Question

Question 3: Identify which of LP problems (1)--(4) has (x1,x2) = (20,60) as its optimal solution. (1) min z = 50xı + 100X2 s.t. 7x1 + 2x2 > 28 2x1 + 12x2 > 24 X1, X2 > 0 (2) max z = 3x1 + 2x2 s.t. 2x1 + x2 < 100 X1 + x2 < 80 X1 <40 X1, X2 > 0 (3) min z = 3x1 + 5x2 s.t. 3x1 + 2x2 > 36 3x1 + 5x2 > 45 X1, X2 20 (4) max z = 4x1 + 5x2 s. t. 4x1 + X2 51 5X1 +56X2 51 50 X1 > 30 X2 > 20 X1, X2 > 0

Answer 1

o min Z 50 x + 100M2 sot 7 X + 2x2 28 241 + 12x2 הל 24 4 Il 70 24 111 12 น Point of intersection 168 42 x + 12 ₂ 2 22 + 12 24 3.6 402, 144 A (0,14) = 1400 320 → optimal B (3-6,14) = c (12,0) = 600

(2) max z 3u, t 21 sot 274 tuz < 100 y tu § 80 4 L 4o O 74 12 2 100 80 26 50 80 21 pred solution i's optimal its Hence (24372) = (20,60) 100 max 2

13) min Z 3 a + 5 H sit 30 + 2% 736 341 + 5% 7 45 1 ใน 7 6 A 18 22 o с 4 12 15 90 11 A (0,18) 45 solution both optimal B (10,3) c (15,0) = 45

14 max Z + 72 40 60 -sh 2 + In 50 y > 30 " 720 ML 2,0 60 ro verere 30 To 에 Since ?? is non- bounded maximize problem hence no solution 1 So The which has (20,60) only LP solution ing problem (2).