Presentation given at ICTAI 13 (http://cecs.wright.edu/atrc/ictai13) by Pierre Schaus:
Variable Objective Large Neighborhood Search: A practical approach to solve over-constrained problems (Pierre Schaus)
5. Minimize sum of violation variables
o1+o2+o3+...
Desired Properties :
A: o1+o2+o3+... should be small
B: o1+o2+o3+... should be balanced
not easy to reached with CP
19. Focus on worse objective at each LNS restart
!
o1
o2
o3
o4
20. Focus on worse objective at each LNS restart
!
o1
o2
o3
o4
21. Focus on worse objective at each LNS restart
!
o1
o2
o3
o4
22. Focus on worse objective at each LNS restart
!
o1
o2
o3
o4
23. In practice …
•
Objective configuration changed at each
LNS restarts.
•
Each objective can be configured
differently
Strong = must decrease
Weak = cannot increase
No = unrestricted
filtering
24. 3 Filtering modes
Weak, Strong, No - Filtering
!
!
o1
o2
o3
o4
Weak Strong No Strong
Strong: must decrease
Weak: cannot increase
No: unrestricted
25. Good idea: o1+o2+o3+o4 in Strong
Ensures at least
same filtering
as original one
o1
o2
o3
o4 o1+o2+o3+o4
Weak Strong No Strong Strong
26. objective value
New solution found during
BnB DFS
restart i
o1 Strong
o2 Weak
o3 No
restart i+1
o1 Weak
o2 No
o3 Strong
time
27. From a LNS-CP Model
to a Variable-Objective LNS Model
CP
Model
objective
Relaxation
configuration
Procedure
++
=
VO-LNS
LNS Model
Model
set objectives into
Strong/Weak/No filtering
mode
29. Artificial Over-Constrained Problem
Random Domain:
5 values on [1..15]
oj
every values 1..15
should appear exactly
once
on each row
oi
violation = sum
of excess &
shortage of each
value
every values 1..15 should
appear exactly once
on each column
30. LNS Relaxations
R1
R2
10% of variables (randomly)
= 10% of variables (randomly)
+ most violated line/column
oi
most violated line*
*tabu mechanism for diversification
31. Objective Configurations
o1 , o2 , ... , o15 , o16 , ... , o30 , otot
C1
C2
o1 , o2 , ... , o30 : No
otot : Strong
o1 , o2 , ... , o30 : Weak
Standard minimization
of sum, does not try
to balance violations
otot : Strong, oi: Strong
The most violated
row/column in current
solutions
oi
35. Real Life - Timetabling
Hospitality Management School
Groups
Weeks
36. Real Life - Timetabling
Hospitality Management School
Groups
Weeks
Groups of 20 students must be scheduled over semester
37. Real Life - Timetabling
Hospitality Management School
PM
Weeks
Groups
AM
Two activities/ Week to decide
Morning & Afternoon
38. Real Life - Timetabling
Hospitality Management School
Weeks
Groups
> 2x A1
= 1x A2
…
For each group: cardinality requirement on activity types
39. Real Life - Timetabling
Hospitality Management School
Weeks
Groups
A1 is scheduled only 4
consecutive times, …
Restriction on activity patterns
40. Real Life - Timetabling
Hospitality Management School
Weeks
Groups
A1 scheduled between 2 and 3 times
A2 scheduled between 1 and 2 times
…
soft-gcc
Demand over activities specific for each weak
41. Real Life - Timetabling
Hospitality Management School
Violations of cardinality
requirements
47. Take away messages
•
Minimizing sum objective (in over-constrained
problems) with CP is hard
•
•
LNS is powerful to diversify search space
Changing dynamically the objective can
•
•
Fasten and improve LNS
Better balance violations
variable objective
LNS