Hybrid Fiber Coaxial Network Design

Abstract

Much interest exists in broadband network services to deliver a variety of products to consumers, such as Internet access, telephony, interactive TV, and video on demand. Due to its cost efficiency, Hybrid Fiber Coaxial (HFC) technology is currently being considered by most Telcos and cable companies as the technology to deliver these products. The topological HFC network design problem as implemented by several major companies is a form of the capacitated tree-star network design problem. We propose a new formulation for this problem and present a heuristic based on hierarchical decomposition of the problem. The proposed solution methodology exploits an Adaptive Reasoning Technique (ART), embedded as a meta-heuristic over specialized heuristics for the subproblems. In this context, we demonstrate the dynamic use of an exact solution technique within ART. The generalizability of the proposed solution methodology is demonstrated by applying it to a second problem, the Traveling Salesman Problem (TSP). Computational results are presented for both the HFC network design problem and the TSP, indicating high-quality solutions expending a very modest computational effort. The proposed solution method is found to be effective, and is shown to be easily adaptable to new problems without much crafting, and as such, has a broad appeal to the general operations research community.

Full text

HYBRID FIBER COAXIAL NETWORK DESIGN
RAYMOND A. PATTERSON
4-30E Business Building, School of Business, University of Alberta, Edmonton, Alberta, Canada T6G 2R6, ray.patterson@ualberta.ca
ERIK ROLLAND
The A. Gary Anderson Graduate School of Management, University of California,
Riverside, California 92521, erik.rolland@ucr.edu
(Received March 1999; revisions received December 1999, July 2000, February 2001; accepted February 2001)
Much interest exists in broadband network services to deliver a variety of products to consumers, such as Internet access, telephony, inter-
active TV, and video on demand. Due to its cost efficiency, Hybrid Fiber Coaxial (HFC) technology is currently being considered by most
Telcos and cable companies as the technology to deliver these products. The topological HFC network design problem as implemented by
several major companies is a form of the capacitated tree-star network design problem. We propose a new formulation for this problem
and present a heuristic based on hierarchical decomposition of the problem. The proposed solution methodology exploits an Adaptive
Reasoning Technique (ART), embedded as a meta-heuristic over specialized heuristics for the subproblems. In this context, we demonstrate
the dynamic use of an exact solution technique within ART. The generalizability of the proposed solution methodology is demonstrated by
applying it to a second problem, the Traveling Salesman Problem (TSP). Computational results are presented for both the HFC network
design problem and the TSP, indicating high-quality solutions expending a very modest computational effort. The proposed solution method
is found to be effective, and is shown to be easily adaptable to new problems without much crafting, and as such, has a broad appeal to
the general operations research community.
1. INTRODUCTION
Hybrid Fiber Coaxial (HFC) networks are currently being
tested and implemented to deliver broadband network
services (such as high-speed Internet access, telephony,
interactive TV, and video on demand) as a bundled prod-
uct. HFC networks are utilized both by large corpora-
tions such as The Roadrunner Group of Time Warner
Inc., GTE, Southern New England Telephone and Tele-
Communications (Watson 1997), and The @Home Corpo-
ration (Watson 1997, Levine 1997); and by small cable TV
operators such as Range TV Cable in Hibbing, MN (serving
6,000 customers) and Friendship Cable in Heath, TX (serv-
ing 8,000 customers) (Brown 1998). These operators of
broadband networks are just the beginning of a rush toward
implementing HFC network topological designs. Gasman
(1997) predicts that by the year 2003, HFC will exist
as a mainstream technology for broadband networks in
North America. At an estimated installation cost of $1,000
per customer (Brightman 1994), finding cheaper topolog-
ical HFC network designs before construction begins in
earnest on nationwide implementation is vitally important
to an industry typically characterized by very thin profit
margins.
In an HFC system, fiberoptic cables run from the head-
end (or central office) to a neighborhood node Optical Net-
work Unit (ONU) (Gupta and Pirkul 1999, Adams 1997,
Watson 1997). Coaxial cable is then run in a bus architec-
ture to serve up to approximately 2,000 customers (Gupta
and Pirkul 1999, Adams 1997, Watson 1997). The capacity
of the bus is approximately 750 MHz, which must be
shared by every customer on the bus. Cox Cable Commu-
nications runs the coaxial cable bus to about 1,000 homes,
whereas GTE and Southern New England Telephone and
Tele-Communications run it to between 200 and 600 homes
(Watson 1997).
A total cost exceeding $100 million to provide HFC
technology is commonplace. Cablevision Systems is spend-
ing $160 million over three years to upgrade their Boston
HFC network to provide high-speed Internet access, digi-
tal cable, and cable telephony (Coleman 1998). MediaOne
spent $200 million to acquire HFC-based telephony (Vit-
tore 1997). MediaOne and American Internet Corp. formed
a joint venture with The Roadrunner Group of Time Warner
to provide Internet service to 100,000 Internet users (Wilder
and Van Beaver 1998). Given an estimated cost of $1,000
per user, this cost also exceeds $100 million. In November
of 1997, A2000 (a joint venture of US West Interna-
tional/United and Philips Communications) committed over
$100 million for HFC technology by Cablespan systems
(Vittore 1997). In June of 1988 AT&T agreed to acquire
Tele-Communications Inc. and in February of 1999 AT&T
agreed to a pact with Time Warner Inc. to provide cable-
based local phone service to over 12 million customers
(including New York City) (Cauley and Blumenstein 1999).
On a much smaller scale, Range TV Cable in Hibbing,
Minnesota—with only 6,000 customers—spent $2.4 mil-
lion on the HFC infrastructure to provide video on demand
(Brown 1998).
Subject classifications: HFC network design: configuration of HFC topology. Tree-star: network architecture. Heuristic search: Solutions to problems in short time.
Area of review: Telecommunications.
Operations Research © 2002 INFORMS
Vol. 50, No. 3, May–June 2002, pp. 538–551 538
0030-364X/02/5003-0538 $05.00
1526-5463 electronic ISSN

Patterson and Rolland / 539
Profit margins in the cable TV industry have traditionally
been very low, and there is little reason to believe that the
profit margins will improve significantly with the introduc-
tion of broadband services. With the significant amounts of
resources being spent on HFC infrastructure, the topolog-
ical design of HFC networks is a timely and very impor-
tant issue. Developing a methodology for finding least-cost
topological HFC network designs is vitally important for
the competitiveness of both cable TV and telephone com-
panies.
In §2 of this paper we discuss HFC network design and
summarize related literature. A mathematical formulation
for HFC design is presented in §3. Section 4 proposes the
problem decomposition. We present the solution procedures
in §5, and discuss the computational results in §6. Our con-
clusions are summarized in §7. In Appendix A we discuss
the details of the Adaptive Reasoning Technique (ART)
solution procedure as applied to HFC network design. In
Appendix B we further demonstrate the generalizability of
the ART solution methodology by applying the adaptive
reasoning technique to the Traveling Salesman Problem.
The appendices can be found at the Operations Research
Home Page http://or.pubs.informs.orgin the Online Col-
lection.
2. RELEVANT LITERATURE
Literature relevant to this paper will be explored in two
steps. First, research related to the HFC problem will be
examined. Second, research related to the solution approach
taken in this paper will be reviewed.
2.1. Research Related to the HFC Problem
Several problems related to the HFC problem have previ-
ously been studied. The local access telecommunications
network design problem (Balakrishnan et al. 1995) is a fun-
damental problem in telecommunication network design.
This problem is one of deciding how to connect customer
demand points to a central switching center (central office,
or headend). In general, full network connectivity can be
achieved by adopting a tree or a star topology (or a combi-
nation of these two), typically with a resource-minimizing
objective. Additional technology can be introduced into this
problem to enable the shared use of high-capacity links.
That is, we often seek to use a high-capacity network struc-
ture that feeds into lower-capacity areas of the network.
The connection points between the high-capacity links and
the remaining links are equipped with electronic equipment
often referred to as concentrators. In general, this network
design problem can be viewed as consisting of four sepa-
rate subproblems (Gouveia and Lopes 1997, Gavish 1991):
(1) the terminal layout problem, where the actual connec-
tions between demand points are determined; (2) the con-
centrator quantity problem, where the task is to determine
the required number of concentrators in order to design an
(cost) efficient network structure; (3) the concentrator loca-
tion problem, where one selects the actual locations for
the concentrators; and (4) the terminal clustering problem,
where each demand point is assigned to a concentrator.
The topological HFC network design problem as imple-
mented by several major corporations (e.g., @Home, Time
Warner) is a form of the capacitated tree-star network
design (CTSD) problem (Gupta and Pirkul 1999). This
paper addresses what we call the hybrid-fiber coax-
ial capacitated tree-star network design (HFC-CTSD)
problem—a version of the local access telecommunications
network design problem. Gupta and Pirkul (1999) label this
problem as the HFC-CATV (Hybrid-Fiber Coaxial Com-
munity Antenna Television) Network Design. The problem
addressed in this paper, as well as by Gupta and Pirkul
(1999), involves a multitap bus architecture, which is accu-
rately characterized by the CTSD topology (see Figure 1).
Thus, we use the term HFC-CTSD to refer to this problem.
2.2. Research Related to Solution Approach
In this research we propose a two-phase algorithm that pro-
duces solutions to all four design subproblems. This paper
first presents a new formulation of the HFC-CTSD prob-
lem. Given the intractability of this problem, we chose
to develop solution techniques for a hierarchically decom-
posed version of the problem formulation. Balakrishnan
et al. (1995) also used hierarchical decomposition to solve
a version of the local access telecommunications network
design problem. Adaptive memory is added to heuristic
solution techniques for the decomposed problems. Through
the Adaptive Reasoning Technique (ART) (Patterson et al.
1999, Rolland et al. 1998, 1999), additional nonredundant
problem constraints are added. The ART algorithm learns
about the impact of the additional nonredundant problem
constraints on the solutions found by the heuristic algo-
rithms. ART adds and drops new nonredundant constraints
as the heuristic algorithm is repeatedly executed. The ART
methodology is a generalized approach that externally mod-
ifies the solution path used by one or more construc-
tive heuristics. Thus, ART self-adapts to specific problem
instances by learning from previous attempts at solving the
problem instance at hand.
Adaptive memory programming techniques, such as tabu
search and ART, belong to the class of “generate-and-test”
search techniques (Glover 1997, Patterson et al. 1999).
Tabu search typically operates with classical transition
neighborhoods (as used in local search), but can also
exploit constructive neighborhoods (Glover 1965, 1977;
Patterson et al. 1999). The memory constructs of the con-
structive tabu search technique (Fleurent and Glover 1998)
and ART are similar.
3. PROBLEM DEFINITION AND FORMULATION
The HFC-CTSD problem is one of connecting customers
in capacitated tree architecture networks to a wide area
network through an Optical Network Unit (ONU) to a

540 / Patterson and Rolland
Figure 1. HFC-CTSD network topology (adapted from Gupta and Pirkul 1999).
central office (or headend). An HFC-CTSD network can be
described in graph theoretic terms as follows:
1. The HFC-CTSD network is represented by a directed
graph with a designated root vertex V∗representing the
central office.
2. A link (fiberoptic or coaxial) between site iand the
central office is represented by an arc between nodes iand
V∗in the graph. The capacity of arc (i V ∗) corresponds to
the capacity of the ONU installed at site i.
3. A coaxial link between sites iand jis represented by
an arc i j or j  i connecting nodes ij ∈V\V∗.
Given this, the input to the problem is a directed graph
G=V  A with a designated root vertex V∗, demand vec-
tor Dwhere dvis the demand at node v, a vector of arc
capacities, and a vector of arc costs. The arcs in Arepre-
sent potential links in an HFC network. Because there are
a variety of ONU sizes, there are parallel arcs between a
node i∈V\V∗and V∗, each with a corresponding cost and
capacity. The output of the problem is a network topology
represented by a subset of the arcs A⊂Athat induce a
directed spanning tree Tof Vrooted in V∗, such that for
each node icontaining an ONU, the following condition is
met: The capacity of arc i V ∗∈Amust be at least as
large as the total demand of the nodes in the subtree of T
rooted in i. Hence, the total demand of the subtree branch
rooted in idoes not exceed the capacity of the ONU placed
at site i. Note that each subtree branch of Trooted in i
contains at most Knodes when uniform demand is used
(where Kdepends on the ONU size). For each node icon-
taining an ONU, the destination of the arc from imust be
V∗. The selected ONUs may have varying capacities and
capacity-dependent costs.
As stated previously, this problem is graphically rep-
resented in Figure 1, and as discussed above, numerous
cable TV companies currently experience a version of this
problem in practice as they are augmenting their services
with Internet and telephony access. Further, the HFC-CTSD
problem was shown in Gupta and Pirkul (1999) to be NP-
complete (by reduction).
We now formulate the HFC-CTSD problem. We assume
that an ONU of any capacity can be located at any demand
node location. Thus, every demand node location also acts
as a potential ONU site. To formulate the HFC-CTSD, we
introduce the following notation:
I set of demand nodes, i=2 n
J the set consisting of the headend (or central office)∪I,
j=1 n
K set of ONU types, k=1 m.

Patterson and Rolland / 541
ONU type k=1 does not use a concentrator, but rather
is a coaxial connection. ONU types 2 through mdo use
concentrators. We further assume that the following data is
available:
didemand at node i.
cijk cost of linking node ito node jusing ONU of type k
kcapacity of ONU type k.
Our decision variables are:
xijk =




1 if arc i j  using ONU type kis included
in the solution
0 otherwise
Auxiliary decision variables are:
fij =the flow on the arc i j
The flow variables capture the flow on arcs (ij). The flow
must be tracked to determine the appropriate concentrator
capacity. A unit of flow from a demand point to node 1
traces out the physical path of links that will be used to
serve that demand. These demand flows are aggregated in
fij . As such, the flow variable is considered to be an aux-
iliary decision variable.
The HFC-CTSD problem formulation is as follows:
Problem HFC-CTSD:
Minimize Z=


n

i=2
n

j=1
j=i
m

k=1
cijk xij k



(1)
Subject to:
n

j=1
j=i
m

k=1
xijk =1∀i∈I (2)
n

j=1
j=i
fij −
n

j=2
j=i
fji =di∀i∈I (3)
fij 
m

k=1
kxijk ∀i∈Ij ∈J and i= j (4)
n

i=2
n

j=2
j=i
m

k=2
xijk =0(5)
xijk ∈01∀i∈Ij ∈Jk ∈K (6)
We assume fixed capacity on each arc of k, but addi-
tional capacity from any node directly to the headend can
be bought by opening an ONU of type k. The costs of these
ONU alternatives differ. The formulation accommodates
nonuniform load requirements (demand) at each node (but
experiments in this paper will focus on uniform demand).
The formulation also accommodates nonuniform arc costs
of cijk for establishing a link between demand nodes iand
jusing an ONU of type k. We here assume that k=1
denotes a demand node that is NOT using an ONU. The
objective function then implicitly minimizes the total cost
of the communication arcs and ONUs that are used. The
flows emanating at the demand nodes are sent to the head-
end. Constraint set (2) implies that the flow from every
demand node, except the headend, is sent to one and only
one other node, using at most one ONU. In this formula-
tion, node 1 is designated as the central office (or headend).
Constraint set (3) implies that the cumulative flow coming
out of every demand node is equal to the local demand
plus the cumulative load going into that node. Constraint
set (4) assures that the cumulative flow coming out of any
node will not exceed the capacity of the outgoing arc, over
all ONU types. When demand is uniform, then the num-
ber of nodes in the subtree is limited by constraint set (4).
Otherwise, the number of nodes allowed in the subtree is
implied by constraint set (4). In constraint set (5) we make
sure that ONUs are only used if the flow coming out of the
node is going directly to the headend. Note that ONU type
k=1 is the same as not using a concentrator, equivalent to
using a coaxial connection. Constraint set (6) ensures the
binary properties of this problem. Note that this mathemati-
cal representation models an access network where there is
no direct communication between noncentral nodes. This is
typical of local access networks, and currently popularized
by the expansion of the cable TV network to accommodate
Internet services.
This formulation of the problem is new. Utilizing a deci-
sion variable with three indices (xijk ) and then adding con-
straint set (5) to eliminate the unwanted variables is a
unique contribution. Without constraint set (5), the model is
very general with the possibility of locating multiple ONUs
on a single subtree. Constraint set (5) prohibits all ONUs
except those which would connect directly to the headend.
Various related versions of this problem have been sug-
gested in the literature. Gupta (1996) and Gupta and
Pirkul (1999) proposed a Lagrangian relaxation of the prob-
lem using additional flow constraints. The results obtained
in this paper indicate potential for significant improve-
ments, as the average gaps between the obtained upper and
lower bound are approximately 10%. Lee et al. (1996) pre-
sented a generalized problem formulation for capacitated
networks with a tree configuration. They test a branch-and-
cut algorithm for networks composed of only 5, 10, and
20 nodes. Research on a multicenter version of the capaci-
tated minimum spanning tree (CMST) is also related to the
HFC-CTSD, particularly when the centers are connected
in starlike fashion (McGregor and Shen 1977, Narasimhan
1990).
4. HIERARCHICAL PROBLEM DECOMPOSITION
In our investigation, problem HFC-CTSD proposed above
could only be solved to optimality (using CPLEX version
4.0, a standard mixed-integer linear-programming tool, on a
Sun Ultra as described below) for small problem instances
(up to 20 nodes). However, the HFC-CTSD problem can be
solved suboptimally by hierarchically splitting the problem

542 / Patterson and Rolland
into two well-known subproblems, and utilizing heuristics
for the subproblems. The first subproblem is the CMST
problem with uniform demand and uniform capacities (see,
for example, Patterson et al. 1999 or Rolland et al. 1999
for a formal definition of the CMST). The CMST plays
an important role in the design of telecommunications net-
works. The CMST problem is NP-complete (Papadimitriou
1978). The CMST has in the past been solved using deter-
ministic heuristics (Kershenbaum and Chou 1974, Esau
and Williams 1966, Altinkemer and Gavish 1986, 1988),
memory-based heuristics (Patterson et al. 1999, Rolland et
al. 1999, Sharaiha et al. 1997, Amberg et al. 1996), cutting-
plane methods (Gouveia and Martins 1995, Hall 1996), and
integer programming formulations (Gavish 1983, Gouveia
1993, 1995). The heuristic approaches are often faster than
the cutting-plane and integer programming approaches, but
the quality of the solutions are typically not as good.
The second subproblem is the capacitated star-star with
concentrators (CSSC) problem. In keeping with the cable
TV nomenclature, concentrators are hereafter referred to
as ONUs. Both demand and capacity are variable (nonuni-
form). This problem is referred to by Soltys et al. (1997) as
the Local Access Optimization problem. This name is a bit
confusing because Balakrishnan et al. (1995) use the term
local access to imply a hierarchical tree structure. Given
this confusion over the precise definition of local access,
we use CSSC to more accurately denote this subproblem.
From the CMST subproblem, each subtree branch from
node V∗is considered to be a metanode. The CSSC prob-
lem is one of connecting the metanodes to a wide area net-
work either directly or through an Optical Network Unit
(ONU) to a central office (or headend). The resulting con-
figuration will be a star-star network. The location of a
potential ONU is deemed to be the location of the node on
each subtree branch that connects directly to the headend
(node V∗). The cost of connecting each metanode (sub-tree
branch) to a potential ONU or headend is the minimum
cost of connecting any node in the subtree branch in ques-
tion to the ONU or headend.
Graph notation can be used to accurately describe the
CSSC problem. Given a graph, GV  A, where Vis the
set of customer demand metanodes (with the associated
demand vector Dv), Ais the set of possible arcs in the graph
(with the associated arc cost vector Ca, an arc capacity of
Kfor each subtree branch rooted from an ONU node, the
associated ONU arc cost vector c
ak and arc capacity of k
for the demand of all metanodes rooted in a particular ONU
and routed via a single arc ato the headend at capacity
level k), and V∗is a designated headend node, the objec-
tive is to find a minimum-cost spanning tree that is rooted
in node V∗through any selected ONUs at any capacity k
where the total demand of each subtree branch rooted in
node V∗does not exceed the capacity of the selected ONU
(k) and a maximum of two arc connections from each
metanode to node V∗are allowed. The selected ONUs may
have varying capacities and capacity-dependent costs.
Because the subproblem solution found to the CMST is
integrated into the CSSC subproblem, both the CMST and
CSSC affect the solutions to the four separate subprob-
lems related to using capacitated trees for designing the
topology of local access networks (terminal layout prob-
lem, concentrator (or ONU) quantity problem, concentrator
(or ONU) location problem, and terminal clustering prob-
lem (Gouveia and Lopes 1997, Gavish 1991)). However,
the CMST primarily impacts the terminal layout problem
and the CSSC primarily impacts the ONU quantity prob-
lem, ONU location problem, and the terminal clustering
problem.
The formulation for the HFC-CTSD problem above
incorporates both the CMST and the CSSC subprob-
lems. The first problem is the classical CMST problem
as described above. This is extracted from the formulation
above by limiting kto a single choice (no concentrators
are used). An illustration of the final solution to the CMST
is depicted in Figure 2. For this illustration we have three
capacity levels and set 1=5
2=10, and 3=15.
Next, the CSSC problem can be superimposed on the
solution configuration given in Figure 2. This is done by
linking each subtree to the headend node either directly, or
via a potential ONU site. Thus, for each subtree, the node
linking the subtree to the headend node is deemed to be a
potential ONU site. In Figure 3, the triangles mark potential
ONU sites.
Each subtree is then represented by a single node, which
is the potential ONU linking the subtree to the headend
(see Figure 4).
The cost of connecting one subtree to the ONU of
another subtree is equal to the minimum cost arc from the
first subtree to the ONU. Of course, capacity must be avail-
able in order to connect a subtree to an ONU. For the
CSSC, the potential ONU sites are now limited to those
selected by the CMST. Note that there is nothing in the
formulation that prevents the solution from reverting to a
CMST solution itself as long as ONU nodes are not needed.
However, if capacity were available and it were cheaper
to link one subtree to another without an ONU, then this
already would have been done by the CMST solution. Thus,
a constraint prohibiting tree formation among non-ONU
subtrees is unnecessary. This can be proven as follows:
Proof. Let the flow into a potential ONU Abe denoted as
dA, and the flow into another potential ONU Bbe denoted
as dB. Further, assume that triangular inequality holds.
Now, if dA+dB>k, where kis equal to the coaxial
line capacity (1from the problem formulation) without
using a higher capacity ONU, then Aand Bcannot be
connected at either Aor Bwithout a higher capacity ONU.
With the inclusion of a higher capacity ONU at either A
or B, then it follows from constraint set (5) that the flow
must go directly from the ONU to node 1 (the headend).
On the other hand, if dA+dB<=k, then Aand Bwould
have been connected into a single subtree in the optimal
CMST solution. Otherwise, the solution would not have
been optimal. 

Patterson and Rolland / 543
Figure 2. Illustration of CMST problem solution.
The final solution to the CSSC problem is depicted in
Figure 5. From this figure, we see that those subtrees that
are not directly connected to the headend are reconnected
to an ONU. The capacity of an ONU (and the load com-
ing through it) may vary between ONUs, as is denoted by
the differently sized trunk lines going from the ONUs to
Figure 3. CMST problem with addition of potential ONU sites.
the headend (we assume for simplicity that the cost of con-
structing these lines is included in the ONU cost).
By recombining the CMST and the CSSC solutions, we
arrive at a feasible solution configuration for the HFC-
CTSD problem. This solution, which may be suboptimal, is
depicted in Figure 6. We see that the HFC-CTSD solution

544 / Patterson and Rolland
Figure 4. Illustration of initial CSSC problem with potential ONU sites.
has individual demand nodes linked together in subtrees
(found in the CMST solution), and these trees are con-
nected to either an ONU or to the headend directly (found
in the CSSC solution).
The next section introduces heuristic solution procedures
for solving the HFC-CTSD problem utilizing the partition-
ing mechanism described above.
5. THE ADAPTIVE MEMORY
SOLUTION PROCEDURE
For most combinatorial optimization problems there exists
a greedy heuristic that produces solutions for the problem.
We know that for Matroid problems (such as the uncapac-
itated version of the Minimum Spanning Tree problem),
a greedy heuristic always produces the optimal solution.
Many greedy heuristics are constructive; that is, they build
up complete solutions from scratch, one solution compo-
nent at the time. While studying the behavior of a greedy
heuristic, such as the Esau and Williams (1966) procedure
for the CMST, one will often see that the greedy heuristic
Figure 5. CSSC solution with selected ONU sites.
in practice only makes a handful of mistakes. These mis-
takes can be divided into two categories: primary mistakes,
that are due to the greedy selection criteria; and secondary
mistakes, that are due to limited available choices imposed
by earlier primary mistakes. The primary mistakes are the
crucial ones, and, given an opportunity, these are the mis-
takes that we would seek to prevent. A similar idea has
previously been proposed for network optimization and
the CMST problem (Karnaugh 1976, Gouveia and Lopes
1997). These approaches, called second-order heuristics,
share some constructs similar to those proposed below, but
lack much of the memory functions we use herein. The
second-order heuristics were found to be successful for the
CMST problem (Kershenbaum et al. 1980), but required
more computational effort than the simpler heuristics (i.e.,
the Esau-Williams’ (1966) procedure).
The Adaptive Reasoning Technique (ART) attempts to
“learn” about the primary mistakes of a heuristic. In the
ART framework, the greedy heuristic is executed repeatedly,
and for each new execution we probabilistically introduce

Patterson and Rolland / 545
Figure 6. Illustration of HFC-CTSD solution with selected ONU sites.
constraints that may prohibit certain solution elements from
being considered by the greedy heuristic. The prohibitions
may last for more than one iteration, and as such, we may
at any time have a collection of active constraints due to
the prohibited solution elements. The length of the prohi-
bition is a function of the cost of the solution element and
a random component. The active constraints are held in a
short-term memory. A long-term memory holds information
regarding which constraints were in the active memory for
the best set of solutions (for example, for the best 10 solu-
tions found so far). We can now augment these two memory
functions with some basic principles of learning. For exam-
ple, we can impose a certain degree of memory loss dur-
ing the execution of the algorithm (some amount of mem-
ory loss is often important, even in human memory, since
it enables us to explore “old territory” in light of newly
discovered evidence). We can also impose a “propensity to
learn” control variable, which can be varied over the exe-
cution time of ART (this is similar to a principle observed
in humans: We are typically less “willing” to learn as we
grow older). The end result of the ART algorithm is a set of
prohibitions that, when used in conjunction with a greedy
algorithm, would enable us to find an optimal, or close to
optimal, solution.
The execution cycle of ART is as follows:
(1) EW heuristic is used to solve the CMST problem,
subject to any additional nonredundant problem constraints
created by ART (the ART method as applied to the CMST
subproblem alone is fully described by Patterson et al.
(1999) and Rolland et al. (1999)).
(2) A solution procedure is applied to solve the CSSC
problem (see below).
(3) The combined HFC-CTSD problem solution found
in steps 1 and 2 is evaluated. Additional nonredundant
constraints are added through the ART mechanism for a
certain number of time periods. Additional nonredundant
constraints whose time duration has expired are removed.
(4) Repeat from step 1 until we have reached a preset
maximum number of iterations.
A depiction of ART as applied to the HFC-CTSD prob-
lem is given in Figure 7. A detailed description of the com-
plete ART heuristic as implemented for the HFC-CTSD
problem is given in Appendix A.
Figure 7. ART components and execution cycle.

546 / Patterson and Rolland
In the context of the HFC-CTSD problem, we need at
least two primary heuristics: one for the CMST and one for
the CSSC problem. We elected to use the Esau-Williams
(EW) heuristic for the CMST. For the CSSC subproblem
we compared the effect of two methods: a simple greedy
procedure, as well as an exact solution approach (AMPL
with CPLEX). In the CSSC subproblem, the set of poten-
tial ONU sites is given by the CMST heuristic. All nodes
(in the CMST solution) that are directly connected to the
headend are deemed to be potential ONU sites.
For the greedy procedure, a cost approximation is used
to decide whether or not to open an ONU. The cost of con-
necting a subtree to an ONU is approximated by the sum
of the minimum cost arc between the subtree and the ONU
plus a fixed-cost allocation (F). The fixed-cost allocation
is based on full-capacity utilization for the largest available
capacity of the ONU:
F=fixed cost of ONU/maximum ONU capacity∗
demand on subtree
When computing the objective function for the HFC-
CTSD, actual costs utilizing minimum ONU necessary to
handle actual demand are used. The greedy CSSC proce-
dure (GCSSC) can now be described as follows:
Procedure GCSSC
1. For the connections between all valid source subtrees
(a valid source subtree is a subtree whose connection to
the headend has not yet been designated as an ONU site,
nor has this subtree previously been connected to another
ONU) and all available ONUs (an available ONU is a node
connected directly to the headend node in the CMST sub-
problem which has either been declared to be an ONU site
or whose subtree has not yet been connected to another
ONU), do:
2. Find the minimum cost connection subject to avail-
able capacity at the largest capacity level (using adjusted
GCSSC costs as described above, only connections to
ONUs with cost savings over connecting directly to the
headend are considered).
3. Record the lowest cost connection. Prohibit this
source subtree from being a potential ONU site and declare
it to be a nonvalid source tree. Declare the subtree with the
“winning” ONU to be an ONU site and a nonvalid source
tree. Repeat from step 1.
4. If no such subtrees are found, then go to step 5.
5. Compute the cost of the new solution, and output
solution configuration.
The GCSSC procedure is fairly simple, and numer-
ous alternative implementations for solving the CSSC are
possible. One alternative is to optimally solve the CSSC
subproblem. The approach is reasonable when the CSSC
subproblem is fairly small. An exact solution method can
be stated as follows:
Procedure EXACT
1. For the connections between all valid source subtrees
(defined above) and all available ONUs, use AMPL with
CPLEX to find an optimal solution to the CSSC subprob-
lem. When used within the ART solution framework, the
AMPL data file is modified at the beginning of each iter-
ation to include all nodes connecting directly to the head-
end as potential concentrators as given by the ART/Esau-
Williams CMST solution.
The AMPL with CPLEX solution procedure is based on
the following formulation of the CSSC problem. Let
I set of subtrees identified in the CMST solution,
i=2 n
J the set consisting of the headend ∪I j =1 n
K set of ONU types, k=1 m.
We further assume that the following data is available:
didemand emanating from subtree i(known from the
CMST solution).
cijk minimum cost of linking subtree ito node jusing
ONU of type k.
kcapacity of ONU type k.
Our decision variables are:
xijk =




1 if arc i j  using ONU type kis included
in the solution
0 otherwise
Problem CSSC
Minimize Z=


n

i=2
n

j=1
j=i
m

k=1
cijk xij k



(7)
Subject to:
n

j=1
j=i
m

k=1
xijk =1∀i∈I (8)
M
m

k=1
xj1k
n

i=2
i=j
m

k=1
xijk ∀j∈I (9)
di+
n

j=2
m

k=1
djxjik 
n

j=1
m

k=1
kikxij k ∀i∈I (10)
n

i=2
n

j=2
j=i
m

k=2
xijk =0(11)
xiik =0∀i∈Ik ∈K (12)
xijk ∈01∀i∈Ij ∈Jk ∈K (13)
where Mis a sufficiently large number.
Constraint set (8) ensures that each subtree will be con-
nected to either an ONU or to the headend. Constraint set
(9) ensures that only open ONUs accept demand from other
subtrees. Constraint set (10) ensures that the cumulative
demand into an ONU plus local demand is less than the
capacity of the selected ONU. Constraint set (11) ensures

Patterson and Rolland / 547
that ONUs are only connected directly to the headend. Con-
straint set (12) simply ensures that no node or ONU can be
linked to itself. The binary nature of the decision variable
is captured in constraint set (13). It should be noted that
the decision analog of this problem has been proven to be
NP-complete (Gupta and Pirkul 1999).
In the next section we present some computational expe-
riences with these algorithms, and in Appendix B we show
the generalizability of ART by applying our algorithm to
the Traveling Salesman Problem.
6. COMPUTATIONAL EXPERIMENTS
To analyze the model and algorithms presented in this
paper, we generated 80 test problems as follows: Xand
Ycoordinates for each demand node were generated ran-
domly in a 100 by 100 square with the headend node in the
center, and the arc cost without use of ONUs (cijk , where
k=1) is equal to the Euclidean distance between the nodes.
The variable arc cost for arcs using ONUs (of level k>1)
are set as follows:
cijk =cij k ∗1+k −1∗01 for k>1
This function is commensurate with recent pricing of con-
centrators, where additional capacity (ports) typically cost
10% more than the smaller capacity. All nodes have unit
demands. Three ONU levels were used, where the lowest
level can be viewed as having no ONU (k=1). Capac-
ities are four for level one, 10 for level two, and 25 for
level three. For each arc, the fixed-cost allocation for the
ONUs are as follows: zero for level one (no ONU), 10 for
level two, and 19 for level three. Four groups of problems
were generated in the first data set: 20 problems with 20
nodes (19 demand nodes and 1 headend node), 20 problems
with 40 nodes (39 demand nodes and 1 headend node), 20
with 60 nodes (59 demand nodes and 1 headend node),
and 20 with 80 nodes (79 demand nodes and 1 headend
node). Optimal solutions were obtained for all the 20-node
problems using CPLEX (version 6). Larger problems could
not be solved to optimality using mixed linear integer-
programming tools (see below), but a Lagrangian relaxation
technique (Gupta and Pirkul 1999) was used to provide
lower bounds as well as a comparative upper bound.
A single processor (300 MHz) SUN Ultra 30 computer
with 128 M of RAM and 1 Gb of swap space, running
Solaris 2.6 CDE version 1.2 was used for all computa-
tions, and all computational times are reported in seconds.
We heuristically solved all 80 problems using the ART
procedure, using the two alternative primary heuristics for
the CSSC subproblem (GCSSC and EXACT) as described
above. CPLEX version 6 was used to solve the EXACT
portion of the heuristic procedures. Details of the compu-
tational results are provided in Tables 1 to 4.
In Table 1 we have reported optimality gaps from five
different solution approaches. The optimal solutions are
reported in column 1, and the CPU times associated with
obtaining these solutions are reported in the last column
(labeled CPLEX). Optimal solutions were obtained using
AMPL with CPLEX, using the HFC-CTSD formulation
presented above. The Esau-William CMST solution is a
feasible solution to the HFC-CTSD problem (using no
ONUs), and these optimality gaps are reported in column
2 (marked EW-CMST). The EW-CMST solution in col-
umn 2 is a baseline solution without ONUs, and must be
greater than or equal to an optimal ONU-based solution.
The CPU times associated with these solutions are always
less than one second. The solution in column 3 is the
ART solution using EW-CMST plus the GCSSC heuris-
tic as the heuristic engine to assign ONUs (marked ART
EW+GCSSC). The associated CPU times are found in col-
umn 7. In column 4 we report the upper bound solution
found using the Gupta and Pirkul (1999) Lagrangian relax-
ation with the associated computational times in column 8.
In column 7 we report the optimality gaps from the solu-
tion constructed by EW-CMST using procedure EXACT to
find optimal ONU assignments for the CSSC sub-problem
(EW +EXACT). The associated computational times are
found in column 9. The EW +EXACT heuristic is used as
the solution engine for the ART heuristic using procedure
EXACT to find improved solutions; results are reported in
column 6 (marked ART EW+EXACT) and the associated
CPU times are in column 10.
Tables 2, 3, and 4 contain upper and lower bounds for
the problems (labeled CPLEX Bounds in columns 1 and
2), and were obtained using AMPL with CPLEX (version
4), with the HFC-CTSD formulation stated in §1. We show
the details for 10 problems each from node sizes 40, 60,
and 80. The CPLEX upper bounds, lower bounds, and CPU
times are provided to demonstrate just how rapidly these
problem instances become intractable (the memory capac-
ity of the Sun Ultra prevented us from getting further results
for these problems). No optimal solutions are known for the
40, 60, and 80 node problems. We report the CPU times in
column 13, indicating the time CPLEX ran before exceed-
ing its memory limitations. The Gupta and Pirkul (1999)
Lagrangian lower bound is shown in column 3. All other
columns in Tables 2, 3, and 4 are identical to those of
Table 1 except that the percentage gaps are based on the
Lagrangian lower bound. We mark the best solution results
for each problem in the tables in italics. The legend for the
tables is presented separately.
From Tables 1, 2, 3, and 4 we see that using the
EW+EXACT heuristic procedure produces better solutions
in significantly less time than the ART with EW+GCSSC
and Lagrangian upper bound heuristics. We also see that
ART with EW +EXACT substantially improves the solu-
tions (as compared to all other methods). Results for the
EW-CMST heuristic (without ONUs) are provided as a
baseline measure. We stress here that the CSTD is a design
problem, and as such the CPU times are not as crucial as for
a real-time optimization problem. However, telecommuni-
cations network planners often solve these problems as part
of case studies where they have to solve a series of related
problems with varying demand patterns and costs. In such

548 / Patterson and Rolland
Table 1. Computational results for 20-node problems.
Optimality Gaps CPU Times
Optimal ART ART ART ART CPLEX
Solution EW-CMST EW +GCSSC LAG UB EW+EXACT EW +EXACT EW +GCSSC LAG EW +EXACT EW+EXACT (optimal)
317 1009% 2.21% 2.52% 0.32% 0.32% 3.00 24.50 <14660 15940
380 474% 4.21% 3.95% 0.79% 0.79% 2.90 21.10 <14300 29280
323 991% 6.81% 2.48% 1.86% 0.31% 3.00 27.90 <14760 16440
314 255% 1.59% 1.59% 1.27% 0.32% 3.00 30.50 <14340 45090
380 026% 0.26% 0.53% 0.00% 0.00% 2.90 29.50 <14750 90720
365 466% 3.84% 3.84% 2.74% 0.82% 2.90 28.70 <15050 53160
292 377% 1.37% 1.37% 2.40% 0.00% 2.90 25.20 <14750 37140
314 1115% 3.18% 1.59% 1.27% 1.27% 3.00 23.11 <14530 3380
329 1033% 2.74% 4.86% 7.29% 1.22% 3.10 30.95 <14890 529530
302 166% 1.66% 1.32% 0.33% 0.33% 2.90 26.39 <14660 440900
365 548% 3.56% 0.82% 5.48% 1.10% 3.00 21.45 <14480 317370
295 847% 8.47% 3.05% 6.44% 5.76% 2.90 25.18 <14990 630
358 447% 1.68% 1.96% 3.63% 0.00% 3.00 33.24 <14690 102140
328 732% 5.49% 1.22% 4.88% 2.13% 3.00 21.38 <15160 4870
323 217% 2.17% 2.17% 2.17% 0.62% 2.90 29.79 <14990 111480
316 949% 4.75% 4.75% 7.28% 2.85% 3.10 26.83 <14680 330
372 1129% 2.42% 2.96% 6.18% 0.27% 3.00 26.03 <14500 64830
330 545% 3.64% 3.03% 1.82% 0.00% 3.10 27.26 <15050 1194750
338 621% 0.59% 0.00% 3.25% 0.00% 3.00 30.47 <14600 40630
261 536% 4.60% 4.60% 0.77% 0.00% 2.90 23.10 <14980 2200
Average: 624% 3.26% 2.43% 3.01% 0.91% 2.98 26.63 <14741 155041
Legend for Tables
EW-CMST: Results from the Esau-Williams CMST heuristics
ART EW +GCSSC: Adaptive Reasoning Technique with EW for CMST and procedure GCSSC for CSSC
LAG UB: Best feasible solution generated by Lagrangian relaxation (Gupta and Pirkul 1999)
LAG LB: Best lower bound generated by Lagrangian relaxation (Gupta and Pirkul 1999)
EW +EXACT: EW for CMST and procedure EXACT for CSSC
ART EW +EXACT: Adaptive Reasoning Technique with EW for CMST and procedure EXACT for CSSC
planning, the trade-off between solution quality and CPU
time becomes more complicated, and the EW +EXACT
solution method is an excellent choice because it is much
faster than the Lagrangian heuristic with comparable (and
often slightly better) solution values.
As seen in the tables above regarding the ART heuris-
tic, if we do not use an exact solution technique for the
CSSC subproblem, the CPU times are severely reduced.
Table 2. Computational results for 40-node problems.
Solution Values Lagrangian Lower Bound Gaps CPU Times
CPLEX Bounds LAG ART LAG ART ART ART
Upper Lower LB EW-CMST EW +GCSSC UB EW+EXACT EW +EXACT EW +GCSSC LAG EW +EXACT EW +EXACT CPLEX
565 448 49870 1610% 1229% 1229% 1189% 728%1720 21790 <1 10290 1823780
936 462 53671 1440% 1235% 937% 807% 639%1820 24430 <1 10700 1861470
1004 424 52398 1432% 897% 840% 764% 573%1680 26060 <19830 1691940
584 480 51396 1421% 1013% 1129% 682% 546%1710 25400 <19260 2450020
588 437 50435 1381% 885% 1183% 865% 746%1730 21920 <19950 2541190
847 489 57278 1610% 1174% 877% 1104% 563%1930 25890 <1 10070 2565590
841 470 52531 2754% 1707% 1422% 1860% 1041%1690 29140 <1 11050 2037040
681 440 49778 1873% 1190% 1230% 1049% 788%1640 27120 <19530 3042988
586 520 54869 1445% 862% 789% 844% 480%1720 28860 <1 11150 3477084
651 485 53231 1272% 577% 708% 671% 182%1840 25680 <1 11030 3454928
Average: 1624% 1077% 1034% 983% 629%1748 25629 <1 10286 2494603
However, the solution quality is also severely affected by
not using an optimal solution method for the CSSC. More-
over, comparing the ART EW +EXACT solution times to
those of CPLEX (for solving the entire CTSD problem)
we see that the heuristic requires only a fraction of the
computational effort required by CPLEX. Note also that
this substantial effort by CPLEX does not even guarantee
that an optimal solution is found with that methodology

Patterson and Rolland / 549
Table 3. Computational results for 60-node problems.
Solution Values Lagrangian Lower Bound Gaps CPU Times
CPLEX Bounds LAG ART LAG ART ART ART
Upper Lower LB EW-CMST EW +GCSSC UB EW +EXACT EW +EXACT EW+GCSSC LAG EW +EXACT EW +EXACT CPLEX
742 610 652025 23.31% 1303% 1319% 951% 766%4970 105360 <1 18990 8162974
1154 518 624143 24.49% 1600% 1456% 1376% 1247%5020 110490 <1 19280 5853411
734 573 622472 31.41% 1278% 1438% 1229% 860%4920 109040 <1 23510 5084951
903 609 664555 28.36% 1376% 1391% 1241% 1000%4910 132780 <1 20700 3116187
743 631 666892 24.01% 616% 901% 856% 496%5070 113990 <1 19710 2308807
1162 579 65452 28.19% 1337% 863% 893% 649%4910 98840 <1 17470 4305454
794 577 604498 25.89% 1762% 1299%1497% 1365% 5020 96950 <1 21830 3773778
1208 617 675951 28.12% 1554% 1066% 1125% 1022%5210 122370 <1 17660 4312363
1021 646 677118 26.12% 1519% 899% 1150% 500%5090 105770 <1 18620 4169509
720 585 610072 27.53% 1540% 1031% 1195% 1015%5260 109600 <1 21440 7057445
Average: 26.74% 1388% 1166% 1151% 892%5038 110519 <1 19921 4814488
(CPLEX stops due to memory restrictions before converg-
ing upon an optimal solution).
In Table 5 we present a summary of all the compu-
tational results. The experimental data sets from Tables
1 through 4 are included, as well as 30 additional prob-
lems. A total of 80 experimental data sets are summarized
in Table 5—20 data sets for each of the 20, 40, 60, and
80-node problems. Average percentages for each method
(over the CPLEX Optimal for the 20-node problems and
the Lagrangian lower bound solution for the 40, 60, and
80-node problems) are reported by problem size in Table 5.
EW +EXACT is a high-quality technique in terms of
both solution quality and computational time. We see that
EW +EXACT on average produces better solutions than
EW-CMST, ART with EW +GCSSC, and the Lagrangian
upper bound. The subtrees identified by the EW-CMST
heuristic contain a very good set of potential ONU sites,
but the simplistic GCSSC heuristic is unable to effectively
select the optimal ONU sites. Even though ART with EW+
GCSSC substantially improves upon the EW +GCSSC
solutions (not shown in the tables), the ART technique is
unable to bring the GCSSC heuristic to optimality.
ART with EW +EXACT finds the best solutions, and
the improvements in the solutions are substantial. ART
with EW +EXACT produces the minimum solution value
Table 4. Computational results for 80-node problems.
Solution Values Lagrangian Lower Bound Gaps CPU Times
CPLEX Bounds LAG ART LAG ART ART ART
Upper Lower LB EW-CMST EW +GCSSC UB EW +EXACT EW +EXACT EW+GCSSC LAG EW +EXACT EW +EXACT CPLEX
1151 687 78339 3352% 1897% 1220% 1195% 978% 12050 297850 <1 40280 4656000
1300 684 74802 3181% 1818% 1337% 1109% 1069% 12280 188460 <1 40900 6018700
1073 728 773325 3642% 2078% 1237% 1328% 966 % 12440 203340 <1 37790 3134500
984 735 787508 3663% 1632% 1670% 1225% 1136 % 11620 365910 <1 41880 5765500
1016 693 734577 4035% 2334% 1190 %1912% 1599% 12040 256490 <1 37930 6413000
1199 749 791573 3972% 2115% 1243% 2140% 1218% 12510 296630 <1 40800 7456800
1074 728 802755 3317% 1635% 1523% 1211% 1162% 11770 296150 <1 39100 4519200
1357 668 788003 3579% 1726% 1650% 1535% 914% 12350 311040 <1 46660 4849300
1270 724 794633 3717% 1553% 1389% 1691% 1099% 12280 281090 <1 36090 2529000
1216 762 819893 3453% 1709% 1392% 1331% 1014% 12180 292070 <1 36880 12814900
Average: 3591% 1850% 1385% 1468% 1115% 12152 278903 <1 39831 5815690
among the heuristic procedures tested in 71 of the 80 test
problems in Table 5 (17 of the 20-node problems, 19 of
the 40-node problems, 19 of the 60-node problems, and 16
of the 80-node problems). ART with EW+EXACT takes
more time than the Lagrangian method for the 20-node
problems, but substantially less time for the larger prob-
lems. ART with EW+EXACT takes more time compared
to ART with EW+GCSSC and also EW +EXACT. The
computational time of ART with EW+EXACT also grows
with the problem size, but not unreasonably.
7. SUMMARY AND CONCLUSIONS
We have proposed a new formulation for the HFC-CTSD
problem (a capacitated tree-star concentrator design prob-
lem). The HFC-CTSD allows a choice among a variety of
ONU options to connect demand nodes through subtrees
with uniform capacities to the headend of a hybrid fiber
coaxial cable TV network. Given the complexity of the
HFC-CTSD problem, we proposed a heuristic based on hier-
archically decomposing the problem into well-known sub-
problems: the CMST and the CSSC. Every demand node is
a potential ONU site, but ONUs are only allowed on arcs
connecting directly to the headend from the CMST subprob-
lem solution. This formulation can easily be accommodated
to allow ONUs of various capacities to be connected in

550 / Patterson and Rolland
Table 5. Summary of expanded computational results.
Percentage Over Best Known Lower Bound CPU Times
Problem ART LAG ART ART ART
Size EW-CMST EW +GCSSC UB EW +EXACT EW +EXACT EW +GCSSC LAG EW +EXACT EW +EXACT
20 624% 326% 243% 301% 091% 298 2663 <14741
40 1728% 1168% 1035% 1067% 637% 1730 25074 <1 10372
60 2719% 1469% 1276% 1254% 928% 5093 112453 <1 20501
80 3531% 1859% 1338% 1534% 1127% 12387 298854 <1 40509
Average: 2151% 1205% 973% 1039% 696% 4877 109761 <1 19031
a treelike fashion by reducing the number of arcs that are
currently prohibited through constraint set (5) in the HFC-
CTSD problem formulation.
The best solution technique found in this research to
solve this problem is the ART methodology, utilizing the
Esau-Williams heuristic for solving the CMST subproblem.
Both a greedy swapping heuristic and an exact solution
procedure for solving the CSSC subproblem were tested.
Computational results of this approach are very favorable.
The contributions of this research are as follows: First,
we present a new formulation for the HFC-CTSD prob-
lem. This problem is a realistic version of local access net-
work design, and is motivated by the redesign of current
cable TV networks in response to providing Internet and
telephony services over such (largely coaxial) networks.
The formulation and solution techniques presented in this
paper address all four subproblems identified in previous
research (Gouveia and Lopes 1997) as the major compo-
nents of local access network design: concentrator quantity
problem, concentrator location problem, terminal clustering
problem, and terminal layout problem.
Second, we propose a method by which we hierarchi-
cally decompose the HFC-CTSD problem, and solve it
using the ART memory-based heuristic solution technique.
ART modifies the CMST subproblem solution. The modi-
fied solution to the CMST subproblem then determines the
potential ONU sites for the CSSC subproblem, which is
then solved. Because ART creates new solutions for the
CMST subproblem by essentially reconfiguring the sub-
trees at each iteration, new potential concentrators are pro-
posed as well as new (closest) connections from the new
subtree configurations to the potential concentrator sites.
Thus, even though ART is directly modifying the CMST
solution, it is implicitly modifying the CSSC solution as
well by manipulating the input to the CSSC subproblem.
This paper contributes to the ART literature by clearly
demonstrating new and inventive ways to utilize the ART
methodology in a hybrid manner with other solution tech-
niques.
Third, we demonstrate the dynamic use of an exact
solution technique (AMPL and CPLEX) within a memory-
based heuristic framework. This is implemented by dynam-
ically rewriting the AMPL CSSC data files for each iter-
ation of the ART heuristic. We have clearly demonstrated
that such a procedure without ART can easily match per-
formance with Lagrangian relaxation in a fraction of the
computational time. And such a procedure with ART sig-
nificantly outperforms Lagrangian relaxation and all other
heuristic methods tested in terms of solution quality.
Fourth, we demonstrated the applicability of the ART
solution method to the well-known Traveling Salesman
Problem (see Appendix B for the TSP implementation
of ART). In Appendix B we showed that with a simple
exchange of the primary heuristic with very minimal modi-
fications to the algorithm, the ART methodology efficiently
solves a large set of well-known TSP instances. For these
problems, we achieved solutions with an average optimality
gap of less than 0.25% (see Table B.1). Optimal solutions
were found for 40% of the test problems. By demonstrating
the ease with which the ART methodology can be modi-
fied to effectively solve the TSP (in addition to solving the
HFC-CTSD problem), this research should serve as a cat-
alyst for designing new memory-based constructive search
heuristics.
APPENDIX
The appendix can be found at the Operations Research
Home Page http://or.pubs.informs.org/pages/collect.html.
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