Linear Inequalities with Objective Functions

In this article, we’ll learn what is the objective function in linear programming problems and how it is used to solve programming problems. The objective function utilises variables. Maximising or minimising the function gives a solution for the function. Well, also see what the objective function in a linear programming problem is. In linear programming problems, it is used to find solutions to maximise profit and minimise cost. We’ll also see terms related to the objective function. Its representation in terms of variables. Theorems related to the objective function and, lastly, its application.

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What is the objective function of a linear programming problem?

The objective function is a real-valued function that is used to solve optimization problems in linear programming. It is represented in terms of linear variables as:

Q = bx+ Ay

(where x and y are variables) and x> 0, y>0

(b and A are constants)

By putting different values in place of x and y, we arrive at different solutions. The most accurate solution is obtained by maximising or minimising the function Q = bx+ Ay. It is done by putting the min or max value of the variables x and y. Hence the variables are referred to as the decision variables.

Most common terms you’ll come across while learning what is the objective function in linear programming problem

To understand the objective function, it is a prerequisite to know the following terms:

a) Optimization problem

It is a problem that functions to maximise or minimise a linear function keeping in mind certain constraints that are determined by a set of linear inequalities.

b) Linear programming

To understand what is the objective function in linear programming problems. It is important to know linear programming first. Linear here refers to the linear mathematical relationship of variables. While programming here refers to putting the program in action. It is used in finding an optimum solution to the given linear function. This linear function is referred to as the objective function.

c) Constraints

In linear programming objective function and objective constraints are to be kept in mind while putting the values of variables x and y in the objective function. These constraints are the simplest representation of inequalities in the objective function. Put so as to represent the non-negative restrictions.

d) Decision variables

For the objective function Q = bx+ Ay, the variables x and y are the decision variables. Decision variables can be the allocation time, resources to be used, or the workforce to be deployed, as these variables can vary according to the situation or requirement.

e) Optimal solution

This is mainly defined in terms of a graph of the objective function. It is also named as a feasible solution. These are basically the points on the feasible region of the objective function graph, which leads to the optimal min or max solution to the objective function.

f) Optimal value

It is that solution of the objective function which leads to a min or max value of the objective function. It can be the maximum resource to be used to obtain maximum output or profit.

Theorems related to Objective function

Two of the most important theorems of object functions are mentioned below:

Theorem 1:

Let’s say F is the feasible region for a linear programming problem,

 let Q= bx + Ay be the objective function.

When Q attains an optimal value, the variables y and x are maintained by the constraints inside the linear inequalities, the optimal value occurs at a point in the corner, which is the vertex within the feasible region.

Theorem 2:

 Let F be the feasible region for a linear programming problem, and

let Q = bx + Ay be the objective function. If F is bounded, then the objective function Q has both a maximum and a maximum value on F, and each of these occurs at a corner point (vertex) of F.

Components of linear programming

There are mainly three components of linear programming. These are as follows: –

1. Decision variables

In an objective function, Q = ax + by, x and y are referred to as the decision variables. Putting different values in place of these decision variables leads to an optimal solution for the linear function

2. Constraints

In the objective function, Q = ax + by, x, and y are variables that have limitations. Such as x> 0, y>0, these limitations put in place are the constraints to be followed.

3. Objective function

It is the real-valued function whose value is minimised or maximised to arrive at an optimal solution keeping in mind the constraints given for the LPP

Application of objective function

The linear programming problems and solutions are used to arrive at a solution to maximise output(profit) and minimise the input (cost and resources used). It is applied in industry, commerce, applied sciences etc.

Conclusion

The objective function in linear programming is used to arrive at an optimal solution by minimising or maximising the objective function. This is done by putting different values in place of variables in the linear equation Q = ax + by. With limitations to the decision variables as x> 0, y>0. The objective function in linear programming is mainly used to maximise the profit by using the maximum output or resources. Also, check out what is the objective function in linear programming problems MCQ, to analyse your preparation.