Weber's location model

The Weber site model is the neoclassical location theory associated with and was in 1909 by the German economist Alfred Weber developed. It is a continuous model that is also known in the literature under the terms Steiner-Weber model or "Fermat problem". It assumes a homogeneous surface that contains an infinite number of possible locations , i.e. H. every point of an area defined as a sales area is regarded as a potential location. Transport costs are central to the approach.

The problem is in its simplest form: taking into account the location of the material, the workforce and the customer, the most cost- effective production location can be found.

The optimal location must be in the triangle of the three presupposed locations. Further assumptions are that the transport costs depend solely on the quantity and the distance. This leads to a mathematical optimization problem . The choice of the location has an impact on whether the starting products are used as weight loss material (such as energy sources) or remain part of the end product as pure material.

Solution

The points of sale of a good on the earth's surface are considered under the following assumptions:

The demand is known for each sales location
The need to manufacture the goods is also known

The aim of the choice of location is the minimum transport costs for both the required materials and the end products to be sold. The transport costs are represented by a constant quantity and are equivalent for materials and finished products. The following objective function can thus be minimized:

${\ displaystyle T = c (a_ {1} r_ {1} + a_ {2} r_ {2} + \ dotsb + a_ {n} r_ {n}) = c \ sum _ {i = 1} ^ {n } a_ {i} r_ {i} \ Rightarrow {\ text {min!}}}$

The objective function contains variables that differ depending on the choice of location. The other variables also have the following meanings: ${\ displaystyle n}$ ${\ displaystyle r_ {i} (i = 1, ..., n)}$

${\ displaystyle c =}$ Transport cost rate per km and t

${\ displaystyle T =}$ Planning period

${\ displaystyle r_ {i} =}$ Transport route

${\ displaystyle a_ {i} =}$ Transport quantity

A simplification of the model is given by the fact that the earth's surface can be segmented by a coordinate system. Thus, a route can also be represented as follows:

${\ displaystyle r_ {i} = {\ sqrt {(x-x_ {i}) ^ {2} + (y-y_ {i}) ^ {2}}}}$

This results in the following, simplified objective function in which the location with the minimum transport costs can be calculated by setting the first partial derivative according to x and y as well as the following positive test of the second partial derivative.

${\ displaystyle T (x, y) = c \ sum _ {i = 1} ^ {n} a_ {i} {\ sqrt {(x-x_ {i}) ^ {2} + (y-y_ {i }) ^ {2}}}}$

However, this method only leads to a valid approximation because the earth can be viewed as a large sphere and the distances can be calculated from this. The weakness of this procedure is the neglect of any differences in height and the assumption that there are direct routes between the points.

literature

Edmund Heinen: Industrial Management . Gabler, Wiesbaden 1990, ISBN 3-409-33150-6 , pp. 240 f .
Thomas Plümer: Logistics and Production . Oldenbourg Wissenschaftsverlag GmbH, Munich 2003, ISBN 3-486-27470-8 , p. 248 f .
Weber Alfred: About the location of the industries. Part 1: Pure theory of the location . 2nd Edition. Tubingen 1909.

Web links

Facility Location Optimizer , a MATLAB-based tool for solving location problems.