Phase 1: Conceptual Understanding:
This problem involves solving a two-variable Linear Programming Problem (LPP). The process requires identifying the feasible region, determining its corner points, and evaluating the objective function at these points to ascertain the maximum value and the nature of the optimal solution.
Phase 2: Detailed Procedure:
1. Determine 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): Setting x=0 yields \(5y=15 \Rightarrow y=3\). The point is (0, 3).
- Point 3 (x-intercept of 5x+2y=10): Setting y=0 yields \(5x=10 \Rightarrow x=2\). The point is (2, 0).
- Point 4 (Intersection of 3x+5y=15 and 5x+2y=10):
Multiplying the first equation by 2 gives \(6x + 10y = 30\).
Multiplying the second equation by 5 gives \(25x + 10y = 50\).
Subtracting the modified first equation from the modified second equation results in \(19x = 20 \Rightarrow x = 20/19\).
Substituting x back into \(5x+2y=10\): \(5(20/19) + 2y = 10 \Rightarrow 100/19 + 2y = 10 \Rightarrow 2y = 10 - 100/19 = 90/19 \Rightarrow y = 45/19\).
The intersection point is (20/19, 45/19).
2. Evaluate Statements:
- Statement (D): The feasible region is unbounded. This statement is incorrect. The region is bounded by the coordinate axes and the two given 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 statement is correct, as determined by the calculations above.
3. Compute 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 is attained at two adjacent corner points: (2, 0) and (20/19, 45/19). When the optimal value occurs at multiple corner points, it also occurs at every point along the line segment connecting them.
- Statement (A): The LPP has a unique optimal solution at (2, 0) only. This statement is incorrect. 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 statement is correct. The maximum value of 10 is unique, and infinite solutions exist on the line segment \(5x+2y=10\) between x=20/19 and x=2.
Phase 3: Conclusion:
The correct statements are (B) and (C). Consequently, the option that encompasses these correct statements is the final answer.