Every linear programming problem involves optimizing a
q,
linear function subject to several non-linear constraints.
non-linear function subject to several linear constraints.
linear regression model subject to several linear constraints.
linear function subject to several linear constraints.
The Correct Answer and Explanation is:
Correct Answer:
Linear function subject to several linear constraints.
Explanation:
Linear programming (LP) is a mathematical technique used to determine the best possible outcome (such as maximum profit or minimum cost) in a mathematical model. The defining characteristics of a linear programming problem are:
- Objective Function (Linear Function):
This is the function that needs to be optimized—either maximized (e.g., profits) or minimized (e.g., costs). It is linear, meaning it involves variables raised only to the first power, with no products or nonlinear operations like squares or square roots.
Example:
Z=5x+3yZ = 5x + 3y, where ZZ is the objective function to be maximized or minimized, and xx and yy are decision variables. - Constraints (Linear Constraints):
These are the restrictions or limitations on the decision variables, and they also must be linear. Constraints are expressed as linear equations or inequalities.
Example:
2x+y≤102x + y \leq 10
x≥0,y≥0x \geq 0, y \geq 0 (Non-negativity constraints)
Why the Other Options Are Incorrect:
- “Linear function subject to several non-linear constraints”
This describes a non-linear programming problem, not linear programming. - “Non-linear function subject to several linear constraints”
This also falls under non-linear programming, because the objective function is not linear. - “Linear regression model subject to several linear constraints”
Linear regression is a statistical technique for predicting the value of a variable. It is not an optimization method in the same context as linear programming.
Conclusion:
Linear programming problems always involve a linear objective function and linear constraints. The power of LP lies in its ability to find optimal solutions efficiently under these conditions using algorithms like the Simplex method or graphical methods for two-variable problems.