Discrete Optimization

Optimisation of the interconnecting network of a UMTSradio mobile telephone system

Matteo Fischetti a,*, Giorgio Romanin Jacur a, Juan Jos�ee Salazar Gonz�aalez 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 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 en-suring secure communications between mobile ter-minals and any other type of user device, eithermobile or fixed. A mobile customer should bereachable at any time and in any location wherethe radio coverage is granted.The connection among mobile terminals (i.e.,

the user’s handheld terminals) and fixed radio base

stations is obtained by means of radio waves.However, a single antenna system cannot cover thewhole service area. In fact, that choice would re-quire high irradiation power both from the fixedand the mobile stations, with consequent possi-ble damage due to the generated electromagneticfield.The above limitations lead to the implementa-

tion of ‘‘cellular systems’’, constituted by severalfixed radio base stations and related antennasystems. Each single radio base station coveragearea is called ‘‘cell’’ and it serves a small region ofvariable size ranging from 10 to 100 m (high userdensity inside business buildings) to 1–20 km (lowuser density areas in the country).

European Journal of Operational Research 144 (2003) 56–67

www.elsevier.com/locate/dsw

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

049-827-7826.

E-mail address: fisch@dei.unipd.it (M. Fischetti).

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

PII: S0377 -2217 (01 )00383 -6

Every fixed radio base station, usually calledbase transceiver station (BTS), is both transmittingand receiving signals on a variable number offrequencies. Depending upon the type of systemconsidered and the radio access scheme, each fre-quency (or carrier) permits the allocation of avariable 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 channelis assigned inside the cell just entered. This featureis commonly called handover. Covering the servedregion with several cells allows for ‘‘frequencyreuse’’, 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 implementingthe 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 universalmobile telecommunication system (UMTS). Thebasic architecture of a UMTS network includesthe 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. Dueto the inherent flexibility featured by next gener-ation BTSs, different network topologies canbe 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 otherside; each CSS is devoted to the management oflocal traffic inside its controlled area, as well asto the connection of the controlled BTSs to theLE.

• LE, which is a switch connecting the BTSsto the network, either directly or throughCSSs.

• Mobility and service data point (MSDP), whichis a database where information about users isregistered; it may be located either together withan 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; itcan access the database to read, write or eraseinformation 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 amultilevel star-type architecture. This is a difficult-to-solve (NP-hard) optimisation problem of crucialimportance in the design of effective and low-costnetworks.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 concerneither different application fields or simpler net-work topologies with respect to the ones studiedhere.As to the literature on various location prob-

lems, we refer the reader to Labb�ee and Louveaux[12] for a recent annotated bibliography. Facilitylocation problems related to the one studied in thepresent paper have been very recently addressed inChardaire et al. [7], where an uncapacitated two-level network design problem is studied, and inKlose [11], where a Lagrangean heuristic based onthe 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 UMTSmultilevel 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 ofthe basic model are presented in Section 4, wherenew families of valid inequalities are introducedalong with the corresponding separation algo-rithms. Computational results on a library of real-world test problems provided by CSELT, a majorresearch laboratory operating with TELECOMItalia, are reported in Section 5. Some conclusionsare finally drawn in Section 6.

M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67 57

2. The UMTS multilevel architecture

In the problem we consider, a certain number ofpotential CSS and LE sites is given, among whichthe planner has to choose those to be actuallyactivated. We consider a three level star-typeUMTS architecture, defined by an upper layermade up of active LEs (chosen in the given set ofpotential LEs), a middle layer made up of activeCSSs (also chosen in the given set of potentialCSSs), and a lower layer made up of the givenBTSs (each of which is required to play the role ofa 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 the17 given BTSs. Note that each activated LEplays the role of the root of a tree spanning adifferent connected component. Moreover, theproblem cannot be decomposed in two indepen-dent subproblems consisting of assigning LEs toCSSs and CSSs to BTSs, respectively, in thatthe choice of the active CSSs and of their trafficload 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-specifiedBTSs the direct connection to an LE can howeverbe forbidden). Every BTS is characterised by itsgeographical location, its carried traffic, the num-ber of channels required, and by its type. The BTSlocation is the result of a complex planning processwhich is not considered in this paper. The BTScarried traffic and number of channels depend onthe expected average number of users served by thecell. More precisely, the traffic is the total trans-mitted information, and the number of channels isthe number of independent simultaneous commu-nications, each supported by a communicationmodule (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). Inthe plain pulse code modulation (PCM) hierarchyeach module collects up to 30 channels at 64 kbpsthus granting a capacity of 2 Mbps. The type de-pends on the connection either to an LE or to aCSS, as seen above.Costs implied by a BTS concern:

• the equipment cost;• the actual connection cost, depending on theconnected CSS or LE; the cost is assumed tobe linear in the number of used modules.

Every CSS is characterised by its type, its location,its traffic capacity, the maximum number of BTSsand modules that can be supported.CSSs may be of two different types, namely

‘‘simple’’ (type 1) or ‘‘complex’’ (type 2), havingdifferent load and cost characteristics.Costs implied by a CSS concern:

• the plant cost, depending on the type of theequipment and on the location;

• the connection cost, depending on the con-nected LE; this cost is linear in the number ofused modules.

Every LE is characterised by its location, its traffic,and by the maximum number of supported PCMmodules.Fig. 1. The three level star-type UMTS architecture.

58 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 ‘‘con-gruence type’’, imposing that any connection ispermitted only between activated sets, or of the‘‘limitation type’’, imposing that the traffic throughany activated set is limited by the given bounds,both in terms of transmitted information and interms of connected modules.The problem then consists of choosing the CSS

and LE to be activated, and the way to connectthem to the BTSs and between each other, so as toproduce a feasible three-level network of minimumcost (a more detailed description is given in thenext section). This combinatorial optimisationproblem is strongly NP-hard, as it generalises theclassical (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 forthe 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 ppotential LE locations.A BTS in location i produces a traffic flow tBTSi

through dBTSi communication channels. Channelsto an LE are packed into ‘‘modules’’ (cables ormicrowave). If Q is the largest number of channelsthat can be arranged in a module, then the BTS inlocation i requires eBTSi :¼ ddBTSi =Qe modules,where dre ¼ minfi 2 N : iP rg denotes the upperinteger part of a given real number r. It isworth observing that Q may in some cases dependon the location that a particular module is con-necting.A CSS in location j of type h 2 f1; 2g can pro-

vide a traffic flow not larger than a given upperbound T CSS-hj , can support a number of modulesnot larger than ECSS-hj , and a number of BTSs not

larger than NCSS-hj .

An LE in location k can provide a traffic flownot larger than a given upper bound T LEk , and cansupport a number of modules not larger than ELEk .

BTS type is pre-defined as basic (it must beconnected 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 f CSS-hj , and the cost to open an LEin location k is f LEk . The fixed cost to activate aBTS in location i and to connect it to a CSS isf BTS-CSSi , whereas the fixed cost is f BTS-LEi in casethe BTS is connected directly to an LE. The fixedcost to lay out one module from the BTS in lo-cation i to a CSS in location j is cBTS-CSSij . The fixedcost to lay out one module from the BTS in lo-cation i to the LE in location k is cBTS-LEik , and thefixed cost to lay out one module from a CSS inlocation j to the LE in location k is cCSS-LEjk .Certain (pre-specified) module connections are

not possible because of the distance or othertechnical limitations.The problem consists in selecting the CSSs and

LEs that must be actually installed and the wayto connect them (and the BTSs) through PCMmodules so as to support all the traffic flows goingfrom the BTSs to the LEs, without violating thegiven bound limits and minimising the sum of thefixed and module costs.Our model is based on the following 0–1 deci-

sion variables:

• yCSS-hj ¼ 1 iff a CSS of type h 2 f1; 2g is openedin location j;

• yLEk ¼ 1 iff an LE is opened in location k;• xBTS-CSSij ¼ 1 iff the BTS in location i is assignedto a CSS in location j;

• xBTS-LEik ¼ 1 iff the BTS in location i is assignedto the LE in location k;

• xCSS-LEjk ¼ 1 iff a CSS in location j is assigned tothe LE in location k.

The model also needs the following nonnegativeinteger variables:

• zCSS-LEjk � number of modules from a CSS in j tothe LE in k

along with the following nonnegative continuousvariables:• wCSS-LEjk � traffic flow from a CSS in j to the LEin k.

M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67 59

The model then reads:

minimiseXmj¼1

Xh¼1;2

f CSS-hj yCSS�hj þ

Xpj¼1

f LEk yLEk

þXni¼1

Xmj¼1

cBTS-CSSij eBTSi

�þ f BTS-CSSi

�xBTS-CSSij

þXni¼1

Xpk¼1

cBTS-LEik eBTSi

�þ f BTS-LEi

�xBTS-LEik

þXmj¼1

Xpk¼1

cCSS-LEjk zCSS-LEjk

subject to

Xmj¼1

xBTS-CSSij þXpk¼1

xBTS-LEik ¼ 1

for i ¼ 1; . . . ; n; ð0Þ

Xni¼1

T BTSi xBTS-CSSij 6

Xh¼1;2

T CSS-hj yCSS-hj

for j ¼ 1; . . . ;m; ð1Þ

Xni¼1

xBTS-CSSij 6

Xh¼1;2

NCSS-hj yCSS-hj

for j ¼ 1; . . . ;m; ð2Þ

Xni¼1

eBTSi xBTS-CSSij 6

Xh¼1;2

ECSS-hj yCSS-hj

for j ¼ 1; . . . ;m; ð3Þ

Xni¼1

dBTSi xBTS-CSSij 6QXpk¼1

zCSS-LEjk

for j ¼ 1; . . . ;m; ð4Þ

zCSS-LEjk 6MjkxCSS-LEjk

for j ¼ 1; . . . ;m; k ¼ 1; . . . ; p; ð5Þ

Xmj¼1

wCSS-LEjk þXni¼1

T BTSi xBTS-CSSik 6 T LEk yLEk

for k ¼ 1; . . . ; p; ð6Þ

Xni¼1

T BTSi xBTS-CSSij ¼Xpk¼1

wCSS-LEjk

for j ¼ 1; . . . ;m; ð7Þ

wCSS-LEjk 6 FjkxCSS-LEjk

for j ¼ 1; . . . ;m; k ¼ 1; . . . ; p; ð8Þ

Xmj¼1

zCSS-LEjk þXni¼1

eBTSi xBTS-CSSik 6ELEk yLEk

for k ¼ 1; . . . ; p; ð9Þ

Xh¼1;2

yCSS-hj 6 1 for j ¼ 1; . . . ;m; ð10Þ

Xpk¼1

xCSS-LEjk ¼Xh¼1;2

yCSS-hj for j ¼ 1; . . . ;m; ð11Þ

yCSS-hj 2 f0; 1g for j ¼ 1; . . . ;m; h ¼ 1; 2;

yLEk 2 f0; 1g for k ¼ 1; . . . ; p;xBTS-CSSij 2 f0; 1g for i ¼ 1; . . . ; n; j ¼ 1; . . . ;m;

xBTS-LEik 2 f0; 1g for i ¼ 1; . . . ; n; k ¼ 1; . . . ; p;xCSS-LEjk 2 f0; 1g for j ¼ 1; . . . ;m; k ¼ 1; . . . ; p;

zCSS-LEjk P 0 and integer

for j ¼ 1; . . . ;m; k ¼ 1; . . . ; p:

Constraints (0) force every BTS to be connectedto either a CSS or an LE. Constraints (1) imposethe limit on the traffic flow provided by a givenCSS, (2) impose that on the number of BTSsconnected to a given CSS, whereas (3) impose thelimit on the number of modules connected to agiven CSS. Inequalities (4) are congruence rela-tions between xCSS-LEjk and zCSS-LEjk variables, alsoused to impose the bound on the number ofmodules connected to a given CSS. Constraints (5)force to zero zCSS-LEjk whenever xCSS-LEjk is zero; valueMjk is a given upper limit on the number of mod-ules between j and k. Constraints (6) are used tobound the traffic flow provided by a given LE,whereas (7) impose that all traffic entering a CSSmust be distributed to an LE. Similarly, (8) forceto zero wCSS-LEjk whenever xCSS-LEjk is zero (valueFjk being a given upper bound on the traffic flow

60 M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67

between j and k), whereas (9) limit the number ofmodules 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/isolatedBTSs, etc.) have to be fixed to 0 and removed fromthe model.

4. Model resolution

The mixed-integer linear programming modelpresented in the previous section revealed verydifficult to solve to proven optimality, even byusing state-of-the-art methods from MathematicalProgramming and Operations Research (see Sec-tion 6 for details). This is mainly due to the in-teraction of two hard substructures, one associatedwith the 0–1 x- and y-variables and the other withinteger z-variables, which notoriously leads tohard-to-solve models.Nevertheless, instances of small size can hope-

fully be solved exactly within acceptable comput-ing time, thus providing useful insights on thestructure of the optimal solutions on real-worldtest problems. Even more importantly, the solu-tion of the linear programming relaxation of themodel – obtained by disregarding the integralityrequirements on the x-, y- and z-variables – can beperformed efficiently in short computing time, andalways provides a lower bound (i.e., an optimis-tic estimate) of the actual minimum cost. Thislower bound is therefore very useful to evaluatethe quality of the approximate/heuristic solutionsprovided 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 heuristicif it is stopped before convergence. The methodfollows the branch-and-cut paradigm, consistingof a tight integration between cutting plane andenumerative techniques. The reader interestedin the branch-and-cut methodology is referredto Padberg and Rinaldi [16], and to Capraraand Fischetti [6] for a recent annotated bibliogra-phy.

The whole package allows for a tight integra-tion with the computer codes currently in use atCSELT, the major Italian research laboratory thatpartially 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, ina format which allows for a graphical display,along with the best lower bound available (eitherthe optimal solution value or the minimum lowerbound associated with the active sub-problems inthe branching queue).

5. Model improvement

A main characteristic of branch-and-cut meth-ods consists on the possibility of improving themodel quality at run time, by introducing into thecurrent model new valid inequalities (i.e., linearconstraints satisfied by all feasible solutions of theproblem at hand) acting as cutting planes. Theselinear inequalities are indeed (valid but) redundantin the original model when the integrality condi-tion on the variables is imposed, but become usefulduring the solution process when the integralitycondition is relaxed.In order to actually embed into the model any

new class of inequalities, one has to be able tosolve the associated separation problem, which canbe formulated as follows:

Given a family F of valid inequalities alongwith a (possibly fractional) solution (x�;y�; z�;w�) of the current model, find a memberof family F which is violated by (x�; y�; z�;w�),or prove that none exists.

We have designed the following main classes ofvalid inequalities, along with the correspondingseparation procedures.

5.1. Logical constraints

xBTS-CSSij 6

Xh¼1;2

yCSS-hj

for i ¼ 1; . . . ; n; j ¼ 1; . . . ;m ð12Þ

M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67 61

(if a BTS i is connected to a certain CSS j, thenCSS j has to be deployed).

xBTS-LEik 6 yLEk for i ¼ 1; . . . ; n; k ¼ 1; . . . ; p ð13Þ(if a BTS i is connected to a certain LE k, then LEk has to be deployed).

xCSS-LEjk 6 yLEk for j ¼ 1; . . . ;m; k ¼ 1; . . . ; pð14Þ

(if a CSS j is connected to a certain LE k, then LEk has to be deployed).We also considered the following trivial con-

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

Xh¼1;2

yCSS-hj 6

Xpk¼1

zCSS-LEjk for j ¼ 1; . . . ;m ð15Þ

(at least one module must be connected to everyactive CSS);

Xpk¼1

yLEk P 1 ð16Þ

(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 : iP rg denotesthe upper integer part of a given real number r.The family of generalised cover inequalities wepropose reads

Xi2C

dBTSi =Q

& ’ Xi2C

xBTS-CSSij

� jCj þ 1

!6

Xpk¼1

zCSS-LEjk

for every C � f1; . . . ; ng; j ¼ 1; . . . ;m: ð17Þ

This family of constraints imposes – in a combi-natorial way – a tight lower bound on the numberof PCM modules connected to a certain CSS.To prove the validity of constraints (17) for our

problem, consider any given CSS j. For everysubset C of BTSs we have two cases:

• not all the BTSs in C are connected to the CSSin j: in this case,

Pi2C x

BTS-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 theCSS in j: in this case we have

Pi2C x

BTS-CSSij ¼

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

Pi2C d

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 cannotbe solved through explicit enumeration. We haveimplemented the following more sophisticatedstrategy.Assume, without loss of generality, that all

traffic demands dBTSi as well as Q are nonnegativeintegers.We consider, in turn, all possible CSSs j ¼

1; . . . ;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, thatwe approach through the following scheme.Let g�j :¼ ð

Ppk¼1 z

CSS-LEjk Þz¼z� denote the right-

hand side value of (17) computed for the solutionðx�; y�; z�;w�Þ to be separated, with respect to theCSS j under consideration. We consider, in se-quence, all possible integer values dP 1 to play therole of

Pi2C d

BTSi =Q

� �, and for each fixed d we

look for a BTS subset C� withXi2C�

dBTSi > Qðd � 1Þ

and such that

fjðdÞ :¼ jC�j �Xi2C�

xBTS-CSSij

!x¼x�

is a minimum: if d � ð1� fjðdÞÞ > g�j , then we havefound a (most) violated generalised cover in-equality, otherwise no such violated inequalityexists 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 0–1 Knapsack Problem (KP), in mini-misation form, in which BTSs i ¼ 1; . . . ; n corre-spond to items, each having a nonnegative costð1� xBTS-CSSij Þx¼x� and a nonnegative weight d

BTSi ,

62 M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67

and one calls for a minimum-cost item subsetwhose global weight is, at least, Qðd � 1Þ þ 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 cantypically remove/fix a large fraction of items fromthe knapsack problem by using standard pre-pro-cessing criteria. In particular, items j with KP costð1� xBTS-CSSij Þx¼x� ¼ 0 can always be selected in theknapsack as they do not deteriorate the objectivefunction value, while contributing in a positiveway to increase the overall weight of the selecteditems. In addition, any item j with cost ð1�xBTS-CSSij Þx¼x� P 1� g�j=d can be removed from theitem set, in that its choice would imply a KP costfjðdÞP ð1� xBTS-CSSij Þx¼x� P 1� g�j=d, hence it can-not lead to a violated generalised cover inequality.This latter reduction criterion typically allowsone to remove a very large fraction of the items(all those with cost ðxBTS-CSSij Þx¼x� 6 g�j =d, includingthose with ðxBTS-CSSij Þx¼x� ¼ 0).According to our scheme, the separation algo-

rithm for generalised cover inequalities requiresthe solution, for each CSS j ¼ 1; . . . ;m, of asequence of knapsack problems with differentknapsack capacities depending on the parameter d.Clearly, all values d6 g�j are not worth tryingas they correspond to KPs with empty item setafter the above reductions (in our separation con-text we always have x� 6 1, hence d6 g�j impliesðxBTS-CSSij Þx¼x� 6 16 g�j=d for all j). On the otherhand, according to our computational experience,values dP g�j þ 1 seldom produce violated cuts.Therefore we decided to only address the cased ¼ dg�j e for all CSSs j with fractional g�j , thussolving, at most, one knapsack problem for eachj ¼ 1; . . . ;m.

6. Computational results

The performance of our branch-and-cut meth-od has been tested on a class of real-life testproblems provided by CSELT. Our main goal wasto evaluate the quality of the heuristic solutionscomputed by CSELT by means of their propri-etary tabu-search method [13], that works as fol-lows.

An initial (possibly infeasible) low-cost partialsolution is first obtained by a simple greedy pro-cedure that allocates every BTS to the CSS or LEwhich minimises the linking cost, without takingcapacity constraints into account. Thereafter, areallocation procedure is applied to try to reducethe degree of infeasibility of the resulting partialsolution. More specifically, if some traffic con-straint happens to be violated at a certain CSS orLE, then the associated BTSs are considered ac-cording to a decreasing sequence of required traf-fic, and reallocated to a different CSS or LE. Asimilar procedure is applied for the violatedmodule constraints, if any. The allocation of CSSsto 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 beenimplemented: (1) inactivation of an active CSS, tobe chosen among the seven less utilised ones, withconsequent reallocation of its associated BTSs atminimum total overall cost; (2) inactivation of anLE, to be chosen among the three less utilisedones, with reallocation of all its associated CSSsand BTSs at minimum total overall cost; (3) acti-vation of a new complex CSS, to be chosen amongseven randomly selected ones, with consequentreallocation of some BTSs; (4) activation of anew LE, to be chosen among three randomly se-lected ones, with consequent reallocation of someCSSs and BTSs; (5) type change of a CSS, i.e.,replacement of a simple CSS by a complex one orvice-versa, possibly followed by a consequentBTSs reallocation; (6) reallocation of a BTS cur-rently allocated to one of the five most utilisedCSSs and LEs; (7) allocation swap between twoBTSs.As customary, the tabu search alternates be-

tween an ‘‘exploration phase’’ characterised by lowpenalties for infeasibilities, and an ‘‘intensificationphase’’ characterised by very high infeasibilitypenalties. Whenever no feasible solution is foundafter 20 moves, diversification is performed byexchanging active CSSs and LEs with non-activeones, while reallocating some BTSs in a vein sim-ilar to that used for the initialisation. The wholeprocedure ends when a predefined maximum

M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67 63

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] linkedwith the commercial LP solverCPLEX 5.0 [10], andwas run on a PC Pentium 133MHz underWindows95. All internally generated cuts of MINTO havebeen deactivated, but we used the MINTO internalprimal heuristics. Moreover, the value of the tabu-search heuristic solution is used as the initial upperbound 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 theremaining constraints, they are generated dynam-ically (i.e., they are separated on-the-fly and ap-pended to the current LP), according to thefollowing scheme. We first separate constraints (1),(4) and (7); if no such cut is violated, we considerconstraints (12)–(14). If none of the above cuts hasbeen 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 inthe 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 bythe CSELT code [13] (Tabu UB), the value ofthe best solution found by the branch-and-boundcode (Best UB), the value of the final lower boundavailable at the end of the enumeration, com-puted as the minimum lower bound associatedwith active nodes in the branching queue (FinalLB), and the percentage gap between the initialtabu-search solution and the final lower bound(gap). The results were obtained by running ourcode on a PC Pentium 133 MHz with a time limitof 2 hours for each instance, which is about 2–3times 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 toeach other, which validates the effectiveness ofboth the tabu-search heuristic and the lower boundprocedures. In addition, in 11 out of the 14 casesin our test-bed the heuristic solution delivered byour branch-and-cut code was strictly better thanthe tabu-search one, i.e., the computing time spentin the enumeration improved both the initial lowerbound and upper bound.More information on the cutting phase of

branch-and-cut code is given in Table 2, wherewe report the actual number of the constraints(0)–(17) that have been generated during thewhole run. According to the table, most of thegenerated 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 7.12

D 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

64 M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67

(12) and (14), whereas constraints (3) and (15) playno 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 theLP relaxation of model (0)–(11) and of model (0)–(16), respectively, are also reported. The tablecolumns have the following meaning:

• Nrows¼maximum number of rows in thesolved LPs;

• Ncols¼maximum number of columns in thesolved LPs;

• LB (0)–(11)¼ root-node lower bound when us-ing model (0)–(11);

• LB (0)–(16)¼ root-node lower bound when us-ing model (0)–(16);

• con¼ number of continuous variables;• 0–1¼ number of binary variables;• int¼ number of (general) integer variables;• mar¼maximum number of rows in an LP, in-cluding Eq. (0);

• #LPsol¼ number of solved LPs;• LP time¼CPU time (over 2 hours) spent withinby LP solver (CPLEX 5.0), in Pentium/133 sec-onds.

• Nodes¼ 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 17 485

B 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 0–1 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

M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67 65

According to Table 3, the additional constraints(12)–(16) did improve the root-node lower boundsignificantly. Moreover, more than 90% of theoverall computing time (7200 seconds) is spentwithin the LP solver, whereas the MINTObranching-tree management and heuristics alongwith our run-time separation procedures, only re-quire a small fraction of the total computingtime.Finally, we compared the performance of our

ad hoc branch-and-cut implementation with thatof the latest versions of powerful commercial MIPsolvers that deploy built-in procedures for theseparation of several classes of general MIP cuts,including the so-called cover, GUB, MIR, flow,and (mixed-integer) Gomory cuts. To this end, foreach instance we generated model (0)–(11) explic-itly and solved it by using, as a black-box, thecommercial 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 preliminarilytuned to achieve the best average performance. Asin the previous experiments, the value of the tabu-search heuristic solution was provided on input toinitialise the current upper bound. However, thetime limit was set to 24 (as opposed to 2) Pentium/133 hours, thus allowing for the exploration amuch larger number of nodes.

The results of the new runs are given in Table 4,where we report the number of generated cuts, thenumber of explored nodes, the final lower boundavailable after the 24-hour computation, and thepercentage gap between the initial tabu-search so-lution and the final lower bound. We do not reportthe Best UB column here, in that CPLEX was ableto 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 ableto 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 largernumber of nodes and generates a considerablenumber of additional cuts (other than cover in-equalities), which produced a significant improve-ment of the final lower bound. Actually, the finallower bound obtained with CPLEX 7.0 (but notwith CPLEX 5.0) after 24 hours compares favor-ably with the one produced by our branch-and-cutimplementation (with CPLEX 5.0) after 2 hours;see column gap in Table 1. However, as alreadyobserved, CPLEX 7.0 was able to improve theinitial upper bound only for instance N. We cantherefore argue that the ad hoc cuts (12)–(16)generated at run-time by our method, besides im-proving the lower bound, are quite effective indriving 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 Gap

A 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 0.71

P 495 1,022,674 15,900,612 4.36 21 75 41 10 11 2,937,534 16,540,890 0.51

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

66 M. Fischetti et al. / European Journal of Operational Research 144 (2003) 56–67

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 anew mixed-integer linear programming problem aswell as several classes of additional constraintsaimed at improving the performance of solutionalgorithms.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 laboratoryoperating with an Italian telephone operator(TELECOM Italia).We have reported our computational experi-

ence on these test instances, showing that themethod 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 fortheir collaboration and helpful suggestions. Thework of the first two authors was also supportedby 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 anonymousreferees for their helpful comments.

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