. , . . , . ³, , 2- - - , g() h(-). ,
W1(; ) = g(y) + h (x-y) (4.1.) .
, max W1 (x; y)
g(y) h(x-y) > 0, . .
. g(y) , - , g(y) -, , .
. . , , g(y), . , 0 < a < 1, 2- : - b(-), 0 < b<1. , + b(-). , , + b(-) = 1 = 1 + (1-1), 0 < y1 < x1. 2- g(y1) + h(x1-y1), W2(x, y, y1) = g(y) + h(x-y) + g(y1) + h(x1-y1), max W2 (x, y, y1) 0 < y < x, 0 < y1 < x1.
N- , N .
WN(x; y; y1;...; yN-1)=g(y)+h(x-y)+g(y1)+h(x1-y1)+... +g(yN-1)+h(xN-1-yN-1) (1.1)
x1 = ay + b(x-y); 0 < y < x
x2 = ay1 + b(x1 - y1) 0 < y1 < x1 (1.2)
xN-1 = ayN-2 + b(xN-2 - yN-2) 0 < yN-2 < xN-2 0 < yN-1 < xN-1
max WN (1.2.), N - . г . , N- . fN(x) ,
N- , ,
N=1, 2,..., x > 0, fN(x) = max WN(x; y; y1;...; yN-1) (1.3)
0 < y < x
f1(x) = max g(y) + h(x-y) (1.4)
0 < y < x
. 1 2- , 2- +b(x-y). , , , + b(-), ( ), f1[ay + b(x-y)]. max
f2(x) = max {g(y) + h(x-y) + f1[ay + b(x-y)]} (1.5)
0 < y < x
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̳ , N-
f(x) = max {g(y) + h(x-y) + fN-1[ay + b(x-y)]} (1.6)
0 < y < x
(1.6.) f1, f2 .., k- k().
.. N- N .
, .. , ,
fN(x) = max [gN(yN) + fN-1(x-yN)] (1.7)
0 < yN < x
f - , N - , - , N- , fN(x) - , N- , , x, yN - , fN; gN(xN) - , N- yN, 0 < yN <x; fN-1 (x-yN) - N-1 , , x - yN.
N- yN = yN*. N-1
fN-1(x-yN*) = max [gN-1(yN-1) + fN-2(x-y*-yN-1)] (1.8)
0 < yN-1 < x-yN*
, N . , y - k- gk(x; y) rk(x; y). , N- .
. gk(x; y) rk(x; y) x > 0
0 < y < x; rk(x; y) 0 < rk(x; y) < ax; < 1 k =1, 2.
fk N(x) N - , k- , . N=1
fk1(x) = max gk(x; y)
0 < y < x
N > 2, ,
fkN(x) = max {[gk(x;y) + fk+1, N-1 [rk(x;y)]}
0 < y < x
. , k = 1; 2;...; N fk(x) , k- N- , . :
k=N
fN(x) = max gN(; y) (1.9)
0 < y < x
k = N - 1,..., 2, 1
fk(x) = max {[gk(x;y) + fk+1 (rk+1(xk;y)] (1.10.)
0 < y < x
1.2. 1 II 5 . ʳ , I, () = 2 () =0,75. ʳ - , II, x(-) = 2(-)2 x(-) = 0,3 (-).
, .
г. ' ' 5 , N=5; k = 1; 2,..., 5, , .
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5- , . 5. (4.9.)
g5(x4; x5) = (5) + (4-5) = y52 + 2(x4-x5)2
f5(x4) = max [y52 + 2(x4 - y5)2].
0 < y < x
5- 4 , max [0; x4].
= 25 - 4(4 - 5) = 0; 5 = 4.
= q4 (4; 5) = 6 > 0; 5 = 4 min.
. : q5 (x4; 0) = 2x42;
q5 (x4; x4) = x42.
, . q5 (x4; 5) max 5 = 0, f5(x4) = 2x42.
, max 5- , , II .
4- , (4.10)
4 - , , 4- 4 I 3-4 II.
4 = 0,754 + 0,3 (3 - 4).
= 24 - 4(3 - 4) + 4(0,75-0,3) [(0,75y4 + 0,39x3 - y4)] = 0;
6,81 y4 - 3,46x3 = 0 y4 = 0,5x3
= 6,81 > 0, min f4(x3) = 1,3 x32.
z4 [0; x3]
z4(x3; 0) = 2x32 + 0,18x32 = 2,18 x32
z4(x3; x3) = x32 + 1,125x32 = 2,125 x32
max z4 4 = 0 f4(x3) = 2,18 x32
, 4 , , , II .
III-
3 - , 3 , 3- 3 I 2-3 II .
3 = 0,753 + 0,3 (2 - 3).
, : min z3=0,52;
z3(x3; 0,52)=1,35x22 3=0 z3(x2; 0)=2,2x22
3 = 2 z3(x2; x2) = 2,23x22
f3(x2) = 2,23 x22
max 3- 2, I- .
II-
2 = 0,752 + 0,3 (1 - 2),
, min z2 = 1,3612
2 = 0,51 z2(x1; 0) = 2,21 x12; z2(x1; x1) = 2,25x12. , f2(x1) = =2,25 x12
, max 2- , I- .
I-
, : z1(x;0)=2,2x2 z1(x; x) = 2,27x2
- I- .
. : I- , - II .
, , , x>0
I - I , 2- 0,75
II - 0,75 I- , 0,752 = 0,56
III - 0,56 I- , 0,75*0,56 = 0,42
IV - 0,42 2- , 0,42*0,3 = 0,126
V - 0,126 2- ,
0,3*0,126 = 0,038.
max 5 f(x) = 2,272.
.
1.3. ' , N . xi - , i- (i = 1; 2; N) qi(xi) , . , , .
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г.
N.
x > 0 k = 1;...; N
:
(1.11)
N -
N = 2; 3;... (1.12)
(1.12-1.13) .
1.4. W N . , , Pi - i- , xi - i- . Ci - i- i = 1;...; N.
г. .
max F (x1; x2;...; xn) = Cixi
ixi < W, xi = 0; 1; 2;...
xi ..
, W - .
1- , x1 = [W/i] - . f1(W) = [W/1]. C1
1- 2- .
max f2(W).
x2 - 2- , 1- W2 - P2x2, C2x2.
max - f1(W - P2x2)
, , N
fN-1 (W - PNxN) - , N-1 , < W - PNxN.
. W = 83 ., : 1 = 24; 2 = 22; 3=16; 4=10; 1=96; 2=85; 3=50; 4=20.
fN(W) fN-1 (W-PNxN), f(W) W 0 < W < 83.
1- . [83/24]=3, 1 = 0; 1; 2; 3.
1 f1(W) .
1 2
W | f1(W) | x1 | W | f2(W) | x2 | x1 | |
0-23 | 0-21 | ||||||
24-47 | 22-23 | ||||||
48-71 | 24-43 | ||||||
72-83 | 44-45 | ||||||
46-47 | |||||||
48-67 | |||||||
68-69 | |||||||
70-71 | |||||||
72-83 |
2- (2 ) 2 = 22 ., 2 = 0; 1; 2; 3 ([W/P2] = [83/22] = 3.
, 1 2. 1
3- 4- , 1-, 2-, 3- 1-, 2-, 3- 4- . 3- , 2, 3.
3 4
W | f3(W) | 3 | x2 | x1 | W | f4(W) | 4 | 3 | x2 | x1 | |
0-15 | 0-9 | ||||||||||
16-21 | 10-15 | ||||||||||
22-23 | 16-21 | ||||||||||
24-31 | 22-23 | ||||||||||
32-37 | 24-31 | ||||||||||
38-39 | 32-37 | ||||||||||
40-43 | 38-39 | ||||||||||
44-45 | 40-43 | ||||||||||
46-47 | 44-45 | ||||||||||
48-63 | 46-47 | ||||||||||
64-67 | 48-57 | ||||||||||
68-69 | 58-63 | ||||||||||
70-71 | 64-67 | ||||||||||
72-83 | 68-69 | ||||||||||
70-71 | |||||||||||
72-81 | |||||||||||
82-83 |
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4 , max 308 ., 3 . 4- 1 .
. W = 83 ., - 0 < W < 83 .. .
: W L1, L2, L3, L4 , C1, C2, C3, C4, max.
, .. .