Exercices de Recherche Opérationnelle : Dualité et Programmation Linéaire

min{−2x1+7x2x1−x2= 3, x1+5x2≤6,−2x1+x2≥
1, x1∈R, x2∈R}
max{2x28x1+ 3x2= 1, x1∈R, x2∈R}
min{x1−3x1≤3, x2≤11, x1≥0}
max 4x1+ 12x2+ 3x3
x1≤1000
x2≤500
x3≤1500
3x1+ 6x2+ 2x3≤6750
x1≥0, x2≥0, x3≥0
x1= 250 x2=
500 x3= 1500

min 4x1+ 5x2+ 2x3+ 6x4
x1+ 3x2+ 2x3−4x4≥5
2x1+x2−3x3+ 5x4≥7
x1, x2, x3, x4≥0
max 5
2x1+ 5x2
x1+ 2x2≤2
x1, x2≥0
λ < 1
5λ
2x1+ 5x2
max 2x1+x2
3x1+ 2x2≥9
x2≤3
3x1−x2≤12
x1, x2≥0

max 3x1+ 5x2
min −9x1+ 3x2+ 12x3
−3x1+ 3x3≥2
−2x1+x2−x3≥1
x1, x2, x3≥0
•
•
•

max 6x1+ 2x2+ 4x3
4x1+ 2x2+x3≤6000
2x1+1
2x2+ 3x3≤4000
x1+x2+x3≤2500
x1, x2, x3≥0
x1x3
max 3x1−4x2
2x1−3x2≤1
3x1−x2≤2
x1, x2≥0
λ λ