1. Location Problems
John H. Vande Vate
Fall, 2002 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. Models Set Customers; Param X{Customer}; Param Y{Customer};
var Xloc >= 0;
var Yloc >= 0;
var Xdist{Customer};
var Ydist{Customer}; 9

10. Constraints DefineXdist1{c in Customer}: Xdist[c] >= X[c]-Xloc;
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. Objective Total Distance: Maximum Distance?
sum{c in Customer}(Xdist[c]+Ydist[c]);
Maximum Distance? 11

12. Minimize The Max? Var Xloc; Var Yloc; Var Xmax >= 0;
var Ymax >= 0;
min objective: Xmax + Ymax;
s.t. DefineXdist1{c in Customer}:
Xmax >= X[c]-Xloc;
DefineXdist2{c in Customer}:
Xmax >= Xloc-X[c];
DefineYdist1{c in Customer}:
Ymax >= Y[c]-Yloc;
DefineYdist2{c in Customer}:
Ymax >= Yloc-Y[c]; 12

13. Min the Max! 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]; 13

14. 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]; 14

15. Solver Model 15

16. Locate a facility... To minimize the max of rectilinear distances
Intuition
Where?
Why? 16

17. Finding the Center(s) 17

18. Assignment #1 NIMBY…. Maximize the Minimum Distance
Can’t say Xdist[c] <= X[c] - Xloc;
Come up with a good formulation 18

19. Outline Rectilinear Location Problems Euclidean Location Problems
Location - Allocation Problems 19

20. Locating a single facility
Distance is not linear
Distance is a convex function
Local Minimum is a global Minimum 20

21. 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... 21

22. 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 22

23. Solver Model 23

24. Convex Minimization Call on Convex Minimization Tool
Minos, Interior Point Methods, …
Typically don’t support discrete variables too… 24

25. 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” 25

26. 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 …. 26

27. Assign Customers to Facilities
Uncapacitated (facilities can be any size)
“Greedy”: Assign each customer to closest facility
Capacitated
Use Optimization 27

28. 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]; 28

29. Set Covering Models 29

30. WesternAir 30

31. 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... 31