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Location Problems

Location Problems. John H. Vande Vate Spring 2006. Rectilinear Location Problems Euclidean Location Problems Location - Allocation Problems. Where to Locate Facilities. On the line, if the objective is to min … The maximum distance traveled The maximum distance left + right

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Location Problems

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  1. Location Problems John H. Vande Vate Spring 2006 1

  2. Rectilinear Location Problems Euclidean Location Problems Location - Allocation Problems Where to Locate Facilities 2

  3. On the line, if the objective is to min … The maximum distance traveled The maximum distance left + right The distance traveled there and back to each customer The item-miles traveled Basic Intuition 1 4 0 3

  4. Travel on the streets and avenues Distance = number of blocks East-West + number of blocks North-South Manhattan Metric Rectilinear Distance 4

  5. Rectilinear Distance 9 5 4 5

  6. To minimize the sum of rectilinear distances Intuition Where? Why? Locate a facility... 6

  7. Solver Model 7

  8. Locate a facility... • To minimize the max of rectilinear distances • Intuition • Where? • Why? 8

  9. Set Customers; Param X{Customer}; Param Y{Customer}; Var Xloc; var Yloc; var Xdist{Customer}>= 0; var Ydist{Customer}>= 0; var dmax; min objective: dmax; s.t. DefineMaxDist{c in Custs}: dmax >= Xdist[c] + Ydist[c]; Min the Max 9

  10. Min the Max Cont’d • DefineXdist1{c in Customer}: Xdist[c] >= X[c]-Xloc; • DefineXdist2{c in Customer}: Xdist[c] >= Xloc-X[c]; • DefineYdist1{c in Customer}: Ydist[c] >= Y[c]-Yloc; • DefineYdist2{c in Customer}: Ydist[c] >= Yloc-Y[c]; 10

  11. Solver Model 11

  12. Like locating a land fill or a prison Want the nearest town as far away as possible Must have some confined region to work in otherwise… Let say the size of the region is M x M Max the Minimum 12

  13. With Min the Max: Xdist[c] >= X[c] – Xloc; Can’t say Xdist[c] <= X[c] - Xloc; Want to say that either Xdist[c] <= X[c] – Xloc OR Xdist[c] <= Xloc - X[c] Find a bound on the X distance between a customer and the facility, call it M Add a variable Left[c] = 1 if X[c] > Xloc and Left[c] = 0 otherwise Xdist[c] <= Xloc – X[c] + 2*Left[c]*M Xdist[c] <= X[c] – Xloc + 2*(1-Left[c])*M NIMBY 13

  14. The Model 14

  15. Set Covering Models 15

  16. WesternAir 16

  17. Rectilinear Location Problems Euclidean Location Problems Location - Allocation Problems Outline 17

  18. Distance is not linear Distance is a convex function Local Minimum is a global Minimum Locating a single facility 18

  19. Total Cost = c1d1(x,y) + c2d2(x,y) + c3d3(x,y) = c1(x1- x)2 + (y1- y)2 +c2(x2- x)2 + (y2- y)2 +c3(x3- x)2 + (y3- y)2 Total Cost/x = c1 (x1 - x)/d1(x,y)+c2 (x2 - x)/d2(x,y)+c3 (x3 - x)/d3(x,y) Total Cost/x = 0 when x = Where to Put the Facility But that depends on x! [c1x1/d1(x,y)+c2x2/d2(x,y)+c3x3/d3(x,y)] [c1/d1(x,y)+c2/d2(x,y)+c3/d3(x,y)] 19

  20. Start somewhere, e.g., pretend dk(x,y) = 1 x = [c1x1+c2x2+c3x3]/[c1+c2+c3] y = [c1y1+c2y2+c3y3]/[c1+c2+c3] That’s the “center of gravity” Step 1: Calculate the real distances, i.e. the dk‘s Step 2: Refine values of x and y x = y = Repeat Steps 1 and 2. ... [c1x1/d1+c2x2/d2+c3x3/d3] [c1/d1+c2/d2+c3/d3] [c1y1/d1+c2y2/d2+c3y3/d3] [c1/d1+c2/d2+c3/d3] Iterative Strategy 20

  21. Solver Model 21

  22. Call on Convex Minimization Tool Minos, Interior Point Methods, … Typically don’t support discrete variables too… Convex Minimization 22

  23. Fixed Number of Facilities to Consider Single Sourcing Two Questions: Location: Where Allocation: Whom to serve Each is simple Together they are “harder” Locating Several Facilities 23

  24. Coopers ALT Heuristic (1963) Put the facilities somewhere Step 1: Assign the Customers to the Facilities Step 2: Find the best location for each facility given the assignments (see previous method) Repeat Step 1 and Step 2 …. Iterative Approach 24

  25. Uncapacitated (facilities can be any size) “Greedy”: Assign each customer to closest facility Capacitated Use Optimization in the assignments This is typically a simple linear integer program Assign the Customers to the Facilities to min the total distance s.t. the Facilities’ capacities Assign Customers to Facilities 25

  26. Location on a Line Rectilinear Location Linear modeling techniques Euclidean Location Convex Minimization Contraction Map Location Allocation Cooper’s ALT Heuristic Summary 26

  27. Location Case Study SGM Service Parts Case Using the ALT Heuristic Next 27

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