Introduction to Location Science - Location Science - page 6

Fig. 1.6 Launhardt's
geometric solution
This problem introduced by Launhardt is exactly what we now call the three-
node Weber problem. However, as pointed out by Pinto ( 1977 ), the problem was
introduced about 10 years before Weber ( 1909 ). Indeed, Launhardt ( 1900 ) proposed
a simple geometric solution scheme for the problem. The solution is obtained as
follows (see Fig. 1.6 ): Consider the triangle ABC defined by the original nodes (the
locational triangle) and select one node, say C. Consider another triangle whose
sides are proportional to the weights w A , w B and w C . 2 Draw a triangle AOB similar
(in the geometric sense) to the wei ght triangle but such that the edge proportional
w C has the same length as edge AB , which is the one opposite to C in the locational
triangle. The new triangle AOB is depicted in Fig. 1.6 . 3 We can now circumscribe
nodes A, B and O, by just touching each point. Finally, a straight line can be drawn
connecting O and C. The intersection between the circle and this line yields the
optimal location for the industrial facility.
This same problem was treated by Weber ( 1909 ) or, to be more accurate, by
the mathematician Georg Pick (1859-1942), who is the author of the appendix
in which the mathematical considerations of Weber's topic are presented. The
problem was solved in a different way but this resulted in the same solution. As
put by Lösch ( 1944 ), the solution to this problem was discovered by Carl Friedrich
Launhardt and rediscovered “one generation later” by Alfred Weber. Nevertheless,
We b e r ( 1909 ), presented a deeper analysis of the problem. He first noted that if
the geometric construction leads to a point outside the original triangle, then the
optimal solution lies on the boundary of the original triangle. Second, he observed
that the pole method, which Launhardt ( 1900 ) believed should work for polygons
2 This triangle is referred to by Weber ( 1909 )asthe weight triangle .
3 Node O was called by Launhardt the pole of the locational triangle.