G52AIP Artificial Intelligence Programming

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G52AIPArtificial Intelligence Programming
Dr Rong Qu

• Constraint Optimisation Problems

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Constraint Satisfaction Problems

• So far
• All solutions are equally good
• In some real world applications, we
• Not only want feasible solutions, but also good
  solutions
• We have different preferences on constraints
• Problems are too constrained that there is no
  solution satisfying all constraints

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Constraint Optimisation Problems

• Real world problems present to be messy
• In some cases a conflict-free solution is needed
• All constraints must be satisfied
• In some cases preferences are given, rather than
  constraints
• Hard and soft constraints
• In some cases, some constraints are more
  important than others
• Constraints with different weights

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Constraint Optimisation Problems

• What are good solutions?
• Objective function
• Problem specific function
• How much are constraints satisfied
• Relatively new research
• Scheduling
• Timetabling
• Resource allocations, etc

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Constraint Optimisation Problems

• Constraint optimisation problems
• Find feasible solutions with the best value of
  objective function
• Constraint optimisation problems (C,f)
• Constraint Satisfaction Problem (C)
• Objective function (f)(maps every solution to a
  numeric value)

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Constraint Optimisation Problems

• A solution ? is preferred to solution ?
• If the value of objective function f under ? is
  less (or larger in maximisation problem) than ?
• An optimal solution of COP (C,f)
• A solution ? of C, such that no other solution of
  C is preferred to ?.

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Constraint Optimisation Problems

• Examples
• Graph colouring
• Minimum number of colours used
• Linear expression
• With constraints for variables in the expression
• Minimise the result of linear expressions

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Branch and Bound (BB)

• General method
• In COP most widely used
• CPLEX in ILOG software
• Based on depth first search
• Branches pruned during the search by a bound

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Branch and Bound (BB)

• Keep the best solution so far
• During the search
• If partial solution (node) are proved cannot
  improve the result
• Prune the branch under the node
• All solutions in branches under the node are
  abandoned

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Branch and Bound (BB)


X
X
Best solution
Best solution
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B B

• Two important factors
• A heuristic function, h
• Estimated objective values for compound labels of
  partial solutions
• A bound, b
• Used to prune brunches with no optimal solutions
• Updated during the search

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B B

• During the search in the search tree
• Before labelling a variable, a value of the
  heuristic function is calculated
• If the heuristic value is greater than the bound
• The whole sub-tree under the node is pruned
• If the heuristic value of a solution is less than
  the existing bound
• Update the bound
• Store the newly found solution
• For a minimisation problem

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B B heuristic

• Heuristic h
• Function maps partial solution to an estimate of
  the objective function value
• Good estimate of the best values of all branches
  under the current node
• If the best value under the branch are worse than
  the current bound, then there is no need to
  explore these branches
• Problem specific

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B B heuristic

• Use heuristic to prune the search tree
• There is no solution in the sub-tree (under the
  node)
• All solution in the sub-tree are not optimal
• So that
• Solutions can be found earlier
• Reduce the search space
• Speed up the search

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B B heuristic

• A good heuristic h is the key to successful BB
• Must underestimate
• Admissible
• Return the lower bound of the heuristic value
• Otherwise the optimal solution may be pruned
• h is the actual cost, h lt h
• For a minimisation problem

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B B heuristic

• A good heuristic h is the key to successful BB
• Must underestimate
• The closer the estimation of heuristic, the
  larger the part of search tree pruned
• h is the actual cost, the closer h is to h the
  better

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B B bound

• Usually set as infinite value at the beginning
• Updated during the search by recording the best
  so far heuristic value
• For a minimisation problem

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B B bound

• Better bound, b
• Helps to find good solutions earlier
• In practice, user can provide a bound
• Satisfied even the solution is not optimal

v

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SEND MORE MONEY - Problem

• S E N D
• M O R E
• M O N E Y
• Cryptarithmetic problem mathematical puzzles
  where digits are replaced by symbols
• Find unique digits the letters represent
  satisfying the above constraints

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SEND MORE MONEY - Model

• Variables
• S, E, N, D, M, O, R, Y
• Domain
• 0, , 9

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SEND MORE MONEY - Model

• Constraints
• Distinct variables, S ? E, M ? S,
• S1000 E100 N10 D
• M1000 O100 R10 E
• M10000 O1000 N100 E10 Y

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SEND MORE MONEY How?

• How would you solve the problem using CP
  techniques?
• Search tree with backtracking
• Constraint propagation
• Forward backward checking
• Combination of above?
• Different problems may find different techniques
  more appropriate

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SEND MORE MONEY - Solution

• 9 5 6 7
• 1 0 8 5
• 1 0 6 5 2
• Is this the only solution?
• Sometimes we want to maximise an objective

S E N D M O R E M O N E Y
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SEND MOST MONEY - Problem

• S E N D
• M O S T
• M O N E Y
• Objective we now want to maximise MONEY

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SEND MOST MONEY - Problem

• Modelling
• What does best mean
• How to find best solution
• Search
• Assign scores for proposed solution, h
• Update the bound, b

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Constraint Optimisation Problems

• Real world problems
• Resource allocation
• Nurse rostering systems in hospitals
• BT services
• Airport scheduling at British Airways
• Flight scheduling, ECLiPSe
• Aircraft allocation, ILOG

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Constraint Optimisation Problems

• Real world problems
• Timetabling
• University course/exam scheduling systems
• Transportation
• ILOG vehicle routing
• Scheduling
• Job shop scheduling

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Summary

• Constraint optimisation problem
• Branch and bound (BB)
• Heuristic h
• Bound b
• Example
• SEND MORE MONEY (CSP)
• SEND MOST MONEY (COP)