8.12 Applications of Geometric Programming 525
Next we chooseX(^2 )to be the optimal solution of OGP 1 [i.e.,X(o^1 pt)] and approx-
imateQ 1 andQ 2 about this point, solve OGP 2 , and so on. The sequence of optimal
solutions of OGPαas generated by the iterative procedure is shown below:
Xopt
Iteration number,α x 1 x 2
0 1.0 1.0
1 0.5385 0.4643
2 0.5019 0.5007
3 0.5000 0.5000
The optimal values of the variables for the CGP arex 1 ∗= 0. 5 andx 2 ∗= 0. 5. It can be
seen that in three iterations, the solution of the approximating geometric programming
problems OGPαis correct to four significant figures.
8.12 Applications of Geometric Programming
Example 8.7 Determination of Optimum Machining Conditions [8.9, 8.10] Geomet-
ric programming has been applied for the determination of optimum cutting speed and
feed which minimize the unit cost of a turning operation.
Formulation as a Zero-degree-of-difficulty Problem
The total cost of turning per piece is given by
f 0 ( X)=machining cost + tooling cost+ handling cost
=Kmtm+
tm
T
(Kmtc+Kt)+Kmth (E 1 )
whereKmis the cost of operating time ($/min),Kt the tool cost ($/cutting edge),
tmthe machining time per piece (min)=πDL/( 12 VF ),T the tool life (min/cutting
edge)=(a/VFb)^1 c/,tcthe tool changing time (minutes/workpiece),ththe handling
time (min/workpiece),Dthe diameter of the workpiece (in),Lthe axial length of the
workpiece (in.),V the cutting speed (ft/min),F the feed (in./revolution),a,b, andc
are constants in tool life equation, and
X=
{
x 1
x 2
}
=
{
V
F
}
Since the constant term will not affect the minimization, the objective function can be
taken as
f (X)=C 01 V−^1 F−^1 +C 02 V^1 c/−^1 Fb/c−^1 (E 2 )
where
C 01 =
Kmπ DL
12
and C 02 =
πDL(Kmtc+Kt)
12 a^1 c/
(E 3 )
