Convert the following LP into its dual form:min 3x1 − x2 + x3s.t.x1 − 2x2 − x3 ≤ 42x1 − x2 + x3 = 8x1 − x2 ≤ 6x1 ≥ 0; x2, x3 free

The dual of a linear programming problem is another linear programming problem. The dual of the given primal problem can be formulated as follows:

The primal problem is:

minimize Z = 3x1 - x2 + x3 subject to: x1 - 2x2 - x3 ≤ 4 2x1 - x2 + x3 = 8 x1 - x2 ≤ 6 x1 ≥ 0; x2, x3 are free

The dual problem is:

maximize W = 4y1 + 8y2 + 6y3 subject to: y1 + 2y2 + y3 ≤ 3 -2y1 - y2 - y3 ≤ -1 -y1 + y2 ≤ 1 y1 is unrestricted in sign; y2 ≥ 0, y3 ≥ 0

Here, y1, y2, y3 are the dual variables corresponding to the constraints in the primal problem. The objective function of the dual problem is formed by the right-hand side values of the constraints in the primal problem. The coefficients of the constraints in the dual problem are the coefficients of the variables in the objective function of the primal problem. The right-hand side values of the constraints in the dual problem are the coefficients of the variables in the objective function of the primal problem. The signs of the inequalities in the dual problem are the opposite of the signs of the inequalities in the primal problem. The sign restrictions on the variables in the dual problem are the opposite of the sign restrictions on the variables in the primal problem.