Chapter 9 Sample+problem+formulations

Chapter 9 Sample+problem+formulations - Let y 1 = 1 if NY...

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Problem 1 (page 502). Let x i = 1 if player i starts x i = 0 otherwise Then appropriate IP is max z = 3x 1 + 2x 2 + 2x 3 + x 4 + 3x 5 + 3x 6 + x 7 s.t. x 1 + x 3 + x 5 + x 7 4 (guards) x 3 + x 4 + x 5 + x 6 + x 7 2 (forwards) x 2 + x 4 + x 6 1 (center) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 = 5 3x 1 + 2x 2 + 2x 3 + x 4 + 3x 5 + 3x 6 + 3x 7 10 (BH) 3x 1 + x 2 + 3x 3 + 3x 4 + 3x 5 + x 6 + 2x 7 10 (SH) x 1 + 3x 2 + 2x 3 + 3x 4 +3x 5 + 2x 6 + 2x 7 10 (REB) x 6 + x 3 1 -x 4 -x 5 + 2 2y x 1 2(1-y) If x 1 =1 (x 1 >0), then y must be equal to 0 , and it will enforce that x 4 + x 5 2. If x 1 =0, then y must be equal to 1, and it will enforce the constraint x 4 + x 5 0. Note that setting M=2 is sufficient here. x 2 + x 3 1 x 1 ,x 2 ,...x 7 ,y are all 0-1 variables Problem 3 (page 503). Let x 1 = Units of Product 1 produced x 2 = Units of Product 2 produced y i = 1 if any Product i is produced y i = 0 otherwise Then the appropriate IP is max z = 2x 1 + 5x 2 - 10y 1 - 20y 2 s.t. 3x 1 + 6x 2 120 x 1 40y 1 x 2 20y 2 x 1 0,x 2 0, y 1 ,y 2 = 0 or 1 Problem 12 (page 503).
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Unformatted text preview: Let y 1 = 1 if NY is used, y 2 = 1 if LA is used, y 3 = 1 if Chicago is used, y 4 = 1 if Atlanta is used. Also y i = 0 if city i is not used. Define x ij = units shipped from warehouse in city i to region j min z = 400y 1 + 500y 2 + 300y 3 + 150y 4 + 20x 11 + 40x 12 + 50x 13 + 48x 21 + 15x 22 + 26x 23 + 26x 31 + 35x 32 + 18x 33 + 24x 41 + 50x 42 +35x 43 st x 11 + x 21 + x 31 + x 41 ≥ 80, x 12 + x 22 + x 32 + x 42 ≥ 70 x 13 + x 23 + x 33 + x 43 ≥ 40, x 11 + x 12 + x 13 ≤ 100y 1 x 21 + x 22 + x 23 ≤ 100y 2 , x 31 + x 32 + x 33 ≤ 100y 3 , x 41 + x 42 + x 43 ≤ 100y 4 , y 1 ≤ y 2 , y 2 + y 4 ≥ 1, y 1 + y 2 + y 3 + y 4 ≤ 2 , All x ij ≥ 0, all y i = 0 or 1....
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