Consider the following statements about a linear programming problem (LPP):
(A) A set of values of the decision variables that satisfies all the constraints and the non-negative restrictions is called a feasible solution.
(B) A feasible solution that maximises or minimises the objective function is called an optimal solution.
(C) The objective function of an LPP may contain a term in \(x^2\).
(D) An LPP always has exactly one optimal solution.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (B) only
In an LPP the objective function \(Z = ax + by\) and all the constraints are linear. A feasible solution satisfies every constraint, and an optimal solution is a feasible solution that gives the best value of Z. An LPP may have no optimal solution, one optimal solution or infinitely many.
For example, maximising \(Z = x + y\) subject to \(x + y \le 4\), \(x, y \ge 0\) gives \(Z = 4\) at (4, 0), at (0, 4) and at every point between them.
So the objective function must be linear, and an LPP can have no, one or infinitely many optimal solutions.
Checking the statements
(A) A set of values of the decision variables that satisfies all the constraints and the non-negative restrictions is called a feasible solution. — Correct. This is the definition of a feasible solution.
(B) A feasible solution that maximises or minimises the objective function is called an optimal solution. — Correct. This is the definition of an optimal solution.
(C) The objective function of an LPP may contain a term in \(x^2\). — Incorrect. The objective function must be linear, of the form \(ax + by\).
(D) An LPP always has exactly one optimal solution. — Incorrect. Maximising \(x + y\) on \(x + y \le 4\) has infinitely many optimal points.