1. John H. Vande Vate Spring, 2001
Location Problems
John H. Vande Vate
Spring, 2001 1

2. Outline Review the Terminal Problem discussed in Daganzo (page 68-71)
An MIP approach
Rectilinear Location Problems
Euclidean Location Problems
Location - Allocation Problems 2

3. Locating Freight Terminals
Examples:
LTL freight terminals
Public transportation
Local transport to consolidation points
Local collection to freight terminals
Walk to bus stop
Trade-off cost of local transport vs impact on “line haul”
More stops = less local transport
More stops = slower line haul 3

4. Aside
Don’t forget service segmentation!
Express Bus vs Local 4

5. Two-Tier System LTL depot Freight terminals Daily delivery
shipments consolidated for cross-country transport
Freight terminals
Local shipments are consolidated for delivery to depot
Daily delivery
Single line-haul vehicle 5

6. On a Line
Along a major highway
Along a bus route
... 6

7. Question: Freight Terminals?
How many freight terminals should we have and where should we put them?
Costs?
Capital and operating costs of terminals
Local transport cost
Stop Cost
... 7

8. The System
Cumulative
demand
Terminals
The depot
location 8

9. Local Travel
Cumulative
demand
The depot
location 9

10. Local Transport Item-miles of local transport10

11. When is this appropriate?
Walking to the bus stop
Local delivery to freight terminal?
Other suggestions?
Weighted distance
Total distance
Maximum Distance …. 11

12. Another Paradigm
local
demand
d(i)
The depot
location 12

13. Optimization Models ={ ={
1 if we collect demand at i at a freight terminal at t
0 otherwise
1 if we open a freight terminal at t
Basic constraints:
ServeEachArea{i in Areas}:
Sum{t in Areas}Send[i,t] = 1;
IsTerminalOpen{i in Areas, t in Areas}:
Send[i,t] <= Open[t];
Send[i,t] ={
Open[t] 13

14. Objectives Item-miles + Stops Total Distance + Stops
sum{a in Areas, t in Areas} d[a]*abs(t-a)*Send[a,t] +
sum{t in Areas} stopcost*Open[t];
Total Distance + Stops
sum{a in Areas, t in Areas} abs(t-a)*Send[a,t] + 14

15. Maximum Distance New Variable: MaxTravel{t in Areas} >= 0
New Constraints:
SetMaxTravel{a in Areas, t in Areas}:
abs(t-a)*Send[a,t] <= MaxTravel[t];
Objective:
sum{t in Areas}(MaxTravel[t]+stopcost*Open[t]) 15

16. Really Travel Left and Right
var MaxTravelL{Areas} >= 0;
var MaxTravelR{Areas} >= 0;
s.t. SetMaxTravelL{a in Areas, t in Areas: a < t}:
(t-a)*Send[a,t] <= MaxTravelL[t];
s.t. SetMaxTravelR{a in Areas, t in Areas: a > t}:
(a-t)*Send[a,t] <= MaxTravelR[t]; 16

17. Role of the Line?
What role does considering the problem on the line play? E.g.,
Along a major highway
Along a bus route
Can we extend model to 2-dimensions? 17

18. Outline Review the Terminal Problem discussed in Daganzo (page 68-71)
An MIP approach
Rectilinear Location Problems
Euclidean Location Problems
Location - Allocation Problems 18

19. Before Moving On... 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 19

20. Rectilinear Distance Travel on the streets and avenues Distance =
number of blocks East-West +
number of blocks North-South
Manhattan Metric 20

21. Rectilinear Distance 9 5 4 21

22. Locate a facility... To minimize the sum of rectilinear distances
Intuition
Where?
Why? 22

23. Locate a facility... To minimize the max of rectilinear distances
Intuition
Where?
Why? 23

24. Models Set Customers; Param X{Customer}; Param Y{Customer};
var Xloc >= 0;
var Yloc >= 0;
var Xdist{Customer};
var Ydist{Customer}; 24

25. 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]; 25

26. Objective Total Distance: Maximum Distance?
sum{c in Customer}(Xdist[c]+Ydist[c]);
Maximum Distance? 26

27. 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]; 27

28. 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]; 28

29. 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]; 29

30. Locate a facility... To minimize the max of rectilinear distances
Intuition
Where?
Why? 30

31. Finding the Center(s) 31

32. Assignment #1 NIMBY…. Maximize the Minimum Distance
Can’t say Xdist[c] <= X[c] - Xloc;
Come up with a good formulation
FOR DISCUSSION IN CLASS NEXT TIME 32

33. Outline Review the Terminal Problem discussed in Daganzo (page 68-71)
An MIP approach
Rectilinear Location Problems
Euclidean Location Problems
Location - Allocation Problems 33

34. Locating a single facility
Distance is not linear
Distance is a convex function
Local Minimum is a global Minimum 34

35. 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... 35

36. 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 36

37. Convex Minimization Call on Convex Minimization Tool
Minos, Interior Point Methods, …
Typically don’t support discrete variables too… 37

38. 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” 38

39. 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 …. 39

40. Assign Customers to Facilities
Uncapacitated (facilities can be any size)
“Greedy”: Assign each customer to closest facility
Capacitated
Use Optimization 40

41. 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]; 41

42. 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... 42