Questions:
Question #1
A company produces three products. The per-unit profit, labor usage, and pollution produced per unit are given in the table 1. At most, 3 million labor hours can be used to produce the three products, and government regulations require that the company produce at most 2 lb of pollution. If we let Xi = units produced of product i then the appropriate LP is
Max z = 6x1+4x2+3x3
4x1+3x2+2x3≤3,000,000
s.t. 0.000003 x1+0.000002x2+0.000001x3 ≤2
x1,x2,x3≥0
a.Explain why this LP is poorly scaled.
b.Eliminate the scaling problem by redefining the units of the objective function, decision variables, and the right hand sides.
Table 1
|
Product
|
Profit($)
|
Labor Usage(Hrs)
|
Pollution (Lb)
|
|
1
|
6
|
4
|
0.000003 lb
|
|
2
|
4
|
3
|
0.000002 lb
|
|
3
|
3
|
2
|
0.000001 lb
|
Question #2
Show that the following LP is unbounded:
Max z = 2x2
x1-x2≤4
s.t. -x1+x2≤1
x1,x2 ≥0
Find the point in the feasible region with
Question #3
Suppose that in solving an LP, we obtain the tableau in Table 2. Although x1 can enter the basis, this LP is unbounded, Why?
|
Z
|
X1
|
X2
|
X3
|
X4
|
rhs
|
|
1
|
-3
|
-2
|
0
|
0
|
0
|
|
0
|
1
|
-1
|
1
|
0
|
3
|
|
0
|
2
|
0
|
0
|
1
|
4
|