Mathematical Model

For the formulation of this localization model is presented below:

I = set of demand nodes i.

J = set of the candidate locations of the facilities j.

dij = distance between the demand node i and its candidate facility located at site j.

hi = demand of node i.

p = number of facilities to locate.

Defining the following decision variables:

χj = 1, if it is located at site j;

0, otherwise.

γij, if the demand of node i is assigned to the facility located in site j;

0, otherwise.

W = maximum distance between a demand node and its facility to which the node is assigned.

The p-center can be formulated as follows:

The objective function (1.1) minimizes the maximum demand-distance between each node of demand and the nearest selected facility.

x The restriction (1.2) establishes that there are p number of facilities to be located.

x The restriction (1.3) requires that each demand node be assigned exactly to only one facility.

x The restriction (1.4) only allows the demand of a node to be assigned to a open facility, this is a selected facility.

x The restriction (1.5) stipulates that the maximum distance between node i and the facility in the site j, denoted by W is larger than the distance between any node i and the facility located on site j.

x The set of restrictions (1.6) and (1.7) establishes the binary nature of the decision variables.