Branch & Bound Another technique for optimization, this time as an and-tree Guaranteed to provide the globally optimum solution to a problem, but may need.

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Presentation transcript:

Branch & Bound Another technique for optimization, this time as an and-tree Guaranteed to provide the globally optimum solution to a problem, but may need to go through many states taking up time and space Use a function fBound to determine which branches can be worked on and which do not need work Tyson Kendon © 2007

0-1 Knapsack Problem (as GA) pi/ci = (5,3,2,1) Tyson Kendon © 2007

Search Control 1 Change any nodes that can change to yes (if there is more than one chose left-most) Otherwise choose the node with the highest fBound Choose the xi to work on where i is smallest Tyson Kendon © 2007

Branch & Bound S0 1 x Item 1 10 profit, 2 cost Env: Maxsol = 0 (!, !, !, !) ? Bound = 16 1 x Item 1 10 profit, 2 cost .5 x Item 2  6 profit, 2 cost Tyson Kendon © 2007

Branch & Bound S0  Env: Maxsol = 0 (!, !, !, !) ? Bound = 16 (0, !, !, !) ? Bound = 12 (1, !, !, !) Bound = 16 Tyson Kendon © 2007

Branch & Bound S1   (!, !, !, !) ? Bound = 16 Env: Maxsol = 0 (0, !, !, !) ? Bound = 12 (1, !, !, !) ? Bound = 16  (1, 0, !, !) ? Bound = 13 (1, 1, !, !) Bound = 16 Tyson Kendon © 2007

Branch & Bound S2    (!, !, !, !) ? Bound = 16 Env: Maxsol = 0 (0, !, !, !) ? Bound = 12 (1, !, !, !) ? Bound = 16   (1, 0, !, !) ? Bound = 13 (1, 1, !, !) ? Bound = 16 (1, 1, !, !) yes Bound = 16 Tyson Kendon © 2007

Branch & Bound S3     Env: Maxsol = 0 (!, !, !, !) Bound = 16 ? Bound = 16  S3 Env: Maxsol = 0 (0, !, !, !) ? Bound = 12 (1, !, !, !) ? Bound = 16   (1, 0, !, !) ? Bound = 13  (1, 1, !, !) yes Bound = 16 (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13 Tyson Kendon © 2007

Branch & Bound S4      Env: Maxsol = 0 (!, !, !, !) Bound = 16 ? Bound = 16  S4 Env: Maxsol = 0 (0, !, !, !) ? Bound = 12 (1, !, !, !) ? Bound = 16   (1, 0, !, !) ? Bound = 13  (1, 1, !, !) yes Bound = 16 (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13  (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 1) Bound = 13 Tyson Kendon © 2007

Branch & Bound S5       Env: Env: Maxsol = 12 Maxsol = 0 (!, !, !, !) ? Bound = 16  S5 Env: Maxsol = 0 Env: Maxsol = 12 (0, !, !, !) ? Bound = 12 (1, !, !, !) ? Bound = 16   (1, 0, !, !) ? Bound = 13  (1, 1, !, !) yes Bound = 16 (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13   (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 1) ? Bound = 13 Tyson Kendon © 2007

Branch & Bound S6        Env: Maxsol = 12 (!, !, !, !) ? Bound = 16  S6 Env: Maxsol = 12  (0, !, !, !) ? Bound = 12 (0, !, !, !) yes Bound = 12 (1, !, !, !) ? Bound = 16   (1, 0, !, !) ? Bound = 13  (1, 1, !, !) yes Bound = 16 (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13   (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 1) ? Bound = 13 Tyson Kendon © 2007

Branch & Bound S7         Env: Maxsol = 12 (!, !, !, !) ? Bound = 16  S7 Env: Maxsol = 12  (0, !, !, !) ? Bound = 12 (0, !, !, !) yes Bound = 12 (1, !, !, !) ? Bound = 16   (1, 0, !, !) ? Bound = 13  (1, 1, !, !) yes Bound = 16  (1, 0, 0, !) yes Bound = 11 (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) ? Bound = 13   (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 1) ? Bound = 13 Tyson Kendon © 2007

Branch & Bound S8          Env: Maxsol = 12 Env: Maxsol = 13 (!, !, !, !) ? Bound = 16  S8 Env: Maxsol = 12 Env: Maxsol = 13  (0, !, !, !) yes Bound = 12 (0, !, !, !) ? Bound = 12 (1, !, !, !) ? Bound = 16   (1, 0, !, !) ? Bound = 13  (1, 1, !, !) yes Bound = 16  (1, 0, 0, !) ? Bound = 11 (1, 0, 0, !) yes Bound = 11 (1, 0, 1, !) ? Bound = 13    (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 1) ? Bound = 13 (1, 0, 1, 1) yes Bound = 13 Tyson Kendon © 2007

Search Control 2 Change any nodes that can change to yes (if there is more than one chose left-most) Otherwise choose the node with the highest fBound Choose the xi to work on where pi is highest Tyson Kendon © 2007

Branch & Bound S0  Env: Maxsol = 0 (!, !, !, !) ? Bound = 16 (!, 0, !, !) ? Bound = 13 (!, 1, !, !) Bound = 12 Tyson Kendon © 2007

Branch & Bound S1   (!, !, !, !) ? Bound = 16 Env: Maxsol = 0 (!, 0, !, !) ? Bound = 13 (!, 1, !, !) ? Bound = 12 (0, 0, !, !) ? Bound = 3 (1, 0, !, !) Bound = 13 Tyson Kendon © 2007

Branch & Bound S2    Env: Maxsol = 0 (!, !, !, !) Bound = 16 ? Bound = 16  S2 Env: Maxsol = 0  (!, 0, !, !) ? Bound = 13 (!, 1, !, !) ? Bound = 12 (0, 0, !, !) ? Bound = 3 (1, 0, !, !) ? Bound = 13  (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13 Tyson Kendon © 2007

Branch & Bound S3     Env: Maxsol = 0 (!, !, !, !) Bound = 16 ? Bound = 16  S3 Env: Maxsol = 0  (!, 0, !, !) ? Bound = 13 (!, 1, !, !) ? Bound = 12 (0, 0, !, !) ? Bound = 3 (1, 0, !, !) ? Bound = 13  (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13  (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 1) Bound = 13 Tyson Kendon © 2007

Branch & Bound S4      Env: Env: Maxsol = 12 Maxsol = 0 (!, !, !, !) ? Bound = 16  S4 Env: Maxsol = 0 Env: Maxsol = 12  (!, 0, !, !) ? Bound = 13 (!, 1, !, !) ? Bound = 12 (0, 0, !, !) ? Bound = 3 (1, 0, !, !) ? Bound = 13  (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13   (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 1) ? Bound = 13 Tyson Kendon © 2007

Branch & Bound S5       Env: Maxsol = 12 (!, !, !, !) Bound = 16 ? Bound = 16  S5 Env: Maxsol = 12  (!, 0, !, !) ? Bound = 13 (!, 1, !, !) ? Bound = 12  (0, 0, !, !) yes Bound = 3 (0, 0, !, !) ? Bound = 3 (1, 0, !, !) ? Bound = 13  (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13   (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 1) ? Bound = 13 Tyson Kendon © 2007

Branch & Bound S6        Env: Maxsol = 12 (!, !, !, !) ? Bound = 16  S6 Env: Maxsol = 12  (!, 0, !, !) ? Bound = 13 (!, 1, !, !) ? Bound = 12  (0, 0, !, !) yes Bound = 3 (0, 0, !, !) ? Bound = 3 (1, 0, !, !) ? Bound = 13  (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13  (1, 0, 0, !) yes Bound = 11   (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 1) ? Bound = 13 Tyson Kendon © 2007

Branch & Bound S7         Env: Env: Maxsol = 13 Maxsol = 12 (!, !, !, !) ? Bound = 16  S7 Env: Maxsol = 12 Env: Maxsol = 13  (!, 0, !, !) ? Bound = 13 (!, 1, !, !) ? Bound = 12  (0, 0, !, !) yes Bound = 3 (0, 0, !, !) ? Bound = 3 (1, 0, !, !) ? Bound = 13  (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13  (1, 0, 0, !) yes Bound = 11    (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 1) ? Bound = 13 (1, 0, 1, 1) yes Bound = 13 Tyson Kendon © 2007

Branch & Bound S8          Env: Maxsol = 13 (!, !, !, !) ? Bound = 16  S8 Env: Maxsol = 13   (!, 0, !, !) ? Bound = 13 (!, 1, !, !) yes Bound = 12 (!, 1, !, !) ? Bound = 12  (0, 0, !, !) ? Bound = 3 (0, 0, !, !) yes Bound = 3 (1, 0, !, !) ? Bound = 13  (1, 0, 0, !) ? Bound = 11 (1, 0, 1, !) Bound = 13  (1, 0, 0, !) yes Bound = 11    (1, 0, 1, 0) ? Bound = 12 (1, 0, 1, 0) yes Bound = 12 (1, 0, 1, 1) ? Bound = 13 (1, 0, 1, 1) yes Bound = 13 Tyson Kendon © 2007