A CD E B 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:

A CD E B 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:

A CD E B 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:

A CD E B (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:

A CD E B 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 …

A CD E B 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 …

A CD E B 2 e1e 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: