What is a Linear Programming?

By  HarshVardhan    164 - 10 January, 23
Linear Programming

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Linear programming is a strategy that assists us in determining the optimal solution for a given problem. An optimum solution is the best possible outcome of a given problem. In basic terms, it is the process of determining how to do something in the best possible way given limited resources. It is the strategy of maximizing resource utilization to get the greatest possible outcome for a certain purpose. Such criteria as lowest cost, biggest margin, or shortest time on such resources have other applications. Amendable programming analysis is used in situations where the optimal values of variables must be found while adhering to particular limitations. Calculus and marginal analysis are ineffective in dealing with these problems.

​​At its core, a linear programming problem consists of three components: an objective function, constraints, and decision variables. The objective function is a linear equation that represents the quantity that we want to optimize, such as profit or cost. The constraints are a set of linear equations or inequalities that represent the limitations or requirements that the solution must meet. The decision variables are the variables that we can change in order to optimize the objective function while meeting the constraints.

Linear Programming Problem

The following are the most important uses of linear programming which are used to solve Linear programming problem.

Manufacturing Problems

These issues are connected to issues in the industry, such as the need for some industries to produce a certain number of units of various products while requiring a set amount of labor, operating hours, and manpower per unit of product, among other things, in order to maximize profit from the sale of these products.

Diet Problems

In order to minimize costs and take into account food availability and costs, it is used to determine how much of various types of elements should be included in the diet.

Transportation Problems

The best strategy to deliver a product from factories or plants located in various places to markets is to identify the lowest transportation schedule possible.

Linear Programming Examples

Here is an example of a linear programming problem that illustrates the basic concepts of the technique:

A company produces two products, A and B, using two resources, labor and materials. Each unit of product A requires 2 hours of labor and 3 units of materials, while each unit of product B requires 1 hour of labor and 2 units of materials. The company has 40 hours of labor and 60 units of materials available. In addition, the company has a demand for at least 10 units of product A and 15 units of product B. The company wants to determine the production levels of product A and B that will maximize profit. The profit for each unit of product A is $5, and for each unit of product B is $4.

Linear Programming examples

The objective function in this case is to maximize the profit, which is represented by the linear equation:

Maximize : 5x + 4y (where x is the number of units of product A produced and y is the number of units of product B produced)

The constraints in this case are:

Labor constraint : 2x + y <= 40 (the company has a maximum of 40 hours of labor available)

Materials constraint : 3x + 2y <= 60 (the company has a maximum of 60 units of materials available)

Demand constraint : x >= 10 (the company has a minimum demand of 10 units of product A)

Demand constraint : y >= 15 (the company has a minimum demand of 15 units of product B)

x, y >= 0 (the production levels of the products must be non-negative)

We can solve this linear programming problem graphically. We could plot the constraints to graph a region where all the solutions will lie in. The solutions will be the points on the boundary of the feasible region that are on the line that maximize the objective function. The optimal solution will be the point where the objective function is maximized and all the constraints are met.

Or we could use the simplex method to solve the problem, starting with an initial feasible solution and then repeatedly making small changes to the production levels of products A and B, guided by the simplex method's rules, until the optimal solution is reached.

In this case, the solution would be to produce 12 units of product A and 20 units of product B, for a maximum profit of $100. With this solution, the company is meeting all the constraints and earning the maximum possible profit.

Linear programming examples could be used in practice to solve problems related to production and resource allocation in a company. The concept, however, could be applied to a wide range of problems like the examples I mentioned earlier.

Terminologies used in Linear Programming Problem

You must have a firm understanding of the fundamental terminology utilized in solving linear programming problems in order to solve them. These terms are listed below:

Objective Function

The issue must have involved a problem with a distinct and definable aim, such as the maximization of profit or the reduction of costs, etc. These examples all come within the category of the objective function.

Decision Variable

Variables, like products and services, that are in competition with one another for a limited amount of resources. The term "decision variable" refers to a set of variables that are connected and have a linear connection that can determine the most ideal solution.

Constraints

The limitations placed on the available resources, such as a limited number of machinery, manpower, etc.

Redundant Constraints

Redundant restrictions are those that are clearly present and yet don't limit the solution to the problem being studied.

Optimum Solution

This is the finest option out of all the ones that could work to assist the problem's goal in the best way.