FacilityLocationProblem < OpsRes

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---+ Facility Location Problems

Facility location problems are a classical example of the use of [[MasterSlaveConstraints][master-slave constraints]].

The following definition of the general facility location problem is from [[https://people.brunel.ac.uk/~mastjjb/jeb/or/facloc.html][J. E. Beasley's OR-Notes]].
<blockquote>
The general facility location problem is: given a set of facility locations and a set of customers who are served from the facilities then:
   * which facilities should be used;
   * which customers should be served from which facilities so as to minimise the total cost of serving all the customers.
</blockquote>

Given the set of facilities %${\cal F}$% and the set of customers %${\cal C}$%, we can define binary (0-1) variables:

   1 %$z_f$% = 1 if facility %$f$% is used, 0 otherwise;
   2 %$y_{fc}$% = 1 if facility %$f$% serves customer %$c$%, 0 otherwise.

Now the following master-slave constraint ensures that facility %$f$% only serves customers if it is used:
%BEGINLATEX{label="eq:facility"}%
\[ \sum_{c \in {\cal C}} y_{fc} \leq |{\cal C}| z_f, f \in {\cal F} \]
%ENDLATEX%

The decision variables, objective function and constraints vary depending on the problem, but similar master-slave constraints are always present in facility location problems.

We will not delve into it here, but a "tighter" formulation would use the following constraints instead of %REFLATEX{eq:facility}%:
%\[ y_{fc} \leq z_f, f \in {\cal F}, c \in {\cal C} \]%

---++ Facility Location Case Studies

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-- Main.TWikiAdminGroup - 22 Apr 2008
Topic revision: r3 - 2018-11-23 - TWikiAdminUser