algorithms

Figure 7.13 Simplex in action.
Initial LP:
max 2x 1 + 5x 2
2x 1 − x 2 ≤ 4 1○
x 1 + 2x 2 ≤ 9 2○
−x 1 + x 2 ≤ 3 3○
x 1 ≥ 0 4○
x 2 ≥ 0 5○
Current vertex: { 4○, 5○} (origin).
Objective value: 0.
Move: increase x 2 .
5○ is released, 3○ becomes tight. Stop at x 2 = 3.
New vertex { 4○, 3○} has local coordinates (y 1 , y 2 ):
y 1 = x 1 , y 2 = 3 + x 1 − x 2
Rewritten LP:
max 15 + 7y 1 − 5y 2
y 1 + y 2 ≤ 7 1○
3y 1 − 2y 2 ≤ 3 2○
y 2 ≥ 0 3○
y 1 ≥ 0 4○
−y 1 + y 2 ≤ 3 5○
Current vertex: { 4○, 3○}.
Objective value: 15.
Move: increase y 1 .
4○ is released, 2○ becomes tight. Stop at y 1 = 1.
New vertex { 2○, 3○} has local coordinates (z 1 , z 2 ):
z 1 = 3 − 3y 1 + 2y 2 , z 2 = y 2
Rewritten LP:
max 22 − 7 3 z 1 − 1 3 z 2
− 1 3 z 1 + 5 3 z 2 ≤ 6 1○
z 1 ≥ 0 2○
z 2 ≥ 0 3○
1
3 z 1 − 2 3 z 2 ≤ 1 4○
1
3 z 1 + 1 3 z 2 ≤ 4 5○
Current vertex: { 2○, 3○}.
Objective value: 22.
Optimal: all c i < 0.
Solve 2○, 3○ (in original LP) to get optimal solution
(x 1 , x 2 ) = (1, 4).
Increase
y 1
{ 3○, 4○}
Increase
x 2
{ 2○, 3○}
{ 1○, 2○}
{ 4○, 5○}
{ 1○, 5○}
217