Objective functions

Objective functions are the most efficient way to solve and represent linear programming optimization problems. It can take the form Z=ax + by where, a and b are limitations, and x and y are variables. x and y are also known as the decision variables and must be maximized and minimized. Alternatively, it can be of the form Z=f(xi).

Generally, we use this function to find the maximum or minimum of any given data. We can write its formula as:

Terms Related to Objective Function:

Some of the most used terms that make understanding of this concept easy and efficient are:

1. Linear Programming: It is the process of achieving the best outcome (maximum profit or lowest cost) in a mathematical form whose elements are represented by linear relationships. Linear programming is a particular case of mathematical programming.

2. Optimization Problem: An optimization problem is a problem in which a linear function has to be maximized or minimized subject to certain limitations set by a collection of linear inequalities.

3. Decision Variables: The decision variables are the variables that will decide my output. For example, in a business, the resources required, people assigned to work, and time allotted are all decision variables. In the form Z=ax+by , x and y are the decision variables.

4. Constraints: The limitations in the set of values that the variables x and y can take are called the constraints of the objective function. The constraints x>0,y>0 are the most basic constraints to show positive numbers. 

5. Feasible Region: A feasible region is an area defined by a set of coordinates that satisfy all the constraints, including non-negative constraints x>0,y>0 of a linear programming problem. It’s an optimization technique.

6. Feasible Solution: It is the value to the variables that lie in the feasible region or on the boundary of the feasible area. It satisfies all restrictions and inequalities.

7. Optimal Value: Optimal Value is a value from the set of feasible solutions with the maximum efficiency—for example, minimum cost, the maximum profit, or the optimal use of resources.

Methods of solving problems with objective functions:

1. Graphical method: The graphical method involves formulating mathematical equations from the given problem then plotting the equations in an x-y plane. The intersecting region gives us a feasible part. 
2. Northwest corner method: First, mathematical equations are formulated from the given problems. The feasible region is marked. The objective function at every corner point of the feasible region is evaluated; N and n are the points’ largest and smallest values. If the feasible area is bounded, then N and n are the maximum and minimum values of the objective function. If the feasible region is unbounded, N is the maximum value of the objective function if the open half part has no common point in the feasible area. In any other case, the objective function will have no solution. n is the minimum value of the objective function if the open half part has no common point in the feasible region. In any other case , the objective function will have no solution
3. Iso – cost method: In this method, we plot lines parallel to the slope of the given equations, plot the graph using the given restrictions and mark the feasible region. Note down the coordinates of the feasible region. Next, find the most suitable Z value and plot the line of this objective function. Finally, draw a line that is parallel to the objective function line. If the objective function is of the maximum cayegory, the line is farthest from the origin and only has one common point in the feasible region. If the objective function is of the minimum category, then this line is nearest to the origin and has one or more common points in the feasible area. Now we can find the optimal solution using the coordinate of the common points.

Theorems Of Objective Function:

The objective function in the linear programming problem has two most important theorems.

1. Suppose R is the feasible region for a linear programming problem and is the objective function. In that case, when R has an optimal value, it must occur at a corner point of the feasible region.

2. Suppose R is the feasible region for a linear programming problem and is the objective function.Under those circumstances, if R is bounded, the objective function Z will have both a maximum and a minimum value for R, and each of these will occur at a corner point of the feasible area.

Types of Linear Programming Problems:

The most commonly used real-life problems of linear programming are :

1. Manufacturing problem: In these types of problems, restrictions like human resources, resources, output, time are given as equations, and we have to find an optimal solution to make a maximum profit with minimum cost.
2. Diet problem: in these types of problems, we are given restrictions like ingredients, cost, time is taken to make, and we have to find an optimal diet with minimal cost and maximum nutrients.
3. Transportation Problem: In these types of problems, we are given route distance, fuel cost, time, and we have to find the cheapest way of transportation.

Conclusion:

In today’s world, everyone thrives on getting the most output with minor input, and optimization gives them the most efficient answer. As a result, the objective function is used across the industry, commerce, management, applied sciences to solve numerous real-life problems.