Facility location optimization and cooperative games.
On April 27, 1802, I gave a shout of joy ... It was seven years ago I proposed
to myself a problem which I have not been able to solve directly, but for which
I had found by chance a solution, and I knew that it was correct, without being
able to prove it. The matter often returned to my mind and I had sought twenty
times unsuccessfully for this solution. For some days I had carried the idea about
with me continually. At last, I do not know how, I found it, together with a large
number of curious and new considerations concerning the theory of probability.
Andre Marie Ampere.
Facility location problems (or plant location problems) are general models that
can be used when a set of clients has to be served by facilities. More precisely, we
are given a set of potential facility locations and a set of clients. The optimization
problem is to select a subset of the locations at which to place facilities and then
to assign clients to theses facilities so as to minimize total cost. Most formulations
considered in this thesis can be viewed as general models that can be applied to a wide
range of context and practical situations. However, as this research has been partly
initiated by the interest of the author in telecommunication network design we will
introduce these models by considering problems in this particular area.
In the context of telecommunication network design an application of discrete
location theory is the optimization of access networks with concentrators. Typically,
we have a number of terminal points that must be connected to a service point. An
obvious solution is to use a dedicated link for each terminal (star network). However, it
is clear that this solution can be very expensive when the number of terminals is large
and when they are far from the service point. Access networks are often constructed by inserting concentrators between the terminals and the service point. Many terminals
are connected to a facility which in turn is connected by a single link to the service
point. The objective is to build a network that will provide the service at minimum
If no extra constraints are involved the mmimum cost network problem can
be expressed as an uncapacitated facility location problem (UFL). If the number of
terminals that can be connected to a concentrator is limited we obtain a so-called
capacitated facility location problem (CFL). CFL can be extended to consider various
types of concentrators with various capacities. This problem is the multi-capacitated
facility location problem (MCFL). MCFL is a straightforward model for low speed
packet switched data networks typical among which are networks connecting sellingpoint
terminals to a database. For other networks, the problem may involve various
traffic constraints. In chapter 1 we present those problems and compare solutions
obtained by Lagrangian relaxation and simulated annealing algorithms.
The architecture mentioned above can be extended with more than one hierarchical
level of concentrator. Unfortunately, we pay for this cost saving through a
decrease of reliability. Therefore, the number of levels is often limited to one or two.
In chapter 2 we study an extension of UFL and CFL to two levels of concentrators.
Obviously, the structure of a network changes according to the way requirements
vary with time. In order to plan investments and to develop strategies, the evolution of
a network has to be determined for several years ahead (typically four or five years). In
this case the main questions to answer are: Where and when to establish concentrators
and of what size? In chapter 3 we study this problem for the dynamic version of UFL.
Now, with the network optimization problem, there naturally arises the problem
of allocating the total minimum cost among customers fairly. Namely, we would like
to allocate the cost in such a way that no subgroup of users would have incentive to
withdraw and build their own network. The standard way to approach such a problem
is by the means of cooperative game theory. In chapter 4 we study the core of location
games derived from UFL and CFL, and in chapter 5 we propose methods to compute
the nucleolus of these games.