European Journal of Operational Research 144 (2003) 56–67
www.elsevier.com/locate/dsw
Discrete Optimization
Optimisation of the interconnecting network of a UMTS
radio mobile telephone system
Matteo Fischetti
a,*
a
, Giorgio Romanin Jacur a, Juan Jose Salazar Gonz
alez
b
DEI, University of Padova, Via Gradenigo 6/a, 35131 Padova, Italy
b
DEIOC, University of La Laguna, Tenerife, Spain
Received 30 November 2000; accepted 18 October 2001
Abstract
In this paper we address a very important optimisation problem arising in the telecommunication field, namely the
design of the interconnecting network of a UMTS radio mobile telephone system. For this NP-hard optimisation
problem we propose a new mixed-integer linear programming model, as well as several classes of additional constraints
meant at improving the performance of solution algorithms and the quality of the lower bounds produced. Afterwards,
we introduce an exact solution procedure in the branch-and-cut framework, and evaluate it on a library of real-life test
problems provided by CSELT, a major research laboratory operating with an Italian telephone operator (TELECOM
Italia). We report on our computational experience on these test instances, showing that the method we propose is
capable of finding tight lower bounds and approximate solutions for real-world instances, within acceptable computing
time.
2002 Elsevier Science B.V. All rights reserved.
Keywords: Communication; Location; Mixed integer linear models
1. Introduction
A mobile radio telephone system aims at ensuring secure communications between mobile terminals and any other type of user device, either
mobile or fixed. A mobile customer should be
reachable at any time and in any location where
the radio coverage is granted.
The connection among mobile terminals (i.e.,
the user’s handheld terminals) and fixed radio base
*
Corresponding author. Tel.: +39-049-827-7824; fax: +39049-827-7826.
E-mail address: fisch@dei.unipd.it (M. Fischetti).
stations is obtained by means of radio waves.
However, a single antenna system cannot cover the
whole service area. In fact, that choice would require high irradiation power both from the fixed
and the mobile stations, with consequent possible damage due to the generated electromagnetic
field.
The above limitations lead to the implementation of ‘‘cellular systems’’, constituted by several
fixed radio base stations and related antenna
systems. Each single radio base station coverage
area is called ‘‘cell’’ and it serves a small region of
variable size ranging from 10 to 100 m (high user
density inside business buildings) to 1–20 km (low
user density areas in the country).
0377-2217/03/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 3 8 3 - 6
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
Every fixed radio base station, usually called
base transceiver station (BTS), is both transmitting
and receiving signals on a variable number of
frequencies. Depending upon the type of system
considered and the radio access scheme, each frequency (or carrier) permits the allocation of a
variable number of channels; in the GSM case,
each frequency carries eight channels.
Whenever a user moves from a cell to an adjacent one during a communication, a new channel
is assigned inside the cell just entered. This feature
is commonly called handover. Covering the served
region with several cells allows for ‘‘frequency
reuse’’, i.e., for the use of the same frequency inside two or more non-interfering cells.
The users’ mobility causes issues related to
the user location detection and to cell change,
which are managed by equipment implementing
the interface between the BTS and the fixed network.
Third generation mobile telecommunication
systems are currently in the course of standardisation in Europe under the name of universal
mobile telecommunication system (UMTS). The
basic architecture of a UMTS network includes
the following devices:
• Mobile terminal (MT) of different types (e.g.,
phone, fax, video, computer).
• Base transceiver station (BTS) interfacing mobile users to the fixed network; a BTS handles users’ access and channel assignment. Due
to the inherent flexibility featured by next generation BTSs, different network topologies can
be undertaken: the BTS can be either directly connected to the switching equipment
(smart BTS) or linked to a BTS controller
(CSS).
• Cell site switch (CSS), which is a switch connected to several BTSs on one side and to a single local exchange (LE) (see below) on the other
side; each CSS is devoted to the management of
local traffic inside its controlled area, as well as
to the connection of the controlled BTSs to the
LE.
• LE, which is a switch connecting the BTSs
to the network, either directly or through
CSSs.
57
• Mobility and service data point (MSDP), which
is a database where information about users is
registered; it may be located either together with
an LE or with a CSS, according to a centralised
or distributed connection management.
• Mobility and service control point (MSCP),
which is a controller to manage mobility; it
can access the database to read, write or erase
information about users, and is generally associated with LEs and MSDPs.
In this paper we address the problem of optimising a UMTS interconnection network having a
multilevel star-type architecture. This is a difficultto-solve (NP-hard) optimisation problem of crucial
importance in the design of effective and low-cost
networks.
The general characteristics of UMTS and related standardisation problems were presented in
[2,3,9,17]; some hints in design and optimisation may be found in [1,4,5,8,14], but they concern
either different application fields or simpler network topologies with respect to the ones studied
here.
As to the literature on various location problems, we refer the reader to Labbe and Louveaux
[12] for a recent annotated bibliography. Facility
location problems related to the one studied in the
present paper have been very recently addressed in
Chardaire et al. [7], where an uncapacitated twolevel network design problem is studied, and in
Klose [11], where a Lagrangean heuristic based on
the relaxation of the capacity constraints is proposed.
The paper is organised as follows. In Section 2
we give a more detailed description of the UMTS
multilevel architecture. A mixed-integer linear programming model is proposed in Section 3, and
a possible solution algorithm in the branch-andcut framework is outlined. Some improvements of
the basic model are presented in Section 4, where
new families of valid inequalities are introduced
along with the corresponding separation algorithms. Computational results on a library of realworld test problems provided by CSELT, a major
research laboratory operating with TELECOM
Italia, are reported in Section 5. Some conclusions
are finally drawn in Section 6.
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M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
2. The UMTS multilevel architecture
In the problem we consider, a certain number of
potential CSS and LE sites is given, among which
the planner has to choose those to be actually
activated. We consider a three level star-type
UMTS architecture, defined by an upper layer
made up of active LEs (chosen in the given set of
potential LEs), a middle layer made up of active
CSSs (also chosen in the given set of potential
CSSs), and a lower layer made up of the given
BTSs (each of which is required to play the role of
a leaf in the star-type structure).
Fig. 1 illustrates a situation where 2 (out of 5)
LEs and 4 (out of 6) CSSs are activated, and define a feasible star-type architecture to serve the
17 given BTSs. Note that each activated LE
plays the role of the root of a tree spanning a
different connected component. Moreover, the
problem cannot be decomposed in two independent subproblems consisting of assigning LEs to
CSSs and CSSs to BTSs, respectively, in that
the choice of the active CSSs and of their traffic
load creates a tight link between the two subproblems.
Each BTS has to be connected to the core network, either through a single active CSS or directly
to a single active LE (for certain pre-specified
BTSs the direct connection to an LE can however
be forbidden). Every BTS is characterised by its
geographical location, its carried traffic, the number of channels required, and by its type. The BTS
location is the result of a complex planning process
which is not considered in this paper. The BTS
carried traffic and number of channels depend on
the expected average number of users served by the
cell. More precisely, the traffic is the total transmitted information, and the number of channels is
the number of independent simultaneous communications, each supported by a communication
module (64 kbit/seconds).
Every CSS is connected to the network through
a single LE.
Channels between a BTS and a CSS or an LE
must be packed into ‘‘modules’’ of a given capacity
(maximum number of channels in a module). In
the plain pulse code modulation (PCM) hierarchy
each module collects up to 30 channels at 64 kbps
thus granting a capacity of 2 Mbps. The type depends on the connection either to an LE or to a
CSS, as seen above.
Costs implied by a BTS concern:
• the equipment cost;
• the actual connection cost, depending on the
connected CSS or LE; the cost is assumed to
be linear in the number of used modules.
Every CSS is characterised by its type, its location,
its traffic capacity, the maximum number of BTSs
and modules that can be supported.
CSSs may be of two different types, namely
‘‘simple’’ (type 1) or ‘‘complex’’ (type 2), having
different load and cost characteristics.
Costs implied by a CSS concern:
• the plant cost, depending on the type of the
equipment and on the location;
• the connection cost, depending on the connected LE; this cost is linear in the number of
used modules.
Fig. 1. The three level star-type UMTS architecture.
Every LE is characterised by its location, its traffic,
and by the maximum number of supported PCM
modules.
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
Costs implied by an LE concern:
• the plant cost, depending on the location.
Feasibility constraints are either of the ‘‘congruence type’’, imposing that any connection is
permitted only between activated sets, or of the
‘‘limitation type’’, imposing that the traffic through
any activated set is limited by the given bounds,
both in terms of transmitted information and in
terms of connected modules.
The problem then consists of choosing the CSS
and LE to be activated, and the way to connect
them to the BTSs and between each other, so as to
produce a feasible three-level network of minimum
cost (a more detailed description is given in the
next section). This combinatorial optimisation
problem is strongly NP-hard, as it generalises the
classical (also strongly NP-hard; see e.g. [12]) Facility Location Problem.
3. A mixed-integer linear programming model
We next introduce a mathematical model for
the problem, based on the following input data.
We consider a set of n BTS locations, a set of m
potential type 1 or 2 CSS locations, and a set of p
potential LE locations.
A BTS in location i produces a traffic flow tiBTS
through diBTS communication channels. Channels
to an LE are packed into ‘‘modules’’ (cables or
microwave). If Q is the largest number of channels
that can be arranged in a module, then the BTS in
location i requires eBTS
:¼ ddiBTS =Qe modules,
i
where dre ¼ minfi 2 N : i P rg denotes the upper
integer part of a given real number r. It is
worth observing that Q may in some cases depend
on the location that a particular module is connecting.
A CSS in location j of type h 2 f1; 2g can provide a traffic flow not larger than a given upper
bound TjCSS-h , can support a number of modules
not larger than EjCSS-h , and a number of BTSs not
larger than NjCSS-h .
An LE in location k can provide a traffic flow
not larger than a given upper bound TkLE , and can
support a number of modules not larger than EkLE .
59
BTS type is pre-defined as basic (it must be
connected to a CSS), or isolated (it must be connected directly to an LE), or free (it can be connected to a CSS or directly to an LE).
The fixed cost required to open a CSS of type h
in location j is fjCSS-h , and the cost to open an LE
in location k is fkLE . The fixed cost to activate a
BTS in location i and to connect it to a CSS is
fiBTS-CSS , whereas the fixed cost is fiBTS-LE in case
the BTS is connected directly to an LE. The fixed
cost to lay out one module from the BTS in location i to a CSS in location j is cBTS-CSS
. The fixed
ij
cost to lay out one module from the BTS in location i to the LE in location k is cBTS-LE
, and the
ik
fixed cost to lay out one module from a CSS in
location j to the LE in location k is cCSS-LE
.
jk
Certain (pre-specified) module connections are
not possible because of the distance or other
technical limitations.
The problem consists in selecting the CSSs and
LEs that must be actually installed and the way
to connect them (and the BTSs) through PCM
modules so as to support all the traffic flows going
from the BTSs to the LEs, without violating the
given bound limits and minimising the sum of the
fixed and module costs.
Our model is based on the following 0–1 decision variables:
• yjCSS-h ¼ 1 iff a CSS of type h 2 f1; 2g is opened
in location j;
• ykLE ¼ 1 iff an LE is opened in location k;
• xBTS-CSS
¼ 1 iff the BTS in location i is assigned
ij
to a CSS in location j;
• xBTS-LE
¼ 1 iff the BTS in location i is assigned
ik
to the LE in location k;
• xCSS-LE
¼ 1 iff a CSS in location j is assigned to
jk
the LE in location k.
The model also needs the following nonnegative
integer variables:
• zCSS-LE
number of modules from a CSS in j to
jk
the LE in k
along with the following nonnegative continuous
variables:
• wCSS-LE
traffic flow from a CSS in j to the LE
jk
in k.
60
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
n
X
The model then reads:
m X
X
minimise
fjCSS-h yjCSSh þ
þ
i¼1
þ
i¼1
þ
cBTS-CSS
eBTS
ij
i
þ
fiBTS-CSS
xBTS-CSS
ij
for j ¼ 1; . . . ; m; k ¼ 1; . . . ; p;
cBTS-LE
eBTS
ik
i
þ
fiBTS-LE
m
X
xBTS-LE
ik
þ
j¼1
p
X
xBTS-LE
ik
TiBTS xBTS-CSS
ij
X
6
k¼1
h¼1;2
yjCSS-h
for j ¼ 1; . . . ; m;
ð10Þ
ð11Þ
xBTS-CSS
ij
6
NjCSS-h yjCSS-h
ð2Þ
X
eBTS
xBTS-CSS
6
i
ij
EjCSS-h yjCSS-h
h¼1;2
for j ¼ 1; . . . ; m;
diBTS xBTS-CSS
6Q
ij
ð3Þ
p
X
zCSS-LE
jk
k¼1
i¼1
for j ¼ 1; . . . ; m;
ð4Þ
zCSS-LE
6 Mjk xCSS-LE
jk
jk
for j ¼ 1; . . . ; m; k ¼ 1; . . . ; p;
n
X
ð5Þ
TiBTS xBTS-CSS
6 TkLE ykLE
ik
i¼1
for k ¼ 1; . . . ; p;
for k ¼ 1; . . . ; p;
2 f0; 1g for i ¼ 1; . . . ; n; j ¼ 1; . . . ; m;
xBTS-LE
2 f0; 1g for i ¼ 1; . . . ; n; k ¼ 1; . . . ; p;
ik
for j ¼ 1; . . . ; m; k ¼ 1; . . . ; p;
zCSS-LE
P 0 and integer
jk
h¼1;2
wCSS-LE
þ
jk
yjCSS-h 2 f0; 1g for j ¼ 1; . . . ; m; h ¼ 1; 2;
xCSS-LE
2 f0; 1g
jk
X
i¼1
j¼1
X
xBTS-CSS
ij
ð1Þ
for j ¼ 1; . . . ; m;
m
X
h¼1;2
p
X
ykLE 2 f0; 1g
TjCSS-h yjCSS-h
h¼1;2
i¼1
n
X
yjCSS-h 6 1 for j ¼ 1; . . . ; m;
xCSS-LE
¼
jk
ð0Þ
for j ¼ 1; . . . ; m;
n
X
ð9Þ
k¼1
i¼1
n
X
eBTS
xBTS-CSS
6 EkLE ykLE
i
ik
i¼1
X
¼1
for i ¼ 1; . . . ; n;
n
X
n
X
ð8Þ
for k ¼ 1; . . . ; p;
cCSS-LE
zCSS-LE
jk
jk
k¼1
xBTS-CSS
ij
zCSS-LE
þ
jk
j¼1
subject to
m
X
ð7Þ
6 Fjk xCSS-LE
wCSS-LE
jk
jk
k¼1
p
m X
X
j¼1
for j ¼ 1; . . . ; m;
j¼1
p
n X
X
wCSS-LE
jk
k¼1
i¼1
fkLE ykLE
p
X
j¼1
j¼1 h¼1;2
n X
m X
p
X
TiBTS xBTS-CSS
¼
ij
ð6Þ
for j ¼ 1; . . . ; m; k ¼ 1; . . . ; p:
Constraints (0) force every BTS to be connected
to either a CSS or an LE. Constraints (1) impose
the limit on the traffic flow provided by a given
CSS, (2) impose that on the number of BTSs
connected to a given CSS, whereas (3) impose the
limit on the number of modules connected to a
given CSS. Inequalities (4) are congruence relations between xCSS-LE
and zCSS-LE
variables, also
jk
jk
used to impose the bound on the number of
modules connected to a given CSS. Constraints (5)
force to zero zCSS-LE
whenever xCSS-LE
is zero; value
jk
jk
Mjk is a given upper limit on the number of modules between j and k. Constraints (6) are used to
bound the traffic flow provided by a given LE,
whereas (7) impose that all traffic entering a CSS
must be distributed to an LE. Similarly, (8) force
to zero wCSS-LE
whenever xCSS-LE
is zero (value
jk
jk
Fjk being a given upper bound on the traffic flow
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
between j and k), whereas (9) limit the number of
modules connected to a given LE. Constraints (10)
impose that no more than one CSS can be activated in a given location, whereas (11) force to
activate every CSS connected to an LE.
Clearly, all variables associated with infeasible
situations (too long connections, basic/isolated
BTSs, etc.) have to be fixed to 0 and removed from
the model.
4. Model resolution
The mixed-integer linear programming model
presented in the previous section revealed very
difficult to solve to proven optimality, even by
using state-of-the-art methods from Mathematical
Programming and Operations Research (see Section 6 for details). This is mainly due to the interaction of two hard substructures, one associated
with the 0–1 x- and y-variables and the other with
integer z-variables, which notoriously leads to
hard-to-solve models.
Nevertheless, instances of small size can hopefully be solved exactly within acceptable computing time, thus providing useful insights on the
structure of the optimal solutions on real-world
test problems. Even more importantly, the solution of the linear programming relaxation of the
model – obtained by disregarding the integrality
requirements on the x-, y- and z-variables – can be
performed efficiently in short computing time, and
always provides a lower bound (i.e., an optimistic estimate) of the actual minimum cost. This
lower bound is therefore very useful to evaluate
the quality of the approximate/heuristic solutions
provided by the practitioners or by ad hoc heuristic procedures.
We have therefore designed an exact solution method, which can also be used as a heuristic
if it is stopped before convergence. The method
follows the branch-and-cut paradigm, consisting
of a tight integration between cutting plane and
enumerative techniques. The reader interested
in the branch-and-cut methodology is referred
to Padberg and Rinaldi [16], and to Caprara
and Fischetti [6] for a recent annotated bibliography.
61
The whole package allows for a tight integration with the computer codes currently in use at
CSELT, the major Italian research laboratory that
partially supported the present research. Our code
reads the input data, in the appropriate format,
possibly along with a heuristic solution. On output, the code returns the best solution found, in
a format which allows for a graphical display,
along with the best lower bound available (either
the optimal solution value or the minimum lower
bound associated with the active sub-problems in
the branching queue).
5. Model improvement
A main characteristic of branch-and-cut methods consists on the possibility of improving the
model quality at run time, by introducing into the
current model new valid inequalities (i.e., linear
constraints satisfied by all feasible solutions of the
problem at hand) acting as cutting planes. These
linear inequalities are indeed (valid but) redundant
in the original model when the integrality condition on the variables is imposed, but become useful
during the solution process when the integrality
condition is relaxed.
In order to actually embed into the model any
new class of inequalities, one has to be able to
solve the associated separation problem, which can
be formulated as follows:
Given a family F of valid inequalities along
with a (possibly fractional) solution (x ;
y ; z ; w ) of the current model, find a member
of family F which is violated by (x ; y ; z ; w ),
or prove that none exists.
We have designed the following main classes of
valid inequalities, along with the corresponding
separation procedures.
5.1. Logical constraints
xBTS-CSS
6
ij
X
yjCSS-h
h¼1;2
for i ¼ 1; . . . ; n;
j ¼ 1; . . . ; m
ð12Þ
62
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
(if a BTS i is connected to a certain CSS j, then
CSS j has to be deployed).
6 ykLE
xBTS-LE
ik
for i ¼ 1; . . . ; n; k ¼ 1; . . . ; p
ð13Þ
(if a BTS i is connected to a certain LE k, then LE
k has to be deployed).
xCSS-LE
6 ykLE
jk
for j ¼ 1; . . . ; m; k ¼ 1; . . . ; p
ð14Þ
(if a CSS j is connected to a certain LE k, then LE
k has to be deployed).
We also considered the following trivial constraints, which proved to be of some use for smallsize instances.
X
yjCSS-h 6
h¼1;2
p
X
zCSS-LE
jk
for j ¼ 1; . . . ; m
ð15Þ
k¼1
(at least one module must be connected to every
active CSS);
p
X
ykLE P 1
ð16Þ
k¼1
(at least one LE must be deployed).
All the above constraints can be efficiently
separated, by enumeration.
5.2. Generalised cover inequalities
Recall that dre ¼ minfi 2 N : i P rg denotes
the upper integer part of a given real number r.
The family of generalised cover inequalities we
propose reads
&
’
!
p
X
X
X
BTS
BTS-CSS
di =Q
xij
zCSS-LE
jCj þ 1 6
jk
i2C
i2C
for every C f1; . . . ; ng; j ¼ 1; . . . ; m:
k¼1
ð17Þ
This family of constraints imposes – in a combinatorial way – a tight lower bound on the number
of PCM modules connected to a certain CSS.
To prove the validity of constraints (17) for our
problem, consider any given CSS j. For every
subset C of BTSs we have two cases:
• not all the BTSs inPC are connected to the CSS
in j: in this case, i2C xBTS-CSS
6 jCj 1, hence
ij
the inequality left-hand side becomes non-positive and the inequality is trivially satisfied;
• all the BTSs in C are indeed connected
to the
P
¼
CSS in j: in this case we have i2C xBTS-CSS
ij
jCj, hence the constraint becomesP
active and
BTS
correctly requires to install at least
=
i2C di
Qe modules to connect CSS j.
The family of generalised cover inequalities contains an exponential number of members. Therefore, the corresponding separation problem cannot
be solved through explicit enumeration. We have
implemented the following more sophisticated
strategy.
Assume, without loss of generality, that all
traffic demands diBTS as well as Q are nonnegative
integers.
We consider, in turn, all possible CSSs j ¼
1; . . . ; m. For each given j, our order of business is
to find a BTS subset C whose associated generalised cover inequality (17) is maximally violated.
This is a hard optimisation problem in itself, that
we approach through
the following scheme.
P
Let gj :¼ ð pk¼1 zCSS-LE
Þz¼z denote the rightjk
hand side value of (17) computed for the solution
ðx ; y ; z ; w Þ to be separated, with respect to the
CSS j under consideration. We consider, in sequence, all
integer
values d P 1 to play the
Ppossible
BTS
role of
d
=Q
,
and
for each fixed d we
i2C i
look for a BTS subset C with
X
diBTS > Qðd 1Þ
i2C and such that
fj ðdÞ :¼
jC j X
i2C !
xBTS-CSS
ij
x¼x
is a minimum: if d ð1 fj ðdÞÞ > gj , then we have
found a (most) violated generalised cover inequality, otherwise no such violated inequality
exists for the given pair (j; d), and we proceed by
considering the next value for d and/or j.
The problem of determining C can now be
viewed as a 0–1 Knapsack Problem (KP), in minimisation form, in which BTSs i ¼ 1; . . . ; n correspond to items, each having a nonnegative cost
ð1 xBTS-CSS
Þx¼x and a nonnegative weight diBTS ,
ij
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
and one calls for a minimum-cost item subset
whose global weight is, at least, Qðd 1Þ þ 1.
This knapsack problem, although NP-hard, can
in practice be solved very quickly through specialised codes (see, e.g, [15]). In addition, one can
typically remove/fix a large fraction of items from
the knapsack problem by using standard pre-processing criteria. In particular, items j with KP cost
ð1 xBTS-CSS
Þx¼x ¼ 0 can always be selected in the
ij
knapsack as they do not deteriorate the objective
function value, while contributing in a positive
way to increase the overall weight of the selected
items. In addition, any item j with cost ð1
xBTS-CSS
Þx¼x P 1 gj =d can be removed from the
ij
item set, in that its choice would imply a KP cost
fj ðdÞ P ð1 xBTS-CSS
Þx¼x P 1 gj =d, hence it canij
not lead to a violated generalised cover inequality.
This latter reduction criterion typically allows
one to remove a very large fraction of the items
(all those with cost ðxBTS-CSS
Þx¼x 6 gj =d, including
ij
BTS-CSS
those with ðxij
Þx¼x ¼ 0).
According to our scheme, the separation algorithm for generalised cover inequalities requires
the solution, for each CSS j ¼ 1; . . . ; m, of a
sequence of knapsack problems with different
knapsack capacities depending on the parameter d.
Clearly, all values d 6 gj are not worth trying
as they correspond to KPs with empty item set
after the above reductions (in our separation context we always have x 6 1, hence d 6 gj implies
ðxBTS-CSS
Þx¼x 6 1 6 gj =d for all j). On the other
ij
hand, according to our computational experience,
values d P gj þ 1 seldom produce violated cuts.
Therefore we decided to only address the case
d ¼ dgj e for all CSSs j with fractional gj , thus
solving, at most, one knapsack problem for each
j ¼ 1; . . . ; m.
6. Computational results
The performance of our branch-and-cut method has been tested on a class of real-life test
problems provided by CSELT. Our main goal was
to evaluate the quality of the heuristic solutions
computed by CSELT by means of their proprietary tabu-search method [13], that works as follows.
63
An initial (possibly infeasible) low-cost partial
solution is first obtained by a simple greedy procedure that allocates every BTS to the CSS or LE
which minimises the linking cost, without taking
capacity constraints into account. Thereafter, a
reallocation procedure is applied to try to reduce
the degree of infeasibility of the resulting partial
solution. More specifically, if some traffic constraint happens to be violated at a certain CSS or
LE, then the associated BTSs are considered according to a decreasing sequence of required traffic, and reallocated to a different CSS or LE. A
similar procedure is applied for the violated
module constraints, if any. The allocation of CSSs
to LEs is performed in a similar way.
During tabu search, every solution is evaluated
by taking into account its overall cost plus nonlinear penalties for violated constraints. The following main tabu-search moves have been
implemented: (1) inactivation of an active CSS, to
be chosen among the seven less utilised ones, with
consequent reallocation of its associated BTSs at
minimum total overall cost; (2) inactivation of an
LE, to be chosen among the three less utilised
ones, with reallocation of all its associated CSSs
and BTSs at minimum total overall cost; (3) activation of a new complex CSS, to be chosen among
seven randomly selected ones, with consequent
reallocation of some BTSs; (4) activation of a
new LE, to be chosen among three randomly selected ones, with consequent reallocation of some
CSSs and BTSs; (5) type change of a CSS, i.e.,
replacement of a simple CSS by a complex one or
vice-versa, possibly followed by a consequent
BTSs reallocation; (6) reallocation of a BTS currently allocated to one of the five most utilised
CSSs and LEs; (7) allocation swap between two
BTSs.
As customary, the tabu search alternates between an ‘‘exploration phase’’ characterised by low
penalties for infeasibilities, and an ‘‘intensification
phase’’ characterised by very high infeasibility
penalties. Whenever no feasible solution is found
after 20 moves, diversification is performed by
exchanging active CSSs and LEs with non-active
ones, while reallocating some BTSs in a vein similar to that used for the initialisation. The whole
procedure ends when a predefined maximum
64
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
number of moves (10,000, in the current implementation) has been performed.
As to our branch-and-cut algorithm, it was implemented in C language using the general-purpose
branch-and-cut framework MINTO 3.0 [18] linked
with the commercial LP solver CPLEX 5.0 [10], and
was run on a PC Pentium 133 MHz under Windows
95. All internally generated cuts of MINTO have
been deactivated, but we used the MINTO internal
primal heuristics. Moreover, the value of the tabusearch heuristic solution is used as the initial upper
bound for the branch-and-cut search.
The cutting-phase generation was implemented
as follows: constraints (0) are handled statically,
i.e., they are present in all solved LPs. As to the
remaining constraints, they are generated dynamically (i.e., they are separated on-the-fly and appended to the current LP), according to the
following scheme. We first separate constraints (1),
(4) and (7); if no such cut is violated, we consider
constraints (12)–(14). If none of the above cuts has
been generated we apply, in sequence, the separation procedures for cuts (2), (3), (5), (6), (8), (9),
(10), (11), (15), (16), and (17); the separation sequence is broken as soon as violated inequalities in
the current family are found.
All instances in our test bed have been provided
by CSELT [13].
Table 1 reports the size of the problem instances
we considered (BTS-CSS-LE), the value of the
initial tabu-search heuristic solution computed by
the CSELT code [13] (Tabu UB), the value of
the best solution found by the branch-and-bound
code (Best UB), the value of the final lower bound
available at the end of the enumeration, computed as the minimum lower bound associated
with active nodes in the branching queue (Final
LB), and the percentage gap between the initial
tabu-search solution and the final lower bound
(gap). The results were obtained by running our
code on a PC Pentium 133 MHz with a time limit
of 2 hours for each instance, which is about 2–3
times larger than the running time of the tabusearch heuristic.
According to the table, the tabu-search solution and the lower bound are quite close one to
each other, which validates the effectiveness of
both the tabu-search heuristic and the lower bound
procedures. In addition, in 11 out of the 14 cases
in our test-bed the heuristic solution delivered by
our branch-and-cut code was strictly better than
the tabu-search one, i.e., the computing time spent
in the enumeration improved both the initial lower
bound and upper bound.
More information on the cutting phase of
branch-and-cut code is given in Table 2, where
we report the actual number of the constraints
(0)–(17) that have been generated during the
whole run. According to the table, most of the
generated cuts are logical constraints of type
Table 1
Upper and lower bound comparison (2-hour time limit on a PC Pentium 133 MHz)
A
B
C
D
E
F
G
H
I
L
M
N
O
P
BTS
CSS
LE
Tabu UB
Best UB
Final LB
Gap (%)
100
95
110
96
105
115
100
110
100
120
90
85
100
85
12
9
14
10
10
15
14
16
25
12
9
10
10
6
4
4
4
5
5
5
5
5
5
4
3
4
3
3
19,850,255
18,917,721
23,215,028
19,088,121
20,683,960
23,975,503
19,840,342
23,220,740
19,838,083
24,927,101
18,179,351
16,981,990
19,850,259
16,624,947
19,850,255
18,915,544
23,214,196
19,087,437
20,680,389
23,967,148
19,840,342
23,220,740
19,835,722
24,925,856
18,178,546
16,981,213
19,849,892
16,624,227
19,606,797.0
18,687,073.3
21,560,353.6
18,847,882.4
20,523,362.4
22,508,426.1
19,580,270.7
21,573,970.3
19,592,028.3
23,559,843.3
17,804,722.9
16,863,167.4
19,603,163.5
16,510,956.5
1.23
1.21
7.12
1.26
0.76
6.09
1.31
7.09
1.23
5.48
2.06
0.70
1.24
0.68
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
65
Table 2
Number of constraints generated during each branch-and-cut run
A
B
C
D
E
F
G
H
I
L
M
N
O
P
(0)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
Total
12
9
14
10
10
15
14
16
25
12
9
10
10
6
12
9
14
10
10
15
14
16
25
12
9
10
10
6
4
2
1
3
2
5
5
4
6
2
2
2
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
12
9
14
10
10
15
14
16
25
12
9
10
10
6
29
14
22
30
18
24
28
20
28
16
20
19
17
16
4
4
4
5
5
5
5
5
5
4
3
4
3
2
12
9
14
10
10
15
14
16
25
12
9
10
10
6
40
21
41
32
30
59
55
57
88
36
20
29
22
12
4
2
3
5
4
5
5
3
3
4
3
3
3
1
0
1
3
0
0
2
0
4
0
2
0
0
0
0
12
9
14
10
10
15
14
16
25
12
9
10
10
6
284
253
324
247
255
348
334
360
522
272
234
237
254
191
0
0
0
0
0
0
0
0
0
0
0
0
0
40
42
27
40
41
41
56
48
59
74
31
24
30
23
14
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
1
1
0
0
17
10
2
12
2
3
16
1
0
10
17
1
0
0
485
379
510
425
407
582
567
593
852
437
369
376
373
308
Table 3
Details on the solved LPs (execution on a PC Pentium 133 MHz)
A
B
C
D
E
F
G
H
I
L
M
N
O
P
Nrows
Ncols
LB (0)–(11)
LB (0)–(16)
Con
0–1
Int
Mar
#LPsol
LP time
Nodes
282
222
308
263
262
362
334
368
531
287
206
223
223
165
1137
810
1414
930
988
1655
1352
1645
2450
1325
753
773
887
597
19,469,735.9
18,448,913.8
21,515,078.2
18,652,790.5
20,455,408.3
22,457,993.3
19,436,571.1
21,519,608.1
19,426,423.6
23,514,555.6
17,549,290.0
16,537,560.2
19,458,326.9
15,869,751.1
19,604,811.5
18,685,814.2
21,559,367.0
18,842,099.9
20,513,526.2
22,506,079.7
19,578,622.2
21,573,374.2
19,591,797.9
23,559,396.9
17,802,542.0
16,861,677.1
19,595,141.0
16,511,182.4
46
30
47
45
41
66
64
69
123
38
24
32
25
18
1047
754
1322
843
911
1523
1226
1509
2204
1250
706
713
840
565
46
30
47
45
41
66
64
69
123
38
24
32
25
18
484
356
485
383
391
554
542
591
813
449
353
373
365
327
9597
12,286
5416
10,637
6627
6366
6202
6107
1911
7780
12,149
9703
9113
8918
6779.86
6671.12
6932.52
6488.30
6327.81
6645.27
6795.24
6803.12
6748.90
6807.39
6924.60
6863.00
6600.37
6521.13
3531
4492
1993
3938
2627
2350
2326
2037
616
2881
4324
3677
3452
3610
(12) and (14), whereas constraints (3) and (15) play
no role in the solution of the instances in our testbed.
Table 3 addresses the size and structure of the
several LPs solved during the MINTO branchand-cut execution; the lower bounds attainable offline (i.e., with no enumeration) when solving the
LP relaxation of model (0)–(11) and of model (0)–
(16), respectively, are also reported. The table
columns have the following meaning:
• Nrows ¼ maximum number of rows in the
solved LPs;
• Ncols ¼ maximum number of columns in the
solved LPs;
• LB (0)–(11) ¼ root-node lower bound when using model (0)–(11);
• LB (0)–(16) ¼ root-node lower bound when using model (0)–(16);
• con ¼ number of continuous variables;
• 0–1 ¼ number of binary variables;
• int ¼ number of (general) integer variables;
• mar ¼ maximum number of rows in an LP, including Eq. (0);
• #LPsol ¼ number of solved LPs;
• LP time ¼ CPU time (over 2 hours) spent within
by LP solver (CPLEX 5.0), in Pentium/133 seconds.
• Nodes ¼ number of evaluated nodes in the
MINTO branch-and-cut tree.
66
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
Table 4
CPLEX 5.0 vs CPLEX 7.0 (24-hour time limit on a PC Pentium 133 MHz)
CPLEX 5.0
A
B
C
D
E
F
G
H
I
L
M
N
O
P
a
CPLEX 7.0
Cov
Nodes
Final LB
Gap
GUB
Cov
Flow
MIR
Gom
Nodes
Final LB
Gap
846
666
924
789
786
1086
1002
1104
1593
861
618
669
669
495
172,462
213,425
174,204
261,253
257,627
147,129
86,216
117,129
18,705
158,881
280,013
306,225
338,307
1,022,674
19,508,813
18,484,704
21,553,849
18,696,094
20,486,598
22,493,827
19,485,359
21,564,126
19,495,698
23,545,762
17,580,268
16,570,887
19,493,983
15,900,612
1.72
2.29
7.16
2.05
0.95
6.18
1.79
7.13
1.73
5.54
3.30
2.42
1.79
4.36
107
71
241
102
113
163
191
224
307
153
113
95
137
21
78
79
173
79
78
133
167
212
320
123
100
86
95
75
61
27
148
59
73
118
153
167
145
99
75
53
65
41
13
13
31
11
10
18
10
18
7
21
17
11
13
10
23
16
22
18
11
26
23
27
30
24
11
16
17
11
1,141,998
2,138,486
208,483
1,683,437
1,678,916
260,439
242,767
129,052
44,114
442,607
1,754,076
4,639
1,766,589
2,937,534
19,706,345
18,792,123
21,649,948
18,915,122
20,630,517
22,549,819
19,684,923
21,629,391
19,632,384
23,634,251
17,914,494
16,980,960a
19,708,894
16,540,890
0.72
0.66
6.74
0.91
0.26
5.95
0.78
6.85
1.04
5.19
1.46
0.00
0.71
0.51
Optimal value for instance N, found by CPLEX 7.0 in 710 seconds.
According to Table 3, the additional constraints
(12)–(16) did improve the root-node lower bound
significantly. Moreover, more than 90% of the
overall computing time (7200 seconds) is spent
within the LP solver, whereas the MINTO
branching-tree management and heuristics along
with our run-time separation procedures, only require a small fraction of the total computing
time.
Finally, we compared the performance of our
ad hoc branch-and-cut implementation with that
of the latest versions of powerful commercial MIP
solvers that deploy built-in procedures for the
separation of several classes of general MIP cuts,
including the so-called cover, GUB, MIR, flow,
and (mixed-integer) Gomory cuts. To this end, for
each instance we generated model (0)–(11) explicitly and solved it by using, as a black-box, the
commercial MIP solver CPLEX in its version 5.0
(the same version used as LP solver within our
branch-and-cut implementation) and in its latest
(greatly enhanced) version 7.0 [10]. The main internal CPLEX parameters have been preliminarily
tuned to achieve the best average performance. As
in the previous experiments, the value of the tabusearch heuristic solution was provided on input to
initialise the current upper bound. However, the
time limit was set to 24 (as opposed to 2) Pentium/
133 hours, thus allowing for the exploration a
much larger number of nodes.
The results of the new runs are given in Table 4,
where we report the number of generated cuts, the
number of explored nodes, the final lower bound
available after the 24-hour computation, and the
percentage gap between the initial tabu-search solution and the final lower bound. We do not report
the Best UB column here, in that CPLEX was able
to improve the initial tabu-search heuristic value –
even with the 24-hour time limit – only in case of
instance N, where version 7.0 (but not 5.0) was able
to converge to an optimal solution.
When comparing the performance of the two
CPLEX versions, we observe that the latest one
(vers. 7.0) is capable of evaluating a much larger
number of nodes and generates a considerable
number of additional cuts (other than cover inequalities), which produced a significant improvement of the final lower bound. Actually, the final
lower bound obtained with CPLEX 7.0 (but not
with CPLEX 5.0) after 24 hours compares favorably with the one produced by our branch-and-cut
implementation (with CPLEX 5.0) after 2 hours;
see column gap in Table 1. However, as already
observed, CPLEX 7.0 was able to improve the
initial upper bound only for instance N. We can
therefore argue that the ad hoc cuts (12)–(16)
generated at run-time by our method, besides improving the lower bound, are quite effective in
driving the branch-and-cut heuristics to find improved feasible solutions.
M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67
7. Conclusions
We have addressed a very important optimisation problem arising in telecommunication, namely
the design of a UMTS interconnecting network.
For this NP-hard problem we have proposed a
new mixed-integer linear programming problem as
well as several classes of additional constraints
aimed at improving the performance of solution
algorithms.
We have also outlined a solution algorithm in
the branch-and-cut framework, and have evaluated it on a library of real-life test problems provided by CSELT, a major research laboratory
operating with an Italian telephone operator
(TELECOM Italia).
We have reported our computational experience on these test instances, showing that the
method we propose produces tight lower and upper bounds.
The method proposed in this paper has also
proved the effectiveness of the tabu-search methodology currently used by CSELT to solve interconnecting network planning issues.
Future direction of work should address the
issue of further improving the lower bound quality, thus allowing for the exact solution of medium- or large-size instances.
Acknowledgements
Work partially supported by CSELT, Torino,
Italy; we thank Chiara Lepschy, Raffaele Menolascino and Giuseppe Minerva from CSELT for
their collaboration and helpful suggestions. The
work of the first two authors was also supported
by MIUR, Italy, while the work of the third author was supported by TIC 2000-1750-CO6-02 and
by PI2000/116, Spain. We thank two anonymous
referees for their helpful comments.
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