Q1) Find the dual of the following Linear Program (LP):

Maximize:
\[ z = -1x_1 - 9x_2 - 2x_3 - 5x_4 \]

Subject to:
Constraint I: \(-6x_1 - 10x_2 + 1x_3 + 9x_4 \leq 44\)
Constraint II: \(+2x_1 - 7x_2 - 9x_3 + 9x_4 \leq 95\)
Constraint III: \(+8x_1 - 4x_2 + 6x_3 + 2x_4 = 149\)
Constraint IV: \(+4x_1 + 8x_2 + 9x_3 + 9x_4 \leq 178\)

Where:
\[ x_1 \leq 0, \quad x_2 \leq 0, \quad x_3 \geq 0, \quad x_4 \geq 0 \]

It's obtained by swapping the coefficients and constants of the objective function and constraints, and changing the direction of optimization. The dual of the given LP is derived as: Minimum w = 44y1 + 95y2 + 149y3 + 178y4 subject to a set of constraints involving y1, y2, y3 and y4.

The given Linear Program (LP) is written in its primal form and we are asked to find the dual of it. According to the dual LP formulation, the objective function of the dual is the right-hand side of the constraints of the primal and the coefficients of constraints in the dual are the coefficients from the objective function of the primal.

Consequently, the dual of the given LP becomes:

Minimize w = 44y1 + 95y2 + 149y3 + 178y4 subject to -6y1 + 2y2 + 8y3 + 4y4 >= -1, -10y1 - 7y2 - 4y3 + 8y4 >= -9, y1 - 9y2 + 6y3 + 9y4 >= -2, 9y1 + 94y2 + 2y3 + 94y4 >= -5, with y1 >= 0, y2 <= 0, y3

unrestricted,y4 >= 0. We use 'Minimize' instead of 'Maximize' since the original LP was a maximization problem. Conversely, if the primal is a minimization problem, then the dual will be a maximization problem.

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