MT 435 Homework Set 7 Due Wednesday, November 19, 2008
1. The simplex algorithm has been used to solve the following linear programming problem:
Minimize z= 10x1+ 20x2+ 15x3+ 21x4+ 5x5subject to :
3x1−10x2+6x3−5x4+2x5= 600
2x1+x2+5x3+2x4−x5= 340
where x1, x2, x3, x4, x5≥0.
The initial and final tableaus are below; the wrow has been suppressed.
x1x2x3x4x5x6x7
x63 -10 6 -5 2 1 0 600
x72 1 5 2 -1 0 1 340
10 20 15 21 5 0 0 0
x53/16 -7/2 0 -37/16 1 5/16 -3/8 60
x37/16 -1/2 1 -1/16 0 1/16 1/8 80
5/2 45 0 67/2 0 -5/2 0 -1500
1. a. What is the minimum of zand the point at which this optimal value is attained?
b. What is the optimal solution point to the dual?
2. For this final tableau, state the matrices Band B−1, and the vector cB.
3. Suppose the constant term c2= 20 is changed to ¯c2= 20 + λ. Over what range on λwill the
optimal solution point found in 1a. remain optimal and, for λin this range, what is the minimum
value of z?
4. Suppose the constant term c5= 5 is changed to ¯c5= 5 + λ. Over what range on λwill the
optimal solution point found in 1a. remain optimal and, for λin this range, what is the minimum
value of z?
5. Suppose the constant term b1= 600 is changed to ¯
b1= 600 + λ. For what range on λwill x5
and x3remain the basic variables for the optimal solution point and, for λin this range, what is
the optimal solution point and the minimum value of z?
6. Find the optimal solution point and the minimum value of zif a variable x6is added to the
original problem, with the objective function (still to be minimized) now z= 10x1+ 20x2+ 15x3+
21x4+ 5x5+ 30x6and the constraints are
3x1−10x2+6x3−5x4+2x5+12x6= 600
2x1+x2+5x3+2x4−x5+10x6= 340
where x1, x2, x3, x4, x5, x6≥0.
