# Question

Consider the following problem. Maximize Z = 8x1 + 24x2, Subject to

And x1 ≥ 0, x2 ≥ 0.

Suppose that Z represents profit and that it is possible to modify the objective function somewhat by an appropriate shifting of key personnel between the two activities. In particular, suppose that the unit profit of activity 1 can be increased above 8 (to a maximum of 18) at the expense of decreasing the unit profit of activity 2 below 24 by twice the amount. Thus, Z can actually be represented as Z(θ) (8 + θ)x1 + (24 – 2θ)x2, where θ is also a decision variable such that 0 ≤ θ ≤ 10.

And x1 ≥ 0, x2 ≥ 0.

Suppose that Z represents profit and that it is possible to modify the objective function somewhat by an appropriate shifting of key personnel between the two activities. In particular, suppose that the unit profit of activity 1 can be increased above 8 (to a maximum of 18) at the expense of decreasing the unit profit of activity 2 below 24 by twice the amount. Thus, Z can actually be represented as Z(θ) (8 + θ)x1 + (24 – 2θ)x2, where θ is also a decision variable such that 0 ≤ θ ≤ 10.

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