1. Location Where to put facilities? Transportation costs
Rates and distances
Volumes to be moved
Other issues
Market presence (speed to market)
Fixed costs

2. 1-Dimensional Intuition
Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
Where to locate?

3. 1-Dimensional Intuition
Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
Where to locate?

4. 1-Dimensional Intuition
Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
Where to locate?

5. 1-Dimensional Intuition
Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
Where to locate

6. What about “Weight” Customers Weight 3 -6 -5 -4 -3 -2 -1 1 2 3 4 5 61 2 3 4 5 6
Where to locate?

7. What about “Weight” Customers Weight 2 -6 -5 -4 -3 -2 -1 1 2 3 4 5 61 2 3 4 5 6
Where to locate?

8. 2-Dimensional Location5 D 4
Manhattan Metric or L1 norm
Distance = 9

9. 2-Dimensions If all the points are the same “weight” D Y D D D1 2 3 4 5 6 7 8 9 10 Y D D D 1 2 3 4 5 6 7 8 9 10
Where to locate? X

10. 2-Dimensions If all the points are the same “weight” Y D1 2 3 4 5 6 7 8 9 10 Y 1 2 3 4 5 6 7 8 9 10
Where to locate? X

11. 2-Dimensions Euclidean Distance or L2 norm Successive Approximations:
X = Average of X’s
Y = Average of Y’s
Calculate distances d1, d2, ...
X = X1/d1+X2/d2+…X4/d4
1/d1 + 1/d2 + …1/d4
Y = Y1/d1+Y2/d2+…Y4/d4
Repeat until movement is small D 1 2 3 4 5 6 7 8 9 10 Y 1 2 3 4 5 6 7 8 9 10 X

12. D D D D

15. “Weights and Rates” If there are associated with each location
Different volumes V1, V2, …, V4
Different transportation rates R1, R2, …, R4
associated with each location
Replace Xi with ViRiXi
Replace Yi with ViRiYi
(Not when calculating distances)

16. Over Emphasis Useful for getting in the neighborhood
One or two iterations generally does this
Ignores lots of (important) details
Availability and cost of sites
Actual transportation network
Reality of freight rates
Non-linear
Often relatively insensitive to distance (LTL)
Dynamics of demand

17. Locating Many Facilities
Select a number of locations
Guess at initial positions
Assign Customers to those locations
Repeat:
Calculate best location to serve assigned customers
Calculate best customers to serve from those locations

18. Locate Distribution Centers
Based on Ford Auto Dealerships in Canada
Parts distribution
4 Distribution Centers
Consider only distance to dealerships
Ignore volume (to keep it simple)
Illustrate approach
Compare with “Actual”

19. Locate Distribution Centers

20. Mixed Integer Linear Programming
Does guarantee the quality of the solution
Computationally more demanding
More Flexible
Technically more demanding

21. Example 13.5 page 498 2 Products 3 Customers Single sourcing
2 warehouses

23. An MIP Model

24. Heuristics Speed the MIP solution Reduce computational demands
More interactive
No guarantee of optimality

25. Some Heuristics Multiple Center of Gravity method for each number
Evaluate other costs after the fact
Inventory
Fixed Costs
Etc.
Successive Elimination
Successive Approximation

26. Successive Elimination
Illustrate with our MIP example
Replace the computationally demanding MIP with sequence of LPs

27. An MIP Model

28. Successive Elimination
Both $ 3,150,000
Remove 1 $ 3,050,000
Remove 2 Not Feasible
With more choices, continue as long as costs reduce…
Does not always find optimum

29. Successive Approximation
Calculate an imputed cost per unit based on anticipated volume through each warehouse
Solve an LP to determine best volumes at these rates
Repeat
Calculate imputed costs per unit based on volumes
Calculate best volumes at imputed costs

30. Covering Models Each site “covers” some customers
Select a best set of sites that cover all customers

31. Western Airlines

32. Solver Model