IEOR151
Lab 7: Location Models
Long He
Dept of Industrial Engineering & Operations Research
Fall 2013
Admin info
y
Midterm
Ń Return on Monday class
Ń Grade will be curved
y
Bring laptop in next lab
Ń Excel with solver installed
Distance
y
1-D: locate over a line
Ń Distance between two points ‫ݔ‬௜ and ‫ݔ‬௝ ݀௜௝ = ‫ݔ‬௜ െ ‫ݔ‬௝
y
2-D: locate over a plane
Ń Euclidian distance
(‫ݔ‬௜ െ ‫ݔ‬௝ )ଶ +(‫ݕ‬௜ െ ‫ݕ‬௝ )ଶ
Ń Manhattan(Metropolitan) distance ‫ݔ‬௜ െ ‫ݔ‬௝ + ‫ݕ‬௜ െ ‫ݕ‬௝
y
3-D
Ń Great-circle distance: locate over a sphere (the
earth) ‫ ݈ܽݎݐ݊݁ܿ × ݏݑ݅݀ܽݎ‬ ݈ܽ݊݃݁
Location problems
y
Network problems
y
Plane problems
Median problem
y
Objective: minimize total demand
weighted travel distance
Ń P-median
Ń Others
Cross-median location problem
Know demand locations (set M)and sizes.
y Find a service location (s) to minimize
total demand weighted metropolitan
distance traveled.
y Formulation
y
෍ ‫ݓ‬௜ ‫ݔ‬௜ െ ‫ݔ‬௦ + ෍ ‫ݓ‬௝ ‫ݕ‬௝ െ ‫ݕ‬௦
௜‫א‬ெ
y
௝‫א‬ெ
Solve 1-median problems in each
dimension respectively.
Cross-median location problem
y
Solve 1-median problem
Ń Step 1: calculate total demand ܹ = σ௜ ‫ݓ‬௜
Ń Step 2: find the location such that (i) total demand on
ଵ
ଵ
left ൑ ܹ; and (ii) total demand on right ൑ ܹ
ଶ
ଶ
1
‫ݓ‬ଵ + ‫ݓ‬ଶ ൑ ෍ ‫ݓ‬௜
2
‫ݓ‬ଶ
௜
‫ݓ‬ଵ
1
‫ݓ‬ଷ + ‫ݓ‬ସ + ‫ݓ‬ହ ൑ ෍ ‫ݓ‬௜
2
௜
‫ݓ‬ଷ
‫ݓ‬ସ
Solution
(include node 2
and 3)
‫ݓ‬ହ
Cross-median location problem
y
Example
Ń Demand locations (in coordinates): A(6,2),
B(8,6), C(5,9), D(3,4).
Ń Demand sizes: A(2000), B(1000), C(3000),
D(2000).
Ń Recommend service location (‫ݔ‬௦ , ‫ݕ‬௦ ).
y
Consider 1-median problems in each
dimension.
Hoteling problem
Two firms A and B locate their hotels
along a street in order to maximize
demand covered.
y Assumptions
y
Ń Demands are uniformly distributed.
Ń Customers go to the closest hotel.
y
Results
Ń Equilibrium: both A and B open their hotels at
the midpoint of the street. (what if not?)