Engineering Optimization: Theory and Practice, Fourth Edition

8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 507

Since lnvis expressed as a function of 4 alone, the value of 4 that maximizes lnv
mustbe unique (because the primal problem has a unique solution). The necessary
condition for the maximum of lnvgives

∂

∂ 4

( nl v)= −11[ln 100−ln( 2 − 11  4 ) ]+( 2 − 11  4 )

11

2 − 11  4

+ [ln 50 8 −ln( 8  4 − 1 )]+( 8  4 − 1 )

(

−

8

8  4 − 1

)

+ 2 [ln 20−ln( 2  4 )]+ 2  4

(

−

2

2  4

)

+ 1 [ln 300−ln( 4 )]+ 4

(

−

1

 4

)

= 0

This gives after simplification

ln

( 2 − 11  4 )^11

( 8  4 − 1 )^8 ( 2  4 )^2  4

− nl

( 100 )^11

( 50 )^8 ( 02 )^2 ( 003 )

= 0

i.e.,

( 2 − 11  4 )^11

( 8  4 − 1 )^8 ( 2  4 )^2  4

=

( 100 )^11

( 50 )^8 ( 02 )^2 ( 003 )

= 2130 (E 3 )

from which the value of∗ 4 can be obtained by using a trial-and-error process as
follows:

Value of∗ 4 Value of left-hand side of Eq. (E 3 )

2 / 11 = 0. 182 0.0

0.15

( 0. 35 )^11

( 0. 2 )^8 ( 0. 3 )^2 ( 0. 15 )

≃ 284

0.147

( 0. 385 )^11

( 0. 175 )^8 ( 0. 294 )^2 ( 0. 147 )

≃ 2210

0.146

( 0. 39 )^11

( 0. 169 )^8 ( 0. 292 )^2 ( 0. 146 )

≃ 4500

Thus we find that∗ 4 ≃ 0. 1 47, and Eqs. (E 2 ) iveg

∗ 1 = 2 − 11 ∗ 4 = 0. 385

∗ 2 = 8 ∗ 4 − 1 = 0. 175

∗ 3 = 2 ∗ 4 = 0. 294

The optimal value of the objective function is given by

v∗=f∗=

(

100

0. 385

) 0 85. 3 (

50

0. 175

) 0 75. 1 (

20

0. 294

) 0 94. 2 (

300

0. 147

) 0 47. 1

= 8. 5 × 2. 69 × 3. 46 × 3. 06 = 242