GEOMETRIC PROGRAMMING

Geometric programming is an optimization technique. Geometric programming is relatively a new technique, for solving linear / non linear optimization problems subject to linear / non linear constraints. Geometric programming has wide applications in many fields of engineering. e.g. (i) Machine Design – Pressure Vessel, Bearing (ii) Metal Cutting (iii) Dam Design – Civil Engineering

Geometric programming can handle easily objective function and constraints with any odd powers of variables. (Conventional N.L.P. methods easily handle only quadratic form of objective functions and constraints). Geometric programming is an efficient technique in complicated cases, where other techniques fail.

Geometric programming is based on “ The arithmetic- geometric mean inequality relationship”.

2121

2xxxx

Many times a problem, not directly solvable by Geometric Programming can be formulated / converted such that G. P. can be easily applicable.

At optimality each term of objective function is a fixed percentage of optimal value of function .

General Geometric Programming problem :

0

1 1

)(T

t

N

n

tanototo

noxCxyMin

m

T

t

N

n

anmtmt

mmtnxCts

1 1

/

m = 1, 2, ...M, xn > 0, n = 1, 2, ....N

Where mt ot m 1 1 1, ,

Cmt > 0, Cot > 0

amtn, aotn are unrestricted in sign.

Tm = Number of terms in mth constraint

To = Number of terms in objective function

T = Total number of terms = Tmm

M

0

If all are positive, then the problem is called

posynomial otherwise sinomial.

The degree of difficulty is defined based on

number of variables and total number of terms in a

problem.

Degree of difficulty = T - (N + 1)

A problem can be

(i) Unconstrained or

(ii) Constrained

Unconstrained problem (Polynomial)

To

t

N

n

an

T

tttto

otno

xCxPCxyMin1 11

)()(

The arithmetic-geometric mean inequality relationship :

32

313

5.12

312

1.22

311.. xxCxxCxxCyMinge

3/13

3/12

3/11321

2/12

2/1121

2121

31

31

31

21

21..

2

xxxxxxIIly

xxxxei

xxxx

For any positive numbers υ1, υ2, ... υT and set of

positive weight w1, w2, ...wT, such that

T

ttw

1

1

T

t

T

t

wttt

tvwv1 1

)(

Now t

otnotn

w

T

t t

N

n

ant

t

N

n

ant

t w

xC

w

xCw

1

11

To be minimized To be maximized

N

n

wa

n

T

t

w

t

tT

t t

N

n

ant

t

oT

ttotno

to

otn

xwC

w

xCw

1

.

11

1 1

If for n = 1, 2, 3, ... N, then a wotn tt

To

0

1

to o

otn

wT

t

T

t t

tN

n

ant w

CxC

1 11

to

wT

t t

t

wCy

0

s/t wtt

To

1

1Normality condition

a wotn tt

To

0

1

and

for n = 1, 2, ... N.

Orthogonality conditions

Application :32

313

5.12

312

1.22

311 xxCxxCxxCyMin

035.11.20333

1

321

321

321

wwwwww

www 321

3

3

2

2

1

1

www

wC

wC

wCy

ywxxC

ywxxC

ywxxC

332

313

25.1

2312

11.2

23

11 gives

x1 = ______

x2 = ______

Constrained Problem :

0,,

1...

1.../

......

321

32121

32111

3210232101

232221

131211

060504030201

xxxwhere

xxxK

xxxKts

xxxKxxxKyMin

AAA

AAA

AAAAAA

If only 2 constraints are considered, then the problem will be of ‘Zero’ degree of difficulty.

2111

0201

211102

02

01

01

2321131106020301

2221121105020201

2121111104020101

0201

...

0....0....0....

1

wwww

KKwK

wKy

AwAwAwAwAwAwAwAwAwAwAwAw

ww

GP

01 02 03 04 05 06

07 08 09 10

1311 12 14

2321 22 24

25 26 2724

01 1 2 3 02 1 2 3

03 1 2 3 4

11 1 2 3 4

21 1 2 3 4

22 1 2 3 4

. . . . . .

. . . .

/ . . . . 1

. . . .

. . . . 1

0

A A A A A A

A A A A

AA A A

AA A A

A A AA

j

Min y K x x x K x x x

K x x x x

s t K x x x x

K x x x x

K x x x x

All x

DGP

222122

22

2022

21

2021

1103

03

02

02

01

01

2221

11

030201

.

....

wwwwherew

wKw

wK

KwK

wK

wKyMax

ww

wwww

0....0......0......0......

1/

2722242114111003

262223211311090306020301

252222211211080305020201

242221211111070304020101

030201

AwAwAwAwAwAwAwAwAwAwAwAwAwAwAwAwAwAwAwAwAwAw

wwwts

2123

13

221 ..3..8..2 xxxxxxZMin

Numericals - GP

(1)

(2) 3 3 2 11 2 1 2 1 2 1 22 . . 4 . . . 8 . .Min Z x x x x x x x x

(3) 1 2 31 2 2 3 1 2 3 1 2 37 . . 3 . . 5. . . . .Min Z x x x x x x x x x x

(4) 1 1 11 2 3 2 3 1 2 1 340 . . . 40 . . 20 . 10 . .Min Z x x x x x x x x x

(5) 1 1 4 4 1.152.75 11.74 * 10/ 71.5 1

Min Z V f V fs t f