In reference to a linear programming problem (LPP), which of the following statements are true?
A. The optimal value of the objective function is attained at the points given by intersections of inequation with the axes only.
B. The objective function of a LPP is a linear function to be optimized.
C. Every LPP admits an optimal solution.
D. The region represented by the inequation system, \( x,y\ge 0 \) , \( x+2y\le 4,3x+y\ge 3,4x+3y\ge 6 \) is bounded in the first quadrant.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → B and D only
By the corner point theorem, the optimal value of an LPP, when it exists, occurs at a corner point of the feasible region. A corner may be the meeting point of two constraint lines, not only a point on an axis.
An LPP need not have an optimal solution: if the feasible region is empty there is no solution, and if it is unbounded the maximum (or minimum) may not exist.
For the system in D, the constraints \( x,y\ge 0 \) and \( x+2y\le 4 \) keep every point inside the triangle with vertices (0, 0), (4, 0) and (0, 2). The region is not empty; for example (4, 0) satisfies all the constraints:
\( \displaystyle 3(4)+0=12\ge 3 \)
\( \displaystyle 4(4)+0=16\ge 6 \)
So the optimum is at a corner point (not only on the axes), an LPP may have no optimal solution, the objective function is linear, and the region in D lies inside a triangle, so it is bounded.
Checking the statements
(A) The optimal value of the objective function is attained at the points given by intersections of inequation with the axes only. — Incorrect. It is attained at a corner point, which may be the meeting point of two constraint lines.
(B) The objective function of a LPP is a linear function to be optimized. — Correct. It has the form Z = ax + by, which is maximised or minimised.
(C) Every LPP admits an optimal solution. — Incorrect. An empty or unbounded feasible region may give no optimal solution.
(D) The region represented by the inequation system, \( x,y\ge 0 \) , \( x+2y\le 4,3x+y\ge 3,4x+3y\ge 6 \) is bounded in the first quadrant. — Correct. It lies inside the triangle with vertices (0, 0), (4, 0) and (0, 2).