The Package Server Location Problem

Arnaud Malapert

, Jean-Charles Régin

and Jean Parpaillon

Laboratoire d’Informatique, Signaux et Systèmes de Sophia-Antipolis (I3S)

CNRS UMR7271, Université Nice Sophia Antipolis, Sophia Antipolis, France

Mandriva, 32 rue Trevise, 75009 Paris, France

Keywords:

Multi-objective Optimization, Mixed Integer Linear Programming, Network Location Model.

Abstract:

In this paper, we introduce a new multi-objective optimization problem derived from a real-world application:

the package server location problem. A number of package servers are to be located at nodes of a network.

Demand for these package servers is located at each node, and a subset of nodes are to be chosen to locate one

or more package servers. Each client is statically associated to a package server. The objective is to minimize

the number of package servers while maximizing the efﬁciency and the reliability of the broadcast of packages

to clients. These objectives are contradictory: the broadcast becomes more efﬁcient as the number of servers

increases. This problem is analyzed as a multi-objective optimization problem and a mathematical formulation

is proposed. In addition, the criteria combination can be speciﬁed via a small dedicated language. Results for

exact multi-objective solution approaches based on mixed integer linear programming are reported.

1 INTRODUCTION

Allocating services to servers of an existing com-

puter network is a complex and difﬁcult task for the

administrators. Indeed, services are interrelated on

nodes of the network under several constraints (node

capacities, incompatibilities between services . . . ).

Moreover, the administrators consider simultaneously

multiple and contradictory objectives or preferences

(cost, performance, reliability, energy consumption).

Thus, a decision tool which can provide solutions

meeting these goals is valuable. Mandriva

, a French

software company, must accomplish such a task in

the deployment of one of their product. This com-

pany is well-known for its public open source dis-

tribution of operating system: Mandriva Linux. It

also offers product solutions for companies to manage

their computer network infrastructure. Pulse2 is an

open-source tool that simpliﬁes application deploy-

ment, inventory, and maintenance of computer net-

works. It helps organizations ranging from a few

computers to more than 100 000 heterogeneous com-

puters spread over more than 1000 facilities to inven-

tory, maintain, update and take full control on their

facilities. On one hand, Pulse2 helps manage net-

works of french governmental agencies or real estate

ofﬁces. On the other hand, it also helps manage net-

http://www.mandriva.com/

work mainly composed of gaming or ATM terminals.

The user can supervise large-scale facilities through a

single web interface console, create and deploy hard

disk images of its computers, perform software and

hardware inventory, and deploy new softwares and

updates. Pulse2 is installed on the top of an existing

infrastructure and provides a cross-platform software

distribution service even for remote locations with a

limited bandwidth. The architecture of Pulse2 is dis-

tributed: Pulse2 is composed of agents communicat-

ing with each other by various protocols (http, ssh,

xml-rpc, etc). An agent is a piece of software that is

replicated and distributed over hosts of the network,

and is usually continuously connected to a central

job scheduler. Distributed agents improve the robust-

ness and the scalability of Pulse2, but have two major

drawbacks: ﬁnding a feasible deployment of agents

on hosts is difﬁcult; upgrading them can cause severe

production downtime. Features of Pulse2 are imple-

mented through client-server models where comput-

ers and agents respectively play the role of clients and

servers. In other words, Pulse2 is a distributed ap-

plication that allows the management by a central au-

thority of application deployment and maintenance in

a heterogeneous network.

An essential functionality of Pulse2 is the man-

agement of software installations and updates, which

can lead to network congestion and failures. A pack-

age server, abbreviated as pserver, maintains the list

Malapert A., Régin J. and Parpaillon J..

The Package Server Location Problem.

DOI: 10.5220/0004282501930204

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 193-204

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

of software packages and serves them to its clients

(computers). Each client is statically associated to a

unique pserver by the central authority. Pulse2 al-

lows to manage groups of clients thanks to its com-

patibility with other resource-manager such as GLPI

For instance, research and administration computers

can have different software suites or security require-

ments.

The deployment of pservers involves strategic de-

cisions that are fundamental since they impact the ef-

ﬁciency and reliability of the broadcast of packages

from pservers to clients when large software pack-

ages are installed or updated (such as an ofﬁce suite)

on hundreds or thousands of clients. An efﬁcient

broadcast reduces the completion time for serving all

clients while avoiding network congestion. A reliable

broadcast reduces the impact of link failures. Note

that the ﬂexibility of the design is important in view

of uncertain scope of future additions to the network.

The main contribution of the paper is twofold. We

describe a new multi-objective optimization problem

derived from a real-world application: the package

server location problem (PSLP). Then, we propose a

ﬂexible and efﬁcient decision support tool based on

operational research methods for multi-criteria opti-

mization. This paper is organized as follows. In Sec-

tion 2, we describe the package server location prob-

lem and discuss an example. In Section 3, we dis-

cuss related works in the ﬁeld of mixed integer lin-

ear programming, cloud management, network de-

sign and facility location, especially server location

in computer networks. In Section 4, we present some

concepts and discuss related works in the ﬁeld of

multi-objective optimization. In Section 5, we intro-

duce a mathematical formulation and discuss in Sec-

tion 6 our method to handle multi-objective optimiza-

tion within an integer programming framework. Fi-

nally, in Section 7, we show the interest of our ap-

proach on a realistic use case and evaluate systemat-

ically the impact of the criteria combination on the

solving process.

2 THE PACKAGE SERVER

LOCATION PROBLEM

Description, Notations, and Deﬁnitions

Tree Network. According to the Mandriva Team,

tree network is the most frequent structure adopted

by its customers. Let N = [1, n] be the indices of

the facilities organized as an oriented tree network

http://www.glpi-project.org/

T = (N, E). The pservers can be located at nodes

(facilities) and the root node is called the central fa-

cility. Each network link (i, j) ∈ E between a pair of

facilities i ∈ N and j ∈ N is characterized by a real

bandwidth B

i j

and a boolean reliability r

i j

. The link

bandwidth and reliability respectively allow to model

network congestion and the probability of failure.

Client-server Model. Each client must be associ-

ated to a single pserver. Clients are located at facilities

and partitioned into g groups, i.e. each client belongs

to a unique group. Let d

be the demand at the facil-

ity i ∈ N for the group s (s = 1, . . . , g). We assume

that groups are treated independently by the central

authority, i.e. packages are never installed or updated

simultaneously for two groups.

For each type k ∈ K(= [1, m]) of pserver, the central

authority determines its maximal number of clients

(its capacity) C

in order to provide the same service

quality. Let M

be the maximal number of pservers of

type k ∈ K which can be located at facility i. The cen-

tral authority determines M

by monitoring the net-

work.

Package Broadcast. A package is broadcasted in

two steps as follows. First, the root pserver located

at the central facility receives the original package

and broadcasts it to other pservers. As soon as they

have received the full copy of the original package,

pservers serve their clients. The broadcast is oriented,

i.e. a pserver can only serve clients located at the

same facility or in its descendants. Let deﬁne a spe-

cial group (s = 0) for which the demand d

is equal to

the number of pservers located at the facility i. Let

S = [0, g] denote the stages where the initial stage

(s = 0) represents the broadcast from the root pserver

to other pservers and next stages represent the broad-

cast from pservers to groups of clients.

Broadcast Model. We consider only the steady

state of the broadcast of the trafﬁc rather than the tran-

sient state, i.e. we do not take into account transmis-

sion times and path latencies (the sum of all transmis-

sion times as well as the queuing inside the facilities).

Indeed, the duration of the transient state is dominated

by the maximal latency which is usually lower than

the total duration of the broadcast. Furthermore, path

latencies are hard to measure and more useful for op-

erational decisions which affect the day-to-day imple-

mentation of these strategic decisions.

We assume that the steady state of the broadcast

of the trafﬁc follows a model such that a) transfers

are synchronous, b) a bandwidth is allocated to each

transfer path in the network, c) the path bandwidth is

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

consumed simultaneously along its links during the

transfer, d) the path bandwidth is shared equally by

clients. Unlike distance, link bandwidths do not sum

up in the path bandwidth. The path bandwidth is lim-

ited by the minimum of link bandwidths, as this is the

fastest any trafﬁc can make through the path. The link

bandwidth is shared by the trafﬁc under consideration.

The total bandwidth allocated to paths sharing a link

must not exceed the link bandwidth.

Figure 1 illustrates the broadcast model where fa-

cilities are represented as nodes, links as arcs between

nodes, and paths as arrows besides the tree. In this

example, the trafﬁc passing through the dashed link

is composed of the paths from its origin, or its ances-

tors, to its destination, or its descendants. The sum

of path bandwidths must not exceed the bandwidth of

the dashed link.

Similarly, the path reliability is limited by the less

reliable link. If the path from the central facility to an-

other facility is reliable, then the facility is said to be

reliable. Note that we assume that local connections

(the pserver and the client are located at the same fa-

cility) are reliable and more efﬁcient than connections

between facilities.

Objectives. In summary, solving the problem boils

down to a) select the number of pservers, b) select

the type of pservers, c) select the location of pservers,

d) associate pservers to clients subject to various ca-

pacity and broadcast constraints. In general, we have

several measures for the quality of the solutions. First,

we aim to minimize the number of pservers or a cost

function associated to their deployment. Then, we

also aim to maximize the reliability of the transfer,

i.e. maximizing the number of pservers located at

reliable facilities and the number of reliable connec-

Figure 1: Broadcast model.

tions between pservers and clients. Besides, we aim

to maximize the efﬁciency of the broadcast of pack-

ages from pservers to clients by minimizing the total

distance or maximizing the total allocated bandwidth.

These objectives are contradictory: the broadcast be-

comes more efﬁcient as the number of pservers in-

creases. That is to say that PSLP, in its generality, is a

multi-objective optimization problem. The ability to

express the kind of trade-off we want depends on the

solving method.

Illustrative Example

Figure 2 gives a complete example composed of a

small instance and several solutions. The instance

contains 6 facilities, 3 levels, 1 type of pservers, and 1

group of 2500 clients. We can install as many pservers

as needed in facilities and each pserver provides 500

connections. The tree network is represented in Fig-

ure 2(a) where each facility i is drawn as a record in

which the left cell contains its index i and the right

cell contains its demand. There are two categories of

link: strong links (bold arrows) are reliable and have

a high bandwidth; weak links (dot-head arrows) are

not reliable and have a low bandwidth.

To solve this problem, Mandriva’s engineers com-

bine solutions obtained by different heuristics to build

a “balanced” solution. First, let’s build two reliable

solutions in which each pserver is connected to the

root pserver located at the central facility by a reli-

able path. The ﬁrst centralized solution consists in

5 pservers located at the central facility (i = 1). In

this case, 400 clients beneﬁt from local connections

while the central facility provides 550 and 1550 con-

nections to its left and right subtrees respectively. Fig-

ure 2(b) shows a solution as reliable but more efﬁcient

in which the facilities 1 and 3 respectively contain 2

and 3 pservers (indicated in the bottom cell). The link

(i, j) is now labeled with the number of connections

provided by the pservers of facility i to clients of the

facility j. The dashed arrows are connections between

a pair of facilities passing through a path rather than

a single link. Thus, the total number of connections

passing through link (1, 2) is the sum of connections

of transfer paths (1, 2) and (1, 4). In this case, 1250

clients beneﬁt from local connections and the central

facility still provides 550 connections to its left sub-

trees. This solution provides more local connections

and the broadcast is more efﬁcient in the right sub-

tree. The trafﬁc passing through the link (1, 3) de-

creases from 1550 to 50 connections while the trafﬁc

passing through (3, 5) and (3, 6) remains unchanged.

Moreover, several transfer paths have been shortened

whereas no one has been lengthened.

On the contrary, the solution in Figure 2(c) favors

ThePackageServerLocationProblem

efﬁciency rather than reliability. The facilities 1, 2,

5 contain a single pserver whereas the facility 3 con-

tains 2 pservers. In this case, the pservers located at

the facilities 2 and 5 are not connected to the cen-

tral facility by a reliable path. However, 1850 clients

beneﬁt from local connections and only 650 connec-

tions pass through the network. The solution in Fig-

ure 2(d) reduces the number of connections passing

through the network from 650 to 550. Note that a fur-

ther analysis is required to determine which of the two

last solutions is preferred.

To conclude, it would be very difﬁcult to express

trade-offs between the reliability and the efﬁciency

using a heuristic because these objectives are contra-

dictory. Indeed, installing pservers in highest levels of

the tree network gives solution more robust, scalable

(if the numbers of clients increases), and reliable, but

possibly less efﬁcient.

3 RELATED WORK

Our problem is related to research about mixed inte-

ger linear programming, network design, cloud man-

agement, and facility location, especially server loca-

tion in computer networks.

Mixed Integer Linear Programming

A mixed-integer program is the minimization or max-

imization of a linear function subject to linear con-

straints. Since more than 50 years, Mixed Integer

Linear Programming (MILP) theory and practice has

been signiﬁcantly developed and is now an indispens-

able tool in Operations Research and Management

Science (Guignard-Spielberg and Spielberg, 2005).

Two reasons for the success of MILP are Linear Pro-

gramming (LP) based solvers and the modelling ﬂex-

ibility of MILP. We now have several extremely ef-

fective state of the art solvers (IBM, 2012; Gurobi

Optimization, 2012) that incorporate many advanced

techniques and, since its early stages, MILP has been

used to model a wide range of applications (Dantzig,

1960). Since then, MILP has been applied in dif-

ferent industrial application areas, such as supply

chain management (Beamon, 1998), production plan-

ning (Wu and Shi, 2011), healthcare (Ferris et al.,

2006), and airline ﬂeet scheduling (Sherali et al.,

2010).

Network Design

In the network design problem (Johnson et al., 1978),

a weighted undirected graph is given. We wish to ﬁnd

a subgraph which connects all the original nodes and

minimizes the sum of the shortest path weights be-

tween all nodes pairs, subject to a budget constraint

on the sum of its edge weights. Usually, the cost of an

edge is a concave function of its capacity. Addition-

ally, there can be a demand between pairs of nodes of

the network that corresponds to sending some product

along a path between these two nodes. The quantities

that pass through an edge are subject to an edge ca-

pacity constraint.

There is a relation between the bandwidth of PSLP

and the edge capacity of the network design problem.

However, it is also quite different because we need to

determine the number of pservers, their types and lo-

cations. Moreover, the demand between pservers and

clients has also to be determined whereas it is given

in the network design problem. Last, the criteria com-

bination used for PSLP is different because links al-

ready exist and it does not make any difference to use

it or not.

Cloud Management

Cloud computing industry faces many decision prob-

lems where operations research could add tremen-

dous value (Iyoob et al., 2012). Indeed, the paradigm

shift from IT as a product to IT as a service and

the accompanying ﬂexibility gives rise to a vast ar-

ray of resource management decisions. The ﬂexibil-

ity and elasticity of cloud computing is enabled by

virtualization, which allows to the control of comput-

ing resources to be physically separated from the re-

sources themselves. There are three major players in

the Cloud IT Supply Chain: a) providers (compute,

storage, and network) b) consumers (individual and

enterprises) c) brokers (equivalent of third-party lo-

gistics).

Cloud providers costs are primarily tied to their

assets and the maintenance of these assets. Some of

these strategic problems are related to PSLP as they

involve capacity, location planning and service levels.

But, they concern physical resources (servers, infras-

tructure, power draw, network) whereas we focus on

the deployment of a distributed application.

The operational problem of assigning virtual ma-

chine to hosts subject to capacity constraints for sev-

eral dimensions (CPUs, memories . . . ) is very differ-

ent because the optimization criteria concern load bal-

ancing or scheduling. Consumer problems concern

the capacity planning and scheduling of virtual ma-

chines in order to minimize subscription costs. Last,

broker problems concern the provider’s choice as well

as inter-provider transformation and migration abili-

ties.

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

1 400

2 200 3 850

4 350 5 500 6 200

(a) Instance

1 400

2 200

200

3 850

4 350

350

6 200

5 500

500 150

(b) Reliable solution

1 400

2 200

3 850

4 350

350

5 500

6 200

200

1 400

2 200

3 850

4 350

6 200

300

5 500

0 150

(d) More efﬁcient solution?

Figure 2: Instance and solution examples.

Facility Location

In network location models (Berman and Krass,

2002), the underlying topology is that of a net-

work, with the possible location of facilities includ-

ing nodes, as well as edges of the network. These

problems often extend the classical p-median and p-

center problems in different directions, including: in-

corporating interactions between facilities; consider-

ing multi-objective models, integrating location and

routing decisions; and considering problems with ro-

bust decision criteria. In the p-center problem, we aim

to minimize the maximum distance between a client

and the facility to which he is allocated. In the p-

median problem, we aim to minimize the sum of the

shortest demand weighted distance between clients

and facilities. In both cases, each client selects the

closest facility.

In discrete location models, the set of potential fa-

cility locations is discrete – often consisting of nodes

of the underlying network. Due to the restricted lo-

cation set, the underlying problem is somewhat eas-

ier – allowing for greater realism in the models. In

stochastic demand/congestion models, this sub-class

of discrete location models allows for demand to be

stochastic and explicitly models congestion (queuing)

at the facilities. In a stochastic environment, demand

for service generated at nodes of the network are ran-

dom variables and the time to service calls is also

probabilistic. The location of servers has been ad-

dressed as stochastic models (Berman and Drezner,

2007; Aboolian et al., 2009) in which servers are

modeled as stochastic queues for which the theory

enables the mathematical analysis of several related

processes, including arriving at the queue, waiting in

the queue, and being served at the front of the queue.

PSLP is strongly related to network location mod-

els because we want to choose locations hosting

pserver in an existing network with opening costs.

However, the optimality criteria are different, espe-

cially when it involves the concept of bandwidth. Fur-

thermore, each client selects the closest facility in fa-

cility location models whereas a client is explicitly as-

sociated to a pserver by the central authority in the

PSLP.

4 MULTI-OBJECTIVE

OPTIMIZATION

Multi-objective optimization (or multi-objective pro-

gramming or "pareto optimization") (Sawaragi et al.,

1985; Steuer, 1986) also known as multi-criteria or

multi-attribute optimization, is the process of simul-

taneously optimizing two or more conﬂicting objec-

tives subject to certain constraints. Many interac-

tive and non-interactive methods have been proposed

for multi-objective mixed integer programming prob-

lems (Alves and Clímaco, 2007). In the last years,

ThePackageServerLocationProblem

there have been many developments devoted to par-

ticular multi-objective combinatorial problems (such

as location, scheduling, knapsack, shortest-path prob-

lems, etc). The researchers’ attention has also focused

on the use of meta-heuristics to solve these problems

such as ant colony optimization, memetic algorithm,

genetic algorithm, particle swarm optimization (Hu

and Eberhart, 2002), and nested partitions (Wu et al.,

2011).

Multi-objective Methods without Trade-off

The simplest approach, the lexicographic optimality,

deﬁnes a ﬁrst kind of trade-off: do not make any

trade-off. It amounts to prefer any improvement of

the most important criterion (objective), even a very

small improvement, to any improvement of the sec-

ond criterion, even a huge improvement. We have

a strong hierarchical preference between criteria. In

other words, the criteria are ordered lexicographi-

cally: a solution s is better than s

if and only if it

is equivalent for criteria 1, . . . , i − 1, and is better for

the i-th criterion. This approach has a great advan-

tage, it does not need the values of the criteria to be

compared. Indeed, the comparison of the values of

different criteria is a difﬁcult task that can be speciﬁc

to a user and a problem. Another approach is simply

to generate all the solutions which may be preferred.

A solution may be preferred if and only if it is not

dominated by another one. Such a non dominated so-

lution is called a pareto-solution. A solution s is bet-

ter than s

if and only if for each criterion, s is at least

equivalent to s

, and there exists a criterion such as

s is strictly better than s

. Intuitively, we cannot im-

prove a criterion of a pareto-solution without a loss

for another criterion. In practice, the set of pareto so-

lutions is huge, and the usual approach is to compute

an approximation of this set.

Multi-objective Methods with Trade-off

The comparison of the values of different criteria is

needed in order to deﬁne a good trade-off. In general,

this comparison is made thanks to a utility function

associated to each criterion.

A utility function u

maps a value v of the criteria z

to a number u

(v). A utility function u

should satisfy

an intuitive property: v

< v

⇒ u

(v1) < u

(v2). The

aim of utility functions is to make possible the com-

parison between different criteria. The utility func-

tions of different criteria are strongly related, in such

a way that the numbers coming from different crite-

ria can be compared: let z

, z

be two criteria and v

be the values of z

, z

in a given solution. We can

know that, in this solution, the criterion z

is more

favored than the criterion z

when u

) > u

Given the values u

, . . . , u

and u

, . . . , u

of the utility

functions for two solutions s and s

, we want now

to deﬁne when s is preferred to s

, noted s >

Many different preference orders >

, and many cor-

responding multi-criteria methods exist. We will con-

sider the following methods.

Weighted Sum. An extremely frequent approach

is to maximize a weighted sum

∑

× u

where

, . . . , u

are weights associated to each criterion.

This approach seems to be quite intuitive and satis-

factory. Nevertheless, despite its general use in many

domains, this approach has some strong drawbacks.

The function

∑

× u

is linear, and, when maximiz-

ing this function for a concave set of solutions, many

good solutions are not reachable. Furthermore, its

sensitivity to the weights is very high.

Maximizing the Minimum. Another frequent ap-

proach is to maximize the minimum value among the

criteria. s >

iff min u

> min u

. This approach

unfortunately produces many indistinguishable solu-

tions, and some of them are not pareto solutions.

Multi-objective Methods with Trade-off and

Dynamic Ordering of the Criteria

The weighted sum method, as well as the method of

maximizing the minimum have clear drawbacks. In-

deed, these two methods can be greatly improved by

integrating in them the following idea: order the cri-

teria depending on the values they take in a solution.

Then, aggregation of the criteria can be done depend-

ing on the rank of the criteria. Let p

, . . . , p

be the

sorted permutation of u

, . . . , u

in increasing order of

values.

Ordered Weighted Average. Let w be a vector of

weights such that w

≥ w

i+1

. The idea is to maximize

∑

× p

. We can see that if w

= w

= . . . = w

, we

get the sum. If w

= 1, w

= w

= . . . = 0, we get the

methods which maximize the minimum. This method

has been introduced by (Yager, 1988).

Leximin/Leximax. A leximin solution is simply a

lexicographic optimal solution on p

, . . . , p

. Lexim-

in/leximax have nice properties, that make them in-

tuitive for the user: a) they produce only pareto so-

lutions; b) they are egalitarist (leximin solutions are

always solutions for the method which maximizes the

minimum); c) they verify the property of reducibility

(if a criteria is set to a value it has in an optimal solu-

tion, we get the same set of optimal solutions). This

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

method has been introduced in social choice com-

munity and then used for multi-objective optimiza-

tion (Ehrgott, 2000).

5 MATHEMATICAL

FORMULATION

Let’s now deﬁne a MILP formulation of the problem.

Let P(i

, i

) = [i

, . . . , i

] denote the unique

path of the tree network T between i

and i

if any.

Location of the Pservers. Let x

denote the num-

ber of pservers of type k ∈ K located at the facility

i ∈ N. Constraints (1) impose a maximal number M

of pservers of type k located at facility i.

≤ M

∀i ∈ N, ∀k ∈ K (1)

Trafﬁc Flow. Let y

denote the number of connec-

tions provided to clients by pservers located at the fa-

cility i ∈ N for the stage s ∈ S. Let y

i j

denote the num-

ber of connections passing through the link (i, j) ∈ E

for the stage s. Constraints (2) enforce that the num-

ber of connections provided by the facility i stays

below the cumulated capacity of its pservers. Con-

straints (3) enforce the ﬂow conservation of connec-

tions at facility j, i.e. incoming (y

i j

) and provided

) connections satisfy the demand (d

) while enough

outgoing connections (y

) remain to serve its descen-

dants.

∑

k∈K

≥ y

∀i ∈ N, ∀s ∈ S (2)

∑

( j,p)∈E

= y

i j

+ y

− d

∀(i, j) ∈ E, ∀s ∈ S (3)

These constraints are illustrated in Figure 3 for a fa-

cility j with three children p

, p

and p

(the stage is

omitted).

Figure 3: Trafﬁc ﬂow.

Client-server Model. Let z

denote the number of

local connections at the facility i for the stage s. Con-

straints (4) enforce that the number of local connec-

tions at the facility i is equal to the demand minus the

number of incoming connections (from its ancestors).

Let z

i j

denote the number of connections from facil-

ities i to j for the stage s, i.e. passing through the

transfer path P(i, j). Constraints (5) enforce that the

number of connections passing through the link (i, j)

is equal to the total number of connections of transfer

paths taking this link.

= d

−

∑

p∈A(i)

∀i ∈ N, ∀s ∈ S (4)

i j

∑

(i, j)∈P(u,v)

∀(i, j) ∈ E, ∀s ∈ S (5)

Broadcast Model. Let b

i j

denote the path band-

width from facilities i to j for the stage s. Con-

straints (6) enforce that the sum of path bandwidths

consumed at the link (i, j) is lower than its capacity.

Let B

min

and B

max

be respectively the minimum and

maximum bandwidth allocated to a single connection

from a pserver to a client. The central authority de-

termines their values depending on the network archi-

tecture (clients, pservers and links). Constraints (7)

and (8) enforce that the bandwidth per connection be-

tween facilities i and j belongs to a given range.

∑

(i, j)∈P(u,v)

≤ B

i j

∀(i, j) ∈ E, ∀s ∈ S (6)

i j

≥ B

min

× z

i j

∀(i, j) ∈ E, ∀s ∈ S (7)

i j

≤ B

max

× z

i j

∀(i, j) ∈ E, ∀s ∈ S (8)

Initial Broadcast. The last constraints handle the

broadcast from the root pserver to other pservers,

i.e. the stage s = 0. Constraint (9) enforces that

the root pserver, which broadcasts packages to other

pservers, is located at the central facility (i = 1). Con-

straints (10) enforce that demand at a facility i for

the stage 0 is equal to its number of pservers. Con-

straints (11) enforce that the root pserver serves all

other pservers.

≥ 1 (9)

∑

k∈K

∀i ∈ N (10)

= z

i j

= b

i j

= 0 ∀i ∈ N\{1}, ∀ j ∈ N (11)

6 COMBINING MULTIPLE

OBJECTIVES

The criteria combination methods introduced in Sec-

ThePackageServerLocationProblem

tion 4 offers much more possible criteria combina-

tions than expected. We refer the reader to (Guttierez

et al., 2011) for an explanation of how to handle mul-

tiple criteria in an integer programming framework.

Their approach is the basis of a ﬂexible system of cri-

teria combination which provides a useful mean to ad-

dress some user’s needs. A key observation is that all

these combinations can be viewed as a variation of the

lexicographic algorithm.

We describe here a general approach for lexico-

graphic optimization that can be implemented on top

of any existing optimization solver. The underlying

optimization solver is only required to be able to max-

imize an objective function F subject to the set of

constraints C. Let max(F,C) be the minimal value

of F among all the feasible solutions of the constraint

system C. This optimal value is computed by calling

the underlying optimization solver. The lexicographic

optimization problem amounts to maximize lexico-

graphically n criteria F

. . . , F

while satisfying a set

of constraints C. A lexicographic optimization prob-

lem P can be solved by solving a sequence of (sin-

gle criterion) optimization problems {P

}

i∈[1,n]

with

an underlying optimization solver. Each problem P

consists of maximizing F

while satisfying the con-

straints C

where: a) C

= C, b) ∀i ∈ [2, n], C

i−1

∪ {F

i−1

= max(F

i−1

)} The solution of the

optimization problem P

is the solution of the lexi-

cographic optimization problem P. The leximin/lexi-

max optimization relies on a similar approach.

A Language for User Deﬁned Combination. We

provide a small dedicated language to let the user de-

ﬁne its own criteria combination. The key idea is to

map, at least, virtually, any possible criteria combi-

nation to an underlying lexicographic order. Indeed,

mono-objective criteria combinations, like a simple

aggregation, ﬁts directly to a lexicographic algorithm

reduced to one criteria and the leximin/leximax or-

der implementation do rely on an underlying lexico-

graphic order implementation. Therefore, the basic

criteria combination is the lexicographic one:

-lexicographic[<lccriteria>{,<lccriteria>}*]

-lex[<lccriteria>{,<lccriteria>}*]

The next level of criteria combination gives access

to the leximin/leximax order:

<lccriteria> ::=

{+,-}leximax[<ccriteria>{,<ccriteria>}*]

{+,-}leximin[<ccriteria>{,<ccriteria>}*]

As a consequence, one, two or more leximin/lex-

imax orders can be lexicographically ordered, mean-

ing that, for instance, the user does not want to or-

der the two ﬁrst criteria, neither he wants to order the

third and fourth criteria, while he clearly wants the set

of the two ﬁrst criteria to be handled before the rest of

the criteria. Note that the leading sign gives the opti-

mization direction, i.e., a leading - for a minimization

and a leading + for a maximization.

The ﬁnal level of combiners gives access to the

agregate and lexagregate combination:

<ccriteria> ::=

{+,-}agregate[<ccriteria>{,<ccriteria>}*]{[

lambda]}?

{+,-}lexagregate[<ccriteria>{,<cccriteria

>}*]{[lambda]}?

This recursive deﬁnition allows the user to de-

ﬁne agregate of agregate or lexagregate. The lex-

agregate combiner uses only one objective function

which could be solved using the lexicographic algo-

rithm. An optional lambda value gives the opportu-

nity to attach a weight to the agregate by means of a

positive integer. All coefﬁcients of the criteria will be

multiplied by this weight. At last, the user can choose

among the predeﬁned criteria which one he wants to

optimize:

<criteria> ::=

{+,-}pserv{[<property>,<value>]}*{[lambda]}?

{+,-}local{[<property>,<value>]}*{[lambda]}?

{+,-}conn{[<property>,<value>]}*{[lambda]}?

{+,-}bandw{[<property>,<value>]}*{[lambda]}?

The language allows to specify a range

property as a single integer value or an inter-

val matching the following regular expression:

<range>::=int(-|-int)?. So, the user can specify

a subset of facilities N

by using [level,<range>]

which restricts to facilities located at given levels of

the network and [reliable,<bool>] which restricts

to facilities for which the path from the central

facility are (non-)reliable. Similarly, the user can

specify a subset of pserver types or stages by using

[type,<range>] and [stage,<range>] respec-

tively. Last, the user can restrict the length of transfer

paths (number of arcs) by using [length,<range>]

and select only (non-)reliable paths by using

[reliable,<bool>]. If a property is not given, then

the condition is satisﬁed.

Predeﬁned Criteria

Several criteria are predeﬁned. Let N

⊆ N, K

⊆ K,

⊆ S and P

⊆ N

respectively be some subsets of

the facilities, pserver types, stages and paths speciﬁed

by the user.

The pserv criterion counts the number of pservers

for a subset of facilities and for a subset of pserver

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

types:

∑

i∈N

∑

k∈K

The local criterion counts the number of local

connections at a subset of the facilities for a subset

of stages:

∑

i∈N

∑

s∈S

The conn criterion counts the number of connec-

tions for a subset of the transfer paths and for a subset

of stages:

∑

(i, j)∈P

∑

s∈S

i j

The bandw criterion sums the path bandwidths for a

subset of the transfer paths and for a subset of stages:

∑

(i, j)∈P

∑

s∈S

i j

Examples. The listing 1 gives aliases for crite-

ria combinations used in Section 7. First, we de-

ﬁne a linear cost function cost_pserv associated

to the number of deployed pservers of each type

by using the agregate combiner. Another agregate

combiner dist_pserv allows to sum the lengths of

transfer paths as in facility location models. The

sum_bandw criterion estimates the total bandwidths

allocated to the broadcast by distinguishing local and

remote connections. The capa_pserv criterion com-

putes the total capacity of the deployment whereas the

bad_pserv criterion returns a vector with the number

of non-reliable, new and “proxy” pservers.

7 EXPERIMENTAL RESULTS

This section presents computational experiments con-

ducted to evaluate our approach. First, we describe a

method for generating realistic instances on which the

experiments have been done. Then, a use case shows

that playing with criteria combination allows to ex-

hibit different solutions which are then improved by

incrementally reﬁning the criteria combination. Last,

the performance of our approach is evaluated depend-

ing on the size of the instances as well as the criteria

combination.

An application provides a MILP platform to solve

PSL problems. It translates a PSL problem and a set

of criteria in one or more integer programs that are

then solved by an underlying integer programming

solver. It has been developed at UNS in C++ and is

available at http://github.com/arnaud-m/opossum un-

der a modiﬁed BSD license. It actually offers in-

terfaces to two integer programming solvers: IBM

Ilog CPLEX (IBM, 2012), GLPK (GLPK, 2012). All

the experiments were conducted on a Linux machine,

with 32GB of RAM and 6 processors (3.2GHz) with

CPLEX 12.3.0 (mono-thread).

Generating Realistic Instances

Mandriva’s team have provided useful information to

generate realistic instances. First, the tree networks

have about four levels. In complex scenario, the av-

erage numbers of facilities and clients are equal to

1000 and 100000 respectively. The characteristics

of a facility are deﬁned by its type (about ten types)

which is associated to a level of the network. We

consider two groups of clients, i.e. the problem has

three stages. We assume that the bandwidth of links

is one of the following: 2MB (low-end DSL), 20MB

(high-end DSL), 100MB (low-end LAN), 1GB (low-

end LAN).

There are ﬁve types of servers which provide dif-

ferent number of connections.The best alternative is

to install package servers on existing Unix servers of

the network, but we can buy and install new Unix

servers (C

= C

= 500). The pservers can also be in-

stalled on Unix (C

= 300) and Windows (C

= 100)

clients, but they provide less connections and are

more expensive. As a last resort, it is possible to acti-

vate the “proxy mode” of any client (C

= 20) which

is thereby temporary transformed, at great expense,

into a pserver with limited capacity.

There are twelve types of facility characterized by

their demands and server capacities as well as some

probabilities used for the network generation. The

generation of the tree network relies on a breadth-ﬁrst

algorithm. For a given node, its children are generated

as follows. First, the number of facilities is drawn ran-

domly for each type of the next level using a binomial

distribution B(n, p). Second, the bandwidth and reli-

ability of the links between the node and its children

are drawn randomly using uniform distributions. Ad-

ditionally, we can enforce that the structure of the net-

work is strictly hierarchical: a link has a lower band-

width and is less reliable than its predecessors. In this

case, the bandwidth probabilities are normalized ac-

cording to the bandwidth of its predecessor in order

to fulﬁll the hierarchical condition.

Use Case

We discuss a realistic use case based on a synthetic

instance for which we apply two scenarios. The net-

work is composed of 769 facilities and 24320 clients.

There is a total of eight available Unix servers and

it is possible to buy at most one new Unix servers

at each facility of levels 0 and 1 (a total of 16 new

ThePackageServerLocationProblem

cost_pserv="-agregate[-pserv[type,0][10],-pserv[type,1][18],-pserv[type,2][6],-pserv[type,3][4],-pserv[type,4][1]]"

dist_pserv="-agregate[-conn[length,1][1],-conn[length,2][2],-conn[length,3][3]]"

sum_bandw="+agregate[-local[stage,1-][5000],-bandw[stage,1-]]"

capa_pserv="+agregate[-pserv[type,0-1][500],-pserv[type,2][300],-pserv[type,3][100],-pserv[type,4][20]]"

bad_pserv="-pserv[reliable,0],-pserv[type,1],-pserv[type,4]"

Listing 1: Aliases of several criteria combinations.

Unix servers). In the ﬁrst scenario, we aim to min-

imize the number of deployments while maximizing

the reliability but also taking into account preferences

between pserver types. In this scenario, the broadcast

model only checks the feasibility constraints, i.e. the

minimal bandwidth allocated to a single client. In the

second scenario, we aim to minimize a cost function

associated to the deployment while maximizing the

broadcast performance.

The user must deploy at least 30 pservers spread

over 16 facilities (-lex[-pserv]). The solution vec-

tor (8, 14, 8, 0, 0) is composed of 8 Unix servers, 14

new Unix servers, 8 Unix clients, 0 Windows clients,

and 0 proxy clients. The spare capacity of a deploy-

ment is deﬁned as the minimal spare capacity among

groups. So, the user determines the maximal spare ca-

pacity equal to 6.2% (-lex[-pserv,capa_pserv])

whereas it was only equal to 3.2% in the previous so-

lution. Now, the user want to increase the reliability

of the deployment modeled by the bad_pserv crite-

ria. The user ﬁrst determines an egalitarist solution

(-lex[-pserv,-leximax[bad_pserv]]) in which

the number of reliable pservers is improved from 22

to 24 and the number of new Unix servers is re-

duced from 14 to 12. The solution vector is now

(8, 12, 10, 0, 0), but the deployment has a low spare

capacity of 0.5%. Then, the user determines an

alternative solution (-lex[-pserv,bad_pserv]) in

which it is preferred to increase the number of reli-

able pservers (from 22 to 25) rather than reducing the

number of new Unix servers (from 14 to 13). This

last solution (8, 13, 9, 0, 0) is preferred to the previous

one because of its better spare capacity (2%). In all

previous solutions, pservers are spread over facilities

of the levels 1 and 2.

The user must deploy 41 pservers spread over 20

facilities for a total cost of 284 (lex[cost_pserv]).

The solution (7, 2, 28, 2, 2) uses all types of pservers

and has a low spare capacity (1.5%). From now

on, pservers are spread over facilities of the levels

1 to 3. Note that the two best solutions of the

ﬁrst scenario have a cost of 356 and 368 respec-

tively. Then, the user determines that improving

the reliability (lex[cost_pserv,bad_pserv]

or lex[cost_pserv,-leximax[bad_pserv]])

only results in increasing the number of re-

liable pservers from 27 to 31 while leaving

the solution vector unchanged. Finally, the

user prefers to improve the broadcast per-

formance (lex[cost_pserv,sum_bandw] or

lex[cost_pserv,dist_pserv]) which changes the

solution vector to (6, 2, 29, 3, 2). The 42 pservers are

spread over more facilities (21) and more levels (1-3)

than in the ﬁrst scenario, but has less spare capacity

(0.8%)

Note that solving times signiﬁcantly depend on

criteria combinations. Indeed, solving times of the

ﬁrst scenario remain lower than 5 seconds whereas

those of the second scenario increases until 5 minutes

(from the second to the fourth criteria combinations).

Intuitively, subproblems with a minimal number of

pservers are easier to solve than those with a minimal

cost because there are less feasible deployments.

This use case has showed that our approach can

provide a ﬂexible and efﬁcient decision support tool

for the deployment of package servers in a tree net-

work. In the next section, we will evaluate the perfor-

mance and beneﬁts of our approach in a systematic

way.

Performance Evaluation

Our approach has been tested on 40 synthetic in-

stances. The number of facilities ranges from 300 to

1700 with the average around 1000. The total demand

(2 groups) ranges from 10000 to 50000. The average

number of existing and new Unix servers is 33.

We discuss here the impact of the criteria combi-

nation on the solving time of the problem. We com-

pare families of combinations which concern simi-

lar aspects of the deployment. Then, we study the

inﬂuence of the criteria ordering before a compari-

son of the primary criterion (see below). Let recall

that the listing 1 page 10 gives aliases for several

criteria combinations. The basic criteria combina-

tion is -lex[(criteria,lccriteria]. The primary

criteria is either -pserv or cost_pserv in order

to restrict the number of pservers. The Secondary

lccriteria are presented gradually and concern the

efﬁciency and the reliability. For each of 24 criteria

combinations, instances were tested with a time limit

set to 1000 seconds per subproblems. The number

of subproblems depends only of the criteria combina-

tion. 835 of 960 problems were optimally solved. The

graphics of Figure 4 show percentages of instances

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

100

1 10 100 1000

% Optimum

Time (s)

dist_pserv

sum_bandw

(a) Distance versus bandwidth

100

1 10 100 1000

% Optimum

Time (s)

lexicographic

leximax

(b) Lexicographic versus leximax

100

1 10 100 1000

% Optimum

Time (s)

rel_perf

perf_rel

100

1 10 100 1000

% Optimum

Time (s)

pserv

cost_pserv

(d) Size versus cost

Figure 4: Experimental results for criteria combinations.

solved optimally as a function of the time in seconds

for groups of criteria combinations.

Distance versus Bandwidth. We compare bi-

criteria combinations related to the efﬁciency of the

broadcast. Figure 4(a) compares the resolution of

problems where lccriteria is either sum_bandw or

dist_pserv with both primary criteria. More than

80% of the dist_pserv problems are solved within

10 seconds whereas only about 30% of sum_bandw

problems. The solver takes advantage of additional

time for solving up to 80% of the sum_bandw prob-

lems within 1000 seconds whereas the resolution of

the dist_pserv problems has reached a ﬂoor around

20 seconds. So, the dist_pserv criterion can be used

for discovering quickly good solution before using the

sum_bandw criterion for solution reﬁnement.

Lexicographic versus Leximax. We compare 4-

criteria combinations related to the reliability of the

broadcast. Figure 4(b) compares the resolution of

problems where lccriteria is either bad_pserv or

-leximax[bad_pserv] with both primary criteria. It

is worthwhile to use leximax to smooth the values

of the criteria. More than 80% of the problems are

solved within 10 seconds for both criteria and the re-

sults are similar.

Reliability versus Performance. Now, we evalu-

ate the impact on the solving time of the relative

ordering between the efﬁciency and reliability crite-

ria. Figure 4(c) compares the resolution of problems

where, for instance, lccriteria is on the one hand

bad_pserv,sum_bandw (rel_perf) and on the other

hand sum_bandw,bad_pserv (perf_rel). The results

indicate that is preferable to improve the reliability of

the deployment, and then its efﬁciency.

Size versus Cost. Last, Figure 4(d) compares the

resolution of problems depending on the primary cri-

terion for all secondary criteria described above. The

results conﬁrm the intuition that pserv subproblems

where the number of pservers is ﬁxed are more con-

strained (and easier) than cost_pserv subproblems

where only the installation cost is ﬁxed. Though,

cost_pserv problems allow to discover more vari-

ous solutions by playing with criteria combinations.

ThePackageServerLocationProblem

8 CONCLUSIONS

We have introduced a new multi-objective optimiza-

tion problem derived from a real-world application:

the package server location problem. We have pro-

posed a ﬁrst mathematical formulation of this multi-

objective optimization problem as well as a method

for the generation of realistic instances. In addition,

a small dedicated language allows virtually to deﬁne

any possible criteria combination.

Results for exact multi-objective solution ap-

proaches based on mixed integer linear programming

have been reported. First, a realistic use case has

shown the efﬁciency and ﬂexibility of our approach

for quick prototyping of deployments with various

structural properties. Second, solving times are really

satisfactory for realistic problem’s sizes. Last, some

criteria combinations are “easier” to solve than others

to obtain solutions with similar characteristics.

In further work, we will compare different linear

programming solvers, and investigate approximations

of the pareto set.

ACKNOWLEDGEMENTS

The authors would like to thank Claude Michel, Mo-

hamed Rezgui (UNS CNRS), and Yvan Manon (Man-

driva). This work was supported by the Agence Na-

tionale de la Recherche (Aeolus project – ANR-2010-

SEGI-013-01). To end, we would like to thank the

anonymous referees for their remarks.

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