Question

Consider the following Linear Programming Problem \( P \): Minimize \( x_1 + 2x_2 \), subject to \[ 2x_1 + x_2 \leq 2, \quad x_1 + x_2 = 1, \quad x_1, x_2 \geq 0. \] The optimal value of the problem \( P \) is equal to:

• A. \( 5 \)
• B. \( 0 \)
• C. \( 4 \)
• D. \( 2 \)

Hint:

For linear programming problems, simplify constraints and evaluate the objective function at feasible points.

Answer 1

The Correct Option is D

Step 1: Formulating the constraints. The constraints are: 1. \( 2x_1 + x_2 \leq 2 \), 2. \( x_1 + x_2 = 1 \), 3. \( x_1, x_2 \geq 0 \). Step 2: Solving using substitution. From \( x_1 + x_2 = 1 \), substitute \( x_2 = 1 - x_1 \) into \( 2x_1 + x_2 \leq 2 \): \[ 2x_1 + (1 - x_1) \leq 2 \quad \Rightarrow \quad x_1 \leq 1. \] Step 3: Objective function. Minimize \( x_1 + 2x_2 \): \[ x_1 + 2(1 - x_1) = 2 - x_1. \] For \( x_1 = 1, x_2 = 0 \), the minimum is \( 2 \). Step 4: Conclusion. The optimal value is \( {(4)} \).