Linear Programming
Minimization of objective function
Grade 12
Question:
<p>Determine the minimum value of \(z = 3x + 2y\), if the feasible region for an LPP is shown in the following figure. (Corner points: \(D(0,10)\), \(C(1,5)\), \(B(4,2)\), \(A(12,0)\))</p>
Step-by-Step Solution
Key Concept: The minimum (or maximum) value of a linear objective function in an LPP always occurs at a corner point of the feasible region. Evaluate z = 3x + 2y at all given corner points and identify the smallest value.
<p><strong>Step 1:</strong> Identify that for a linear objective function, the optimal value occurs at a corner point of the feasible region.</p><p><strong>Step 2:</strong> Evaluate z = 3x + 2y at each corner point:</p><p>• At D(0, 10): z = 3(0) + 2(10) = 20</p><p>• At C(1, 5): z = 3(1) + 2(5) = 3 + 10 = 13</p><p>• At B(4, 2): z = 3(4) + 2(2) = 12 + 4 = 16</p><p>• At A(12, 0): z = 3(12) + 2(0) = 36</p><p><strong>Step 3:</strong> Compare all values: 20, 13, 16, 36. The minimum value is 13, occurring at corner point C(1, 5).</p><p>∴ Minimum value of z = <strong>13</strong></p>
Correct Answer: 13