Chapter 4

[Page 782 ( continued )]

1.	Model must be resolved; Z = 43,310, do not implement; no; x 1 = x 2 = x 3 = x 4 = 112.5
3.	No effect; $740; x 1 = 22,363.636, x 3 = 43,636.364, x 4 = 14,000
5.	Add slack variables for 3 warehouses constraints; coefficients in objective function$9 for s 1 , $6 for s 2 , $7 for s 3 , solution does not change
7.	(a) max. Z = 190 x 1 + 170 x 2 + 155 x 3 ; s.t. 3.5 x 1 + 5.2 x 2 + 2.8 x 3 500, 1.2 x 1 + 0.8 x 2 + 1.5 x 3 240, 40 x 1 + 55 x 2 + 20 x 3 6,500, x i 0; (b) x 1 = 41.27, x 2 = 0, x 3 = 126.98, Z = 27,523.81
9.	(a) min. Z = 4 x 1 + 3 x 2 + 2 x 3 ; s.t. 2 x 1 + 4 x 2 + x 3 16, 3 x 1 + 2 x 2 + x 3 12, x i 0; (b) x 1 = 2, x 2 = 3, Z = $0.17
11.	(a) max. Z = 8 x 1 + 10 x 2 + 7 x 3 ; s.t. 7 x 1 + 10 x 2 + 5 x 3 2,000, 2 x 1 + 3 x 2 + 2 x 3 660, x 1 200, x 2 300, x 3 150, x 1 0, x 2 0, x 3 0; (b) x 1 = 178.57, x 3 = 150, Z = 2,478.57
13.	(a) max. Z = 7 x 1 + 5 x 2 + 5 x 3 + 4 x 4 ; s.t. 2 x 1 + 4 x 2 + 2 x 3 + 3 x 4 45,000, x 1 + x 2 6,000, x 3 + x 4 7,000, x 1 + x 3 5,000, x 2 + x 4 6,000, x 1 0, x 2 0, x 3 0, x 4 0; (b) x 1 = 5,000, x 2 = 1,000, x 4 = 5,000, Z = 60,000
15.	(a) max. Z = 1,800 x 1 a + 2,100 x 1 b + 1,600 x 1 c + 1,000 x 2 a + 700 x 2 b + 900 x 2 c + 1,400 x 3 a + 800 x 3 b + 2,200 x 3 c ; s.t. x 1 a + x 1 b + x 1 c = 30, x 2 a + x 2 b + x 2 c = 30, x 3 a + x 3 b + x 3 c = 30, x 1 a + x 2 a + x 3 a 40, x 1 b + x 2 b + x 3 b 60, x 1 c + x 2 c + x 3 c 50, x ij 0; (b) x 1 b = 30, x 2 a = 30, x 3 c = 30, Z = 159,000
17.	(a) min. Z = 1.7( x t 1 + x t 2 + x t 3 ) + 2.8( x m 1 + x m 2 + x m 3 ) + 3.25( x b 1 + x b 2 + x b 3 ); s.t. .50 x t 1 - .50 x m 1 - .50 x b 1 0; - .20 x t 1 + .80 x m 1 - .20 x b 1 0, - .30 x t 2 - .30 x m 2 + .70 x b 2 0, - .30 x t 2 + .70 x m 2 - .30 x b 2 0, .80 x t 2 - .20 x m 2 - .20 x b 2 0, .50 x t 3 - .50 x m 3 - .50 x b 3 0, .30 x t 3 - .70 x m 3 - .70 x b 3 0, - .10 x t 3 - .10 x m 3 + .90 x b 3 0, x t 1 + x m 1 + x b 1 1,200, x t 2 + x m 2 + x b 2 900, x t 3 + x m 3 + x b 3 2,400; x ij 0 (b) x t 1 = 600, x t 2 = 180, x t 3 = 1,680, x m 1 = 600, x m 2 = 450, x m 3 = 480, x b 1 = 0, x b 2 = 270, x b 3 = 240, Z = 10,123.50
19.	(a) max. Z = .02 x 1 + .09 x 2 + .06 x 3 + .04 x 4 ; s.t. x 1 + x 2 + x 3 + x 4 = 4,000,000, x 1 1,600,000, x 2 1,600,000, x 3 1,600,000, x 4 1,600,000, x 2 - x 3 - x 4 0, x 1 - x 3 0, x 1 0, x 2 0, x 3 0, x 4 0; (b) x 1 = 800,000, x 2 = 1,600,000, x 3 = 800,000, x 4 = 800,000, Z = 240,000
21.	(a) max. Z = 7.8 x 11 + 7.8 x 12 + 8.2 x 13 + 7.9 x 14 + 6.7 x 21 + 8.9 x 22 + 9.2 x 23 + 6.3 x 24 + 8.4 x 31 + 8.1 x 32 + 9.0 x 33 + 5.8 x 34 ; s.t. 35 x 11 + 40 x 21 + 38 x 31 9,000, 41 x 12 + 36 x 22 + 37 x 32 14,400, 34 x 13 + 32 x 23 + 33 x 33 12,000, 39 x 14 + 43 x 24 + 40 x 34 15,000, x 11 + x 12 + x 13 + x 14 = 400, x 21 + x 22 + x 23 + x 24 = 570, x 31 + x 32 + x 33 + x 34 = 320, x ij 0; (b) x 11 = 15.385, x 14 = 384.615, x 22 = 400, x 23 = 170, x 31 = 121.212, x 33 = 198.788, Z = 11,089.73
23.	(a) min. Z = 1.7 x 11 + 1.4 x 12 + 1.2 x 13 + 1.1 x 14 + 1.05 x 15 + 1.0 x 16 + 1.7 x 21 + 1.4 x 22 + 1.2 x 23 + 1.1 x 24 + 1.05 x 25 + 1.7 x 31 + 1.4 x 32 + 1.2 x 33 + 1.1 x 34 + 1.7 x 41 + 1.4 x 42 + 1.2 x 43 + 1.7 x 51 + 1.4 x 52 + 1.7 x 61 ; s.t. x 11 + x 12 + x 13 + x 14 + x 15 + x 16 = 47,000, x 12 + x 13 + x 14 + x 15 + x 16 + x 21 + x 22 + x 23 + x 24 + x 25 = 35,000, x 13 + x 14 + x 15 + x 16 + x 22 + x 23 + x 24 + x 25 + x 31 + x 32 + x 33 + x 34 = 52,000, x 14 + x 15 + x 16 + x 23 + x 24 + x 25 + x 32 + x 33 + x 34 + x 41 + x 42 + x 43 = 27,000, x 15 + x 16 + x 24 + x 25 + x 33 + x 34 + x 42 + x 43 + x 51 + x 52 = 19,000, x 16 + x 25 + x 34 + x 43 + x 52 + x 61 = 15,000, (b) x 11 = 12,000, x 13 = 25,000, x 14 = 8,000, x 15 = 2,000, x 33 = 2,000, x 34 = 15,000, Z = $80,200; (c) x 16 = 52,000, Z = $52,000
25.	(a) add x ss 150, x ww 300, x cc 250; x nc = 700, x nw = 0, x sw = 150, x ss = 150, x es = 900, x wc = 250, x ww = 300, x cc = 250, x cw = 250, x ws = 50, Z = 20,400; (b) changes demand constraints from 1,200 to = 1,000; x nc = 400, x nw = 300, x sw = 150, x ss = 150, x ec = 50, x es = 850, x wc = 300, x ww = 300, x cc = 250, x cw = 250, Z = 21,200
27.	(a) max. Z = 2 x 1 + 4 x 2 + 3 x 3 + 7 x 4 ; s.t. x 2 + x 4 300, 6 x 1 + 15 x 2 1,200, 5 x 3 + 12 x 4 2,400, x i 0; (b) x 1 = 200, x 3 = 480, Z = 1,840
29.	(a) max. Z = 35 x 1 + 20 x 2 + 58 x 3 ; s.t. 14 x 1 + 12 x 2 + 35 x 3 35,000, 6 x 1 + 3 x 2 + 12 x 3 20,000, x 1 0, x 2 0, x 3 0; (b) x 1 = 2,500, Z = 87,500
31.	(a) min. Z = 15,000 x 1 + 4,000 x 2 + 6,000 x 3 ; s.t. x 3 / x 2 2/1, 25,000 x 1 + 10,000 x 2 + 15,000 x 3 100,000, (15,000 x 1 + 3,000 x 2 + 12,000 x 3 )/ (10,000 x 1 + 7,000 x 2 + 3,000 x 3 ) 2/1, (15,000 x 1 + 4,000 x 2 + 9,000 x 3 )/(25,000 x 1 + 10,000 x 2 + 15,000) x 3 .30, x 2 7, x 1 0, x 2 0, x 3 0; (b) x 2 = 2.5, x 3 = 5.0, Z = 40,000; (c) x 4 , no effect
33.	(a) ; s.t. x 1 + y 1 + 20,000 - c 1 = 60,000, c 1 + y 2 + 30,000 - c 2 = 60,000 + y 1 , c 2 + y 3 + 40,000 - c 3 = 80,000 + y 2 , c 3 + y 4 + 50,000 - c 4 = 30,000 + y 3 , c 4 + y 5 + 80,000 - c 5 = 30,000 + y 4 , c 5 + y 6 + 100,000 - c 6 = 20,000 + y 5 , x 1 + y 6 c 6 , x 1 , y i , c i 0; (b) x 1 = 70,000, y 3 = 40,000, y 4 = 20,000, y 1 = y 2 = y 5 = y 6 = 0, c 1 = 30,000, c 5 = 30,000, c 6 = 110,000, Z = $10,700; (c) x 1 = 90,000, y 3 = 20,000, c 1 = 50,000, c 2 = 20,000, c 5 = 50,000, c 6 = 130,000, Z = $9,100
35.	(a) max. Z = .7 x cr + .6 x br + .4 x pr + .85 x ar + 1.05 x cb + .95 x bb + .75 x pb + 1.20 x ab + 1.55 x cm + 1.45 x bm + 1.25 x pm + 1.70 x am ; s.t. x cr + x cb + x cm 200, x br + x bb + x bm 300, x pr + x pb + x pm 150, x ar + x ab + x am 400, .90 x br + .90 x pr - .10 x cr - .10 x ar 0, .80 x cr - .20 x br - .20 x pr - .20 x ar 0, .25 x bb + .75 x cb - .75 x pb - .75 x ab 0, x am = 0, .5 x bm + .5 x pm - .5 x cm - .5 x am 0, x ij 0; (b) x cm = 125, x ar = 300, x cr = 75, x bb = 300, x pm = 125, x ab = 100, Z = 1,602.50 [Page 783]
37.	(a) min. Z = 40 x 1 + 65 x 2 + 70 x 3 + 30 x 4 ; s.t. x 1 + x 2 = 250, x 3 + x 4 = 400, x 1 + x 3 = 300, x 2 + x 4 = 350, x 1 0, x 2 0, x 3 0, x 4 0; (b) x 1 = 250, x 3 = 50, x 4 = 350, Z = 24,000
39.	39.(a) max. Z = 175 (7 x 1 ); s.t. 8 x 1 + 5 x 2 + 6.5 x 3 3,000, x 1 + x 2 + x 3 120, 90(7 x 1 ) 10,000, 7 x 1 - 12 x 2 = 0, 12 x 2 - 10 x 3 = 0, 7 x 1 - 10 x 3 = 0, x 1 0, x 2 0, x 3 0; (b) x 1 = 15.9, x 2 = 9.3, x 3 = 11.1, Z = 19,444.44
41.	(a) min. Z = 3 x 13 + 4 x 14 + 5 x 12 + 2 x 34 + 7 x 45 + 8 x 25 ; s.t. x 13 + x 14 + x 12 = 5, x 45 + x 25 = 5, x 13 = x 34 , x 14 + x 34 = x 45 , x 12 = x 25 , x ij 0; (b) x 14 = 5, x 45 = 5, Z = 55,000
43.	(a) min. Z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ; s.t. 3 x 1 + 2 x 2 + 2 x 3 + x 4 = 700, x 3 + 2 x 4 + x 5 = 1,200, x 2 + x 5 + 2 x 6 = 300, x i 0; (b) x 2 = 50, x 4 = 600, x 6 = 125, Z = 775; (c) min. Z = 4 x 1 + x 2 + 2 x 3 + 0 x 4 + 6 x 5 + 5 x 6 ; s.t. 3 x 1 + 2 x 2 + 2 x 3 + x 4 700, x 3 + 2 x 4 + x 5 1,200, x 2 + x 5 + 2 x 6 300, x i 0; x 2 = 300, x 4 = 600, Z = 300
45.	(a) max. Z = 4 x 1 + 8 x 2 + 6 x 3 + 7 x 4 - 5 y 1 - 6 y 2 - 7 y 3 ; s.t. s = 2,000, x 1 s , s 1 = s - x 1 + y 1 , s 1 10,000, x 2 s 1 , s 2 = s 1 2 x 2 + y 2 , s 2 10,000, x 3 s 2 , s 3 = s 2 - x 3 + y 3 , s 3 10,000, x 4 s 3 , x i 0, y i 0, s i 0; (b) x 1 = 0, x 2 = 10,000, x 4 = 10,000, y 1 = 8,000, y 2 = 10,000, s = 2,000, s 1 = s 2 = s 3 = 10,000, Z = 50,000
47.	(a) max. Z = 130 x 1a + 150 x 1b + 90 x 1c + 275 x 2a + 300 x 2b + 100 x 2c + 180 x 3a + 225 x 3b + 140 x 3c + 200 x 4a + 120 x 4b + 160 x 4c ; s.t. x 1a + x 1b + x 1c = 1, x 2a + x 2b + x 2c = 1, x 3a + x 3b + x 3c = 1, x 4a + x 4b + x 4c = 1, 1 x 1a + x 2a + x 3a + x 4a 2, 1 x 1b + x 2b + x 3b + x 4b 2, 1 x 1c + x 2c + x 3c + x 4c 2, x ij 0; (b) x 1a = 1, x 2b = 1, x 3b = 1, x 4c = 1, Z = 815; (c) max. Z = 130 x 1a + 150 x 1b + 90 x 1c + 275 x 2a + 300 x 2b + 100 x 2c + 180 x 3a + 225 x 3b + 140 x 3c + 200 x 4a + 120 x 4b + 160 x 4c ; s.t. x 1a + x 1b + x 1c 1, x 2a + x 2b + x 2c 1, x 3a + x 3b + x 3c 1, x 4a + x 4b + x 4c 1, x 1a + x 2a + x 3a + x 4a = 1, x 1b + x 2b + x 3b + x 4b = 1, x 1c + x 2c + x 3c + x 4c = 1, x ij 0; x 2a = 1, x 3b = 1, x 4c = 1, Z = 660
49.	Z values: A = 1.000, B = 1.000, C = 1.000; all 3 efficient
51.	(a) min. Z = x ; s.t. 150 x = 650 + y 1 , 150 x + y 1 = 450 + y 2 , 150 x + y 2 = 600 + y 3 , 150 x + y 3 = 500 + y 4 , 150 x + y 4 = 700 + y 5 , 150 x + y 5 = 650 + y 6 , 150 x + y 6 = 750 + y 7 , 150 x + y 7 = 900 + y 8 , 150 x + y 8 = 800 + y 9 , 150 x + y 9 = 650 + y 10 , 150 x + y 10 = 700 + y 11 , 150 x + y 11 500; (b) Z = x = 4.45, y 1 = 18.18, y 2 = 236.36, y 3 = 304.54, y 4 = 472.72, y 5 = 440.91, y 6 = 459.09, y 7 = 377.27, y 8 = 145.45, y 9 = 13.63, y 10 = 31.81, y 11 = 0
53.	(a) max. Z = y ; s.t. y x 1 = 0, y 2 x 2 = 0, y 2 x 3 = 0, 10 x 1 + 8 x 2 + 6 x 3 960, 9 x 1 + 21 x 2 + 15 x 3 1,440, 2 x 1 - 3 x 2 - 2 x 3 60, - 2 x 1 + 3 x 2 + 2 x 3 60, x i 0, y 0; (b) x 1 = x 2 = x 3 = y = 20; (c) remove balancing requirement, x 1 = x 2 = x 3 = y = 32
55.	max. Z = 850 x 1 + 600 x n + 750 x s + 1,000 x w ; s.t. x 1 + x n + x s + x w = 18, x 1 + x n + x s + x w + y 1 + y n + y s + y w = 60, 400 y 1 + 100 y n + 175 y s + 90 y w 9,000, 10 y 1 25, 5 y n 10, 5 y s 10, 5 y w 10, x 1 6, x n 6, x s 6, x w 6, x i 0, y i 0; x 1 = 6, x n = 0, x s = 6, x w = 6, y 1 = 14.44, y n = 10, y s = 7.56, y w = 10, Z = 15,600 (multiple optimal)
57.	(a) max. Z = .85 x 1 + .90 x 2 - y 1 - y 2 ; s.t. x 1 5,000 + 3 y 1 , x 2 4,000 + 5 y 2 , .60 x 1 + .85 x 2 + y 1 + y 2 16,000, x 1 .3( x 1 + x 2 ), x 1 .6( x 1 + x 2 ), x 1 0, x 2 0, y 1 0, y 2 0; (b) x 1 = 5,458.128, x 2 = 12,735.63, y 1 = 152.709, y 2 = 1,747.126, Z = 14,201.64
59.	(a) min. Z = SS (ranking) x ij , s.t. S x ij hr., S x ij = project hr., S (hourly rate) x ij budget; (b) x A3 = 400, x A4 = 50, x B4 = 250, x B5 = 350, x C4 = 175, x C7 = 274.1, x C8 = 50.93, x D2 = 131.7, x D7 = 15.93, x E1 = 208.33, x E8 = 149.07, x F1 = 291.67, x F2 = 108.3, x F6 = 460, Z = $12,853.33
61.	x 1 = 0, x 2 = 4, x 3 = 18.4, x 4 = 6.4, x 5 = 24.8, y 1 = 72.22, y 2 = 72.44, y 3 = 64.95, y 4 = 62.34, y 5 = 52.24, y 6 = 38.9, y 7 = 28.53, y 8 = 43.35, Z = $360,196
63.	min. Z = S x ij (priority ij ) s.t. available slots j , U11B: 3-5M, U11G: 3-5T, U12B: 3-5T, U12G: 3-5M, U13B: 3-5T, U13G: 3-5M, U14B: 5-7M, U14G: 3-5M, U15B: 5-7T, U15G: 3-5M, U16B: 5-7T, U16G: 5-7T, U17B: 5-7M, U17G: 5-7T, U18B: 3-5T, U18G: 5-7M, Z = 27
65.	1-D (1 hr.), 2-E (1 hr.), 3-F (2 hr.), 3-H (6 hr.), 4-I (1 hr.), 5-J (1 hr.), A-5 (8 hr.), B-4 (5 hr.), C-6 (8 hr.), G-7 (2 hr.), Z = 35 hr.; 6 crews originate in Pittsburgh, 4 in Orlando
67.	min. Z = .41 x 14 + .57 x 15 + .37 x 24 + .48 x 25 + .51 x 34 + .60 x 35 + .22 x 46 + .10 x 47 + .20 x 48 + .15 x 56 + .16 x 57 + .18 x 58 ; s.t. x 14 + x 15 24,000, x 24 + x 25 18,000, x 34 + x 35 32,000, x 14 + x 24 + x 34 48,000, x 15 + x 25 + x 35 35,000 ( x 14 + x 24 + x 34 )/2 = x 46 + x 47 + x 48 , ( x 15 + x 25 + x 35 )/2 = x 56 + x 57 + x 58 , x 46 + x 56 = 9,000, x 47 + x 57 = 12,000, x 47 + x 58 = 15,000, x ij 0; x 14 = 24,000, x 24 = 18,000, x 34 = 6,000, x 35 = 24,000, x 47 = 12,000, x 48 = 12,000, x 56 = 9,000, x 58 = 3,000, Z = $39,450