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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?)