Facility location

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Facility location
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Service facility location

• Geographic representation
• Location problems are based on how geography is
  modeled.
• In 2D, location can be modeled as xy coordinate
  plane, or in a network.
• Two ways of measuring distances between two
  points with coordinates (xi, yi) and (xj, yj)
• Euclidean distance
• Metropolitan metric distance

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Service facility location

• Number of facilities
• Single facility location problem can be solved
  analytically with relative ease.
• Multi-facility location problems are difficult
  because of demand assignment at nodes. Also for
  some services, the type of facilities may vary.
• Optimization criteria
• Objective function in the location problem
  different for different services. Depending on
  the owner of the facility, the objectives could
  be
• Maximization of utility, profit, social benefit
  Minimization of travel times, cost.

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Single dimension single facility location problem

• Suppose we wish to find location for a single bus
  stop that will serve all the boys-hostels from
  Cauvery to Mandakini.
• Fix a arbitrary point on the road as origin.

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Single facility location problem

• Result says that the bus stop should be located
  at the median w.r.to demand density.
• This method is called cross-median approach.
• The result can be extended to 2D using either
  form the distance measurement (Euclidean or
  Metropolitan).

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Single facility location problem

• Since x and y coordinates are independent of each
  other. We can solve two sub-problems (one for
  each axis).
• Thus the optimal location of the service facility
  will have
• xs at the median value of wi ordered in
  x-direction
• ys at the median value of wi ordered in
  y-direction
• Because each (or both) of the coordinates may be
  unique or lie within a range, the optimal
  location may be at a point, on a line or within a
  region.

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Example
Median weight (7135)/2 8
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Example Solution
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Example Solution
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Example Solution

• Non-unique optimal solution (2,2) and (3,2).
• Hence the line segment joining these two optimal
  solution is also an optimal location. Weighted
  distance of all the location from any of these
  optimal point will be same. Verify!

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Single facility location with Euclidean metric
distance
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Single facility location by center of gravity
method

• An intuitive but non-optimal approach, because it
  does not minimize the travel distances from each
  location to the service facility.
• Center of gravity solution could serve as a
  starting point for the previous, rather tedious,
  method.

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Single facility location by center of gravity
method

• The optimal solution of the Euclidean method will
  always result in a point and will rarely match
  the center of gravity solution.
• Example of Euclidean method for location problem
  FedEx

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Locating a retail outlet

• Objective profit maximization
• Countably finite number of alternative locations
  evaluated to find the most profitable site.
• Method developed by David L. Huff.
• Basic idea Attractiveness of a facility is
  directly proportional to its size and inversely
  proportional to the distance.

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Locating a retail outlet
Shopping center may have the parameter value 2
whereas for regular grocery store, this could be
10.
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Locating a retail outlet

• The probability that the customers might be
  attracted to other stores is captured by
  calculating such probability
• Now we can calculate the total consumer
  expenditure for product class k at each facility
  j.

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Locating a retail outlet
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Example
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Example

• From the previous discussion, assume that a
  facility is located at optimal point (2,2).
  Assume that we want to check profitability of
  location (3,2).
• Assume
• Ci weight100.
• Each customer order represents an expenditure of
  approximately Rs 10 and ? 2.
• The new location at (3,2) is twice as big as the
  current one at (2,2).
• Travel distances are given in the next table

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Example
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Example
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Example
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Example
Proposed site will have higher market share than
the current location. Hence, the proposed
location is better.
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Multiple facilities Location set covering problem
2
9
20
40
40
30
1
30
7
20
35
20
8
30
30
3
15
30
4
25
6
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Site within 30 miles could be served by a node.
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5
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Multiple facilities Location set covering problem
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Maximal covering problem

• The set covering problem minimizes the number
  of facilities located under the constraint that
  all demand locations need to be served.
• However, we have not considered budgetary
  constraints.
• These budgetary constraints might not allow us to
  serve all demand locations.
• In these cases, the problem is to locate the
  budgeted facilities to maximize the serviced
  demand.
• These problems are called maximal covering
  location problems.
• On the other hand, when the objective is to
  minimize the total weighted distance traveled
  from all demand centers to the opened facilities,
  the problem is called a p-median problem.

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Maximal covering problem

• In addition to maximizing the serviced
  population, additional constraints may be imposed
  within the maximal covering problem.
• For example, for emergency facility locations, it
  may be desirable to locate facilities in order to
  maximize the population which can be served
  within five minutes while insuring the entire
  population can be served within ten minutes.
• This secondary or reduced service time
  requirement is included by means of mandatory
  closeness constraints.

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Maximal covering problem

• Let there be J 1,2,n demand centers with known
  locations and demands pj. In addition, let there
  be I 1,2,m possible sites, where a maximum of
  K facilities may be opened.
• Let the maximum allowable response time (or
  distance) be R, which is specified.
• If the minimum of facilities required to serve
  all the demand centers within the given response
  time (or distance) exceeds the allotted (K),
  then not all demand centers may be served and we
  attempt to serve the maximum population.
• Let tij be time (or distance) from ith site to
  jth demand center.
• The problem formulation then becomes

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Maximal covering problem

• For pj 1 for all js, the problem is same as
  the set covering problem and will minimize the
  of facilities while serving all demand centers.
  For demand distribution not uniform, the optimal
  set of facilities will serve maximum population.
• Mandatory closeness constraints The demand
  centers be located no more than T time units (or
  distance) from an open facility, where T gt R.
• We have to include these additional mandatory
  closeness constraints in our problem.

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Multiple facilities Bowman-Stewart formula
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Multiple facilities Bowman-Stewart formula