Objective Function in Linear Programming Problem (LPP)

The objective function in a Linear Programming Problem (LPP) represents the mathematical expression that needs to be optimized. It defines the goal or objective of the problem. The objective can be either minimized or maximized, depending on the problem.

The Nature of Objective Function

The objective function in a Linear Programming Problem is always linear. This means that the relationship between the variables in the objective function is linear, with a constant coefficient for each variable.

Reasoning behind the Correct Answer: Option C (Linear)

The objective function in a Linear Programming Problem is always linear for several reasons:

1. Linear Relationships: The objective function is constructed using linear relationships between the variables. Each variable is multiplied by a constant coefficient and summed together. There are no higher-order terms such as squares, cubes, or higher powers.

2. Linearity Assumption: Linear Programming is based on the assumption of proportionality and additivity. This means that the impact of changing the variables is assumed to be linear. Nonlinear relationships would violate the assumptions of the model.

3. Mathematical Simplicity: Linearity allows for mathematical simplicity in solving Linear Programming Problems. Linear functions are easy to differentiate and optimize. Nonlinear functions, on the other hand, introduce complexities and may require advanced mathematical techniques to solve.

4. Graphical Interpretation: Linear functions can be easily represented and interpreted graphically. The objective function can be plotted on a graph as a straight line, and the optimal solution can be found by identifying the highest or lowest point on the line within the feasible region.

5. Applications: Many real-world problems can be modeled as Linear Programming Problems, and their objective functions often exhibit linear relationships. Examples include resource allocation, production planning, transportation problems, and inventory management.

In conclusion, the objective function in a Linear Programming Problem is always linear. It is constructed using linear relationships between the variables, follows the assumptions of linearity and proportionality, and allows for mathematical simplicity and graphical interpretation.