For each of the following linear programming models, give your recommendation on

Question

For each of the following linear programming models, give
your recommendation on which is the more efficient way (probably)
to obtain an optimal solution: by applying the simplex method
directly to this primal problem or by applying the simplex method
directly to the dual problem instead. Explain.
(a) Maximize Z = 10x1 - 4x2 + 7x3,
subject to
3x1 - x2 + 2x3 25
x1 - 2x2 + 3x3 25
5x1 + x2 + 2x3 40
x1 + x2 + x3 90
2x1 - x2 + x3 20
and
x1 0, x2 0, x3 0.

(b)Maximize Z = 2x1 + 5x2 + 3x3 + 4x4 + x5, subject to x1 + 3x2 + 2x3 + 3x4 + x5 6 4x1 + 6x2 + 5x3 + 7x4 + x5 15 and xj 0, for j = 1, 2, 3, 4, 5.

Explanation / Answer

Again, the dual is closely related to the primal and vice versa. With the first problem, once we construct the dual, we see that the dual will only have 3 functional constraints and hence in doing the simplex method, we would have to do less linear algebra grunt work. For the second problem, the primal problem would probably be easier to solve since here we would only have 2 constraints and the row reductions, aka linear algebra grunt work, would be easier. However, if we created the dual of the second problem, we could solve the problem graphically which is really really easy.

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