Question

For the LPP, Min \(Z= 5x + 7y\) subject to \(x≥0, y≥0; 2x+y≥8, x+2y≥ 10,\) the basic feasible solutions are:

• A. (0, 0), (10, 0), (2, 4) and (0, 8)
• B. (10, 0), (2, 4) and (0, 8)
• C. (0, 0), (0, 10), (2, 4) and (8, 0)
• D. (0, 10), (4, 2) and (8, 0)

Answer 1

The Correct Option is B

To solve the Linear Programming Problem (LPP), we need to find the feasible solutions that satisfy the given constraints. The objective function is to minimize \(Z = 5x + 7y\) subject to the constraints \(x \ge 0\), \(y \ge 0\), \(2x + y \ge 8\), and \(x + 2y \ge 10\). First, we determine the points where the constraint lines intersect the axes and each other:

• Step 1: Intersection Points on the Axes
  
  • For \(2x + y = 8\):
    • If \(x = 0\), \(y = 8\).
    • If \(y = 0\), \(x = 4\).
  • For \(x + 2y = 10\):
    • If \(x = 0\), \(y = 5\).
    • If \(y = 0\), \(x = 10\).
• Step 2: Intersection of the Constraint Lines
  • Solve \(2x + y = 8\) and \(x + 2y = 10\) simultaneously:
    • Multiply the second equation by 2: \(2x + 4y = 20\).
    • Subtract the first equation: \(3y = 12 \Rightarrow y = 4\).
    • Substitute \(y = 4\) into \(2x + y = 8\): \(2x + 4 = 8 \Rightarrow x = 2\).

    The intersection point is \((2, 4)\).

• Step 3: Feasible Region and Basic Feasible Solutions
  Considering non-negativity constraints, the feasible region is bounded by the lines intersecting at \((10,0)\), \((0,8)\), \((2,4)\), as well as along the axes (subject to non-negativity). Therefore, basic feasible solutions include points forming the corners or intersections in this region: \((10,0)\), \((2,4)\), \((0,8)\).

Thus, the correct options for the given LPP are: (10, 0), (2, 4), and (0, 8).