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Московский государственный индустриальный университет

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Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

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				A B
			A	B
		(x, y)		Ox
	Oy
	Oy
A	(0, 0)						B
(x1, y1)	x1 > 0, y1 > 0 y = y(t)
(x1, y1)	x1 > 0, y1 > 0 y = y(t)
		A B				= 2gy
	y = ty(x)				dt	= 2gy
					ds

A(0, 0) B(x1, y1) :

		x1
1		x1	1 + y
1			1 + y		2
t[y(x)] = √			√			dx −→ inf; y(0) = 0, y(x1) = y1.
	2g
	2g			y
		0

m x0

x(t) mx¨ = f f [f1, f2]

T −→ inf,	mx¨ = f,	f [f1, f2],
x(0) = x0,	x˙ (0) = v0,	x(T ) = x˙ (T ) = 0.

t[y(x)]

ai			i

1 i m

bj	j	1 j n

cij	i	j

xij i j

			i
			m	n
				cij xij
			=1 j=1
n		R+
xij		R+
	xij ai
j=1
i
m	xij = bj
=1
m		n	n		m
			j	ai,
		cij xij −→ inf,	xij	ai,	xij = bj , xij R+.
i=1 j=1			=1		i=1

bj j aij j

i	ci	i

	xi	i
	n	n
	i		bj , xi 0.
	cixi −→ inf,	aij xi
	=1	i=1

z = c0 + c1x1 + c2x2 + . . . + cnxn.

xi (xi ≥ 0)

	a1,1x1 + . . . + a1,nxn	= b1
	a2,1x1 + . . . + a2,nxn	= b2
	a2,1x1 + . . . + a2,nxn	= b2
	. . .

		= bm.
am,1x1 + . . . + am,nxn		= bm.

zmax

z → max x1 . . . xn

≤≥

n Rn

x1, x2, . . . , xk k

	x1	= a1,k+1xk+1 + . . . + a1,nxn + b1
		= a2,k+1xk+1 + . . . + a2,nxn + b2
x2
. . .

		= ak,k	+1xk+1 + . . . + ak,nxn + bk
xk

z = c0 + ck+1xk+1 + . . . + cnxn

	x1	=	b1
		=	b2
x2
. . .

		=	bk
xk

(b1, b2, . . . , bk, 0 . . . 0)

z = c0

		x1 + a1,k+1xk+1 + . . . + a1,nxn							= b1
			+ a2,k+1xk+1 + . . . + a2,nxn						= b2
	x2		+ a2,k+1xk+1 + . . . + a2,nxn						= b2
			. . .
				z
				z					= bk
	xk		+ ak,k+1xk+1 + . . . + ak,nxn						= bk
z − (ck+1xk+1 + . . . + cnxn) = c0


		x1		x2	. . .	xk	xk+1	. . .		xn
x1							a1,k+1			a1,n	b1
x2							a2,k+1			a2,n	b2

xk							ak,k+1			ak,n	bk
−	z						c			c	c
−							k+1			n	0

−z

b1, b2 . . . bk

z = c0

−z

−z cj

cj xj xi

alj > 0	j	cj < 0
alj > 0		i
aij > 0	bi/aij

bi	= min	bl	alm > 0.
aij		alm

					i	aij
	aim =		aim	(m = 0 . . . n)
	aim =		aij	(m = 0 . . . n)
			aij
					j		aij
a	= a	lm −	a	a	(m = j, l = i)		( )
lm		lm −	im ·	lj

z zmax → ∞

z = 2x1 − x4

x1 + x2 = 20

x2 + 2x4 ≥ 5

−x1 − x2 + x3 ≤ 8

x5 x6

x1 + x2 = 20

x2 + 2x4 − x5 = 5

−x1 − x2 + x3 + x6 = 8

x1 x2 x3

x1 x2 x3 x4 x5 x6

−z

x1 x2 x3 x4 x5 x6

−z

x1 x2 x3 x4 x5 x6

−z

x1 x2 x3 x4 x5 x6

−z

−	z	c	=	−	3	x4
−	a2,4 = 2	4		−
	a2,4 = 2
						x2
	x4					a2,4

x1 x2 x3 x4 x5 x6

−z


	x1	x2		x3	x4	x5		x6
x1
x4
x3
−z									37 21
						x2	x4


	x1	x4	x3		x2	x5		x6
x1
x4
x3
−z									37 21

−z

zmax = 37 12

x2		=	0
	x1	=	20




		=	28
x3		=	28
	x4	=	5/2
	x5	=	0
x6		=	0

(20, 0, 28, 5/2, 0, 0)

x1 x2

z = 12x1 + 12x2 + 18x3 + 18x4 + 18x5 + 30x6 + 30x7 + 30x8

0, 4x1 + 0, 9x2 + 0, 5x3 + 0, 3x4 + 0, 7x6 + 0, 9x8 ≤ 250

0, 5x1 + 0, 5x3 + 0, 2x4 + 0, 5x5 + 0, 3x6 + 1, 4x7 ≤ 450

0, 3x1 + 0, 5x2 + 0, 4x3 + 1, 5x4 + 0, 3x5 + 1, 0x7 + 0, 9x8 ≤ 600

y
D	C
F	G
E	H
A	B	x

•a

•b

•av

•bv

•O(xv0, yv0)

EF GH

ABCD

ABCD M hx

AB N hy AD hx = Ma

hy =	b		M 1 M 2 N 1 N 2
hy =	N		M 1 M 2 N 1 N 2
	N
		= M 2 = 0	av = 0 bv = 0 N 1 = N 2 = M 1 =
		= M 2 = 0		av > 0 bv > 0
		0 < N 1 < N 2 < N M 1 < M 2 < M		av > 0 bv > 0
		0 < N 1 < N 2 < N M 1 < M 2 < M
		C av > 0 bv > 0		DC BC
		M 2 = M	0 < N 1 < N N 2 = N 0 < M 1 < M
		M 2 = M
		AD av > 0 bv > 0		AD
			N 2 = N	M 1 = 0
		CD
		CD