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HW7_sol

# HW7_sol - IE 335 Operations Research Optimization Solutions...

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IE 335 Operations Research - Optimization Solutions to Homework 7 Spring 2011 Problem 28 7-12.(b) Primal problem: maximize 44 x 1 - 3 x 2 + 15 x 3 + 56 x 4 (1) subject to x 1 + x 2 + x 3 + x 4 = 20 v 1 (2) x 1 - x 2 0 v 2 (3) 9 x 1 - 3 x 2 + x 3 - x 4 25 v 3 (4) x 1 ,...,x 4 0 (5) Dual problem: minimize 20 v 1 + 25 v 3 (6) subject to v 1 + v 2 + 9 v 3 44 (7) v 1 - v 2 - 3 v 3 ≥ - 3 (8) v 1 + v 3 15 (9) v 1 - v 3 56 (10) v 1 is unrestricted in sign (URS) (11) v 2 0 (12) v 3 0 (13) 7-12.(d) Primal problem: minimize 5 x 1 + x 2 - 4 x 3 (14) subject to x 1 + x 2 + x 3 + x 4 = 19 v 1 (15) 4 x 2 + 8 x 4 55 v 2 (16) x 1 + 6 x 2 - x 3 7 v 3 (17) x 1 is URS (18) x 2 ,x 3 0 (19) x 4 0 (20) Dual problem: 1

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maximize 19 v 1 + 55 v 2 + 7 v 3 (21) subject to v 1 + v 3 = 5 (22) v 1 + 4 v 2 + 6 v 3 1 (23) v 1 - v 3 ≤ - 4 (24) v 1 + 8 v 2 0 (25) v 1 is URS (26) v 2 0 (27) v 3 0 (28) 7-12.(f) Primal problem: maximize 19 y 1 + 4 y 2 - 8 z 2 (29) subject to 11 y 1 + y 2 + z 1 = 15 v 1 (30) z 1 + 5 z 2 0 v 2 (31) y 1 - y 2 + z 2 4 v 3 (32) y 1 ,y 2 0 (33) z 1 ,z 2 is URS (34) Dual problem: minimize 15 v 1 + 4 v 3 (35) subject to 11 v 1 + v 3 19 (36) v 1 - v 3 4 (37) v 1 + v 2 = 0 (38) 5 v 2 + v 3 = - 8 (39) v 1 is URS (40) v 2 0 (41) v 3 0 (42) 7-12.(h) Primal problem: maximize 10 x 3 + 10 x 4 (43) subject to x 1 + x 2 + x 3 + x 4 = 80 v 1 (44) x 1 - 2 x 2 0 v 2 (45) x 2 - 2 x 3 0 v 3 (46) x 3 - 2 x 4 0 v 4 (47) x 1 ,x 2 0 (48) x 3 ,x 4 is URS (49) Dual problem: 2
minimize 80 v 1 (50) subject to v 1 + v 2 0 (51) v 1 - 2 v 2 + v 3 0 (52) v 1 - 2 v 3 + v 4 = 10 (53) v 1 - 2 v 4 = 10 (54) v 1 is URS (55) v 2 ,v 3

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HW7_sol - IE 335 Operations Research Optimization Solutions...

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