1. Advanced LP models
column generation

2. min
max

3. min
max OK OK ?

4. Bin packing example
bins of size K
items of different types
items of type i have size s(i) and there are n(i) of them
problem
put all items into the bins minimizing the number of bins

8. + = 3 1 2 2 4 4 2 + = 4
feasible patterns + = 8

9. x 2 1 3 4 5 6 7 + = 3 8 4 + 1 1 2 4 3 x 2 1 3 4 5 6 7

10. min S x i 1 2 1 3 x = 2 2 4 3 4 4 4 4 2 8 x
integer

11. min
max OK OK ?

12. 1 2 4 3
< 1 .. 1 2 1 y 2 3 1 2 2 4 3 4 4 4 2

13. y < + 1 2 2 y < + a y < + a s < K + max a y + < 13 1
< + 1 2 2 y 3 1
< + a 2 3 1 y
< +
for all feasible patterns a 2 3 1 s
< K +
max a 2 3 1 y +
< 1

14. otherwise add pattern maximizing knapsack2 3 1 y + a 2 3 1 s
< K + a 1 a 2 a 3
integer
knapsack !
if max optimality !
<
otherwise add pattern maximizing knapsack

15. example
bin capacity = 20
3 types:
type 1: size = 7; quantity = 50
type 2: size = 5; quantity =100
type 3: size = 3; quantity = 70 25 24 4
53 bins x
starting patterns
y= ( / /2 )
generated patterns
y= ( 1/3 1/ /6 )
y= ( )

16. however, change quantities to 52 97 71
solution is : x = 26 1
x =
x = 0
x = 2 3 4
53.1
how to get an integer solution?

17. V(j)=optimal value for a knapsack of capacity j
max a 2 3 1 y +
< 1 a 2 3 1 s
< K + a 1 a 2 a 3
integer
V(j)=optimal value for a knapsack of capacity j
V(j) = max { V(j - s(i)) + y(i) : i= 1, ... , n}
V(j)
>
V(j - s(i)) + y(i)
i= 1, ... , n; j=1,...,K
V(K)
< 1
V(j)
>

18. min
max

19. max S b(i) y(i)
V(j)
>
V(j - s(i)) + y(i)
i= 1, ... , n; j=1,...,K
V(K)
< 1
V(j)
>
however this is the dual problem we need the patterns which are in the primal
so let’s make the dual of the above problem

20. It turns out that the dual is a flow problem
nodes
capacity values (0,1,2,...,K)
arcs
each possible filling from
capacity j to capacity j+s(i)
dual variables
flows on arcs

21. 4 29 24 25 1 2 3 4 5 6 7 8 9 10 11 12
25+4 13 14 15 16 17 18 19 20

22. 4
25 times the pattern 25 1 2 3 4 24 4 5 4 6 25 7 8 24 9 10 11 12
25+4 13 29 4 14 15 24
25+4 16 17 18 24 19 20

23. 4
24 times the pattern 1 25 2 24 3 4 4 5 4 6 7 25 8 24 9 10 11 12
25+4 13 29 4 14 15 24 16
25+4 17 18 24 19 20

24. 4
4 times the pattern 1 25 2 4 3 24 4 5 4 6 7 25 8 24 9 10 11 12
25+4 4 13 29 14 15 24
25+4 16 17 18 24 19 20