This document discusses factors to consider when determining optimal locations for various facilities and services. Key location factors include transportation costs, demand levels at different locations, available supply, and distance metrics. Common distance metrics include rectilinear, Euclidean, squared Euclidean, and Chebyshev distances. An example problem demonstrates how to minimize the weighted sum of distances between a new facility and existing demand points using different distance measures.

1. Location decisions
• Where to locate factories?
• Where to locate warehouses?
• Where to locate a crane (when it is not in use?)
• Where to locate an elevator (when it is not in
use?)
• Where to locate an ambulance depot?
• Where to locate a garbage dump?
• Where to locate the photocopyingmachine in the
departmentoffice?
• Where to locate bus stops?

2. Factors influencing location
• Costs of moving people and material to some
location
– Demand at some location, could be weighted demand
– Supply available at some points
– Cost per unit distance
• Route choice
– How is the material moved? What route do people
select?
• Distancemeasure
• What are the possible locations (feasible
locations)?

3. Distance Measures
• Rectilinear distance(L1 norm)
– d(X, Pi) = |x - ai| + |y - bi|
• Straight line or Euclidean distance (L2 norm)
– d(X, Pi) =
• Squared Euclidean
• Chebyshev distance (L∞ norm)
– d(X, Pi) = max{|x - ai|, |y - bi|}
• Minkowski distance
– d(X, Pi)= ∑ 𝑥 − 𝑎%
&
'
(
(x - a ) + (y - b )
i
2
i
2
X = (x, y)
Pi = (ai, bi)
Pi = (ai, bi)
Pi = (ai, bi)
X = (x, y)
X = (x, y)

4. Distance measures - continued
• Rectilinear/Manhattan/Mannheim/Chandigarh
– Facility allocation in Grid like cities and warehouses (Look at maps of a few
cities)
• Euclidian distances (as the crow flies)
• Squared euclidean also used
• Building of cell-phone towers
• Chebyshev distance
– Used to calculate the time taken for movement of overhead crane (crane
can move on the x and y axes at the same time but at the same speed
along each axis)
• Network distance

5. Mannheim, in
Baden-Württemberg
Germany
From
Openstreetmap.org

6. Chandigarh
From
Openstreetmap.org

7. Example
• 5 demand locations on the plane, e.g. machines on a shop-
floor
– P1 = (1,1), P2 = (6,2), P3 = (2,8), P4 = (3,6) and P5 = (8,4)
• New general purpose machine to be located on the shop-
floor
• Number of demands (trips per day) from new machine to
existing machines
– 10, 20, 25, 20 and 25
• Cost of movement proportional to distance
• Where should new machine be located?
• In general, given locations (𝑎𝑖,𝑏𝑖) and weights 𝑤𝑖, where
should we locate the facility to minimize sum of weighted
distance to all 𝑖’s? [Try for different distance measures]