Linear programming: what it is, how to use it and how to do it

The linear programming is a valuable tool for economic decision-making as it enables the search for optimal solutions to complex problems with multiple variables.

As companies strive to become more efficient and competitive in a globalized market, linear programming has become an indispensable technique in organizational management.

What is Linear Programming?

Linear programming is a mathematical technique used to optimize the performance or efficiency of a system. This technique is widely used in the business world to solve planning, resource allocation and decision-making problems.

A linear programming problem attempts to find the maximum or minimum value of an objective function, such as: B. maximizing a company's profits or minimizing the production costs of a product. The objective function has constraints that must be met, such as: B. the budget available to the company or the amount of resources available to produce the product.

Uses of Linear Programming

Linear programming is used in a variety of areas such as: B. in business, engineering, operations management and enterprise resource planning.

For example, it can be used to optimize resource allocation within a company, plan the production of goods and services, maximize efficiency in allocating transportation routes, or optimize the distribution of products in a market.

The Importance of Linear Programming

Linear programming is important because it allows you to make objective decisions, optimize processes and resources, increase efficiency and find innovative solutions.

These are some of the reasons why you should consider using line scheduling:

1. decision making: Linear programming allows you to make data-driven, objective decisions. Because it uses mathematical models that clearly represent the situation to be solved and allow you to find the best possible solution.
2. Synchronise: Linear programming is used to optimize processes and resources in a variety of areas, such as: B. in production, sales, planning and project management. By finding the optimal solution, profits can be maximized or costs can be minimized.
3. Efficiency: Linear programming enables more efficient use of resources as it enables optimal planning and allocation of resources. This enables cost reductions and greater process efficiency.
4. Innovation: Linear programming makes it possible to solve complex problems and find innovative solutions. This is particularly important in areas such as engineering, science and technology, where innovative solutions are required to achieve progress.

What are linear programming methods?

Linear programming problems can be solved using techniques such as the simplex method or the Lagrange multiplier method. These techniques can be used to find the optimal solution to the problem efficiently.

Below you can learn more about the methods for solving linear programming problems:

Graphical method

This method is useful when working with linear programming problems with only two variables. In this method, the constraints and the objective function are plotted on a Cartesian plane and the intersection of the constraints is sought to find the optimal solution.

Simplex method

This is one of the most commonly used methods for solving multi-variable linear programming problems. This method creates a table with the variables and constraints and performs a series of iterations to find the optimal solution.

Lagrange multiplier method

This method is used when the linear programming problem contains equality constraints. In this method, a Lagrangian function is constructed and Lagrange multipliers are used to find the optimal solution.

Feasible Regions Method

This method is applied when the linear programming problem contains inequality constraints. In this method, the space of variables is divided into several feasible regions and each of these regions is tested to find the optimal solution.

What are the steps for linear programming?

Here are the general steps for linear programming:

• Define the problem: The first step is to define the problem you want to solve. It is important to clearly define the goal and the conditions to be met.
• Determine the variables: The variables are the unknowns that you want to find in the problem. It is important to determine which variables are relevant to the problem and name them.
• Formulate the objective function: The objective function is a mathematical equation that represents the goal of the problem, either maximizing or minimizing a value. The objective function should relate to the identified variables and be linear.
• Specify the constraints: The constraints are the restrictions that must be met when solving the problem. These constraints must relate to the identified variables and be linear. In addition, the constraints must be in the form of inequalities or equations.
• Representation of the problem in the form of a linear system of equations: Once the objective function and constraints are defined, they can be represented in the form of a linear system of equations.
• Solving the system of linear equations: There are several methods for solving systems of linear equations, one of the most common is the simplex method. This method can be used to find the optimal solution that meets the boundary conditions and optimizes the objective function.
• Interpretation of the solution: Once the optimal solution is found, it is important to interpret it in order to make informed decisions and evaluate the effectiveness of the model. The model may need to be adjusted and resolved if the results do not meet expected goals.

These are the general steps of linear programming. Each problem is unique and may require specific adjustments, but these steps provide a general guide for solving linear programming problems.

Example of a linear programming problem

Here is a simple example of a linear programming problem:

Suppose a farmer has 100 acres of land on which he wants to grow wheat and barley. The cost of growing wheat is $20 per acre and the cost of growing barley is $10 per acre. The farmer wants to maximize his profits and knows that wheat produces a profit of $50 per acre and barley produces a profit of $30 per acre. Additionally, the farmer knows he can only grow 75 acres of wheat due to irrigation restrictions. How many acres must he plant with wheat and barley to maximize his profits?

To solve this linear programming problem, we can apply the simplex method. First we have to formulate the objective function and the constraints:

Objective function: Profit maximization = 50x + 30y (where “x” is the number of wheat areas and “y” is the number of barley areas).

Additional conditions:

• Land restriction: x + y ≤ 100
• Cost constraint: 20x + 10y ≤ C (where C is the available budget)
• Irrigation restriction: x ≤ 75

Next, we construct a simplex table to solve the problem:

In the first line of the table we enter the coefficients of the objective function. The first column contains the constraints and the other columns contain the coefficients of the individual variables in each constraint. The RHS (right side) is the value of each constraint.

We then convert the constraints into equations and solve them to get the values ​​of “x” and “y”:

Land restriction: x + y = 100.

Cost limit: 20x + 10y = C

Irrigation constraint: x = 75

We can simplify the table by replacing the constraints with the values ​​of x:

We then use the simplex method to find the optimal solution. After a few iterations, we find that the optimal solution is to plant 75 acres of wheat and 25 acres of barley, maximizing the farmer's profit at $3.750.

This is a simple example of how a linear programming problem can be solved using the Simplex method to maximize a farmer's profit from growing wheat and barley on his land.

Conclusion

In summary, linear programming is a powerful mathematical tool for solving optimization problems in a variety of fields, used to maximize or minimize a linear function under certain constraints.

Linear programming requires accurate and reliable data to function properly. Therefore, it is important to have appropriate systems in place to collect and analyse relevant and accurate data so that informed and accurate decisions can be made. Additionally, linear programming can be used to analyse large data sets and identify patterns and trends that are not visible to the naked eye, which can be of great use in strategic decision making.

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KEYWORDS OF THIS BLOG POST

linear programming | linear | Coding