Optimisation of the interconnecting network of a UMTS radio mobile telephone system

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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 Jos

ee Salazar Gonz

a

alez

b

aDEI, University of Padova, Via Gradenigo 6/a, 35131 Padova, Italy bDEIOC, 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, 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

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 1–20 km (low user density areas in the country).

www.elsevier.com/locate/dsw

*

Corresponding author. Tel.: 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: S 0 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 system considered 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 calledhandover. 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 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 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. 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 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, and a 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.

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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 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 ofactiveLEs (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.

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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 ofnBTS locations, a set ofm

potential type 1 or 2 CSS locations, and a set ofp

potential LE locations.

A BTS in locationiproduces a traffic flowtBTS

i

through dBTS

i communication channels. Channels

to an LE are packed into ‘‘modules’’ (cables or microwave). IfQis the largest number of channels that can be arranged in a module, then the BTS in location i requires eBTS

i :¼ dd

BTS

i =Qe modules,

where dre ¼minfi2N :iPrg denotes the upper integer part of a given real number r. It is worth observing thatQmay in some cases depend on the location that a particular module is con-necting.

A CSS in locationjof typeh2 f1;2g can pro-vide a traffic flow not larger than a given upper bound TCSS-h

j , can support a number of modules

not larger thanECSS-h

j , and a number of BTSs not

larger thanNCSS-h

j .

An LE in locationkcan provide a traffic flow not larger than a given upper boundTLE

k , and can

support a number of modules not larger thanELE

k .

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 typeh

in location jisfCSS-h

j , and the cost to open an LE

in location k is fLE

k . The fixed cost to activate a

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

fBTS-CSS

i , whereas the fixed cost is f

BTS-LE

i 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 locationjiscBTS-CSS

ij . The fixed

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

ik , and the

fixed cost to lay out one module from a CSS in locationjto the LE in location kiscCSS-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 deci-sion variables:

• yCSS-h

j ¼1 iff a CSS of typeh2 f1;2gis opened

in locationj;

• yLE

k ¼1 iff an LE is opened in location k;

• xBTS-CSS

ij ¼1 iff the BTS in locationiis assigned

to a CSS in locationj;

• xBTS-LE

ik ¼1 iff the BTS in location iis assigned

to the LE in locationk;

• xCSS-LE

jk ¼1 iff a CSS in location jis assigned to

the LE in locationk.

The model also needs the following nonnegative integer variables:

• zCSS-LE

jk number of modules from a CSS injto

the LE ink

along with the following nonnegative continuous variables:

• wCSS-LE

jk traffic flow from a CSS injto the LE

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The model then reads: minimise X m j¼1 X h¼1;2 fjCSS-hyjCSShþX p j¼1 fkLEykLE þX n i¼1 Xm j¼1 cBTS-CSSij eBTSi þfiBTS-CSSxBTS-CSSij þX n i¼1 Xp k¼1 cBTS-LE ik e BTS i þfBTS-LE i xBTS-LE ik þX m j¼1 Xp k¼1 cCSS-LEjk zCSS-LEjk subject to Xm j¼1 xBTS-CSSij þX p k¼1 xBTS-LEik ¼1 fori¼1;. . .;n; ð0Þ Xn i¼1 TiBTSxBTS-CSSij 6 X h¼1;2 TjCSS-hyjCSS-h forj¼1;. . .;m; ð1Þ Xn i¼1 xBTS-CSSij 6 X h¼1;2 NCSS-h j y CSS-h j forj¼1;. . .;m; ð2Þ Xn i¼1 eBTSi xBTS-CSSij 6 X h¼1;2 ECSS-j hyjCSS-h forj¼1;. . .;m; ð3Þ Xn i¼1 diBTSxBTS-CSSij 6QX p k¼1 zCSS-LEjk forj¼1;. . .;m; ð4Þ zCSS-LEjk 6MjkxCSS-LE jk forj¼1;. . .;m; k¼1;. . .;p; ð5Þ Xm j¼1 wCSS-LEjk þX n i¼1 TiBTSxBTS-CSSik 6TLE k y LE k fork¼1;. . .;p; ð6Þ Xn i¼1 TiBTSxBTS-CSSij ¼X p k¼1 wCSS-LEjk forj¼1;. . .;m; ð7Þ wCSS-LEjk 6FjkxCSS-LE jk forj¼1;. . .;m; k¼1;. . .;p; ð8Þ Xm j¼1 zCSS-LEjk þX n i¼1 eBTSi xBTS-CSSik 6ELE k y LE k fork¼1;. . .;p; ð9Þ X h¼1;2 yCSS-h j 61 forj¼1;. . .;m; ð10Þ Xp k¼1 xCSS-LEjk ¼ X h¼1;2 yCSS-j h forj¼1;. . .;m; ð11Þ yjCSS-h2 f0;1g forj¼1;. . .;m; h¼1;2; ykLE2 f0;1g fork¼1;. . .;p; xBTS-CSS ij 2 f0;1g fori¼1;. . .;n; j¼1;. . .;m; xBTS-LEik 2 f0;1g for i¼1;. . .;n; k¼1;. . .;p; xCSS-LEjk 2 f0;1g forj¼1;. . .;m; k¼1;. . .;p;

zCSS-LEjk P0 and integer

forj¼1;. . .;m; k¼1;. . .;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 z

CSS-LE

jk variables, also

used to impose the bound on the number of modules connected to a given CSS. Constraints (5) force to zerozCSS-LE

jk 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-LE

jk whenever x

CSS-LE

jk is zero (value

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betweenjandk), 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 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 Sec-tion 6 for details). This is mainly due to the in-teraction of two hard substructures, one associated with the 0–1x- andy-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 providcomput-ing useful insights on the structure of the optimal solutions on real-world test problems. Even more importantly, the solu-tion of the linear programming relaxasolu-tion 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 code reads 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 associatedseparation 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 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 6 X h¼1;2 yjCSS-h for i¼1;. . .;n; j¼1;. . .;m ð12Þ

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

xBTS-LEik 6yLE

k for i¼1;. . .;n; k¼1;. . .;p ð13Þ

(if a BTSiis connected to a certain LEk, then LE

khas to be deployed).

xCSS-LEjk 6yLE

k forj¼1;. . .;m; k¼1;. . .;p

ð14Þ

(if a CSSjis 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. X h¼1;2 yCSS-j h6X p k¼1 zCSS-LEjk forj¼1;. . .;m ð15Þ

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

Xp

k¼1

yLE

k P1 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 ¼minfi2N : iPrg denotes the upper integer part of a given real number r. The family of generalised cover inequalities we propose reads X i2C dBTS i =Q & ’ X i2C xBTS-CSS ij jCj þ1 ! 6X p k¼1 zCSS-LE jk for everyC f1;. . .;ng; j¼1;. . .;m: ð17Þ

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 subsetCof BTSs we have two cases:

• not all the BTSs inCare connected to the CSS in j: in this case, Pi2CxBTS-CSS

ij 6jCj 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 have Pi2CxBTS-CSS

ij ¼

jCj, hence the constraint becomes active and correctly requires to install at least Pi2CdBTS

i =

Qemodules to connect CSSj.

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 demandsdBTS

i as well asQare nonnegative

integers.

We consider, in turn, all possible CSSs j¼

1;. . .;m. For each givenj, our order of business is to find a BTS subsetCwhose 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 g j :¼ ð

Pp

k¼1z CSS-LE

jk Þz¼z denote the

right-hand side value of (17) computed for the solution

ðx;y;z;wÞ to be separated, with respect to the CSS j under consideration. We consider, in se-quence, all possible integer valuesdP1 to play the role of Pi2CdBTS

i =Q

, and for each fixed d we look for a BTS subsetC with

X

i2C

diBTS>Qðd1Þ

and such that

fjðdÞ:¼ jCj X i2C xBTS-CSSij ! x¼x

is a minimum: ifd ð1fjðdÞÞ>gj, then we have

found a (most) violated generalised cover in-equality, otherwise no such violated inequality exists for the given pair (j;d), and we proceed by considering the next value fordand/or j.

The problem of determining C can now be

viewed as a 0–1Knapsack Problem (KP), in mini-misation form, in which BTSs i¼1;. . .;n corre-spond to items, each having a nonnegative cost

ð1xBTS-CSS

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and one calls for a minimum-cost item subset whose global weight is, at least,Qðd1Þ þ1.

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, itemsjwith KP cost

ð1xBTS-CSS

ij Þx¼x¼0 can always be selected in the

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

ij Þx¼xP1gj=d can be removed from the

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

fjðdÞPð1xBTS-CSSij Þx¼xP1gj=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 costðxBTS-CSS

ij Þx¼x6gj=d, including

those withðxBTS-CSS

ij Þx¼x ¼0).

According to our scheme, the separation algo-rithm for generalised cover inequalities requires the solution, for each CSS j¼1;. . .;m, of a sequence of knapsack problems with different knapsack capacities depending on the parameterd. Clearly, all values d6g

j are not worth trying

as they correspond to KPs with empty item set after the above reductions (in our separation con-text we always have x61, hence d6g

j implies

ðxBTS-CSS

ij Þx¼x616gj=d for all j). On the other

hand, according to our computational experience, values dPg

jþ1 seldom produce violated cuts.

Therefore we decided to only address the case

d¼ dgje for all CSSs j with fractional gj, thus solving, at most, one knapsack problem for each

j¼1;. . .;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 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 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 among seven 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-purpose branch-and-cut frameworkMINTO 3.0[18] linked with the commercial LP solverCPLEX 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 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, 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 2–3 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 7.12 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

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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:

• Nrows¼maximum number of rows in the solved LPs;

• Ncols¼maximum number of columns in the solved LPs;

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

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

• con¼number of continuous variables;

• 0–1¼number of binary variables;

• int¼number of (general) integer variables;

• mar¼maximum number of rows in an LP, in-cluding Eq. (0);

• #LPsol¼number of solved LPs;

• LP time¼CPU time (over 2 hours) spent within by LP solver (CPLEX 5.0), in Pentium/133 sec-onds.

• Nodes¼number of evaluated nodes in the MINTO 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

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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)Gomorycuts. 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 our branch-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 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 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 MIRGom 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 a

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7. Conclusions

We have addressed a very important optimisa-tion problem arising in telecommunicaoptimisa-tion, 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 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 and by PI2000/116, Spain. We thank two anonymous referees for their helpful comments.

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