Set%20Based%20Search%20Modeling%20Examples%20II

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Set Based Search Modeling Examples II

• M. Reza Zakerinasab
• mrzakeri_at_ucalgary.ca
• Please include CPSC433 in the subject line of
  any emails regarding this course.
• Slides originally created by Andrew M Kuipers.

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Example 0-1 Knapsack

• 0-1 Knapsack Problem
• For a given list of n items
• with weights W ltw1, , wngt
• and values V ltv1, , vngt
• we want to maximize the value of a knapsack with
  capacity C by either placing, or not placing,
  items from I into C

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Example 0-1 Knapsack

• Facts
• The items in our knapsack, which we can simply
  represent by their index
• F 1, , n
• States
• A state is a set of items, where the sum of the
  weight of the items is less than or equal to
  capacity C
• S F ? F ??f?F wf ? C

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Example 0-1 Knapsack

• Extension Rules
• Ext A ? B ?s?S ? A?S ((s-A) ? B) ? S
• This basic extension rule definition allows us to
  perform all sorts of set manipulation, so long as
  the result is a valid state
• ie adding and removing either single or multiple
  items from the knapsack, as long as the result
  weighs less than the maximum capacity C

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Example 0-1 Knapsack

• Example
• W 20, 7, 13, 5
• V 50, 30, 25, 15
• C 25
• s0
• s1 2, 3 by A ? B 2, 3
• s2 2, 3, 4 by A ? B 4
• s3 1, 4 by A 2, 3 ? B 1

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Example 0-1 Knapsack

• Problems with this Model
• The control must select between a large number of
  possible extension rules to apply
• Only a single possible solution being
  manipulated how can we compare solutions?
• How might we define the goal?
• While we have correctly modeled the problem, this
  model does not lend well to a search process
• Remember for a given problem, there are many
  ways to construct a model

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Example 0-1 Knapsack GA

• Another approach Genetic Algorithm
• Brief Definition A genetic algorithm (GA) is a
  search heuristic that mimics the process of
  natural evolution.
• Simplified Steps
• Encode the problem,
• Iteratively, apply operators, compare solutions

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Example 0-1 Knapsack GA

• GA Operators Brief Introduction
• Mutation
• Crossover

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Example 0-1 Knapsack GA

• Genetic Algorithm Approach
• I ?(w1, v1), , (wn, vn)?, C capacity
• where w((wi, vi)) wi and v((wi, vi)) vi
• Facts
• F i1, , im (?j1..m w(ij) ? C)

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Example 0-1 Knapsack GA

• States
• S ? 2F
• do we need to get any more specific?

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Example 0-1 Knapsack GA

• States
• S ? 2F
• do we need to get any more specific? YES!
• S F ? 2F ??f?F wf ? C
• Extension Rules
• Ext A ? B ?s?S ? A?S (s-A)?B?S ?
• (Mutation(A,B) ?
• Combination(A,B)

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Example 0-1 Knapsack GA

• Mutation(A,B) ?
• A P ? B (P K) ? J
• where K ? P , J ? I and (K ? J) ?
• Remove some subset K from the fact P randomly
• Replace it with some set of items J that does not
  contain any items in K
• We dont want to replace the fact, just create a
  new fact that is a mutation of original fact.

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Example 0-1 Knapsack GA

• Mutation Example
• W 3, 9, 7, 6, 11
• V 4, 7, 3, 9, 8
• C 25
• A 1, 3, 4 P
• K 1, 4 ? P
• J 2, 5 ? I
• (K ? J) ?

B (P K) ? J B 2, 3, 5
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Example 0-1 Knapsack GA

• Combination(A,B) ?
• A P, Q ? B K
• where
• K ? (P ? Q) ?
• (K ? P) ? (K ? Q) ?
• min(P,Q) ? K ? max(P, Q)
• Use existing facts P and Q to generate a new fact
  K, which is a combination of P Q, yet not equal
  to P or Q, and of size between that of P and Q

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Example 0-1 Knapsack GA

• Combination Example
• W 3, 9, 7, 6, 11
• V 4, 7, 3, 9, 8
• C 25
• P 1, 3
• Q 2, 3, 4
• P ? Q 1, 2, 3, 4
• K 2, 3 ? (P ? Q)

B 2, 3
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Example 0-1 Knapsack GA

• Actually, whats happening in this mutation /
  combination scenario???

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Example 0-1 Knapsack GA

• Using this mutation / combination scenario, we
  act as follows
• We select some random solutions which are valid
  states.
• In a loop
• We compare the solutions to find our best
  solution.
• We apply mutation on some random solutions from
  our solution set or combination on random pairs.
• This loop stops when some certain condition
  satisfies.
• We find an enough good solution.
• Our best solution does not improves after some
  specific time.