Geometric Programming Lecture
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Transcript of Geometric Programming Lecture
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