Engineering Optimization: Theory and Practice, Fourth Edition

8.9 Primal and Dual Programs in the Case of Less-Than Inequalities 517

λ 12 =λ 01 −λ 03 −λ 11 (E 7 )

λ 12 = 2 λ 11 −λ 02 (E 8 )

From Eqs. (E 7 ) nd (Ea 8 ) we have,

λ 12 =λ 01 −λ 03 −λ 11 = 2 λ 11 −λ 02

3 λ 11 −λ 02 +λ 03 =λ 01 (E 9 )

Adding Eqs. (E 5 ) nd (Ea 9 ) we obtain,

λ 21 = 4 λ 11 − 2 λ 01 (E 10 )

= 3 λ 02 − 2 λ 01 from Eq.(E 6 )

λ 11 =^34 λ 02 (E 11 )

Substitution of Eq. (E 11 ) n Eq. (Ei 8 ) ivesg

λ 12 =^32 λ 02 −λ 02 =^12 λ 02 (E 12 )

Equations(E 11 ) (E, 12 ) and (E, 7 ) iveg

λ 03 =λ 01 −λ 11 −λ 12 =λ 01 −^34 λ 02 −^12 λ 02 =λ 01 −^54 λ 02 (E 13 )

By substituting forλ 03 , Eq. (E 4 ) ivesg

λ 02 = 8 λ 01 − 4 (E 14 )

Using this relation forλ 02 , the expressions forλ 03 , λ 11 , λ 12 , andλ 21 can be obtained
as

λ 03 =λ 01 −^54 λ 02 = − 9 λ 01 + 5 (E 15 )

λ 11 =^34 λ 02 = 6 λ 01 − 3 (E 16 )

λ 12 =^12 λ 02 = 4 λ 01 − 2 (E 17 )

λ 21 = 4 λ 11 − 2 λ 01 = 22 λ 01 − 21 (E 18 )

Thus the objective function in Eq. (E 2 ) an be stated in terms ofc λ 01 as

v(λ 01 )=

(

1

λ 01

)λ 01 (

2

8 λ 01 − 4

) 8 λ 01 − 4 (

10

5 − 9 λ 01

) 5 − 9 λ 01

×

(

30 λ 01 − 51

6 λ 01 − 3

) 6 λ 01 − 3 (

40 λ 01 − 02

4 λ 01 − 2

) 4 λ 01 − 2

( 5 )^22 λ^01 −^21

=

(

1

λ 01

)λ 01 (

1

4 λ 01 − 2

) 8 λ 01 − 4 (

10

5 − 9 λ 01

) 5 − 9 λ 01

×( 5 )^6 λ^01 −^3 ( 01 )^4 λ^01 −^2 ( 5 )^22 λ^01 −^21

=

(

1

λ 01

)λ 01 (

1

4 λ 01 − 2

) 8 λ 01 − 4 (

10

5 − 9 λ 01

) 5 − 9 λ 01

( 5 )^32 λ^01 −^71 ( 2 )^4 λ^01 −^2