1. Location-allocation models We have a number of existing facilities Each existing facility has a demand w j We have to place m new facilities And we have to decide how the existing facilities are allocated to the new facilities

2. Location-allocation models Existing facilities:

3. And how do we allocate supply? How do we locate the new facilities? Location-allocation models

4. One-dimensional location- allocation by dynamic programming 123456789101112131415 22221 Example of heuristic procedure

5. One-dimensional location- allocation by dynamic programming 123456789101112131415 22221 Example of heuristic procedure Optimal value = 2 + 2 + 9 = 13

6. One-dimensional location- allocation by dynamic programming 123456789101112131415 22221 Example of heuristic procedure Optimal value = 3 + 1 + 1 + 7 = 12

7. One-dimensional location- allocation by dynamic programming Use Dynamic programming method instead Assume that a j < a j+1 j = 1,…,n-1

8. One-dimensional location- allocation by dynamic programming i: stages (number of new facilities which have not been located) s: states (index of first facility which have not been allocated to a new facility)

9. One-dimensional location- allocation by dynamic programming i: stages (# new facilities not located) s: states (first facility not allocated to new) m – i located Stage i i - 1 not located ≥ m - i≥ i - 1 m – i + 1 ≤ s ≤ n – i + 1if i < m s = 1if i = m

10. One-dimensional location- allocation by dynamic programming 123456789101112131415 22221 Example 1 of dynamic programming (1)(2)(3)(4)(5)

11. One-dimensional location- allocation by dynamic programming 123456789101112131415 212½1½3 Example 2 of dynamic programming 2½4 (1)(2)(3)(4)(5)(6)(7)

12. Two-facility with euclidean distance A B

13. A B A B

14. A B A B

15. A B C Three collinear points

16. Two-facility with euclidean distance A B C A B C A B C A B C A B C A B C

17. A B C Three collinear points

18. Two-facility with euclidean distance