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:)Programmation linéaire(
:)Transport(
:)Affectation(
:
1-::
: .
:)(.
.
2-:) (:
- :) (---)(-)(.
3-:
.:1-) :()Max ()Min(2-)XY(...:3- :)(4- :5-:.
4-:
..
:1-)( :
:n
jjj XCxf
1
Opt
:Opt :)Optimalité()Max ()Min.(
Cj :.Xj :) (.
j :) ()n.(2-:) (
:ij b,,Xaij
:aij:)(
.bi :.i :)m(.j :)n.(
:
):(
,,:ijij
ijij
ijij
bXa
bXa
bXa
:.
-)Max:(ijij bXa
-)Min:(ijij bXa
: (=) :ijij bXa.
3-) ( : :0jX.
:)Max()Min(
:Max Z=2X+Y :Min Z= X + 5Y::
)(...2X + Y 10.X+Y 40.Y = 10.
X 10.X,Y 0.X + Y = 30.
:X,Y 0.
5-:
) ( .:
5-1-):(
. :
1-.2-.3-)X ()Y.(4-.5-.6-)
)Max ()Min( .(-)Max:(
: :B
T . : CD .60
48
.86.
:)(
::
- :
:X :) ()B.(Y :) ()T.(
-:1- :)Max(
:Max Z = 8 X + 6 Y
2- : .
:)C(: ...602Y +4X
)D ( ... :484Y +2X3- :BT :X,Y 0
:Max Z = 8X + 6Y
4X + 2Y 602X+ 4Y 48
X, Y 0
)(.:-) :1 ...(4X + 2Y = 60
)2 ...(2X + 4 Y = 48-XY)X ()Y (:
X = 0 Y = 30Y = 0 X = 15X = 0 Y = 12Y = 0 X = 24
)X ()Y (:
)1 ()2 ()A B C D (
C)1 ()2.(:
-A)00:(Z = 00 < 60(A)0 < 48
(0, 0) = (0, 0)-B)012:(Z = 72
24 < 6048 = 48
(0, 12) (0, 0).-D)150:(120 =Z
60<2448 =48
(1)
(2)
)0,0 ()15,0(-C): (
C)1 ()2 ()126 (:132 =Z
60 =6048 =48
)0,0 (>)12,6()Z ( )0 (
132=Z)C ( . :126
132.
: )MaxMin.(
-)Min:(:X =12
Min Z = 18X + 10Y)1(Y =6)1(484X + 8YX =10)2( 12X + 10Y 12)2(Y =12)3(10X + 15Y 150X =15
X,Y 0)3(Y =10
A = (1) (2) = (8.57, 1.71) Z = 171.36.B = (2) (3) = (3.75, 7.5) Z = 142.5.D (12, 0) Z = 216.C (15, 0) Z = 270.
B.
-:1-):(
)Z (.
:Max Z = 2X + 4Y)1 (X = 8
X 8 (1))2 (Y = 3Y 3 (2))3(X=10
3X + 6Y 30 (3) Y = 5 X,Y 0
A(0, 0) Z = 0 B(0. 3) Z = 12 E(8, 0) Z =16 0 < 8 0 < 8 8 = 8 0 < 3 3 = 3 0 < 3 0 < 30 18 < 30 24< 30 (0, 0) = (0, 0) (0, 3) (0, 0) (8,0) (0,0)
C(?, ?) = (2) (3) D (?, ?) = (1) (3) = (4, 3) = (8, 1)Z = 20 Z = 204 < 8 8 = 83 = 3 1 < 330 = 30 30 = 30(4, 3) > (0, 0) (8, 1) > (0, 0)
CDZ.
2-):(.
: MaxZ=6X + 4Y X = 10 (1) X 10 (1) Y = 12 (2) Y 12 (2) X, Y 0
) ( .
3-) (:) (
.: Max Z = 12X + 8YX = 4(1)
Y = 6 6X + 4Y 24 (1) (2) X = 8 2X + 4Y 16 (2) Y = 4 Y 7 (3) (3) Y = 7 (X, Y) (0, 0)
)1 ()2 ( )3()3 ( .
)1 ()2 (.
4-:.
: X1 = 10 MaxZ=2X1 + 5X2
X2 = 10 (1) X1 + X2 10 (1) X1 = 24 2X1 + 3X2 48 (2) X2 =16 (2) X1, X2 0X1
X2
20
10
16
10 20 24
.
5- (=):)(
.(=) ) (.
: 10 = X Max Z = 6X + 8Y 15 = Y (1) 30X + 20Y 300 (1) 22 = X 5X + 10Y 110 (2) 11 = Y (2) 4X – Y = 0 (3) 4X = Y (3) X = 2 (4) X = 2 (4) X,Y 0
)1 ()2 ()3 ()2 ()4 ()X = 2Y = 8 (Z =76.
:) () (:
-:-.-.
- :-)XY (....-).(-100%.
5-2-:
:1-.2-) (
.:
2X + 3Y = 40 (1) Max Z = 4X + 5Y 3X + Y = 50 (2) 2X + 3Y 40 (1)
: 3X + Y 50 (2) X = 15.71 X, Y 0Y = 2.85 Z = 77.09 .
:-.- :
-).(-100%.
5-3-):(
:1-) (.2-.3-.
:
)Cj(
)Xi, ei, Ai(bjVCk
)aij(
)()ei, Ai(
) ()C( =Z
-)Max:(:.
Max Z = 8X + 6Y 4X + 2Y 60 2X + 4Y 48 X, Y 0
:1-)(::
:-) ()ei(
).(:4X + 2Y + e1 = 604X + 2Y 60
2X + 4Y +e2 = 482X + 4Y 48-)Ai(
M . .:Max -MAiMin +MAi
:MaxZ = 8X + 6Y + 0 e1 + 0 e2
4X + 2Y + e1 = 602X + 4Y + e2 = 48
X, Y, e1, e2 0
:8 6 0 0X Y e1 e2bjVCk
4 2 1 02 4 0 1
6048
e1
e2
00
- 8 - 6 0 0Z = 0ZC:
jk
jk
CaijCC
bCZ
2-:Max0 C
3- : :
-Max) ()C ( ..
-)MaxMin(bj)kaij (K
:Kij
j
ab
Min.
- ..-:
bj: =–×
ZC .
.:
1 21
41 0
1 3 21 1
1518
Xe2
80
0 -2 2 0Z = 120
1 0 31
61
0 1 61
31
126
XY
86
0 0 35
32Z = 132
C0126132
)e1 = 0, e2 = 0bj .(
35
31B
61T .3
2
61B3
1T ..
:)V ( 1
)0.(
-)Min:().(
: Min Z = X1 + 3X2
X1 + X2 4 (1) 2X1 + X2 6 (2) X2 = 3 (3) X1, X2 0
Max.:
Min Z = X1 + 3X2 + 0e1 + 0e2 + MA1 + MA2. X1 + X2 – e1 +A1 = 4 (1) 2X1 + X2 + e2 = 6 (2) X2 + A2 = 3 (3) X1, X2, e1, e2, A1, A2 0
)1 :(X1 + X2 – e1 = 4 X1 + X2 > e1 )Ai.(
+ ei =Z.- ei =.
)3 :(Ai
V.-MaxMinMin :
:C..
1 3 0 0 M MX1 X2 e1 e2 A1 A2bjVCk
1 1 -1 0 1 02 1 0 1 0 0
0 1 0 0 0 1
463
A1
e2
A2
M0M
M-1 2M-3 -M 0 0 0Z = 7M1 0 -1 0 1 -1
2 0 0 1 0 -1 0 1 0 0 0 1
133
A1
e2
X2
M03
M-1 0 -M 0 0 3-2MZ = M+91 0 -1 0 1 -1
0 0 2 1 -2 1 0 1 0 0 0 1
113
X1
e2
X2
103
0 0 -1 0 1-M 2-MZ = 10
)Ai .(Ai
.C
) (1) (X13X21
10 .
11X12 .
Ai .
6-:
..
.
.6-1-)Dualité()(:
6-1-1- ):() ( .:
-MaxMin.- ) (.-.-.- XiYi .
:) (
Min Z=m
1i
bi yi Max Z =n
1j
cj xj
aji yi cj aij xj bi
aji yi = cj aij xj = bi
yi 0 xj 0:
Min Z= 2A+3B+4C Max Z= 10x+20y A+5B+6C 10 x+2y 2 2A+B+7C 20 5x+y 3 A,B,C 0 6x+7y 4 x,y 0
:) ( )(
.
:Max Z =x+2y
3x+4y 205x+6y 30x+y 10x,y 0
Max
)- (: - (5x+6y) -30-5x -6y -30
:Max z=x+2y3x+4y 20-5x-6y -30x+y 10x,y 0
:Min Z=20A-30B+10C
3A-5B+1C 14A-6B+1C 2A,B,C 0
:.
6-2-1-:
:-Cbj.-:-Z.- +) -
-.(+- )Ck( ....
:Min)-.(
: Min Z=40Y1+90Y2 Max Z=X1+3X2
2Y1+Y2 1 2X1+X2 40 Y1+4Y2 3 X1+4X2 90 Y1, Y2 0 X1, X2 0Min Z=40Y1+90Y2+0e1+0e2+MA1+MA2 Max Z=X1+3X2+0e1+0e2
2Y1+Y2-e1+A1=1 2X1+X2+e1=40 Y1+4Y2-e2+A2=3 X1+4X1+e2=90
X1, X2, e1, e2 0Y1, Y2, e1, e2, A1, A2 0 )(
: 1 3 0 01 x2 e1 e2bjVCk
2 1 1 01 4 0 1
4090
e1
e2
00
-1 -3 0 0Z=0
47 0 1 - 4
1
41 1 0 4
12
35
245
e1
x2
03
y1 y2 e1 e2 A1
A2
1 0 - 74 + 7
1
0 1 + 71 - 7
27
1
75
y1
y2
4090
0 0 -10 -20Z=70
6-1-3-:
Min Z= 20Y1+30Y2+40Y3 Max Z = 2X1+3X2+4X3
5Y1+8Y2+11Y3 2 5X1+6X2+7X3 20 (1) 6Y1+9Y2+12Y3 3 8X1+9X2+10X3 30 (2) 7Y1+10Y2+13Y3 4 11X1+12X2+13X3 40 (3) Y1, Y2, Y3 0 X1, X2, X3 0
"" :
X3, X2, X1 .)Max Z .(
)X3, X2, X1(
)1()2()3.(
:-
X2,X1 X3
-X3, X2, X1
:203040.
- :.
:" """
"" )Minc.(:
- )Min C.(
- 41 0 0 4
32135 Z=1 0 7
4 - 71
0 1 - 71
72
1020
x1
x2
13
0 0 71
75Z=70
Yi )(.
- X3,X2,X1
X1, X2, X3.
- : .
6-2-: ) ( ) :
(...:-)Cj(.- ) ( )bj.(- )aij( .
) (.-
)C MaxCMin(.
-)(.
.-
: :aij :aij=0.
:
-:ij*j
*j a
e
X
-:j
jhj X
ea
: (*))Xj ()ej.(
(+))bj (.:
Max Z= 2A+4B+3C 3A+4B+2C 60 2A+B+2C 40 A+3B+2C 80 A, B, C 0
) (:2 4 3 0 0 0A B C e1 e2 e3bjVCk
31 1 0 3
1 - 31 0
65 0 1 - 6
13
2 0
35 0 0 + 3
2 - 31 1
320
350
380
BCe3
430
611 0 0 6
53
2 0Z= 3230
1-:) (
ABC :)BC ()A.(
1-1-:AA=0 .
A)A (2A )A (A: )2( +
A611 :2
611
623.
A 623
A.1-2-:
BC)(:-
-C--)(--.
B:)B
31 1 0
31
31 0
Z 611 0 0
65
32 0
31 +
611 0
611 .
13
633
1 + 0 0 0
31 +
65 0
31
65
65 .
13
25
31 +
32 0
31
32 2
- 633 - 2
5 0 2
+++++++++++++++++++---------------------------------------------------
[0, 2]B [0+4,2+4]B
46.C:
) C 65 0 1
61
32 0
Z 611 0 0
65
32 0
65 +
611 0
511
1 + 0 0 0 .
61 +
65 0 5 .
32 +
32 0 -1.
511 -1 0 5
------------------------------------- +++++++++++++++++++
[0, 5]C[0+3,5+3] .
C38.2-) (:
)bj ( :
-bj
-ei) (--- .
)1e:(bj e1
320
31
320 +
31 0 -20
350
61
350 -
61 0 100
380
32
380 -
32 0 40
-20 0 40 100
0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - - - - - - - - - -
[-20, 40][-20+60,40+60][40,100]
.
)2e:(bj e2
320
31
320 -
31 0 20
350
32
350 +
32 0 -25
380
31
380 -
31 0 80
-25 0 20 80
- - - - - - - - - - - - - - - - - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[-25, 20][-25+40,20+40][15,60]
.)3e:(
380)(.
.
380
803
803
160.
7 -:
1-:) (
)AiV(.
2-):(
ij
j
a
b
) (.
3-):(
..
4-:C
.
Z.
:
.
8-) ( :)1<X (
:-
.-)
(.
-) ( :.
9 -: ) ( ) (:
*:- ).(
-.-.-.
*:-
.-.- .
.-.-.
)Transport(
:1-:
.
.
) (...
) (....
2-::
---) ()(-
-)Max ()Min(-) :.(...
3-:)n ()m(
:1-:)Max ()Min(
:n
iij
m
jij XCxOptf
1 1)(
2-::
- :n
i
m
jji ba
1 1
- :m
jiij aX
1
- :n
ijij bX
1
3-:Xij 0.:Cij :.
Xij :)i ()j.(ai :) ()i(.bj :)j(.
4-:
)m+n ()m ×n ( .
:
)(B3B2B1
)Bi()Ai(
3
11
jjX
C13X13
C12X12
C11X11A1
3
12
jjX
C23X23
C22X22
C21X21A2
3
1jijX
2
1i
2
13
iiX
2
12
iiX
2
11
iiX)(
) () ()(
:1-) (:
Xij = Min (ai, bj)2-.
::
)X1, X2, X3 ( .)Y1, Y2, Y3.(
:-X1)Y1, Y2, Y3 ()884 (-X2)Y1, Y2, Y3 ()162416 (-X3)Y1, Y2, Y3 ()24168 (.
)X1, X2, X3 ()778256 ( .
)Y1, Y2, Y3 ()4110272 (.:
)(::
)()()72 (Y1)56 (X1
)102 (Y2)82 (X2
)41 (Y3)77 (X3
:Cij =)i ()j.(Xij =)i ()j.(Z =.ai =)i.(bj =)j.(n =3.m =3.
3
1
3
1i jijij XCMinZ
3,2,1,3
1iaX
jiij
3,2,1,3
1jbX
ijij
.0
3
1
3
1
ij
i jji
X
ba
: =.
48
8
16
1624
16
24
8
)bjai(Min =Xij ) (:
1-:
) ()bjai(Min =Xij .
Y3Y2Y1Yi
Xi
56 0884
56X1
82 66 01624
6616
16X2
77 41 024
4116
368
X3
21541 0102 36 072 16 0
) ().(
Z:Z = 56 (4) + 16 (16) + 66 (24) + 36 (16) + 41 (24) = 3624
::
:-:-.-:-.
-.
2-) (:
) (
..
:-).(-.
Y3Y2Y1Yi
Xi
56 0884
56X1
82 41 25 016
4124
2516
16X2
77 02416
778
X3
21541 0102 77 072 16
.
Z = 56 (4) + 16 (16) + 25 (24) + 41 (16) + 77 (16) = 2968 :Z.
:- :.- :.
3-) (:) (
.
Y3Y2Y1Yi
Xi
56 088456X1
82 41 016
4124
4116
X2
77 61 02416
618
16X3
21541 0102 41 072 16 0
Z = 56 (4) + 41 (24) + 41 (16) + 16 (8) + 61 (16) = 2968 ZZ
.:
- :.- :.
4-::
Y3Y2Y1Yi
Xi
56 0884
56X1
82 41 016
4124
4116
X2
77 612416
618
16X3
21541 0102 41 072 16 0
Z = 56 (4) + 41 (24) + 41 (16) + 16 (8) + 61 (16) = 2968 Z
Z
. :
.
.5-)Vogel:(
:
---.
:.
Y3Y2Y1Yi
Xi
56 088
564
X1
82 41 016
4124
4116
X2
77 5 02416
58
72X3
21541 0102 46 5 072 04 8 8
8 8 8 8 8
)8 ( .
.Z:Z = 56 (8) + 41 (24) + 41 (16) + 72 (8) + 5 (16) =2744
Z = 2744. :
.
.
5-:
:1-)I ()J (
:Ii + Jj = Cij, Xij > 0
2-)ijE (:0ijX,ijC–jJ+iI=ijE
Min0 Eij
Max0 Eij.
0 < Eij.4-0 < Eij)Eij
|Eij|( )(
).(:
--+-+-.
: =min Xij/(Xij - )
Xij-.5-)0 Eij.(
::
Y3Y2Y1Yi
i jXi 412456884
560X1
-44-0
8216
4124
4116
12X2
000
772416
618
164X3
16-0+0
2154110272
:ijE.Eij
. :56.=
Y3Y2Y1Yi
Xi 080
5688
564
0X1
8-04-
8216
4124
4116
16X2
000
772416
58
728X3
16-00
2154110272
0 Eij :56 X1Y241X2Y241X2Y372X3Y15
X3Y3 :Z = 56(8) + 41(24) + 41(16) + 72(8)+ 5(16)
:I1 = 0
I,J.
= 2744 .
.
:ijE)(.
6-)Max:(
) (...:
- .
Min.-
:IJ :Ii + Jj = -Cij.
Eij :Eij = Ii + Jj + Cij:0 < Eij
)Max Eij (.0 Eij.
:Max:-
Y3Y2Y1Yi
Xi
10246
X1
20073
X2
30101010
Y3Y2Y1Yi
Xi
10753
X1
20926
X2
30101010
Max Min
-.
7-:1-):(
=:
-.-.2-):(
)Xij ()Xij < m + n - 1 ( .
IJ . 0,0000001=:
ij =Xi ±X.:.
3-: .) (
.4-:=0ijE
.
8-::
--
---.
9-:
.:
:)MaxMin :(
- :Min Z= 2X1Y1 + 3X1Y2 + 4X1Y3 + 5X2Y1 + 6X2Y2 + 7X2Y3
-:-:X1Y1 + X1Y2 + X1Y3 = 10
X2Y1 + X2Y2 + X2Y3 = 20-:X1Y1 + X2Y2 = 10
X1Y2 + X2Y2 = 10X1Y3 + X2Y3 = 10
-:X1Y1, X1Y2, X1Y3, X2Y1, X2Y2, X2Y3 0:
::- :iiij YXCMinZ- :iiij YXCMinZ
-:-:- :jii bYX- :jii bYX
Y3Y2Y1Yi
Xi
10432
X1
20765
X2
30101010
- :jii bYX- :jii bYX
- :0iiYX- :0iiYX
10-:.
)Affectation(
:1-:
.. :44.
2-::
-.-.-.-.
3-::
-)MaxMin:(m
jij
n
iij XCOptZ
1 1
- :
1- :11
i
m
jij aX
2-:1$
1j
iij bX
3-Xij10:Xij = X2ij
- :Xij 0
: .
4-::
4-1-:
).(
:333!:3! = 3 2 1 = 6
6. :
.).(4-2-:
.:1- .2-.3-.
:4)M1M2M3M4 .(
4)1= 2=3=4= .(
)10:( :4321
6784M1
5693M2
2365M3
4142M4
:
1-:
.
3001011323302310
3031014323602340
4142236556936784
2-:0
. .
- .
3001011323302310
.3-:
:-) (
.-.-).(-.
.
3002011412201200
3002011412201200
3001011323302310
1
01M2M
M3M4:Z = 8 + 3 + 2 + 1 =14
5 -)Max:().(...
:1-MaxMin
.2-Min.:
Max Min
801234567
198765432
800011011
801012012
900000000
6 -:6-1-:
.:
011140210064
111421064
.
6-2-: .Min
Max.6-3-:
. :0000
0000
6-4-:
)- Xij (Xij
.:
6232620142215016
6111482103124
.
–) -12 () -12(