Consider the Linear Programming Problem P: \[ \text{Maximize } c_1x_1 + c_2x_2 \] subject to: \[ a_{11}x_1 + a_{12}x_2 \leq b_1, \] \[ a_{21}x_1 + a_{22}x_2 \leq b_2, \] \[ a_{31}x_1 + a_{32}x_2 \leq b_3, \] \[ x_1 \geq 0, \, x_2 \geq 0, \] where \(a_{ij}, b_i, c_j\) are real numbers (i = 1, 2, 3; j = 1, 2). Let \[ \begin{bmatrix} p \\ q \end{bmatrix} \] be a feasible solution of P such that \( p c_1 + q c_2 = 6 \), and let all feasible solutions \[ \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \] of P satisfy \( -5 \leq c_1x_1 + c_2x_2 \leq 12 \). Then, which one of the following statements is NOT true?
Consider the Linear Programming Problem P: \[ \text{Maximize } c_1x_1 + c_2x_2 \] subject to: \[ a_{11}x_1 + a_{12}x_2 \leq b_1, \] \[ a_{21}x_1 + a_{22}x_2 \leq b_2, \] \[ a_{31}x_1 + a_{32}x_2 \leq b_3, \] \[ x_1 \geq 0, \, x_2 \geq 0, \] where \(a_{ij}, b_i, c_j\) are real numbers (i = 1, 2, 3; j = 1, 2). Let \[ \begin{bmatrix} p \\ q \end{bmatrix} \] be a feasible solution of P such that \( p c_1 + q c_2 = 6 \), and let all feasible solutions \[ \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \] of P satisfy \( -5 \leq c_1x_1 + c_2x_2 \leq 12 \). Then, which one of the following statements is NOT true?
Show Hint
- P has an optimal solution
- The feasible region of P is a bounded set
- If \[ \begin{bmatrix} y_1
y_2
y_3 \end{bmatrix} \] is a feasible solution of the dual of P, then \( b_1y_1 + b_2y_2 + b_3y_3 \geq 6 \) - The dual of P has at least one feasible solution
The Correct Option is B
Solution and Explanation
- (B) The feasible region of P is a bounded set: Since the constraints do not guarantee that the feasible region is bounded (the constraints are only inequalities), it is possible for the feasible region to be unbounded. Hence, this statement is false.
- (C) If \( \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix} \) is a feasible solution of the dual of P, then \( b_1y_1 + b_2y_2 + b_3y_3 \geq 6 \): This is the condition for a feasible solution in the dual problem, so it is true.
- (D) The dual of P has at least one feasible solution: Every linear programming problem has a dual, and the dual problem will always have at least one feasible solution. Therefore, this statement is true. Thus, the correct answer is (B).
Final Answer: (B) The feasible region of P is a bounded set.
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