3. Consider the following LP [19pts] max 5x1 + 8x2 + 6x3 subject to x1 + x2 + x3 ≤ 10 − x1 + 2x2 + 4x3 ≤ 10 4x1 + 4x2 + 2x3 ≤ 8 x1, x2, x3 ≥ 0 Answer the following questions “manually” without using CPLEX, while using your knowledge of the optimal basis from problem 1.
(a) Using the Dual Theorem, what is the optimal dual solution?
(b) By how much can the objective coefficient for x1 change while keeping the same optimal basis (in other words, what is ∆)?
(c) By how much can the objective coefficient for x2 change before the optimal basis (in other words, what is ∆)?
(d) By how much can the rhs of the 2nd constraint change while keeping the same optimal basis (in other words, what is the allowable ∆)?
(e) Assume we add an activity corresponding to the decision variable x4, with objective coefficient c4 = 10, and constraint coefficients 2, 4, and 1 corresponding to the 1st, 2nd , and 3rd constraints. Will there be a change to the optimal basis?
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