cpsc 433 artificial intelligence set based search modeling examples ii andrew kuipers...

CPSC 433 Artificial Intelligence

Set Based Search Modeling Examples II

Andrew [email protected]

Please include [CPSC433] in the subject line of any emails regarding this course.

http://www.ucalgary.ca/


Example: 0-1 Knapsack

0-1 Knapsack Problem

• For a given list of n items:– with weights W = <w1, …, wn>

– and values V = <v1, …, vn>

• we want to maximize the value of a knapsack with capacity C by either placing, or not placing, items from I into C




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 | fF’ wf C }




Extension Rules

Ext = { A B | sS AS | ((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



Example: 0-1 KnapsackExample:

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 }




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



Example: 0-1 Knapsack GA

Another approach: Genetic Algorithm

Brief Definition: A genetic algorithm (GA) is a search model that mimics the process of natural evolution.

• A population of potential solutions (individuals)• Mutate and combine to create new individuals• Use a fitness function to evaluate individuals




GA Operators Brief Introduction

Mutation

Crossover




Genetic Algorithm Approach

I = (w1, v1), …, (wn, vn), C = capacity

where w((wi, vi)) = wi and v((wi, vi)) = vi

Facts

F = { { i1, …, im } | 1 j m ij I (j=1..m w(ij)) C }




StatesS 2F

* do we need to get any more specific?

Extension Rules

Ext = { A → B | sS AS | (s-A)BS (Mutation(A,B) Combination(A,B) }




Mutation(A,B) A = { P } B = { P, (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 don’t want to replace the fact, just create a new fact

that is a mutation of original fact.



Example: 0-1 Knapsack GAMutation Example:

I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) C = 25

A = { (3,4), (7,3), (6,9) } = P

K = { (3,4), (6,9) } P

J = { (9,7), (11,8) } I

(K J) =



Example: 0-1 Knapsack GAMutation Example:

I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) C = 25

A = { (3,4), (7,3), (6,9) } = P

K = { (3,4), (6,9) } P

J = { (9,7), (11,8) } I

(K J) = B = { P, (P – K) J }= { P, { (7,3), (9,7), (11,8) } }




Combination(A,B) A = { P, Q } B = { P, Q, 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



Example: 0-1 Knapsack GACombination Example:

I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) C = 25

P = { i1, i3 }

Q = { i2, i3, i4 }

P Q = {i1, i2, i3, i4 }

K = { i2, i3 } (P Q)



Example: 0-1 Knapsack GACombination Example:

I = (3, 4), (9, 7), (7, 3), (6, 9), (11,8) C = 25

P = { i1, i3 }

Q = { i2, i3, i4 }

P Q = {i1, i2, i3, i4 }

K = { i2, i3 } (P Q)B = {P, Q, { i2, i3 } }




How does search proceed in a GA?

• What does a search instance look like?

• Should a search control only select the best individuals in the state?

• Population control, genocide


cpsc 433 artificial intelligence set based search modeling examples ii andrew kuipers...

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