The objective function

The Objective Function

An objective function from word problems

In linear programming, the objective function is the mathematical expression that needs to be maximized or minimized, such as profit or cost. It is usually derived from a real-life word problem.

Linear programming components

Any linear programming problem has the following key components:

1. Objective: The goal to be achieved, such as maximizing profit or minimizing cost.
2. Alternative courses of action: Different options or decisions that can help achieve the objective.
3. Available resources: These are usually limited and must be used efficiently (e.g., time, money, materials, labor).
4. Mathematical representation: Both the objective and its limitations (constraints) must be expressible as linear equations or linear inequalities.

Therefore, linear programming aims at finding the best use of the available limited resources.

Programming refers to the application of mathematical methods to obtain the best possible solution to a problem.

Steps to solve linear programming problems

1. Read the problem carefully — Read several times if needed to fully understand the situation.
2. Define the variables — Use letters like x and y to represent quantities of interest (e.g., number of products, hours of labor, etc.).
3. Formulate the mathematical model — Using the variables defined, write:
   • The objective function (what you want to maximize or minimize).
   • The constraints (the limitations), expressed as inequalities.
4. Plot the constraints on a graph — Represent the feasible region that satisfies all constraints.
5. Identify the corner points (vertices) of the feasible region.
6. Evaluate the objective function at each corner point — Determine which one gives the maximum or minimum value (depending on the goal).
7. Draw a conclusion — State which values of x and y give the optimal solution and interpret the result in the context of the problem.