Engineering Optimization: Theory and Practice, Fourth Edition

798 Answers to Selected Problems

(b) 20 (c) 19 (d) 14 (e) 14 5.17(a)2.7814

(b)2.7183 5.18(a)2.7183 (b)2.7289 (c)2.7183

5.200.25 5.210.001257 5.220.00126 5.240.00125631

CHAPTER 6

6.1Min.f=P 0 ( 0. 5 u^21 + 0. 5 u^22 −u 1 u 2 −u 2 )

6 .2f ̃ 1 = 7. 0751 , f ̃ 2 = 47 .8087 wheref ̃=

3 fρl^4

Eh^2

6.4x 1 = 56. 567 , x 2 2 = 5. 974 6.5x 1 ∗ = 4. 5454 , x∗ 2 = 5. 4545

6.7f= 4250 x^21 − 0001 x 1 x 2 − 5002 x 1 x 3 + 5001 x^22 − 005 x 2 x 3 + 7505 x 32

− 0001 x 1 000 − 2 x 2 − 0003 x 3 ,X∗= { 0. 3241 , 0. 8360 , 0. 3677 }

6.9 X∗≈ { 1 , 1 } 6 .12 X∗= { 0. 9465 , 2. 0615 , 2. 9671 }

6.14f (z 1 , z 2 ) =− 5 + 1. 0429 z 1 − 0. 7244 z 2 + 0. 5 z^21 + 0. 5 z^22

6.16(a) yes (b)no 6.19(a) 60,002.0 (b)241.3729

6.30 X 1 = { 2 ,− 1 ,− 8 }X 2 = { 2 ,− 0. 7 ,− 8 }X 3 = { 2. 26 ,− 0. 85 ,− 8 }

X 4 = { 2. 15 ,− 0. 74 ,− 7. 755 } 6.35 X 2 = { 5. 57 , 0 }, f 2 >f 1

6.38x∗ 1 = 1 , x 2 ∗= 1 6 .45 X 5 = { 2. 0869 , 1. 7390 }, f 5 = − 8. 3477

6.47 X∗= {− 2 , 1 , 4 } 6 .48x∗= 1. 1423 , y∗= 0. 8337

6.50x∗ 1 = 1. 698105 , x 2 ∗= 0. 883407 6.52 X∗= { 5 ,− 8 }

6 .55(a)no (b)yes

CHAPTER 7

7.1 X∗= { 2 , 3 },f∗= − 50

7.6(a)Min.f= 12 x^21 + 03 x 22 − 8 x 1 x 2 − 22 x 1 + 06 x 2 − 87 , x 2 + 2 = 0 ,

x 1 +x 2 ≤ 0

(b)Min.f= 18 x 1 8 − 6 x 2 − 07 , x 2 + 2 = 0 , x 1 +x 2 ≤ 0

7 .8 X∗= { 1. 74558 , 1. 95265 }, f∗= − 9. 23478

7.11Max.f= 3. 5483 d^4 w, 2. 2227 × 10 −^6 d^4 − 1 ≤ 0 , 0. 2223 d^2 w − 150 ≤ 0 ,

d≤ 25 7.13− 8 s 1 + 4 s 2 < 0 , s 1 + 2 s 2 ≤ 0 , −s 1 ≤ 0

7 .15 X∗= { 0. 75 , 4. 56249 }, f∗= 0. 25391

7.18 X∗= { 3 , 3 },f∗= 81 7.21x 1 ∗= 4 cm 2 , x∗ 2 =x 3 ∗= 2 cm 1

7.23(a)φk= 2 x−rk

(

1

2 −x

+

1

x− 10

)

,