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.
CPSC 433 Artificial Intelligence
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
CPSC 433 Artificial Intelligence
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 | fF’ wf C }
CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack
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
CPSC 433 Artificial Intelligence
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 }
CPSC 433 Artificial Intelligence
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
CPSC 433 Artificial Intelligence
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
CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
GA Operators Brief Introduction
Mutation
Crossover
CPSC 433 Artificial Intelligence
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 } | 1 j m ij I (j=1..m w(ij)) C }
CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
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) }
CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
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.
CPSC 433 Artificial Intelligence
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) =
CPSC 433 Artificial Intelligence
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) } }
CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
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
CPSC 433 Artificial Intelligence
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)
CPSC 433 Artificial Intelligence
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 } }
CPSC 433 Artificial Intelligence
Example: 0-1 Knapsack GA
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