1. **Problem Statement:** We have a primal linear program (P): $$\max w = 5x_1 + 8x_2$$ subject to $$x_1 + x_2 \leq 6$$ $$x_1 + 2x_2 \leq 8$$ $$x_1, x_2 \geq 0$$ The final simplex tableau for (P) is given as: $$\begin{array}{c|cccc|c} x_1 & 1 & 0 & 2 & -1 & 4 \\ x_2 & 0 & 1 & -1 & 1 & 2 \\ \text{z-row} & 0 & ? & 2 & 3 & 36 \\ \end{array}$$ Problem (D) is the dual of (P). --- 2. **(a) Phases required to solve (D):** - The dual (D) has constraints corresponding to variables in (P). - Since (P) has \(\leq\) constraints with slack variables, the dual will have \(\geq\) constraints or equalities. - The starting tableau of (D) will have artificial variables if constraints are equalities or \(\geq\). - Here, (D) has two constraints (dual variables \(u_1, u_2\)) and two slack variables in (P). - Because (D) has two artificial variables (due to \(\geq\) constraints), solving (D) requires **phases 1 and 2**. **Answer:** phases 1 and 2, because the starting tableau of (D) has two artificial variables. --- 3. **(b) Shadow price of 1st constraint of (P):** - Shadow prices correspond to the values of dual variables at optimum. - From the final tableau, the coefficients of slack variables \(y_1, y_2\) in the objective row give shadow prices. - The coefficient of \(y_1\) in the z-row is 2. **Answer:** 2 --- 4. **(c) Shadow price of 1st constraint of (D):** - The shadow price of the 1st constraint of (D) corresponds to the value of \(x_1\) in the final tableau. - From the tableau, the value of \(x_1\) in the solution column is 4. **Answer:** 4 --- 5. **(d) Value of \(u_2\) in optimum solution of (D):** - The dual variables correspond to slack variables in (P). - From the tableau, the value of \(y_2\) (dual slack) in the solution column is 2. - Hence, \(u_2 = 2\). **Answer:** 2 --- 6. **(e) Interpretation of '?' (reduced cost \(r_{x_2}\)) in final tableau:** - The reduced cost \(r_{x_2}\) relates to the slack of the 2nd constraint in (D). - The shadow price of \(x_2\) is the value of \(u_2\) which is 2. - Complementary slackness implies if shadow price is positive, slack must be zero. - Therefore, \(r_{x_2} = 0\). **Answer:** \(r_x = 0\) --- **Summary of answers:** (a) phases 1 and 2, because the starting tableau of (D) has two artificial variables (b) 2 (c) 4 (d) 2 (e) \(r_x = 0\)