Question

You are given the final simplex tableau for the dual problem. Give the solution to the primal problem and the solution to the associated dual problem. Problem: Minimize C = 3x + 4y subject to x + 4y ≥ 8 x + y ≥ 5 2x + y ≥ 7 x ≥ 0, y ≥ 0 Final tableau: u v w x y P Constant 0 1 7 3 4 3 − 1 3 0 8 3 1 0 − 1 3 − 1 3 1 3 0 1 3 0 0 2 4 1 1 16 Give the solution to the primal problem. The minimum is C = at (x, y) = . Give the solution to the associated dual problem. The maximum is P = at (u, v, w) = .

Answer 1

The solution to the primal problem can be derived from the final simplex tableau by evaluating the constants in the rows of the basic primal variables (x, y). The solution to the dual problem is found in the 'Constant' column for non-basic variables (u, v, w), with the value of the dual objective function P at the bottom right corner.

Step-by-step explanation:

In linear programming, once you have the final tableau of the dual problem, you can find the solution to the primal problem by looking at the values assigned to the primal variables in the tableau. According to the final simplex tableau given for the dual problem, we can identify the solution to the primal and dual problems.

The primal variable x corresponds to the constant in the row where the column for x has a 1 and all other primal variables have 0. For variable y, it's the same process. In the final tableau given, it appears that the coefficients for x and y in the 'P' row are not zero, which is typically the case in standard simplex tableaus. However, we'll use the provided tableau assuming all simplifications were previously made. Substituting the provided constant values into the inequalities of the original problem allows you to calculate the values for x and y.

Similarly, the solution to the dual involves the variables u, v, and w, which can be found in the 'Constant' column of the tableau under the non-basic variables. The value of the dual objective function P is in the bottom right corner of the tableau.