Introduction to Location Science - Location Science - page 7

with more than three sides, does not necessarily yield the optimal solution when
more than three nodes are involved. A practical algorithm for solving the problem
with an arbitrary number of nodes was proposed by Weiszfeld ( 1937 ). 4 The iterative
procedure proposed in this work was recently revisited in depth by Plastria ( 2011 ).
A synthesis of the first steps towards inserting location theory into an economic
context is due to Lösch ( 1944 ). The importance of this work stems from the fact that,
for the first time, location theory and the theory of market areas were connected.
This work constitutes the first explicit recognition of the strong link that is often
observed between these two areas.

1.3 Towards a New Science

The 1960s set the foundations of Location Science as new scientific area. We first
witnessed the natural extension of the Weber problem to the multi-facility case.
This was done, among others, by Miehle ( 1958 ) and Cooper ( 1963 ). In particular,
the latter work introduced the planar p-median problem for which each demand
node must be served by one out of p new facilities to be located. This became
a fundamental problem in Location Science, which still attracts the attention of
the scientific community (see the recent papers by Brimberg and Drezner 2013 ,
Brimberg et al. 2014 , and Drezner et al. 2014 ).
The seminal papers by Hakimi ( 1964 , 1965 ) opened new important research
directions. Hakimi ( 1964 ) introduced the concept of absolute median of a graph:
a single facility is to be located anywhere in a network so as to minimize the sum
of the distances of the nodes of the network to the facility. The author proved that
there always exists an optimal solution for which the absolute median is a vertex
of the graph. It is also in this paper that the concept of absolute center was first
introduced: a single facility has to be located (anywhere in the network) in order to
minimize the maximum distance between the facility and all the vertices. This work
was extended to the multi-facility case by Hakimi ( 1965 ): now, p facilities are to
be located. The vertex-optimality property is still valid for the resulting p-median
problem. This property is of major importance because it means that many network
location problems can be cast into a discrete setting which, in turn, leads to the
possibility of using integer programming and combinatorial optimization techniques
for tackling these problems.

4 The author is now known to be Andrew Vázsonyi (1916-2003).