Question

Given LP:
Max \(z = 20x_1 + 6x_2 + Px_3\), subject to: 
\(8x_1 + 2x_2 + 3x_3 \le 250\), 
\(4x_1 + 3x_2 \le 150\), 
\(2x_1 + x_3 \le 50\), 
\(x_1,x_2,x_3 \ge 0.\) 
Optimal solution: \(x_1^*=0,\;x_2^*=50,\;x_3^*=50.\) 
Optimal dual variables: \(y_1^*=0,\;y_2^*=2,\;y_3^*=8.\) 
Find the value of parameter \(P\) such that the solution remains optimal (round off to one decimal place).

Hint:

When a variable is positive in the optimal solution, its reduced cost must be zero—use this directly to determine unknown objective coefficients.

Answer 1

Correct Answer: 7.9 - 8.1

Complementary slackness for variable \(x_3^* = 50 > 0\): \[ \text{Reduced cost of } x_3 = 0. \] Reduced cost formula for maximization: \[ \bar{c_3} = P - (3y_1 + 0y_2 + 1y_3) = 0. \] Substitute the dual values: \[ P - (3(0) + 0(2) + 1(8)) = 0, \] \[ P - 8 = 0, \] \[ P = 8. \] Final Answer: \(8.0\)