Solution

The correct answer is Min P = 8u + 10v + 15w

Explanation:

To identify the objective function for the dual problem of the given linear programming problem (LPP), we need to understand the primal and the dual relationship.

Primal Problem: [ \(\text{Maximize } Z = 30x + 50y\) ] Subject to: [ \(2x + 3y \leq 8\) ] [ \(3x + 2y \leq 10\) ] [ \(2y \leq 15 \quad \text{(which simplifies to } y \leq 7.5 \text{)}\) ] [ \(x, y \geq 0\) ]

Dual Problem:

From a primal problem in standard form (Maximize objective function, subject to ≤ constraints), the dual problem will be: \(\text{Minimize } P = 8u + 10v + 15w\) Subject to:

• [ \(2u + 3v \geq 30\) ]
• [ \(3u + 2v + 2w \geq 50\) ]
• [ \(u, v, w \geq 0\) ]

• The coefficients of the objective function in the primal (30 and 50) become the right-hand side constants in the dual constraints.
• The right-hand side coefficients in the primal constraints (8, 10, and 7.5/15) become the coefficients in the dual objective function.
• The matrix of coefficients in the primal constraints is transposed to become the coefficients in the dual constraints.

Given this, the correct dual objective function to minimize is: Min P = 8u + 10v + 15w