Annals of Operations Research 6(1986)23-34 23
DISCRETE STOCHASTIC LOCATION MODELS
F.V. LOUVEAUX
faculth Universitaires Notre-Dame de la Paix, Namur, Belgium
Abstract
In this paper, we study how the two classical location models, the simple plant
location problem and the ;>-median problem, are transformed in a two-stage sto-
chastic program with recourse when uncertainty on demands, variable production
and transportation costs, and selling prices is introduced. We also discuss the rela-
tion between the stochastic venion of the SPLP and the stochastic version of the
p-median.
Keywords and phrases
Stochastic programming with recourse, p-median, simple plant location.
1. Introduction
The classical facility location problem consists in finding the optimal location
and size of facilities to be established among a given set of possible sites in order to
meet supposedly known demands specified at a given set of locations with the objective
of minimizing total costs consisting of fixed costs for establishing facilities and variable
production and transportation costs.
In the static uncapacitated case, extensively studied since Kuehn and Hamburger
[10] and for which we can refer to the recent surveys by Krarup and Pruzan [11] and
Cornuejols et al. [3] , the main issue is on the location choice since sizes are obtained
as the sum of the demands served from each open location. Consequently, the
established capacities are fully utilized.
In the dynamic context discussed by Manne [14], the time-phasing of the
decisions becomes important. The dynamic uncapacitated facility location problems
were introduced by Roodman and Schwarz [16] and in a slightly different form by
Wesolowsky and Truscott [18]. Van Roy and Erlenkotter [17] propose a dual-based
e J.C. BaltzeT A.G., Scientific Publishing Company

24 F. V. Louveaux, Discrete stochastic location models
procedure that extends approaches developed by Bilde and Krarup [2] and Erlenkotter
[4] for static uncapacitated problems. Their method assumes that capacities are fully
used in each period. They propose to solve capacitated problems by extensions similar
to those introduced by Guignard and Spielberg [7] in the static case.
This paper addresses the stochastic facility location problem in which demands,
variable production and transportation costs as well as selling prices can be random.
Uncertainty in demands induces that full utilization of capacities becomes infeasible
and the requirement that demands should be met in all circumstances becomes un-
realistic. This explains why a selling price is introduced, since optimal decisions on the
size of facilities will result from a trade-off between the cost of increasing the capacity,
the net profit of selling goods and the probability of the various demand levels.
Other work on stochastic location problems is mainly concerned with optimal
location on networks, see e.g. Handler and Mirchandani [8], including reallocation
decisions, see e.g. Berman and Leblanc [1 ] or Louveaux and Thisse [13], or is based
on dominance assumptions that transform the problem into two simpler problems, see
e.g. Jucker and Carlson [9].
Franca and Luna [6] propose to apply Bender's decomposition to the stochastic
transportation problem introduced by Williams [19] in which the shipments are
decided before the random events are observed.
In this paper, we present a stochastic model for the simple plant location prob-
lem and for the /7-median problem in terms of a two-stage stochastic program with
recourse, and we study the relations existing between the two models.
2. A private sector model
The deterministic model of the uncapacitated facility location problem, also
known as the simple plant location problem, is the following program:
(SPLP) minimize 2 = Z fx + Z Z c..y.. (1)
subject to Z 7.. = 1 / G / (2)
.~ X. <Q iSI, / / (3)
y.. > 0 iel, fGJ (4)
. E iO. l} fej, (5)X.