Engineering Optimization: Theory and Practice, Fourth Edition

526 Geometric Programming

If the maximum feed allowable on the lathe isFmax, we have the constraint

C 11 F≤ 1 (E 4 )

where

C 11 =Fm−ax^1 (E 5 )

Since the total number of terms is three and the number of variables is two, the degree

of difficulty of the problem is zero. By using the data

Km= 0. 10 , Kt= 0. 50 , tc= 0. 5 , th= 2. 0 , D= 6. 0 ,

L= 8. 0 , a= 140. 0 , b= 0. 29 , c= 0. 25 , Fmax= 0. 005

the solution of the problem [minimizef given in Eq. (E 2 ) subjectto the constraint

(E 4 ) can be obtained as]

f∗= 1 $.03 per piece, V∗= 23 ft 3 /min, F∗= 0. 0 05 in./rev

Formulation as a One-degree-of-difficulty Problem

If the maximum horsepower available on the lathe is given byPmax, the power required

for machining should be less thanPmax. Since the power required for machining can

be expressed asa 1 Vb^1 Fc^1 ,wherea 1 , b 1 , andc 1 are constants, this constraint can be

stated as follows:

C 21 Vb^1 Fc^1 ≤ 1 (E 6 )

where

C 21 =a 1 Pm−ax^1 (E 7 )

If the problem is to minimizef given by Eq. (E 2 ) ubject to the constraints (Es 4 ) nda

(E 6 ) it will have one degree of difficulty. By taking, Pmax= 2. 0 and the values ofa 1 ,

b 1 , andc 1 as 3.58, 0.91, and 0.78, respectively, in addition to the prev ious data, the

following result can be obtained:

f∗ 1 =$.05 per piece, V∗= 902 .0 ft/min, F∗= 0. 0 05 in./rev

Formulation as a Two-degree-of-difficulty Problem

If a constraint on the surface finish is included as

a 2 Vb^2 Fc^2 ≤Smax

wherea 2 ,b 2 , andc 2 are constants andSmaxis the maximum permissible surface rough-

ness in microinches, we can restate this restriction as

C 31 Vb^2 Fc^2 ≤ 1 (E 8 )