1. John H. Vande Vate Spring 2005
Location Problems
John H. Vande Vate
Spring 2005 1

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

3. Basic Intuition 1 4 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 1 4 3

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

5. Rectilinear Distance 9 5 4 5

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

7. Solver Model 7

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

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]; 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. Locate a facility... To minimize the max of rectilinear distances
Intuition
Where?
Why? 12

13. Finding the Center(s) 13

14. NIMBY Maximize the Minimum Distance
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] <= X[c] – Xloc + 2*Left[c]*M
Xdist[c] <= Xloc – X[c] + 2*(1-Left[c])*M 14

15. The Model 15

16. Outline Rectilinear Location Problems Euclidean Location Problems
Location - Allocation Problems 16

17. Locating a single facility
Distance is not linear
Distance is a convex function
Local Minimum is a global Minimum 17

18. Where to Put the Facility
Total Cost = S ckdk(x,y)
= S ck(xk- x)2 + (yk- y)2
Total Cost/x = S ck (xk - x)/dk(x,y)
Total Cost/x = 0 when
x = [Sckxk/dk(x,y)]/[Sck/dk(x,y)]
y = [Sckyk/dk(x,y)]/[Sck/dk(x,y)]
But dk(x,y) changes with location... 18

19. Iterative Strategy Start somewhere, e.g.,
x = [Sckxk]/[Sck]
y = [Sckyk]/[Sck]
as though dk= 1.
Step 1: Calculate values of dk
Step 2: Refine values of x and y
x = [Sckxk/dk]/[Sck/dk]
y = [Sckyk/dk]/[Sck/dk]
Repeat Steps 1 and 19

20. Solver Model 20

21. Convex Minimization Call on Convex Minimization Tool
Minos, Interior Point Methods, …
Typically don’t support discrete variables too… 21

22. 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” 22

23. Iterative Approach 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 …. 23

24. Assign Customers to Facilities
Uncapacitated (facilities can be any size)
“Greedy”: Assign each customer to closest facility
Capacitated
Use Optimization 24

25. Dealers sourced by multiple ramps
Under the old system dealerships could be fed from multiple destination ramps. This proved to be costly and more difficult to manage. 25

26. An Allocation Model 26

27. Allocation Model Var x{Custs, Facs} binary; minimize AllocationCost:
sum{c in Custs, f in Facs} C[c,f]*x[c,f];
s.t. AssignEachCust{c in Custs}:
sum{f in Facs} x[c,f] = 1;
s.t. FacilityCapacity{f in Facs}: sum{c in Custs}D[c]*c[c,f] <= Cap[f]; 27

28. Set Covering Models 28

29. WesternAir 29

30. The Rest of the Story If there is If there is labor content…
Value Added: E.g., BMW Assembly Plant
High Value items: E.g., Intel EU distribution center
If there is labor content…
Competition…
Service vs Cost... 30