Linear Programming — Questions & Answers

Sample questions from Linear Programming

1. Q1. Maximise Z = 7x + 9y subject to 5x + 4y <= 41, 4x + 5y <= 41, x >= 0, y >= 0. The maximum value of Z is:

   • A.41
   • B.287/5
   • C.656/9
   • D.369/5✓
   Solution

   The lines meet at (41/9, 41/9) where Z = 656/9 = 72.9. Corner values are 0 at (0,0), 287/5 = 57.4 at (41/5,0), 656/9 at (41/9,41/9) and 369/5 = 73.8 at (0,41/5). The largest is 369/5, so the optimum is at the axis vertex (0, 41/5), not at the intersection.

2. Q2. If the constraint set of an LPP is x + y <= 8, x + y >= 8, x >= 0, y >= 0, then the feasible region is:

   • A.Empty
   • B.Unbounded
   • C.The segment joining (8,0) and (0,8)✓
   • D.The whole triangle with vertices (0,0), (8,0), (0,8)
   Solution

   The two inequalities together force x + y = 8. Combined with x >= 0, y >= 0 this is exactly the line segment from (8,0) to (0,8) — a bounded, non-empty, degenerate feasible region.

3. Q3. A workshop assembles x fans and y coolers with 6x + 4y <= 96 (labour), 2x + 5y <= 80 (wiring) and x + y <= 20 (packing). Profit is Rs 400 per fan and Rs 350 per cooler. The maximum profit is:

   • A.Rs 22000/3
   • B.Rs 5600
   • C.Rs 6400
   • D.Rs 7400✓
   Solution

   6x + 4y = 96 meets x + y = 20 at (8,12); x + y = 20 meets 2x + 5y = 80 at (20/3,40/3). Z = 400x + 350y is 0, 6400, 7400, 22000/3 = 7333.3 and 5600 at (0,0), (16,0), (8,12), (20/3,40/3) and (0,16). The maximum Rs 7400 is at 8 fans and 12 coolers.

4. Q4. For the LPP: Maximise Z = 3x + 2y subject to x + y ≤ 4, x ≥ 0, y ≥ 0, the maximum value of Z is:

   • A.6
   • B.12✓
   • C.8
   • D.10
   Solution

   B is correct. Corner points: (0,0): Z=0. (4,0): Z=12. (0,4): Z=8. Maximum Z=12 at (4,0).

5. Q5. The corner points of the feasible region of an LPP are (2,3), (7,3), (7,9) and (2,11). The maximum value of Z = 4x + 5y is:

   • A.83
   • B.43
   • C.63
   • D.73✓
   Solution

   Z at (2,3) = 8 + 15 = 23, at (7,3) = 28 + 15 = 43, at (7,9) = 28 + 45 = 73, at (2,11) = 8 + 55 = 63. The largest of these is 73 at (7,9).

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