Question

Which of the following statements are correct in reference to the linear programming problem (LPP):
Maximize Z = 5x + 2y
subject to the following constraints
3x + 5y \(\le\) 15,
5x + 2y \(\le\) 10,
x \(\ge\) 0, y \(\ge\) 0.
(A) The LPP has a unique optimal solution at (2, 0) only.
(B) The feasible region is bounded with corner points (0, 0), (2, 0), (20/19, 45/19) and (0, 3).
(C) The optimal value is unique, but there are an infinite number of optimal solutions.
(D) The feasible region is unbounded.
Choose the correct answer from the options given below:

• A. (A) and (D) only
• B. (A), (B) and (C) only
• C. (A), (C) and (D) only
• D. (B) and (C) only

Hint:

A key indicator of multiple optimal solutions in an LPP is when the slope of the objective function line is the same as the slope of one of the boundary lines of the feasible region. Here, the slope of Z is -5/2, which is the same as the slope of the constraint line \(5x+2y=10\).

Answer 1

The Correct Option is D

Objective: Solve a two-variable Linear Programming Problem (LPP) by determining the feasible region, identifying its vertices, and evaluating the objective function at these vertices to find the maximum value and the nature of the optimal solution. Procedure: 1. Identify Corner Points of the Feasible Region: The feasible region is defined by the constraints: \(3x + 5y \le 15\), \(5x + 2y \le 10\), \(x \ge 0\), and \(y \ge 0\).

• Vertex 1 (Origin): (0, 0).
• Vertex 2 (y-intercept of 3x+5y=15): Setting x=0 yields \(5y=15 \Rightarrow y=3\). Vertex is (0, 3).
• Vertex 3 (x-intercept of 5x+2y=10): Setting y=0 yields \(5x=10 \Rightarrow x=2\). Vertex is (2, 0).
• Vertex 4 (Intersection of 3x+5y=15 and 5x+2y=10):
  • Multiply the first equation by 2: \(6x + 10y = 30\).
  • Multiply the second equation by 5: \(25x + 10y = 50\).
  • Subtract the modified first equation from the modified second: \(19x = 20 \Rightarrow x = 20/19\).
  • Substitute x into \(5x+2y=10\): \(5(20/19) + 2y = 10 \Rightarrow 100/19 + 2y = 10 \Rightarrow 2y = 10 - 100/19 = 90/19 \Rightarrow y = 45/19\).
  • Vertex is (20/19, 45/19).

2. Evaluate Statements Regarding the Feasible Region:

• Statement (D) "The feasible region is unbounded." This is false. The region is bounded by the coordinate axes and the two constraint lines in the first quadrant.
• Statement (B) "The feasible region is bounded with corner points (0, 0), (2, 0), (20/19, 45/19) and (0, 3)." This is true, as determined in step 1.

3. Calculate Objective Function Z = 5x + 2y at Each Vertex:

• At (0, 0): Z = \(5(0) + 2(0) = 0\).
• At (0, 3): Z = \(5(0) + 2(3) = 6\).
• At (2, 0): Z = \(5(2) + 2(0) = 10\).
• At (20/19, 45/19): Z = \(5(20/19) + 2(45/19) = (100+90)/19 = 190/19 = 10\).

4. Analyze Optimality: The maximum value of Z is 10. This maximum occurs at two adjacent vertices, (2, 0) and (20/19, 45/19). If the optimal value is achieved at multiple vertices, it is also achieved at all points on the line segment connecting them.

• Statement (A) "The LPP has a unique optimal solution at (2, 0) only." This is false. While (2,0) is an optimal solution, it is not the only one.
• Statement (C) "The optimal value is unique, but there are an infinite number of optimal solutions." This is true. The maximum value of Z is 10 (unique), and there are infinite solutions along the line segment \(5x+2y=10\) between x=20/19 and x=2.

Conclusion: The correct statements are (B) and (C). Therefore, the option that encompasses these correct statements is the correct choice.