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A CD E B 2 3 4 1 1 1 6 B 2 C 4 C D + 1 D 6 D B + 3 D E + 1 E B + 1 maximize B+C+D+E B = 2 C = 4 D = 4 E = 3 max: 13 B 0 C 0 D 0 E 0 Distance from A to:Solution:
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A CD E B 2 3 4 1 1 1 6 B 2 C 4 C D + 1 D 6 D B + 3 D E + 1 E B + 1 maximize B+C+D+E B 2 C 4 C – D 1 D 6 D – B 3 D – E 1 E – B 1 maximize B+C+D+E Upper bound constraints:Rewritten slightly:
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A CD E B 2 3 4 1 1 1 6 B 2 C 4 C – D 1 D 6 D – B 3 D – E 1 E – B 1 maximize B+C+D+E (B)e 1 2e 1 (C)e 2 4e 2 (C – D)e 3 e 3 (D)e 4 6e 4 (D – B)e 5 3e 5 (D – E)e 6 e 6 (E – B)e 7 e 7 e 1 0 e 2 0 e 3 0 e 4 0 e 5 0 e 6 0 e 7 0 Upper bound constraints: Non-negative multipliers: Multiplied constraints:
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A CD E B 2 3 4 1 1 1 6 (B)e 1 2e 1 (C)e 2 4e 2 (C – D)e 3 e 3 (D)e 4 6e 4 (D – B)e 5 3e 5 (D – E)e 6 e 6 (E – B)e 7 e 7 B(e 1 – e 5 – e 7 ) + C(e 2 + e 3 ) + D(–e 3 + e 4 + e 5 + e 6 ) + E(–e 6 + e 7 ) 2e 1 + 4e 2 + e 3 + 6e 4 + 3e 5 + e 6 + e 7 Multiplied constraints:Summed constraints:
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A CD E B 2 3 4 1 1 1 6 e 1 – e 5 – e 7 = 1 e 2 + e 3 = 1 –e 3 + e 4 + e 5 + e 6 = 1 –e 6 + e 7 = 1 B(e 1 – e 5 – e 7 ) + C(e 2 + e 3 ) + D(–e 3 + e 4 + e 5 + e 6 ) + E(–e 6 + e 7 ) 2e 1 + 4e 2 + e 3 + 6e 4 + 3e 5 + e 6 + e 7 B + C + D + E 2e 1 + 4e 2 + e 3 + 6e 4 + 3e 5 + e 6 + e 7 Summed constraints: If … Then …
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A CD E B 2 3 4 1 1 1 6 e 1 – e 5 – e 7 1 e 2 + e 3 1 –e 3 + e 4 + e 5 + e 6 1 –e 6 + e 7 1 B(e 1 – e 5 – e 7 ) + C(e 2 + e 3 ) + D(–e 3 + e 4 + e 5 + e 6 ) + E(–e 6 + e 7 ) 2e 1 + 4e 2 + e 3 + 6e 4 + 3e 5 + e 6 + e 7 B + C + D + E 2e 1 + 4e 2 + e 3 + 6e 4 + 3e 5 + e 6 + e 7 Summed constraints: Moreover, if … Then still …
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A CD E B 2 e1e1 3 4 1 1 1 6 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e 1 0 e 2 0 e 3 0 e 4 0 e 1 – e 5 – e 7 1 e 2 + e 3 1 –e 3 + e 4 + e 5 + e 6 1 –e 6 + e 7 1 e 5 0 e 6 0 e 7 0 minimize 2e 1 + 4e 2 + e 3 + 6e 4 + 3e 5 + e 6 + e 7 e 1 = 3 e 2 = 1 e 3 = 0 e 4 = 0 e 5 = 0 e 6 = 1 e 7 = 2 min: 13 A new linear program (the dual) Solution:
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