Difference: FacilityLocationProblem (1 vs. 3)

Revision 32018-11-23 - TWikiAdminUser

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META TOPICPARENT name="MasterSlaveConstraints"

Facility Location Problems

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  -- TWikiAdminGroup - 22 Apr 2008
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Revision 22008-04-22 - TWikiAdminUser

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META TOPICPARENT name="MasterSlaveConstraints"

Facility Location Problems

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

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 We will not delve into it here, but a "tighter" formulation would use the following constraints instead of (?? REFLATEX{eq:facility} not defined in eqn list ??):
\[ y_{fc} \leq z_f, f \in {\cal F}, c \in {\cal C} \]
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Facility Location Case Studies

Results from OpsRes web retrieved at 22:09 (GMT)

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\usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return...
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\usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return...
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Number of topics: 3
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 -- TWikiAdminGroup - 22 Apr 2008

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Revision 12008-04-22 - TWikiAdminUser

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META TOPICPARENT name="MasterSlaveConstraints"

Facility Location Problems

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

The following definition of the general facility location problem is from J. E. Beasley's OR-Notes.

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.

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:

  \sum_{c \in {\cal C}} y_{fc} \leq |{\cal C}| z_f, f \in {\cal F} (1)

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 (1):

\[ y_{fc} \leq z_f, f \in {\cal F}, c \in {\cal C} \]

-- TWikiAdminGroup - 22 Apr 2008

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