linear
;
formulate
the model
p Decisionvar obj function constraints
,
,
Lecture 2
-
Decision variables
x
y
=
=
premise
the unit of
vientmis
fit
To
subject
unit of standard
the
=
,
250x+caox
=
to
20x +
10x
3x
constraint
30y
+
+
=
64
3300
1088
34
x
,
y
=
360
0
Decision variables
unit of bowl
x
=
y
Decision variables
unit of
x
=
I
unit of
unit of
product
product
product
function
objective
18x
Maximize z
=
+
produced
produced
produced
I
I
I
10y
+12z
4x
Maximize
3y
Yz
x
,
+
y
27
,
z?0
unit of mug
produced
produced
function
z
=
40x +
soy
constraints
x
constraints
+
objective
=
4x
+
+
2y=40
120
34
=
x
,
y
=
0
;
z
is
profit
per
day
Decision variables
x
Xe
x
xx
number of
=
number of rice
:
bag
a
powder in
a
in
d
bran
number of Vitamin in
=
Objective
in
corn
number of fish
:
bag
a
bag
bag
function
Minimize
z
x + 1
=
,
.
SX2
+ 1 5X
.
,
2
+
.
5X4
Constraints
x
+
,
x
xz
+
,
0 4(x ,
x,
x +
,
x320
,,
+
.
51x ,
.
19x ,
x2
+
x2 , X3
x
2
1
+
x2
+
2
,
x
x42200
.
xy20
Xax+Xy
+
X420
+
+
,
x4)
+
x
,
x,
x
2x2
+
y
+
x4)
+x4)
2x
,
0 64 , -8
+
.
4x
-
0 47
.
0 5x , +0 57- 0
+
0
,
,
.
.
+
- 3x
.
.
973-0
20
.
.
-
0
4xy20
5X-85X420
2
14-0 1x2
.
.
-
-
0
.
1x420
Decision variables
Xij=
i
j
Objective
of ingredient
Amount
1
=
,
2
and
,
Original
=
in coffee ;
i
3
/O)
and
concentrated
functions
maximize
z
1200 (X
=
-
z
500x
700x
=
+
10
-500
,
0
15002Xes+ Xec
+
400x
-
,
800x20
+
,
x e0 +Xa0
+
600x
-
e0
,0
Constraints
E
ment"
Ingredients
E
inconcentistt
Amount of
as
1
3
0
0
:
ingredients
Xyc
?
:
6 [X
.
x2x0
:
0
Xij
,
+Xzotsob
10
.
,
+Xec+Xsc)
,
12x ,
+xx
+
Xsc)
+xz X3
xottec
x
:
42x
.
<
Restriction
4 X
.
iii
Xec ?
:
2
<10 ?
+
!
300
he
zot Xa
=
190
he
+Xy
!
200
↳g
0
+
L10UX
600X
-
ex
1000X
,
+
+
Xad
-
30
60032
1100x2e+ 900X3c
Decision variables
ts
current account
investing in bank note
investing in stock market
money investing in gold
=
x
=
functions
Maximize
subject
in
-amount of
money
amount of
money
amount of
x1
Objective
money investing
amount of
x:
z
to
=
0 02X
.
,
+8 04x2
0 054
+
.
.
3
:
x
+
,
xz
+
xz
x4
+
↑
,
+
=
1 500 000
,
,
x
I s00 ,
x4
=100 000
,
Xy
x2
=
x3
xz x32
+
Restriction
:
X
,,
x2 Xs
,
,
1000000
I
I
=
x
wou
500 000
,
300 000
,
700 008
,
=
0
+0
.
01x4
Decision variables
X
:
X
:
Xs
Objective
in
newspaper
The number
of ads
in
radio
The number
of ads
in
TV
functions
to
z
radio
x
=
,
S
Yes
9
:
Restriction
+200000x
, +30000x ,
,
<IS
XI
q
TV
100 000x
,
=
:
newspaper
Budget
of ads
:
maximize
subject
The number
1000x
:
X
↑
o
,
+
.,
3000x
x2 , x3
,
+25000x3
=
a
=
300 000
,
S
I
+
-
10
-1
in
24
Decision variables
X.
X
:
44
:
xs
objective
The number
of workers in shift
The number
of workers
The number
of workers in
:
Xs
:
shift
in
shift ?
function
Minimize z
X
=
+x2 + xs Xy
+
,
+
x
+
,
X6
subject to
Period
ex-4
4-
8
-
X
:
:
8
-
12
(e4-8)
(4-12)
The number of workers in shift 2
(8-167
The number of workers in shift 3
:
1s
of workers in shift /
The number
:
x
12
:
16
16-28
,
x
73
+
+
,
x4
:
,,
X
x4
+
+
,
3
=
x2 =S
:
20 -24
x
+
xy
:
X6
+
,
Xs
x6
10
=
6
=
10
=
8
I
x20x3 , x4 /3
,
,
46
=
0
(12-202
(16
(20
-
-
24)
-
43
b
-
1
-
is
10
-
1
-
so
8 1
-
if
factory produced
more
capacit
than warehouse
~use
:
in
factory
=in warehouse
secision variables
xij
=
produced in factory
thu number of unit
delivered to warehouse
1
objective
=
1
j
2
,
,
=
1 2 3
,
;
,
function
Minimize z
3x
=
+4x
,
,
2+8x+ Skait fret 6xze
subject to
Factory
Factory
1
2
Warehouse
capacity
<11
:
:
+
X12 +X13
XeltXeetxes
q!
xij
1000
=
=
7000
Xi+Xer
x12 + X22
x13
=0
+
X23
4008
6008
=
=
2000
:
-
Decision variables
xij
objective
Assign workeri to
:
jobj
;
i
=
1
,
2 3
,
,
4
,
function
Maximize z
=
9x
+
Id
7x t
10X +10x
,
,,
,
Stedt 7xeitsxat70x2s
9X36
9X3; 8x3 1035
+
+
8xxyat6x4s
744d+
subject
to
i
:
e
xid+Xit flat
3 xedt
Ex
"
worker
'
B
4
:
+
is
Yeat Xes
xsae
2
xyd+i+xate
↑
=I
=
I
=I
=
I
suga : "
xij
20
department
,
work
vvw
j
=
Di 1
,
A
,
u
Decision variables
x
objective
unit
=
,
produced
in
month :
function
minimizez
so x-esoxcees+
=
(1 1 C1102X
.
,
+x2-e0cees+
1
.
12 ,
Close
,
subject
to
capacity
Demand
:
:
Yes
X
x
x
Restriction
X
I
,
;
eso
-250
,
-
,
?
250
0
+
+
x2
=
280
X2 - 200
x ,+
+
+
x
=
,
xz
300
=
-
450
X
,
+
xzts
=
Eso
e
Decision variables
Xij
objective
Amount of
=
i
1 2 3
,
,
=
function
Mavinize z
component
j
,
=
5
P,
,
grade ;;
in
;
e
e3(XIstXestX3g) + 20(X , p X2p Xp3
12 (x1s
+xp+X , e
10(X2s+Xp xxe)
=
+
+
-
-
18x2e 18xge)
14(X3s+ X3p+ Yze)
189X
+
-
+
+
1
e
+
subject to
component
rications
E
E
1
XistXiptXie
x2 , +xiptX29
:
e
:
3
+
+
super
X
:
ivm
:
:
E
5
P
e
:
0 3
<IP
=
0 4(X
.
<0
+Xp+Xse)
25(Xi
-
X2stYs
x10 x2p+ 230
+
:
+
.
xist
:
+kestXas
(X1 , +Xcs X3s)
.
x sp
Demand
3500
=0 X ,
,
x 29
pier
2700
!
x3 j x30 x32
:
4500
=
+
xyp
+
3000
=3008
xietxetYe=
Restriction
3008
xij
=
0
X3p)
Extra
:
Xie
x
?
O 6
.
0 1
ee =
.
Xietkzetkiel
CX1etxce+Xse)
dis
linear model given
e
:
graph to find
optimal solution
Lecture 3
returne !
20X ,
30x2
+
x-intercept
y-intercept
10x ,
672
+
Isoline
180
160
140
Let
-
&
A
⑧
IVO
-
68
40
250x
=
,
z
B
=
,
=
0
feasible
·
0
x
,
;
,
=
28x ,
do do o due Du ir ! is
Profit
G
⑧
2902
A
20 , 1087
B
(30 , 903
33608
C
190 , 303
31200
D
<108
07
30x2
+
+
,
(3)
,
360
07
x-intercept
<120 , 03
1103
y-intercept
10
,
,
03
10 1805
,
38
3x
223x10
C
(x , 47
=
50
=
,
co-region
Point
3x2
1088
(108
intercept
3x
-
,
+
+290x2
x
,
Find B
-
-
o
3x ,
14500
=
x2
·
80
7
X
-
120
;
-
(165
=
x-intercept
y
3300
=
31906
27000
-
;
2)
=
3300-21)
360
=
,
30x
;
,
+
-
30x)
10X
=
,
x1
x2
=
=
(2)
=
3680
-
333
300
38
90
opHimal
solution
139907
,
1203
3x
x
+
,
,
x
=
x
x
Let
z
=
0
,
=
+
=
0
0
0
60
=
X,
xz
x
,
,,
=
50
=
,
X
x,
X
=
,
3750
x
=
=
0
,
0
,
xz
X
=
=
So
75
,
⑧
⑧
⑧
·
⑧
eoptimal solution
;
⑧
~
⑧
lu su du du du de Bu du du hir iz iss
i
W
~
Point
⑧
(x , 47
2902
A
B
220 187
,
C
(0
D
,
20)
Profit
G
100
30
=
M
=
(0 , 60)
=
(80 , 07
=
)
=
(0 , 302
)(100 , 03
30
=
,
80
=
=
100
=
,
xz
,
=
,
2x)
0
240
=
xz
,
,
x2
x
=
+
,
x
&
0
=
x2
472
30
=
=
(2 , 40)
)(30 0)
,
not in exarc
~
Unbounded Solutions
find
max
there is
no
cannot
6
2,
2x ,
=
x
0
=
,
X= 0
,
+
x2
3x2
=
2
10
8
0
X
,
12
,
3
=
0
=
=
0
=
x
=
0
=
x
3
3x ,
x
3x 6
220 ,
x2
=
↑
-
-
volimit
-
=
-
=
-
7
6
2x
limits
~
X1=312
,
as
in
-
323 ,
=
6Xz
+
=
2
03
!
in in
!I
⑧
-
12
=4
-
r
-
,
> X1
27
e(0 2)
,
=
xx
4
-
·
3
-
L
more
24 07
,
limitless
r
& 9750
x1
90x
=
=
0
,
xz
=
0
, +79x2180
x2
,
X
=
=
,
30
160
75
146
10
100
+
,
x,
=
x2
3x
3x9
,
=
0
+
x
x2
=
0
=
=
=
0
0
2x2
Xe
=
,
-X
2x2
=
308
>x
=
=
=
=
20 1003
190
(150
=
,
,
,
X,
ro
03
10
=
80 (80 ,
cover
feasible region
-
-
·
-
can
=
Find solution
cmultiple solutions
3
-
·
28020 , 2007
=
isoline
-
40-
488
-xz
=>
60
100
-
50
↑
2x
Multiple optimal solutions
2008
is do
joe
or
inoivolso
03
↳
160
80
while
x
=
,
0
20 802
,
-
~
2so
profit
line
Infeasible solutions
not in
z
2x
Let
x
+
34
y
2x
Let X
+
x
0
=
3y
15
=
B
48
=
Decision variables
Xij
i
objective
Maximize z
number of
=
y
,
,
j
,
rooms
:
:
12x1
+
130X
X6ntX6s-
180
X6n+4,
100
0
,
x
n
1
y
Xis- e 576n-2 5X6 ,
65 + XIs
0
.
= 0
.
.
-
3X6n
-
0
xij
.
5X1n
?
0
=
0
0
=
=
0
,
=
,
,
Point
/n +145X
15XJut 15x1s+ 18x6,
+
.
x
Let
to
Xin+XIs-15Xgn- 1 X6 =
+
option ;;
4 S
,
=
77SX6n+190X6s
Construction costs
Number of
room
6x - sy
function
=
subject
6 1
=
with
;
=
6 300
,
;
x
x
2x
8
=
=
20
Is
-
0
(x
⑧
2)
,
y)
B
⑧
profit
8)
24
57
46 S
B
(7
C
(15 , 07
75
P
20 , 07
2
,
=
6
x)
-
40 (2)
-
=
10
optimal
↓
ja "
20
,
Sy
30
; 2y
⑧
A
5
+
=
=
=
Do
0
.
3y
10
=
4 S
--
=
+
↑ I
A
=
3X
=
-
-
y
=
0
2x
20
0
y
=
38
=
0
=
(2
0
=
z
y
~
y
Let
&
18
=
,
Sy
+
x
y
,
0
=
5x
30
=
0
=
=
exam
.
solution
Y
-
z
e 1
.
40x ,
=
profit
.
z
coefficient
of
30x
+
=
30x ,
↓
by
objective/change right hand
10
=>
20x2
+
of
how would it affect
constraints
①
z
Let
407 , +50X
x,
↑
x,
+
Let
2x2
x
,
4x
+
,
Let
0
=
3x2
x
,
=
,
0
=
=
x2
=
0
=
2
they x2
40
=
then
,
x2
=
x,
20
40
=
x
,
=
30
,
10
120
0
0
10
X2
=
=
)x
,
=
40
30
=
=
=
(0402
A
-
-
ctimal
B
D
is is
230 0)
,
,
,
2x
+
,
3x2
10 , 201
(24 , 8)
C
(30, 8)
p(0 , 0)
800
1360
1200
E
⑧
ass
profit/ine
~
z
⑧
=
Let
z
↑
x
48
30
z = 40x ,
100x + 50x2
,
=
,
x2
-
Let
3008
=
=
0
0
=
) xz
=
7x
=
60 (0 , 607
=
,
30430
,
30
-
-
F
-
+100x
<optimal solution
is is
o
<change !
O
D
=
,
x2
=
0
0
optimal solution
-
is is so
,
2008
·
10
10
D
x
0)48-
F
10
↑
z
=
o
=
)
=
eslope
xz
xx
=
20
=
,
20 207
,
30 (50
,
c)
x2
x
2)
-
=
,
(2)
160-3)
48
=
=
1
-
120
; SX,
o
~
=
,
Profit
B
40
=
4x + 8x
;
-
A
solution
+
(3) 22)
x,
-
F
240 03
(13 x4
40
,
(10 , 00
50
=
y
(9283
=
20
x
4X
200
-
=
x2
40
=
0
=
x2
B;
200
=
8
24
slope
slope
of constraints
of constraint
slope
↑
clockwise Iner constraint
30
-
F
10
10
D
2
-
counter clockwise Paw constraint
A
-
optimal
B
I
I
I
10
20
I
Suc
solution
exc
4x
+342
,
:
2
,
1 I
!
1
-
58
120
slope=11
;
slope
+50x
,
2,
2
=
coefficient of
:
1
=
slope
;
-
X
,
the
in
function
this
within
-
the
range
,
optimal
solution will be unchanged
es ?
,?
↑
25
,
=
40x
:
;
40
=
-
211
-
x +
of objective function
-
y
:
&?
I
SO2 ,
3
=
..
so
22
⑧
:
c
66 67
.
coefficient of
X
,
the
in
function
0
-
~
ass
profit/ine
1 I
-
~
-
-
-
,
4
-
=80
I2
:
so
: 302280
u
'Urweruku
18
~
6x
=
(3
,
07
,
,
+
3x2
20
,
6)
-
I
22 =
4
-
4 3
.
,
12 !
-
C
24
. .4 ?
...
!
C
?
I
D
2220
O ?
2
z
24 3
.
?
-
40
z
Let
407 , +50X
x,
↑
=
40
=
then
0
x,
20
Ates
Binding
10
constraints
:
10
interest
slope
slope
of constraints
of constraint
:
exc
x +
,
4x
:
2
25 ?
...
c
=
40x
:
+
,
,
40
=
=
3x2
; SX,
x2
x
800
10 ,
I
201
(30, 8)
2)
-
=
,
,
Profit
-
120
4x + 8x
;
-
(24 , 8)
(2)
160-3)
48
8
=
24
=
1
1360
1200
p(0 , 0)
E
⑧
o
~
ass
2
profit/ine
slope= I
-
;
+50x
2x
(3) 22)
A
solution
slope=11
;
120
,
optimal
B
is is
40
=
+342
,
of objective function
slope
-
-
intersectio
no
->
A
D
give point
Non-binding
⑧
-
+
,
(13 x4
40
x,
-
F
:
,
(10 , 00
50
=
y
30
x
4X
200
=
2
they x2
0
=
x2
B;
200
=
slope
;
66 67
=
.
,
: 302280
Notes
stack
:
less than
:
surplus
:
greater
~
↑
remaining
equal constraints
equal
extra
til than
x
,
+2x2
?
greater
58
123I
pain5
48
30
-
40
=
< X
+ 2x
,
&
⑧
-
-
10
D
⑧
=> x
than
,
slack
+
,
902
x
+
,
40
1 + 2x2
40
Vurnuves
40
=
crew
=
Is constraints
2180 07
,
,
solution
=
texi-surplus
108
,
x
;
=
,
0
x2
,
=
40
4x
W
W
↳
↳
-
optimal
B
i no sto so
S
=
,
=
30
⑧
so
70
~
80
ass
90
100
profit/ine
120
120
=
120
remaining
32
~
28
=91 280
=
20
profit
..
0
2008
profit/py
Profil
=
4020) + S02402
=
=
solution
3x2
es
100
<
remaining
...
+
,
100
2x2
80
W
A
-
-
2x2
W
F
10
M
slope
point
last
+
= 40
requirement
↓ 20
same
2x2
+
,
x,
=>
constraints
than
minimum
What if
<
resourc
=
2000-1360
80
-
40
= 16 Idual values
by $16 , hur unit
.
4x + 3x =120
,
-
,
and constraint
optimal solution
4(0)
↑
Profit
Dual value
y
-
30
10
D
0(40)
=
+
=
502201
120
-
=
=
1000
Profit
6
Dual value
-
optimal
B
is is
?
slope
↑
=
402407 +0230)
=
1608
=
-
-
1360
=
=
1600
6
120
160
?
solution
o
1
:
2x
=-
z
18 , 01
,
=
6(03
Dual value
slope=-ja
~
6
optimal solution
9
?
⑧
-
y
-
!
nonbinding
=
6(0)
=
0
e
16
232
,
Binding constraint
z
↑
2
3
-
32
=
24
=
24-24
=
32
-
428)
+
3(8)
+
-(6 , 07 , [0 ,8)
...
4x2216
+
,
e(83
I
10 , 47
j
92
160
=
⑧
constraint
slope
0(37
+
60
60
240 03
,
-
168
A
-
solution
4(40)
66
1360 -1000
-
F
10
3220)
+
optimal
co 202
,
-
constraint
-
+
3(8)
:
24
on
y
constrainst
dr
2
4x
;
10 , 4
6
=
= 24
+ 3x2
4201
;
z
ja~
,
+
3(4)
6(0)
+
347
qual value
optimal solution
12
=
12
=
24
=
-
12
1
=
24-12
⑧
-
↑
y
12
...
2-
=
920
!
-
36
ro
z
=
4550
=
7
(7 , 87
32
counter Clockwise
Co 9)
e
"
.
.
15 0
,
,
(914)
,
Clockwise
28r
⑧
*
~14
-
20
X
(12 1 , 07
(26
.
[0 26 837
,
,
.
483 33
↑16
1
.
⑧
?
C
=
,
115) 43
.
optimal solution
~
8
&
A
↳
>
is Loc o be
I
-
~
I
1
↑
Let
27
=
(ctx ,
14500
0
=
=
xy28
=
188
20 (0)
,
(xz
7x ,
=
=
50
160
38(18 , 02
-
140
-
r
r
120
A
100
88
68
20x
Let
,
+30X2
x,
=
x2
10x ,
Le
x
+
,
x2
3x ,
+
0
=
=
=
=
Let
=
<108 ,
108
constraint
1
3
;
=
,
32
32
Sz
:
20x
(3)
3x
an
u
+
3300
;
30x1 30x ,
; 70x ,
=
-
360
=
3xy
②
mix
=
+
(3) -213
=
300
0
102383
240
+
63903
=
300 + 540
=
? 1080
840
slacks
0
Find 2 and 22
Find
;
counterclockwise
clockwise
! -
-I
C
,
I
2 , = 290237
290(23
3
C, =
:.
-
22
Ce
=
C
.
193 33
.
!
C
=
,
200
,?
290
250
Tz
-
258733
(2
250(3)
=
3
2
3
193 33
-
Ce
=
.
(2
375
250
!
>
&
+38x2
,
,
(2) x10
21290)
=
=
(1
>
jo du it brirido
solution
Optimal
366
i =8
W
ed
,
C
·
-
(6)(165 07
180 (0 1887
=
=
342
20
,
-
1088
x x)
0
-
,
>x2
=
C0 110)
110
=
=
6Xe
+
40
=
xx)
0
a
-
&
3300
x2
=>
0
=
=
⑧
-
22
?
=
250
375
③
->
(1)
Isoprofi ↓line
(2)
-
3600 133
-
-
-
+
x
,
=
38
,
x2
=
90
(30, 90)
Constraint
1
↑
Find
188
160
140
an
1
20x ,
;
more
;
from
(1
20x
r
z
129.
100
a
20
3300
1207
,
30(120)
+
3600
=
3600
=
30x2
+290(120)
down to
34800
=
.:
Dral value
From (k
20x
-
·
C
z
-
20(98)
;
+
,
250 (903
=
jo du it brirido
②
30(303
+
2708
2700
+
2902303
31200
=
Dual value
...
33600-31200
=
3300 2700
-
③
4
=
2
more
left
from
10x ,
;
6x2
+
(30 , 903
;
(2)
z
6
move
120
A
88
-
a
right
:
20
=3600
1080-2)
+
+
,
6(907
=
6x2
2502303
840
840
+
290(103
=
33608
infinity
:dual value
-
60=
40
,
-
r
100
=
9
;
187307
10x
=
?
2700
...
constraint
4
I
=
>
&
u
=
3042
W
ed
140
34800-33600
=
290 , 303
-
60=
40
=
3600
-
move
88
+
,
eso(01
=
30
2007
;
-
to
up
-
+30x2
=
·
: 840
-
·
C
-
ed
W
u
jo du T t brid so
②
original and
;
=
9
!
B
new
=
33600
-
3300
constraint
Find /
3
3x +3x2
,
;
left
more
;
3(8)
6
140
3x
-
r
7
120
88
+
,
20
32(10)
3x2
=
=
more
1103
,
330
=
366
right (7962 3e3
.
3 [787
3x ,
330
es0207 +7902118)
=
319007
=
+
3[63 337
=
+ 3x2
=
=
.
402
35866 57
.
.
-
a
Dual value
i
-
33600-31900
:
368
60=
40
C0
(70 , 63 323)
A
100
+
368
=
=
56 67
.
Dral value
=
56 67
.
336
-
-
·
C
340293400
...
-
ed
W
u
jo du T t brid so
②
③
⑧ bral value
constraints
Pral
;
value
34800-33600
:
3600
Dual
value
constraint s ;
;
Dual value
Oral value
=
:
4
Dual value
=
4
for
2700
0
=
=
35600-31900
368
Dual value
-
4
3300
33600-31280
=
3380
constraint e
-
=
=
-
-
.
67
:Dual
338
33863-33600
488
36
=
368
=
36 67
.
value
for 93
=
36 61
.
an
408