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

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

3. 3 3 Basic Intuition 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 0 1 4

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

5. 5 5 Rectilinear Distance 5 4 9

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

7. 7 7 Solver Model

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

9. 9 9 Min the Max 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];

10. 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];

11. 11 Solver Model

12. 12 Max the Minimum 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

13. 13 NIMBY 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

14. 14 The Model

15. 15 Set Covering Models

16. 16 WesternAir

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

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

19. 19 Where to Put the Facility Total Cost = c 1 d 1 (x,y) + c 2 d 2 (x,y) + c 3 d 3 (x,y) = c 1  (x 1 - x) 2 + (y 1 - y) 2 +c 2  (x 2 - x) 2 + (y 2 - y) 2 +c 3  (x 3 - x) 2 + (y 3 - y) 2  Total Cost/  x = c 1 (x 1 - x)/d 1 (x,y)+c 2 (x 2 - x)/d 2 (x,y)+c 3 (x 3 - x)/d 3 (x,y)  Total Cost/  x = 0 when x = [c 1 x 1 /d 1 (x,y)+c 2 x 2 /d 2 (x,y)+c 3 x 3 /d 3 (x,y)] [c 1 /d 1 (x,y)+c 2 /d 2 (x,y)+c 3 /d 3 (x,y)] But that depends on x!

20. 20 Iterative Strategy Start somewhere, e.g., pretend d k (x,y) = 1 –x = [c 1 x 1 +c 2 x 2 +c 3 x 3 ]/[c 1 +c 2 +c 3 ] –y = [c 1 y 1 +c 2 y 2 +c 3 y 3 ]/[c 1 +c 2 +c 3 ] –That’s the “center of gravity” Step 1: Calculate the real distances, i.e. the d k ‘s Step 2: Refine values of x and y x = y = Repeat Steps 1 and 2.... [c 1 x 1 /d 1 +c 2 x 2 /d 2 +c 3 x 3 /d 3 ] [c 1 /d 1 +c 2 /d 2 +c 3 /d 3 ] [c 1 y 1 /d 1 +c 2 y 2 /d 2 +c 3 y 3 /d 3 ] [c 1 /d 1 +c 2 /d 2 +c 3 /d 3 ]

21. 21 Solver Model

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

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

24. 24 Iterative Approach 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 ….

25. 25 Assign Customers to Facilities 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

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

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