Free Essay

Math for Economics Nyu

In:

Submitted By wangchao1994
Words 2142
Pages 9
foIntro to Linear Programming

Motivation:
A baker has 150 kg flour, 22 kg sugar, and 27.5 kg of butter to make the following two type of cakes:

1 dozen of this cake needs
3 kg flour
1 kg sugar
1 kg butter

1 dozen of this cake needs
6 kg flour
0.5 kg sugar
1 kg butter

Profit from 1 dozen is $20

Profit from one dozen is $30

How many dozen from each cakes should the baker make to maximize his profit?

Geometric solution:

Geometric solution:

The contour of z with the highest function value touches the feasible set at the point B.
To find the coordinates of B we intersect the flour and butter border:
3x1+6x2=150
x1+x2=27.5 and get x1=5 and x2=22.5. Hence the maximum profit is 775 dollars.

x1 C x2 Ä 450 x1 x2

x1 x1 x2

0

x2

0

2x1 C 3x2
3x1 C 2x2 Ä 1200 x1 C 2x2 Ä 800 x1 C x2 Ä 450
0; x2 0:

Setting upx a problem:
1

x1

x2

3;500x1 C 3;000x2

.x1 ; x2 /

x1 C x2 Ä 450 x1 x2

x1 x1 0

12;000x1 C 4;000x2

x2 x2 0

12;000x1 C 4;000x2

2x1 C 3x2
3x1 C 2x2 Ä 1200
4;000x1 C 4;000x2 x1 C 2x2 Ä 800 x1 C x2 Ä 450
0; x2 1;000x1 C 5;000x2
0:

x1

x1

x2

x1

32;000
20;000
x1

3;500x1 C 3;000x2
12;000x1 C 4;000x2
4;000x1 C 4;000x2
1;000x1 C 5;000x2
0; x2 0: x1 Solution:

48;000

0

x2

0

48;000
32;000
20;000 x2 3;500x1 C 3;000x2

.x1 ; x2 /

12;000x1 C 4;000x2
12;000x1 C 4;000x2

48;000

4;000x1 C 4;000x2

32;000

1;000x1 C 5;000x2 x1 x1

x2
3;500x1 C 3;000x2
12;000x1 C 4;000x2
4;000x1 C 4;000x2
1;000x1 C 5;000x2
0; x2 0:

20;000 x1 0

48;000
32;000
20;000

x2

0

b1
6 : 7
D4 : 5
:
bm

c1
6 : 7
D4 : 5
:
cn

Rm

2 3 x1 6 : 7
D4 : 5
:
xn

n

Generalizing:
2 3 b1 6 : 7
D4 : 5
:
bm

Rm

2

Rn

3

m

Rn

f .x11; : : : ; xn / D c1 x1 C c2 x2 C c 6 : 7
D4 : 5
:
cn

Rn

m

a x C a12 x2 C a22 x2 C

Ca1n xn Ä b1
Ca2n xn Ä b2

:
:
:

a c1 x C c2 a C C f .x1 ; : : : ; xn / D m1x1 1 C x2m2 x2 C cn xn a11 x1 C a12 x2 C xj Ca1n xn Ä b1 j
0
a21 x1 C a22 x2 C
Ca2n xn Ä b2
:
:
:
am1 x1 C am2 x2 C
Camn xn Ä bm

Camn xn Ä bm

D 1; : : : ; n
T

f. / D

xj

A Ä

j D 1; : : : ; n

0

C cn xn

A D aij

n

2 11 1
3
x1 x C a 6 : 21 1
7
D 4 : 5 Rn
:
xn

n

A D aij

n

T

f. / D

A Ä

F
F

f. / D

f. / D

f. /
F

f. /

2F

h. /

h. /

h. /

F
2F

h. /

ai1 x1 C

b
Ci

Cai1 xn ai n x 1

ai n xn Ä

ai1 x1

C ai n xn

bi

ai n xn Ä bi

ai1 x1 ai1 x1 C

C ai n xn D bi

ai1 x1 C
C ai n xn Ä bi ai n xn Ä bi

bi

C ai n xn D bi

In a linear programming what can a x C Ca x Äb go wrong? a x a x Ä b ai1 x1 C ai1 x1

i1 1
F
F i1 1

in n

i

in n

i

F
5x
xÄ3
Ä 4
0

F

5x xÄ3 5x x Ä 4
5x
xxx 3 3 0
ÄÄ

x Äx3ÄÄ 4 x x 4

x

x

x

xÄ3

x

xÄ3

x

x x x

0

3

4

xÄ3

4

x

0

5x x 0

x

x

0

x

3

x

4

4

5x xÄ3 x 0 x 5x

x

3

x x F

3

5x

x

5x xÄ3 x 0

4

5x

x

5x

3

0

x x 0

5x x Ä 3
5x
xÄ3 0 x xx Ä 3
0

5x

0

0

x

5x x xÄ3 x xÄ 4 x x0 3
Ä

x

F

F

F 1F

F
F
FF

F

F
1 1
F

F

1

F
F

1

F f .x1 ; x2 / D 2x1 C 3x2

F

F
F

x1
Ä 30 x2 Ä 20 x1 C 2x2 Ä 54 f .x ; x / D 2x1 C 3x2 x1 0;1x2 2 0:

Practice:

f .x1 ; x2 / D 2x1 C 3x2

Ä f .x1 ; x2 / Dx1 1 C 3x2 30
2x

f .x1 ; x2 / D 2x1 C x2 2 20
3x Ä x1 x1 x1 0; x2

54 xx C 2x2 ÄÄÄ 30
1x1
30
1
0: xx2 Ä 20
2 Ä 20
C 2x Ä 54 x x1C 2x2 2 Ä 54 1 C 2x2 D 54 x1 0; x
0; x22

0:
0:

x1 x2 D 20

x1
Ä 30 x2 Ä 20 x1 C 2x2 Ä 54
0; x2 0:

.14; 20/

x1 D 30

x2 D 12

1

.14; 20/
C 2x2 D 54

C 2x2 D 54
2x D 54
2

x2 D 20

x D 20 x22 D 20

.14; 20/
.14; 20/

.14; 20/ x1 Dx1 C 2x2 D 12
30
x2 D 54

x1xDD 30 x2 x 12 12
30
DD
1
2
1

x2 D 20

x1 D 30

x2 D 12

1

F

1

F

F

Practice: f .x1 ; x2 / D 2x1 C 3x2

f .x1 ; x2 /x D 2x1 C 3x2
2

Ä 30 x2 Ä 2020)
(0,
x1 C 2x2 Ä 54
0; x2 0: x1 x1

Ä 30 x2 Ä 20
(30, 12) x1 C 2x2 Ä 54
0; x2 0:

x1

x(14, 20)
1

(0, 0)

(30, 0)

(x 1, x 2)

2x 1 + 3x 2

(0, 0)
(30, 0)
(30, 12)
(14, 20)
(0, 20)

0
60
96
88
60

x1

.14; 20/

2x2 D 54

x2 D 20

f .x ; x /

x1 C 2x2 D 54

.x1 ; x2 /

2
.14; 20/ D 112 x x1 .30; 12/
D 30

x2 D 20

2

.30; 12/

f .x1 ; x2 /

x1 D 30

x2

x2

x2 D 12

1
(0, 20)

(x 1, x 2)
(30, 12)

(0, 0)

2x 1 + 3x 2

(0, 0)
(30, 0)
(30, 12)
(14, 20)
(0, 20)

(14, 20)

0
60
96
88
60

f (x 1 , x 2 ) = 96 x1 x1

(30, 0)

f (x 1 , x 2 ) = 30

f (x 1 , x 2 ) = 60

f .x1 ; x2 ; x3 / D 2x1 C 3x2 C 4x3

f .x1 ; x2 /
.30; 12/

.x1 ; x2 / x f .x1 ;1x2 / 0; x2

.30; 12/

x1 C x2 C x3 Ä 50 x1 C 2x2 C 4x3 Ä 80
0; x3 0:

R3 x2 R3

f ( x 1 , x 2 ) = 96

Introduction to Duality Theory x1 f ( x 1 , x 2 ) = 30

f ( x 1 , x 2 ) = 60

f ; x ; / D 2x C 3x C 4x
An economist.xwill xoftenCask:50
What happens to the optimal solution if the availability x C x x Ä of the resources changes?Ä 80 x C 2x C 4x
1

2

2

3

1

0; x2

1

1

x1

2

3

2

3

0; x3

3

0:

R3

Back to baker’s example: Suppose the baker came across an extra kilo of flour
R
that had been hidden away in storage.
How much would this extra kilo add to his maximum profit?
How much would an extra kilo of sugar contribute to profit?
How much would an extra kilo of butter contribute to profit?
3

Answer: One extra kilo of flour will shift the point B up:
We need to solve
3x1+6x2=151
x1+x2=27.5
We get x1=14/3 and x2=137/6 hence the new optimum profit increases by u1*= 10/3 dollars.
Similar computation will show that :
One extra kilo of sugar doesn’t affect the optimum profit u2*=0
One extra kilo of flour increases the optimum profit by u3*=10 dollars.

Properties of marginal profits: u1*, u2*, u3*
Multiply the inequality constraints by these special numbers

The Dual Problem
Suppose the baker gets tired of running business. An entrant wants to take over and buy all the ingredients. The baker intend to charge u1 dollars for 1 kg flour u2 dollars for 1 kg of sugar u3 dollars for 1 kg of butter
Because 1 dozen of Cake A requires 3 kg flour, 1 kg of sugar and 1 kg of flour and the baker makes 20 dollars in profit from 1 dozen of cakes he wants
3u1+1u2+1u3 ≥ 20
Similarly,
6u1+0.5u2+1u3 ≥ 30
Presumably, the entrants wants to buy the baker’s resources as inexpensively as
Possible. Hence we try to solve the following problem min z=150u1+22u2+27.5u3

The Dual to the Baker’s Problem x3 x3

x1 x1 M D 2x1 C x2
M D 2x1 C x2

x x11 x4 x4 2x1 C x2
2x1xCCx2x C x
2
D 8
1
2
3
x1 C 2x2 C x3
D 8
3x1 C 2x2
C x4 D 24
3x C 2x
C x4 D 24
0; x2 1 0: 2
0; x2 0:
2x1 x2 C M D 0
2x1 x2 C M D 0

x
2x 1
2 11
1
6 ✐
3
64 ✐
4 3
2
2

x2 x3 x2 x3 2
1
2
1
2
0
2
0
1
0
1
0
2

x4 x4 0
0
1
1
0
0
2

x x x1 x2 2 x3x3 x4x4
221
8
1
00 8 3 1 1 1 3
6
3
3
66
66 1 2 2
0 1 1
64
0
1
3 3
3 3
4
1
2
00 1 3 0 0 2 3
3
3 x1 8 x1 DD 8

M
3
M
0 3
8
0
8 7
0 24 5
7
0 24 5
1
0
1
0

MM 3 3
0 0 16 16 7
7 7
7
8
0 0 87 7
5 5
1 16 16
1

x2x2 D 0
D0

Generalizing duality

Rn n
R
A Ä
A Ä
g. / D T T
g. / D

m
RRm
n
RRn

m mn n
A A
T
f . f/.D D T
/

Rm

AT T
A

P
P
f. / D T T f . ÄD
/
A
A Ä

P
P
g. / D T
g. / D
AT T
A

ci

ci

xi

T

xi bj A Ä

AT

Rm

yj

bj

Practice:

AT

A Ä
A Ä

A

x1 x1 yj

bj

yj

bj

yj

T

5x1 C 7x2
2x1 C 3x2 Ä 25
5x17x 7x2 4x Ä 16
C C
1
2
2x1 C 3x2 Ä 25 x1 C 9x2 Ä 21
7x C 4x Ä 16
0; x21 0: 2

x1 C 9x2 Ä 21
0; x25y10: 16y2 C 21y3
2
C

2y C 7y C

y

5
7

25y1 C1 16y2 C 221y3 3
3y C 4y C 9y3
2y1 C1 7y2 C 2 y3 5 y1 0; y2 C y3 C 9y
3y1 0; 4y2 0: 3 7 y1 0; y2 0; y3 0:
T

h. / D

T
T
h. / A Ä
D
T
A Ä

F. / D

:
:
T

F .. / AT /T T
D
. AT /T

:
:

G. / D

T

G. A/ D T
Ä
A Ä

:

:

AT

A Ä

Duality Theorems:

P
P
F
F
NN

A D

1

A D

1

x1

FF
NN

y1

5x1 C 7x2
2x1 C 3x2 Ä 25
7x1 C 4x2 Ä 16
1
F xF C 9x2 Ä 21
0; x2 0: P P P
P

F

f .N /.N / g.Ng.N / f D D/
25y C 16y2 C 21y3
P P P P1 f .N2y1/ C g.N2/ C y3
D D f/.N g.N /7y
3y1 C 4y2 C 9y3

0; y2

S

0; y3

0:

Sf g
S
f g
S

h. / D
AT Ä

F. / D
. AT /T

f g

N
N

N

5
7

F

P

N

S D f 2 Rm W D A m f g

P

S Df 2R W DA

g

T

:
T

G. / D
A Ä

:
T

:

Duality Theorems:
P

F

P

F
F

N

1

F
N

A D

P
P
f .N / D g.N /
P
P f .N / D g.N /

N

f g

S f g

S

N

S D f 2 Rm W D A

g

g

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