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).
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).
Show Hint
When a variable is positive in the optimal solution, its reduced cost must be zero—use this directly to determine unknown objective coefficients.
Updated On: Jan 13, 2026
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Correct Answer: 7.9 - 8.1
Solution and Explanation
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: \(8.0\)
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