Optimization_1

Engineering Analysis

Advanced Engineering Mathematics

E. Kreyszig

Chapter 22



Continued



g(t) = 0 to find t

Method of steepest descent



Continued

Example

x = x1i + x2j

f = (/ x i + / y j)f



New x1 = (1-2t)x1

New x2 = (1-6t)x2

63

13.9

3.08

0.68

f (x)=(x12+3x22)

t = (x12+9x22)/(x12+54x22)

z(t) = (1-2t)x1i + (1-6t)x2j

f = x12 + 3x22

0

1

2

3

x1

x2



Continued

Linear programming



Example

Continued

Objective function to be maximized

Constraints



Feasibility region

2x1+ 8x2 = 60

5x1 + 2x2 = 60

z = 40x1+ 88x2

(10, 5)

Max hourly revenue



2x1+ 8x2 60

5x1+ 2x2 60

Inequalities

Slack variables

O

C

B

A
Vertices A, B, C and O are basic feasible solutions (defined as having n-m zero variables) Vertex B is optimal solution Any point inside blue region represents a feasible solution


Continued
All bj nonnegative. If a bj < 0, multiply equation by -1 x1, .., xn include slack variables ( for which cj of these variables in f are zero)


Continued

1.psd


Continued

Vertex O

2.psd


Continued

Operation O1

Operation O2

3.psd4.psd


Vertex A

5.psd6.psd


Row 1 + 20 Row 2

Row 3 2.5 Row 2

x1=30 x2=0 x3=0 x4= -90 z=1200

Basic variables

0 72 20 0 1200

0 -18 -2.5 1 -90

T1

7.psd


Continued

Operation O1

Operation O2

Basic variables

9.psd10.psd11.psd


Vertex B

O

C

B

A

12.psd13.psd14.psd

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