Knapsack Problem (????)

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Knapsack Problem (????)

• ???
• 20090427

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The Knapsack Problem

• Def ?N????????,??
• ?????? (w1, w2, , wn) ??? (p1, p2, , pn)
• ????????????W
• ?????????????
• ??????????

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Knapsack Problem????

• Fractional Knapsack Problem
• ?????????
• ????????(Greedy Approach)
• 0/1 Knapsack Problem
• ??????
• ?????????(Dynamic Programming)
• ???????????? P150

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????????

• ??????(0/1)
• ?N?????????V???,?????????????i???????ci,???wi?
  ?????????????????????????????,????????
• ??????
• ?N?????????V?????i??????ni???,?????ci,???wi?
  ?????????????????????????????,????????

5

• ??????
• ?N?????????V?????i???????ci,???wi????????????,
  ??????????,???????????????????????????????????,???
  ?????
• ????????
• ????????????????????????,i???j,??????i,??????j??
  ?????,??????????????????,???????????,????????????
  ???

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• ??????
• ????S???,N???,?????w1, w2, , wn,?N???????????,???
  ??????????,?????????S?

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????

• ????? ???? ????
• ??????? 0/1 ????
• ?????? 0/1 ????

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????? ???? ????

• ???????c???????,?n?????????????,????i?????????wi?
  pi??????????????????????,??????????????
• ??
• 5 20 //???????????
• 6 3 //???????????
• 2 5 //
• 3 8
• 10 6
• 7 4

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• ??
• 0
• 1
• 1
• 1
• 0.714286

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????

• ?????????????,????????????????????????????????????
  ????,?????????

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????

• //?????????

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(No Transcript)
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(No Transcript)
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????????
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2-???????0/1????

• ?????????????????
• (1)????????????????????
• (2)???????????????,????????????
• (3)?????????????,?????????
• ???????????????,??????????????????????????????

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?0/1?????,??i???????,????????,?xi????i?????
??,??xi0?,????i???????,xi1?,????i?????????????,?
???????????
??,????????????????2.1,???????2.2????????X(x1,
x2, , xn)?
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??0/1???????????? ?(x1, x2, ,
xn)???0/1??????????,?( x2, , xn)????????????
????,?(y2, , yn)????????????,? ??,
???(x1, y2, , yn)???0/1?????(x1, x2, ,
xn)????,???????
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0/1???????????????(x1, x2, ,
xn),?????xi??????xi1??xi0???xi-1???,????(x1, ,
xi-1),???xi?,???????????? (1)???????????i,?xi0,?
?????? (2)??????????i,?xi1,????????vi?
??????????????????xi??????????V(i,
j)????i(1in)???????????j(1jC)???????????,?????
????????
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• ??????

1??0???
2 ?????0
3 ? i ???????? ?????????
(?2.3)
(?2.4.1)
(?2.4.2)
5 ??? i ???????
4 ???? i ??????
20
?2.3?????i????????0?????0????????j???,???????0?
?2.4???????????i?????????????,????i?????????????
?i-1??????????????,???i????????????????i????????
?????,????????? (1)????i???????,?????????????i-1?
???????j-wi??????????i??????vi (2)???i?????????,?
?????????????i-1????????j?????????????,???????????
??i????????j?????????
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??,?5???,??????2, 2, 6, 5, 4,?????6, 3, 5,
4, 6,??????10?
????????,???(n1)(C1)????V,Vij????i??????
??j????????????
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????,????1???,???????????????????? ????,????2???,
???????????????????? ????,???n??????,V(n,C)??????
C??????n??? ?????????
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?????????????? ?V(n,C)?????,??V(n,C)gtV(n-1,C),???
n????????,?n-1?????????C-wn??????,?n??????????,?n
-1?????????C?????????,?????1????????????????,?????
?
(?6.13)
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x11
x21
x30
x40
x51
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?????? ?n???????????wn?,???????vn?,?
????C,??Vn1C1??????,??Vij???i????????j??
?????????,??xn?????????,???????0/1?????????
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?????? 0/1 ????(????)
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????????
??????????????,?????????? ??,???????????????
,????? ??????? X11010111
??????????????,????????,? ?????
?????????
? ???? ? ???? ? ????
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????? ????????????????,?????(?? ?),???????????,
???????????
??????????(???) ????????????????????????? ??????
???Pc
xi ?????i????,
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????
1)?????????? 2)???????????,???????S - mid ??
???? sum ?f(xi) 3) ??????? N,0lt N lt sum 4)
?????????S - mid ???????????? 5)
??(3)?(4),???????????
?? ???6????????????????Pc?? ??
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???? ?????????
?????????
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10??????????????
????????
????? 1 2 3 4 5 6 7 8 9 10
??? 8 2 17 7 2 12 11 7 3 7
???? 0.1 0.02 0.22 0.09 0.02 0.16 0.14 0.09 0.03 0.09
????? 8 10 27 34 36 48 59 66 69 76
????????
??? 23 49 76 13 1 27 57
??????? 3 7 10 3 1 3 7
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?????
??????,??????????
???????????(?????),?????????,
????,?????????(?????).
??????? A 11010001
B 01011110
??(two-point crossover)??,??(multi-point
crossover)??
33
?????
????????????,???????.???????,???????? ??????????,
1??0?0??1.???????????,?????????,???????,???????.
??????????Pm???????????????????????? ????,?????

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????????? ??(??) ??????????,???
???????,??t???P(t) ???????????????
????P(t1)? ?? ???P(t)??????????
??,??????,????? (??????)???????
????? ?? ???P(t)???????,?????(??????)???
?????????????????????
???????
??
??????t
????
???? ?? ??
??t1???t
??t1
N
Y
??
?????????
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??????
36

• ????
• ???????

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?????????

• ????????
• ???AB???????N????,????????????????,???????????,?
  ???????????????????????1??????
• ????????????,????????????????????,???????

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??

• ???K,???????
• ???????????????A B,1ltA,Blt100,???????A?????,B????
  ??????????M N,1ltN, Mlt100,???????????????????????
  2???1ltXiltA,1ltYiltB?????(N S E
  W)???????,?????????????????

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• ??M??????????
• ??????
• lt?????gt lt??gt lt??gt
• ???3???
• L??90?
• R??90?
• F??????
• 1lt ?? lt 100

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??

• ????????,????
• ?????(Robot i crashes into the wall)
• Robot i crashes into robot j. (i??????)
• OK(??????)

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????

• ??????????????,????????????????????,?????????,??0
  
• ?????????????,??03,??(????)4?????????,?????,?????
  4???????????????
• ???????????,??????,????????????????,??????

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????

• 1
• 5 4
• 2 2
• 1 1 E
• 5 4 W
• 1 F 7
• 2 F 7

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????

• Robot 1 crashes into the wall

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(No Transcript)
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????

• ??20090429(??) 16401815
• ??????????
• ??
• ?????????0/1????5???????????3,2,1,4,525,20,15,
  40,50,?????6?
• ??????????
• ????????????????(??)???????,??????????????????????
  ????????????,???????(2?),??????????,??????????????
  ??,?????????(1?),?????????????

48

• service.in
• 20 3 //????,???????
• 7
• 4 2 10 9 16 15 17
• Service.out
• 9 //????
• 6 //???
• 4 1 //?4??????????1??
• 1 4
• 2 7
• 3 10
• 6 13
• 5 16
• ??????????