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Discrete Optimization

Optimisation of the interconnecting network of a UMTS

radio mobile telephone system

Matteo Fischetti a,*, Giorgio Romanin Jacur a, Juan Josee Salazar Gonzaalez b

a DEI, University of Padova, Via Gradenigo 6/a, 35131 Padova, Italyb 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 computingtime.

2002 Elsevier Science B.V. All rights reserved.

Keywords:Communication; Location; Mixed integer linear models

1. Introduction

A mobile radio telephone system aims at en-

suring secure communications between mobile ter-

minals and any other type of user device, eithermobile 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 users handheld terminals) and fixed radio base

stations is obtained by means of radio waves.

However, a single antenna system cannot cover the

whole service area. In fact, that choice would re-

quire high irradiation power both from the fixed

and the mobile stations, with consequent possi-ble damage due to the generated electromagnetic

field.

The above limitations lead to the implementa-

tion 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 120 km (low

user density areas in the country).

European Journal of Operational Research 144 (2003) 5667

www.elsevier.com/locate/dsw

* Corresponding author. Tel.: +39-049-827-7824; fax: +39-

049-827-7826.

E-mail address: [email protected](M. Fischetti).

0377-2217/03/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.

PII: S0 3 7 7 -2 2 1 7 (0 1 )0 0 3 8 3 -6
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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 systemconsidered and the radio access scheme, each fre-

quency (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 adja-

cent 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 in-

side 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 net-

work.

Third generation mobile telecommunication

systems are currently in the course of standardi-

sation 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 mo-

bile users to the fixed network; a BTS han-

dles users access and channel assignment. Due

to the inherent flexibility featured by next gener-

ation BTSs, different network topologies can

be undertaken: the BTS can be either di-

rectly connected to the switching equipment

(smart BTS) or linked to a BTS controller

(CSS). Cell site switch (CSS), which is a switch con-

nected to several BTSs on one side and to a sin-

gle 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.

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 centralisedor 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 asso-

ciated with LEs and MSDPs.

In this paper we address the problem of opti-

mising a UMTS interconnection network having a

multilevel star-type architecture. This is a difficult-

to-solve (NP-hard) optimisation problem of crucial

importance in the design of effective and low-cost

networks.

The general characteristics of UMTS and re-

lated standardisation problems were presented in

[2,3,9,17]; some hints in design and optimisa-

tion may be found in [1,4,5,8,14], but they concern

either different application fields or simpler net-

work topologies with respect to the ones studied

here.

As to the literature on various location prob-

lems, we refer the reader to Labbee and Louveaux

[12] for a recent annotated bibliography. Facilitylocation problems related to the one studied in the

present paper have been very recently addressed in

Chardaire et al. [7], where an uncapacitated two-

level network design problem is studied, and in

Klose [11], where a Lagrangean heuristic based on

the relaxation of the capacity constraints is pro-

posed.

The paper is organised as follows. In Section 2

we give a more detailed description of the UMTS

multilevel architecture. A mixed-integer linear pro-

gramming model is proposed in Section 3, anda possible solution algorithm in the branch-and-

cut 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 algo-

rithms. Computational results on a library of real-

world 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.

M. Fischetti et al. / European Journal of Operational Research 144 (2003) 5667 57

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2. The UMTS multilevel architecture

In the problem we consider, a certain number of

potential CSS and LE sites is given, among whichthe 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 ofactive 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 de-

fine 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 indepen-

dent 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 sub-

problems.

Each BTS has to be connected to the core net-

work, 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 num-ber 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 trans-

mitted information, and the number of channels is

the number of independent simultaneous commu-

nications, 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 de-

pends 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 con-

nected LE; this cost is linear in the number of

used modules.

Every LE is characterised by its location, its traffic,

and by the maximum number of supported PCM

modules.Fig. 1. The three level star-type UMTS architecture.

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Costs implied by an LE concern:

the plant cost, depending on the location.

Feasibility constraints are either of the con-

gruence 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]) Fa-

cility 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 ofn BTS locations, a set ofmpotential type 1 or 2 CSS locations, and a set ofp

potential LE locations.

A BTS in location iproduces a traffic flow tBTSithrough dBTSi communication channels. Channels

to an LE are packed into modules (cables or

microwave). IfQ is the largest number of channels

that can be arranged in a module, then the BTS in

location i requires eBTSi : ddBTSi =Qe modules,

where dre minfi2 N :iP rg denotes the upperinteger 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 con-

necting.

A CSS in location jof type h2 f1; 2g can pro-vide a traffic flow not larger than a given upper

bound TCSS-hj , can support a number of modules

not larger than ECSS-hj , and a number of BTSs not

larger than NCSS-hj .

An LE in location k can provide a traffic flow

not larger than a given upper bound TLEk , and can

support a number of modules not larger than ELEk .

BTS type is pre-defined as basic (it must be

connected to a CSS), or isolated (it must be con-

nected directly to an LE), or free (it can be con-

nected to a CSS or directly to an LE).The fixed cost required to open a CSS of type h

in location j is fCSS-hj , and the cost to open an LE

in location k is fLEk . The fixed cost to activate a

BTS in location i and to connect it to a CSS is

fBTS-CSSi , whereas the fixed cost is fBTS-LEi in case

the BTS is connected directly to an LE. The fixed

cost to lay out one module from the BTS in lo-

cationito a CSS in locationjis cBTS-CSSij . The fixed

cost to lay out one module from the BTS in lo-

cation ito the LE in location k is cBTS-LEik , and the

fixed cost to lay out one module from a CSS in

location jto the LE in location kis cCSS-LEjk .

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 01 deci-sion variables:

yCSS-hj 1 iff a CSS of type h2 f1; 2g is openedin location j;

yLEk 1 iff an LE is opened in location k; xBTS-CSSij 1 iff the BTS in location i is assigned

to a CSS in location j;

xBTS-LEik 1 iff the BTS in location iis assignedto the LE in location k;

xCSS-LEjk 1 iff a CSS in location jis assigned to

the LE in location k.

The model also needs the following nonnegative

integer variables:

zCSS-LEjk number of modules from a CSS in jtothe LE in k

along with the following nonnegative continuous

variables:

wCSS-LEjk traffic flow from a CSS in jto the LEin k.


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The model then reads:

minimise Xm

j1Xh1;2

fCSS-hj yCSShj X

p

j1

fLEk yLEk

Xni1

Xmj1

cBTS-CSSij eBTSi

fBTS-CSSi

xBTS-CSSij

Xni1

Xpk1

cBTS-LEik eBTSi

fBTS-LEi

xBTS-LEik

Xmj1

Xpk1

cCSS-LEjk zCSS-LEjk

subject to

Xmj1

xBTS-CSSij Xpk1

xBTS-LEik 1

for i 1;. . .;n; 0

Xni1

TBTSi xBTS-CSSij 6

Xh1;2

TCSS-hj yCSS-hj

for j 1;. . .;m; 1

Xni1

xBTS-CSSij 6Xh1;2

NCSS-hj yCSS-hj

for j 1;. . .;m; 2

Xni1

eBTSi xBTS-CSSij 6

Xh1;2

ECSS-hj yCSS-hj

for j 1;. . .;m; 3

Xn

i1

dBTSi xBTS-CSSij 6QX

p

k1

zCSS-LEjk

for j 1;. . .;m; 4

zCSS-LEjk 6MjkxCSS-LEjk

for j 1;. . .;m; k 1;. . .;p; 5

Xmj1

wCSS-LEjk Xni1

TBTSi xBTS-CSSik 6T

LEk y

LEk

for k 1;. . . ;p; 6

Xni1

TBTSi xBTS-CSSij

Xpk1

wCSS-LEjk

for j 1;. . .;m; 7

wCSS-LEjk 6FjkxCSS-LEjk

for j 1;. . .;m; k 1;. . .;p; 8

Xmj1

zCSS-LEjk Xni1

eBTSi xBTS-CSSik 6E

LEk y

LEk

for k 1;. . .;p; 9

Xh1;2

yCSS-hj 6 1 for j 1;. . .;m; 10

Xpk1

xCSS-LEjk Xh1;2

yCSS-hj for j 1;. . .;m; 11

yCSS-hj 2 f0; 1g for j 1;. . .;m; h 1; 2;

yLEk 2 f0; 1g for k 1;. . . ;p;

xBTS-CSSij 2 f0; 1g for i 1;. . .;n; j 1;. . .;m;

xBTS-LEik 2 f0; 1g for i 1;. . .;n; k 1;. . .;p;

xCSS-LE

jk

2 f0; 1g for j 1;. . .;m; k 1;. . . ;p;

zCSS-LEjk P 0 and integer

for j 1;. . .;m; k 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 rela-

tions between xCSS-LE

jk and zCSS-LE

jk variables, alsoused to impose the bound on the number of

modules connected to a given CSS. Constraints (5)

force to zero zCSS-LEjk wheneverxCSS-LEjk is zero; value

Mjk is a given upper limit on the number of mod-

ules 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-LEjk whenever xCSS-LEjk is zero (value

Fjkbeing a given upper bound on the traffic flow


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between jand k), whereas (9) limit the number of

modules connected to a given LE. Constraints (10)

impose that no more than one CSS can be acti-

vated in a given location, whereas (11) force toactivate 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 Sec-

tion 6 for details). This is mainly due to the in-

teraction of two hard substructures, one associated

with the 01x- and y-variables and the other with

integer z-variables, which notoriously leads to

hard-to-solve models.

Nevertheless, instances of small size can hope-

fully be solved exactly within acceptable comput-

ing time, thus providing useful insights on the

structure of the optimal solutions on real-worldtest problems. Even more importantly, the solu-

tion of the linear programming relaxation of the

model obtained by disregarding the integrality

requirements on thex-,y- andz-variables can be

performed efficiently in short computing time, and

always provides a lower bound (i.e., an optimis-

tic 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 heu-

ristic procedures.We have therefore designed an exact solu-

tion 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 bibliogra-

phy.

The whole package allows for a tight integra-

tion with the computer codes currently in use at

CSELT, the major Italian research laboratory that

partially supported the present research. Our codereads the input data, in the appropriate format,

possibly along with a heuristic solution. On out-

put, 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 meth-

ods 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 condi-

tion 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 memberof familyFwhich 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-CSSij 6Xh1;2

yCSS-hj

for i 1;. . .;n; j 1;. . .;m 12


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(if a BTS i is connected to a certain CSS j, then

CSS jhas to be deployed).

xBTS-LEik 6yLEk for i 1;. . .;n; k 1;. . .;p 13

(if a BTS iis connected to a certain LE k, then LE

khas to be deployed).

xCSS-LEjk 6yLEk for j 1;. . .;m; k 1;. . .;p

14

(if a CSS jis connected to a certain LEk, then LE

khas to be deployed).

We also considered the following trivial con-

straints, which proved to be of some use for small-

size instances.

Xh1;2

yCSS-hj 6Xpk1

zCSS-LEjk for j 1;. . .;m 15

(at least one module must be connected to every

active CSS);

Xpk1

yLEk P 1 16

(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 minfi2 N : iP rg denotesthe upper integer part of a given real number r.

The family of generalised cover inequalities we

propose reads

Xi2C

dBTSi =Q

& Xi2C

xBTS-CSSij

jCj 1

!6Xpk1

zCSS-LEjk

for every C f1;. . .;ng; j 1;. . . ;m: 17

This family of constraints imposes in a combi-

natorial 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 Cof BTSs we have two cases:

not all the BTSs inCare connected to the CSS

in j: in this case,

Pi2CxBTS-CSSij 6 jCj 1, hence

the inequality left-hand side becomes non-posi-

tive and the inequality is trivially satisfied;

all the BTSs in C are indeed connected to the

CSS in j: in this case we haveP

i2CxBTS-CSS

ij jCj, hence the constraint becomes active andcorrectly requires to install at least

Pi2Cd

BTSi =

Qe modules to connect CSS j.

The family of generalised cover inequalities con-

tains an exponential number of members. There-

fore, 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 dBTSi as well asQ are nonnegative

integers.

We consider, in turn, all possible CSSs j1;. . .;m. For each given j, our order of business isto find a BTS subset C whose associated genera-

lised cover inequality (17) is maximally violated.

This is a hard optimisation problem in itself, that

we approach through the following scheme.

Let gj : Pp

k1zCSS-LEjk zz denote the right-

hand side value of (17) computed for the solution

x;y;z;w to be separated, with respect to the

CSS j under consideration. We consider, in se-quence, all possible integer values dP 1 to play the

role ofP

i2CdBTSi =Q

, and for each fixed d we

look for a BTS subset C withXi2C

dBTSi >Qd1

and such that

fjd : jCj

Xi2CxBTS-CSSij

!xx

is a minimum: ifd 1fjd> gj , then we havefound a (most) violated generalised cover in-

equality, otherwise no such violated inequality

exists for the given pair (j; d), and we proceed byconsidering the next value for d and/or j.

The problem of determining C can now be

viewed as a 01 Knapsack Problem (KP), in mini-

misation form, in which BTSs i 1;. . . ;n corre-spond to items, each having a nonnegative cost

1xBTS-CSSij xx and a nonnegative weight dBTSi ,


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and one calls for a minimum-cost item subset

whose global weight is, at least, Qd1 1.This knapsack problem, although NP-hard, can

in practice be solved very quickly through specia-lised 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-pro-

cessing criteria. In particular, items jwith KP cost

1xBTS-CSSij xx 0 can always be selected in theknapsack 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 1xBTS-CSSij xx P 1g

j=d can be removed from the

item set, in that its choice would imply a KP cost

fjdP 1xBTS-CSSij xx P 1gj=d, hence it can-

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 xBTS-CSSij xx 6gj=d, including

those with xBTS-CSSij xx 0).According to our scheme, the separation algo-

rithm for generalised cover inequalities requires

the solution, for each CSS j 1;. . .;m, o f asequence of knapsack problems with different

knapsack capacities depending on the parameter d.

Clearly, all values d6gj are not worth tryingas they correspond to KPs with empty item set

after the above reductions (in our separation con-

text we always have x 6 1, hence d6gj implies

xBTS-CSSij xx 6 16gj=d for all j). On the other

hand, according to our computational experience,

values dPgj 1 seldom produce violated cuts.Therefore we decided to only address the case

d dgj e for all CSSs j with fractional gj , thus

solving, at most, one knapsack problem for each

j 1;. . . ;m.

6. Computational results

The performance of our branch-and-cut meth-

od 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 propri-

etary tabu-search method [13], that works as fol-

lows.

An initial (possibly infeasible) low-cost partial

solution is first obtained by a simple greedy pro-

cedure that allocates every BTS to the CSS or LE

which minimises the linking cost, without takingcapacity 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 con-

straint happens to be violated at a certain CSS or

LE, then the associated BTSs are considered ac-

cording to a decreasing sequence of required traf-

fic, 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 non-

linear penalties for violated constraints. The fol-

lowing 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) acti-

vation of a new complex CSS, to be chosen amongseven randomly selected ones, with consequent

reallocation of some BTSs; (4) activation of a

new LE, to be chosen among three randomly se-

lected 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 cur-

rently allocated to one of the five most utilised

CSSs and LEs; (7) allocation swap between two

BTSs.As customary, the tabu search alternates be-

tween 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 sim-

ilar to that used for the initialisation. The whole

procedure ends when a predefined maximum


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number of moves (10,000, in the current imple-

mentation) has been performed.

As to our branch-and-cut algorithm, it was im-

plemented in C language using the general-purposebranch-and-cut frameworkMINTO 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 tabu-

search 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 dynam-

ically (i.e., they are separated on-the-fly and ap-

pended 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 separa-

tion procedures for cuts (2), (3), (5), (6), (8), (9),

(10), (11), (15), (16), and (17); the separation se-

quence is broken as soon as violated inequalities in

the current family are found.

All instances in our test bed have been providedby 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 boundavailable at the end of the enumeration, com-

puted 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 23

times larger than the running time of the tabu-

search heuristic.

According to the table, the tabu-search solu-

tion 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)

BTS CSS LE Tabu UB Best UB Final LB Gap (%)

A 100 12 4 19,850,255 19,850,255 19,606,797.0 1.23

B 95 9 4 18,917,721 18,915,544 18,687,073.3 1.21

C 110 14 4 23,215,028 23,214,196 21,560,353.6 7.12D 96 10 5 19,088,121 19,087,437 18,847,882.4 1.26

E 105 10 5 20,683,960 20,680,389 20,523,362.4 0.76

F 115 15 5 23,975,503 23,967,148 22,508,426.1 6.09

G 100 14 5 19,840,342 19,840,342 19,580,270.7 1.31

H 110 16 5 23,220,740 23,220,740 21,573,970.3 7.09

I 100 25 5 19,838,083 19,835,722 19,592,028.3 1.23

L 120 12 4 24,927,101 24,925,856 23,559,843.3 5.48

M 90 9 3 18,179,351 18,178,546 17,804,722.9 2.06

N 85 10 4 16,981,990 16,981,213 16,863,167.4 0.70

O 100 10 3 19,850,259 19,849,892 19,603,163.5 1.24

P 85 6 3 16,624,947 16,624,227 16,510,956.5 0.68


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(12) and (14), whereas constraints (3) and (15) play

no role in the solution of the instances in our test-

bed.

Table 3 addresses the size and structure of the

several LPs solved during the MINTO branch-

and-cut execution; the lower bounds attainable off-

line (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 maximum number of rows in thesolved LPs;

Ncols maximum number of columns in thesolved LPs;

LB (0)(11) root-node lower bound when us-ing model (0)(11);

LB (0)(16) root-node lower bound when us-ing model (0)(16);

con number of continuous variables; 01 number of binary variables; int number of (general) integer variables; mar maximum number of rows in an LP, in-

cluding Eq. (0);

#LPsol number of solved LPs; LP time CPU time (over 2 hours) spent within

by LP solver (CPLEX 5.0), in Pentium/133 sec-

onds.

Nodes number of evaluated nodes in theMINTO branch-and-cut tree.

Table 2

Number of constraints generated during each branch-and-cut run

(0) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) Total

A 12 12 4 0 12 29 4 12 40 4 0 12 284 0 42 0 1 17 485B 9 9 2 0 9 14 4 9 21 2 1 9 253 0 27 0 0 10 379

C 14 14 1 0 14 22 4 14 41 3 3 14 324 0 40 0 0 2 510

D 10 10 3 0 10 30 5 10 32 5 0 10 247 0 41 0 0 12 425

E 10 10 2 0 10 18 5 10 30 4 0 10 255 0 41 0 0 2 407

F 15 15 5 0 15 24 5 15 59 5 2 15 348 0 56 0 0 3 582

G 14 14 5 0 14 28 5 14 55 5 0 14 334 0 48 0 1 16 567

H 16 16 4 0 16 20 5 16 57 3 4 16 360 0 59 0 0 1 593

I 25 25 6 0 25 28 5 25 88 3 0 25 522 0 74 0 1 0 852

L 12 12 2 0 12 16 4 12 36 4 2 12 272 0 31 0 0 10 437

M 9 9 2 0 9 20 3 9 20 3 0 9 234 0 24 0 1 17 369

N 10 10 2 0 10 19 4 10 29 3 0 10 237 0 30 0 1 1 376

O 10 10 1 0 10 17 3 10 22 3 0 10 254 0 23 0 0 0 373

P 6 6 2 0 6 16 2 6 12 1 0 6 191 40 14 0 0 0 308

Table 3

Details on the solved LPs (execution on a PC Pentium 133 MHz)

Nrows Ncols LB (0)(11) LB (0)(16) Con 01 Int Mar #LPsol LP time Nodes

A 282 1137 19,469,735.9 19,604,811.5 46 1047 46 484 9597 6779.86 3531

B 222 810 18,448,913.8 18,685,814.2 30 754 30 356 12,286 6671.12 4492

C 308 1414 21,515,078.2 21,559,367.0 47 1322 47 485 5416 6932.52 1993

D 263 930 18,652,790.5 18,842,099.9 45 843 45 383 10,637 6488.30 3938

E 262 988 20,455,408.3 20,513,526.2 41 911 41 391 6627 6327.81 2627

F 362 1655 22,457,993.3 22,506,079.7 66 1523 66 554 6366 6645.27 2350

G 334 1352 19,436,571.1 19,578,622.2 64 1226 64 542 6202 6795.24 2326

H 368 1645 21,519,608.1 21,573,374.2 69 1509 69 591 6107 6803.12 2037

I 531 2450 19,426,423.6 19,591,797.9 123 2204 123 813 1911 6748.90 616

L 287 1325 23,514,555.6 23,559,396.9 38 1250 38 449 7780 6807.39 2881

M 206 753 17,549,290.0 17,802,542.0 24 706 24 353 12,149 6924.60 4324

N 223 773 16,537,560.2 16,861,677.1 32 713 32 373 9703 6863.00 3677

O 223 887 19,458,326.9 19,595,141.0 25 840 25 365 9113 6600.37 3452

P 165 597 15,869,751.1 16,511,182.4 18 565 18 327 8918 6521.13 3610


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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 re-

quire 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) explic-

itly 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 ourbranch-and-cut implementation) and in its latest

(greatly enhanced) version 7.0 [10]. The main in-

ternal CPLEX parameters have been preliminarily

tuned to achieve the best average performance. As

in the previous experiments, the value of the tabu-

search 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 so-

lution and the final lower bound. We do not report

theBest UBcolumn here, in that CPLEX was able

to improve the initial tabu-search heuristic value

even with the 24-hour time limit only in case ofinstance 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 in-

equalities), which produced a significant improve-

ment 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 favor-ably 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 im-

proving the lower bound, are quite effective in

driving the branch-and-cut heuristics to find im-

proved feasible solutions.

Table 4

CPLEX 5.0 vs CPLEX 7.0 (24-hour time limit on a PC Pentium 133 MHz)

CPLEX 5.0 CPLEX 7.0

Cov Nodes Final LB Gap GUB Cov Flow MIR Gom Nodes Final LB GapA 846 172,462 19,508,813 1.72 107 78 61 13 23 1,141,998 19,706,345 0.72

B 666 213,425 18,484,704 2.29 71 79 27 13 16 2,138,486 18,792,123 0.66

C 924 174,204 21,553,849 7.16 241 173 148 31 22 208,483 21,649,948 6.74

D 789 261,253 18,696,094 2.05 102 79 59 11 18 1,683,437 18,915,122 0.91

E 786 257,627 20,486,598 0.95 113 78 73 10 11 1,678,916 20,630,517 0.26

F 1086 147,129 22,493,827 6.18 163 133 118 18 26 260,439 22,549,819 5.95

G 1002 86,216 19,485,359 1.79 191 167 153 10 23 242,767 19,684,923 0.78

H 1104 117,129 21,564,126 7.13 224 212 167 18 27 129,052 21,629,391 6.85

I 1593 18,705 19,495,698 1.73 307 320 145 7 30 44,114 19,632,384 1.04

L 861 158,881 23,545,762 5.54 153 123 99 21 24 442,607 23,634,251 5.19

M 618 280,013 17,580,268 3.30 113 100 75 17 11 1,754,076 17,914,494 1.46

N 669 306,225 16,570,887 2.42 95 86 53 11 16 4,639 16,980,960a 0.00

O 669 338,307 19,493,983 1.79 137 95 65 13 17 1,766,589 19,708,894 0.71P 495 1,022,674 15,900,612 4.36 21 75 41 10 11 2,937,534 16,540,890 0.51

a Optimal value for instance N, found by CPLEX 7.0 in 710 seconds.


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7. Conclusions

We have addressed a very important optimisa-

tion problem arising in telecommunication, namelythe 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 evalu-

ated it on a library of real-life test problems pro-

vided by CSELT, a major research laboratory

operating with an Italian telephone operator

(TELECOM Italia).

We have reported our computational experi-

ence on these test instances, showing that the

method we propose produces tight lower and up-

per bounds.

The method proposed in this paper has also

proved the effectiveness of the tabu-search meth-

odology currently used by CSELT to solve inter-

connecting network planning issues.

Future direction of work should address the

issue of further improving the lower bound qual-

ity, thus allowing for the exact solution of medi-um- or large-size instances.

Acknowledgements

Work partially supported by CSELT, Torino,

Italy; we thank Chiara Lepschy, Raffaele Men-

olascino 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 au-

thor was supported by TIC 2000-1750-CO6-02 andby PI2000/116, Spain. We thank two anonymous

referees for their helpful comments.

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