-
for buffer allocation) with a piecewise linear function in the
constrained model.
Finally,
suppose the planner specifies commodity-dependent
buffer
requirements as follows: the selected route for each commodity <p,q> must have
excess capacity equal to a fraction 8
Q
of the projected
demand d
.
The hop-
constrained model can accomodate this requirement by using a modified
value of d
r
1
demand
*(l+8 a ) for each commodity <p,q> (or equivalently a modified variable
r4
cost for each commodity).
The hop-constrained formulation's
dependent proportional
powerful
buffers,
change the buffer specifications
model
fixed buffers,
volume-
and commodity-dependent buffers provides
set of capabilities to the
constraints to generate various
ability to
network planner.
Just as
network configurations,
(i.e.,
we
we
a
parameterize the hop
can also parametrically
change the parameters b—,
(3-,
y-,
and 8
).
For
each specification, the hop-constrained model identifies a cost-effective design that
serves as the basis for secondary enhancements and detailed performance
evaluation.
-23
Modeling alternate routes
5.2
The hop
and hence the
constraints indirectly influence the density of the topological design,
availability of alternate routes for various commodities.
(contingency) capacities on these alternate routes,
strategies described earlier
and /or
model
explicitly
to select a
design that has adequate capacity to accomodate
least a prespecified fraction of various
is
or
select (as before) a
more
we
will consider
do not have any
whenever
that the
all its
In this
fail
restrictions.
Also, for convenience,
used when one
For purposes of
(or are highly congested).
common.
We
links are operational.
primary and secondary routes that are
links in
secondary route has no hop
demand d pq
alternate routes.
capacities for a secondary route that will be
on the primary path
links
discussion,
they
and plan
also select
demands along
at
primary route for each commodity; this primary route
the preferred communication path
must
enhanced model.
in Section 3 also solves the relaxation of this
Suppose we wish
we
As we
the contingency flows.
Fortunately, the decomposition
constraints in the hop-constrained formulation.
scheme,
can employ the buffering
approach requires additional decision variables and
illustrate next, the latter
method proposed
we
To plan
arc-disjoint,
we assume
this
i.e.,
that the
Let 6.^, (0 < A-,, < 1) be the fraction of the
P4
FH
secondary route must accomodate.
To incorporate
alternate routing,
we change
the original hop-constrained
formulation as follows:
•
add
a
Let
new
wW
and
•
set of
=
1 ii
binary routing variables, say,
commodity <p,q> uses
w^ for each commodity <p,q>.
(directed) arc
(i,j)
for
its
secondary route,
otherwise;
include the w-variables in the forcing constraints
(2.4) of
formulation
[HCDP]
as follows:
zM
+
zW
+
The new forcing
design
if it
wM
+
wW
<
y
for all
tj
constraints specify that edge
<p,q> e K,
{i,j}
all
(i,j)
e E.
must be included
(2.4')
in the
belongs to either the primary or secondary route for commodity
is at most 1, edge {i,j} cannot simultaneously belong to both
<p,q>. Since y^
the primary
and secondary
two routes are
•
add the
arc-disjoint;
cost term
V
V
routes; thus, constraints (2.4') ensure that these
and
q;
<t>r>Q
d™ w P9
to the objective function to
<p,q>
account for the spare capacity on the secondary route.
(i,j)
24
(2.4').
we
solve this enhanced formulation,
To
The Lagrangian problem decomposes
again dualize the forcing constraints
into
an edge selection subproblem and a
routing subproblem for each commodity.
The edge
as before, while the routing subproblems
now
To
variables.
subproblem
selection
is
the
same
involve determining both the z and
we
find the optimal values of the z-variables
w
use the hop-constrained
shortest path algorithm as before; a regular shortest path solution provides the
optimal w-values (since
constraints).
Observe
we assumed
that the
select exclusive arcs for the
constraints
(2.4')
that
Lagrangian subproblem solution
primary and secondary paths (since
which ensure
arc-disjoint paths).
we
solution for local improvement,
might:
primary route for each commodity, and
when
We
links of the
might also consider
(i)
Thus,
may not
we have
necessarily
dualized
to obtain a feasible starting
use the z-solution to select the
generate alternate routes by applying a
(ii)
some information from
shortest path algorithm (perhaps, using
the residual network
secondary routes do not have hop
the w-solution) to
primary route are removed.
a stronger relaxation that involves
simultaneously
generating the hop-constrained primary routes and arc-disjoint secondary routes to
minimize
total variable costs.
Effectively, the
problem formulation contains an
additional set of strengthening constraints
z Pfl
+
Z
W
W P9
+
+
w rfl
<
i
which are not dualized (and hence appear
that this arc-disjoint routing
an enumerative technique
to
modified
to
<p,q> e K,
all
is
e E,
(i,j)
in the routing subproblem).
We suspect
not polynomially solvable, necessitating
to find the optimal routing.
develop a variant of the K-shortest path algorithm
(1969)),
6.
subproblem
f or all
One
interesting possibility
(see, for
is
example, Dreyfus
account for hop constraints on the primary routes.
Conclusions
In this paper,
we have proposed
a hop-constrained network design
support long-term network planning decisions. For
effective solution
method
this
model,
we demonstrated
to
we developed an
that can solve relatively large problems.
varying the hop restrictions,
model
By systematically
the method's ability to generate
several alternative cost-minimizing designs with different performance
characteristics.
Using hop constraints as a surrogate
for service restrictions greatly reduces the
complexity of the model, and enables us to generate provably near-optimal solutions
relatively quickly using
an optimization-based approach. In contrast, representing
-25
delay restrictions explicitly requires numerous assumptions about
service rates, failure rates,
etc.,
Modeling connectivity
model.
network must be 3-connected)
traffic arrivals,
and might possibly introduce
non-linearities in the
restrictions (for instance, the
requirement that the
is
explicitly incorporate both delay
equally complex. Consequently, models that
and connectivity
restrictions
can often be solved
only by heuristic methods that do not provide any bounds on the quality of the
Because
solutions.
we
can solve the hop-constrained model effectively, our
approach provides a useful design tool
models might also apply
for
performing sensitivity analyses. Similar
to other large-scale
planning problems in logistics and
manufacturing systems design.
Several extensions and variants of the basic model merit further
investigation.
First,
would be
it
interesting to develop
multiplier adjustment schemes (such as dual ascent)
improvement methods
to further
to the
design of
current transmission and switching capacity.
for
network expansion planning
an existing network,
we
is
make
model
demands
of the planning horizon,
must account
for
new networks
that
have no
Adapting the model and solution
an important next
the problem
adversely aftect algorithmic performance.
multi-period setting,
and
heuristic initialization
To account
step.
for
require additional capacity constraints in the problem
formulation; these constraints
of the hop-constrained
and
test alternative
improve the quality of the upper and lower
bounds. Second, our model applies
method
and
is
more
Finally,
difficult to solve
developing multi-period versions
a fruitful area for further
for various
and might
development. In the
commodities are specified
for each period
and network expansion costs change over time. The model
two types
of tradeoffs: fixed design costs versus routing costs in
each period, and expanding the network beyond current requirements in
anticipation of future
demand
in
order to exploit economies of scale. The basic hop-
constrained network design model that
the building block for these
Acknowledgements:
insightful
We
comments and
we have developed
in this
paper can serve as
and other model extensions.
would
like to
thank the referees and editors for their
helpful suggestions to correct, clarify and
presentation.
-26
improve the
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The
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—
FIGURE
1:
Piecewise Linear, Concave Cost Function
Total
cost
I
F.
.
13
13
—
F
ij
13
Traffic
on edge
(ij)
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