Organic Chemistry: Some Basic Principles And Techniques

This topic is from the chapter of Linear Programming in which a function’s minimum and the maximum value are evaluated with the help of defined constraints. Constraints are called inequalities of any problem. Solutions to these inequalities are known as feasible regions which consist of some set of points. These sets of points intersect the constraints when they are traced in the graph. After intersecting all the constraints of the solutions then they either result in a polygonal area or covered area or a bounded region which is known as a bounded feasible region. And if the area is open or the region is not bounded then it is either called as unbounded feasible region or an infeasible region. 

Many real-world issues can be traced and solved with the help of linear programming to predict the current trend and future trends. These types of problem-solving techniques are widely used by manufacturing companies for the calculations of quantity predictions. 

Now, let’s dive deep into the topic by understanding the definitions, usefulness, and examples of each term independently.

Feasible and Infeasible Region

In linear programming, feasible and infeasible regions play a major role to identify the area for which the linear programming equation is defined. The feasible region is defined as the values which satisfy a given constraint (minimum and maximum value or range of values for an LP equation). On the other side, if the constraints are not satisfied by the LP equation, then it falls under the infeasible region.

In Mathematics, linear programming is the best-utilized technique for optimizing operations with few constraints. The main purpose of linear programming is to maximize or minimize the numerical value. It consists of linear functions which stand subject to limitations in the form of linear equations or in the form of inequalities.

Linear programming is utilized for getting the considerable optimal explanation for a problem with given conditions. In linear programming, we develop our real-life problem into a mathematical technique. It involves an objective function, linear inequalities subject to limitations.

Bounded and Unbounded Region

A bounded region can be defined as if the LP equations completely lie in given constraints which can be easily covered through the given feasible vectors. On the other hand, if the values of the feasible vectors are too large then it is said to be in an unbounded region.

Let us see the examples for different regions

In practice there are three types of regions possible which are as follows:

1.   Bounded feasible region.
2.   Unbounded feasible region.
3.   Infeasible region.

How to find the feasible region?

The feasible region of the graph contains all the points that satisfy all the inequalities present in the system. 

To draw the graph of the feasible region, first, we have to draw a graph of each and every inequality in the present system. Then we have to find the area where all the graphs overlap. That is the required feasible region.

Let’s look at all the three types of regions with examples.

Example of Bounded Feasible Region

Consider the following situation:

So, in this case, we can clearly evaluate the first condition by inserting ‘0’ in the ‘x’ variable to get ‘y’ as ‘5’ and on the other hand ‘0’ in the ‘y’ variable to get ‘x’ as ‘10’. It will give us two coordinates i.e., A (0,5) and B (10,0).

By plotting the A and B in the graph along with the two more statements which state that ‘x’ and ‘y’ are greater than ‘0’ (clearly states that the region is in 1st quadrant) creates a bounded feasible region.

Example of Unbounded Feasible Region

Consider the following situation:

So, in this case, we can see only one condition is given which is for ‘x’ coordinate. There is no information about the other coordinate. So, the region is feasible but there are no boundary conditions to completely satisfy the same. As ‘y’ can have too many solutions.

Example of Infeasible Region

Consider the following situation:

In the following case, the ‘x’ coordinate is greater than ‘4’ but again there are no defined conditions for the other coordinate. So, this is a type of infeasible region as too large values come under this region.