Location-allocation models
• We have a number of existing facilities
• Each existing facility has a demand wj
• We have to place m new facilities
• And we have to decide how the existing
facilities are allocated to the new facilities
Location-allocation models
Existing facilities:
Location-allocation models
How how
And
do we
dolocate
we
allocate
the
new supply?
facilities?
One-dimensional locationallocation by dynamic programming
Example of heuristic procedure
2
2
2
1
2
3
2
4
5
6
1
7
8
9
10 11 12 13 14 15
One-dimensional locationallocation by dynamic programming
Example of heuristic procedure
Optimal value = 2 + 2 + 9 = 13
2
2
2
1
2
3
2
4
5
6
1
7
8
9
10 11 12 13 14 15
One-dimensional locationallocation by dynamic programming
Example of heuristic procedure
Optimal value = 3 + 1 + 1 + 7 = 12
2
2
2
1
2
3
2
4
5
6
1
7
8
9
10 11 12 13 14 15
One-dimensional locationallocation by dynamic programming
• Use Dynamic programming method
instead
• Assume that aj < aj+1
j = 1,…,n-1
One-dimensional locationallocation 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)
One-dimensional locationallocation by dynamic programming
• i: stages (# new facilities not located)
• s: states (first facility not allocated to new)
Stage i
m – i located
i - 1 not located
≥m-i
≥i-1
m–i+1≤s≤n–i+1
if i < m
s=1
if i = m
One-dimensional locationallocation by dynamic programming
Example 1 of dynamic programming
2
2
2
1
2
3
(1) (2) (3)
2
4
5
6
(4)
1
7
8
9
10 11 12 13 14 15
(5)
One-dimensional locationallocation by dynamic programming
Example 2 of dynamic programming
2
1
2½
1
2
3
(1) (2) (3)
1½
4
5
(4)
2½ 4
6
7
8
(5) (6)
3
9
10 11 12 13 14 15
(7)
Two-facility with euclidean distance
A
B
Two-facility with euclidean distance
A
A
B
B
Two-facility with euclidean distance
A
A
B
B
Two-facility with euclidean distance
Three collinear points
A
C
B
Two-facility with euclidean distance
A
A
C
B
A
C
C
B
C
B
B
A
A
C
C
B
A
B
Two-facility with euclidean distance
Three collinear points
A
C
B
Two-facility with euclidean distance