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

Step 1: Understanding the Concept:
This problem requires solving a two-variable LPP by finding the feasible region, identifying its corner points, and then evaluating the objective function at these points to find the maximum value and determine the nature of the optimal solution.
Step 2: Detailed Explanation:
1. Find the 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 \(y \ge 0\).
- Point 1 (Origin): (0, 0).
- Point 2 (y-intercept of 3x+5y=15): Set x=0, \(5y=15 \Rightarrow y=3\). Point is (0, 3).
- Point 3 (x-intercept of 5x+2y=10): Set y=0, \(5x=10 \Rightarrow x=2\). Point is (2, 0).
- Point 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 new first from the new second: \(19x = 20 \Rightarrow x = 20/19\).
Substitute x back into \(5x+2y=10\): \(5(20/19) + 2y = 10 \Rightarrow 100/19 + 2y = 10 \Rightarrow 2y = 10 - 100/19 = 90/19 \Rightarrow y = 45/19\).
Point is (20/19, 45/19).
2. Analyze the Statements:
- (D) The feasible region is unbounded. This is false. The region is bounded by the axes and the two lines in the first quadrant.
- (B) The feasible region is bounded with corner points (0, 0), (2, 0), (20/19, 45/19) and (0, 3). This is true, as calculated above.
3. Evaluate the Objective Function Z = 5x + 2y at Corner Points:
- Z at (0, 0) = \(5(0) + 2(0) = 0\).
- Z at (0, 3) = \(5(0) + 2(3) = 6\).
- Z at (2, 0) = \(5(2) + 2(0) = 10\).
- Z at (20/19, 45/19) = \(5(20/19) + 2(45/19) = (100+90)/19 = 190/19 = 10\).
4. Analyze Optimality:
The maximum value of Z is 10. This maximum value occurs at two adjacent corner points, (2, 0) and (20/19, 45/19). When the optimal value is achieved at more than one corner point, it is also achieved at every point on the line segment connecting them.
- (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 unique.
- (C) The optimal value is unique, but there are an infinite number of optimal solutions. This is true. The unique maximum value is 10, and there are infinite solutions on the line segment \(5x+2y=10\) between x=20/19 and x=2.
Step 3: Final Answer:
The correct statements are (B) and (C). Therefore, option (D) is the correct choice.