Modeling with finite domain constraints II

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Modeling with finite domain constraints II

• Kris Kuchcinski
• kku_at_ida.liu.se

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Outline

• Advanced modeling example for scheduling
• elliptic wave filter example
• use of cumulative constraints for task scheduling
  and bin packing problems
• The role of forward checking and look-ahead
  methods during search
• Meta-search heuristics
• credit search
• limited discrepancy search

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Elliptic wave filter example
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Precedence constraints
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Modeling with Sicstus I
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Modeling with Sicstus II
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Experimental results
Results for 2 adders and 2 multipliers
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Modeling with CHIP
CPU time 0.23 s
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Use of cumulative constraint for task scheduling
Resource
Resource limit
Time
cumulative(T1, T2, T3, D1, D2, D3, 1, 1, 1,
ResourceLimit)
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Use of cumulative constraint for binpacking
problems
Time
Time limit
Resource
cumulative(R1, R2, R3, 1, 1, 1, D1, D2, D3,
TimeLimit)
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Branch and Bound algorithm

• Branch and Bound consists of the following steps
• finds a bound for the cost function after each
  decision,
• cut out all the branches of the decision tree
  which give results worse than the bound (branch),
• update the bound if a better value for the
  performance index is found.

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Typical branch and bound search(TSP problem)
1 L ³ 0
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Search with restart(CHIPs min_max)
1 L ³ 0
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When backtracking is initiated during search?

• by constraint violation Prolog standard,
• unavoidability of constraint violation in the
  next step- Forward Checking (FC),
• unavoidability of constraint violation in future
  steps- (Full) Look Ahead (LA) or Maintaining Arc
  Consistency (MAC).

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Forward Checking and Look Ahead
Not yet instantiated variables
Currently instantiated variable
Instantiated variables
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Forward Checking algorithm
procedure AC-FC(CurrentVar) Q (Vi,
VCurrentVar) in arcs(G), i gt CurrentVar consiste
nt true while not Q empty consistent select
and delete any arc (Vk, Vm) from Q if
arc_consistent(Vk, Vm) then consistent not
Dk empty endif endwhile return consistent end
AC-FC
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Look Ahead algorithm
procedure AC-LA(CurrentVar) Q (Vi,
VCurrentVar) in arcs(G), i gt CurrentVar
consistent true while not Q empty
consistent select and delete any arc (Vk,
Vm) from Q if arc_consistent(Vk, Vm)
then Q Q È (Vi,Vk) such that (Vi,Vk) in
arcs(G), i ¹ k, i ¹ m, i gt
CurrentVar consistent not Dk empty
endif endwhile return consistent end
AC-LA
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Meta-heuristics - motivation

• Large optimization problems can run for a long
  time without providing any solution,
• The search using branch and bound can be trapped
  in the beginning of the search tree and can not
  explore the rest of the search space,
• There exist known heuristics and meta-heuristics
  for selected problems.

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Credit search
credit(T, 8, credits 10, my_delete, my_in
domain, 3, backtracks part(1,2)),