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# Linear Programming : Scaling, Unbounded LPs and Feasible Region

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
s.t.

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
s.t.

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
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#### Solution Summary

Linear Programming, Scaling, Unbounded LPs and Feasible Region are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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