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1 15 Comparison and Evaluation of Dynamic Programming and Genetic Algorithm for a Knapsack Problem

2 n 2 n (Dynamic Programming : DP) (Genetic Algorithm : GA) 2 i

3 Abstract Comparison and Evaluation of Dynamic Programming and Genetic Algorithm for a Knapsack Problem KAKETANI Kou Knapsack problem is described as follows: there are some items with given value and volum. A knapsack has given capacity. Selected a proper subset of items to maximize the total value of selected items such that the total volume of the selected items is less than or equal to hte capacity of the knapsack. One of the easy methods for solving the problem is to investigate whole subsets of items, however, there are 2 n subsets where n is the number of items. Therefore, this brute force method is an exponentially time complexity. By this research, in order to solve a problem is two solution methods, Dynamic Programming(DP) and Genetic Algorithm(GA), are taken up, those solution methods are compared and evaluated. key words Knapsack Problem, Dynamic Programming Genetic Algorithm ii

4 iii

5 iv

6 STEP STEP STEP STEP STEP v

7 DP GA vi

8 1 ( ) NP- [1] NP- [2]

9 2

10 2 2.1 W p i w i n i(1 i n) J {1,2,, n} w j W p j [1] j J p j J w j j J W p j j J j J j J 2.2 3

11 ,

12 3 3.1 (Dynamic Programming) [1] [2] W w i 5

13 v[k, c](k = 1, 2, 3,, n; c = 0, 1, 2,, W ) v[k, 0] = 0 (1 k n) (3.1) p 1 v[1, c] = { 0 (0 c < w1 ) p 1 (w 1 c W ) (3.2) v[k, c] v[k 1, c] w k v[k-1,c] w k v[k 1, c w k ] + p k v[k, c] v[v[k 1, c], v[k 1, c w k ] + p k ] 6

14 3.2 v[k, c] = { v[k 1, c] (c < wk ) v[v[k 1, c], v[k 1, c w k ] + p k ] (w k c) (3.3) 3.4 k 3.4 k 3.4 c w k v[k 1, c w k ] + p k = v[k, c] (3.4) w j W p j j J j J p j j J 7

15 4 4.1 [2] 4.2 8

16 4.2 No Yes 4.1 9

17 GA [3] n l n = 4 l = W p i w i 10

18 [2310] W 8 p 0 1 p 1 5 p 2 4 p 3 2 w 0 1 w 1 3 w 2 2 w 3 5 STEP p [p2, w2] 4.3 STEP1-1 STEP w 2 w p 2 p [p3, w3] 4.4 STEP1-2 STEP STEP1-2 5 w [2310]

19 [p1, w1] 4.5 STEP [2310] 1 2 STEP2-1,2-2 1 STEP1-1,1-2 STEP STEP2-2 5 w w [p1, w1] 4.6 STEP2-3 STEP STEP2-3 5 w ST EP 2 3 p 0 2 [2310] 7 12

20 [p0, w0] 4.7 STEP A 111 B 75 C 60 D 39 E 15 E 5% D 13% C 20% A 37% B 25%

21 4.3 M M/2 M ( ) ( ) 1 (Order Crossover : OX) (Partially Mapped Crossover : PMX) (Cycle Crossover : CX) (Uniformed Order Crossover : UOX) 4 2 P 1, P 2 P 1 O 1 P 1 P 2 O 1 P 2 O

22 4.3 P1 : P2 : O1 : * * * P2 O1 : P 1, P 2 P 2 O 1 P 1 P 2 P 2 O 1 O P1 : P2 : O2 : * * * P1 : P2 : O2 : * P1 : P2 : O2 : P 1 1 O 1 O 1 1 P 2 1 P 1 O 1 P 2 15

23 P1 : P2 : O1 : 4 * 1 * 3 * P1 : P2 : O1 : P1 : P2 : O1 : * P 1 O P 2 O 2 O 1 O P1 : P2 : T : P1,P2 P1 : P2 : O1 : * * 1 * 3 2 O2 : 1 2 * 5 * * O1 : O2 :

24 [4] P1 : P1 : P1 : P1 :

25 4.3 P1 : P1 : Fittness( ) Generation( ) T Fittness T Generation

26 ( ) = ) ( ) (5.1)

27 DP (sec) , , , , , , , , , ,10, ,100, ,100,

28 GA (sec) , , , , , ,

29 ,000 1,000,000 10,000, ,000, , 10 6, 10 7, = 2, = 12, = 100, = 625, (DP) ,201 (GA) ,081 DP (sec) GA (sec)

30

31 24

32

33 [1] [2] : [3] GA [4] [5] T. C. R. [ 2 ]

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