Question:
Consider the following Linear Programming Problem $ P $: Minimize $ x_1 + 2x_2 $, subject to
$ 2x_1 + x_2 \leq 2 $,
$ x_1 + x_2 = 1 $,
$ x_1, x_2 \geq 0 $.
The optimal value of the problem $ P $ is equal to:
Consider the following Linear Programming Problem $ P $: Minimize $ x_1 + 2x_2 $, subject to
$ 2x_1 + x_2 \leq 2 $,
$ x_1 + x_2 = 1 $,
$ x_1, x_2 \geq 0 $.
The optimal value of the problem $ P $ is equal to:
Show Hint
For linear programming problems, simplify constraints and evaluate the objective function at feasible points.
Updated On: Apr 11, 2025
- \( 5 \)
- \( 0 \)
- \( 4 \)
- \( 2 \)
Show Solution
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The Correct Option is D
Solution and Explanation
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 \( {(4)} \).
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