# Linear programming

Linear programming |
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**Linear programming** is method to achieve optimal solutions in the decision-making process.

It is very often used due to the simple and clear calculation algorithm, and also ability to produce impressive graphs of decision-making issues. In a business dominate rational management principle, which assumes that the resources should be used to the maximum extent. If those resources can be described quantitatively we use a mathematical model. The disadvantage of linear programming is that not everything can be expressed with numbers.

## Mathematical Model

To apply linear programming in the decision-making process, one should develop a mathematical model that will contain:

**goal function**(otherwise known as the criterion function) this is the most important part of the model, because it illustrates the objectives of managers,**decision variables**describe the tools and resources that are available,**constraints**these are obstacles that may arise during the implementation of decision described by goal function.

## Solution of a developed model

It is important that the model had a linear form. Solving the problem we obtain acceptable solutions that satisfy the constraints. If model has two variables, it is easy to solve using **geometrical method** in the Cartesian coordinate system:

- after determining the set of feasible solutions we draw it in the system,
- Then you need to find a function that will have at least one point in common with the set of feasible solutions

In this way we determine the appropriate value of the goal function, which can be used to make a decision.

Another known method for solving the model is **simplex method**, which consists of:

- transformation of constraints (which are very often in the form of inequality) to the system of linear equations, by using the so-called. balance variables
- then decision variables are calculated that represent the best decision fitted to the established model

If the decision depends on many variables, the calculation requires a computer with a specially prepared software. An important element is also a sensitivity analysis that is the answer to the question of how to change the selected parameter being equal, so that the optimal solution remains in equilibrium.

**See also:**

## Examples of Linear programming

**Production Scheduling**: Linear programming can be used to optimize production schedules in order to maximize profits. For example, a factory may have multiple lines of production that can produce different products. Linear programming can be used to determine the most efficient production schedule to maximize profits.**Resource Allocation**: Linear programming can be used to allocate resources in an optimal manner. For example, a company may need to allocate its financial resources among various projects in order to maximize its profits. Linear programming can be used to determine the optimal allocation of these resources.**Routing Problems**: Linear programming can be used to solve routing problems. For example, a truck may have to deliver goods from one location to another. Linear programming can be used to determine the most efficient route that minimizes the time and cost of delivery.**Facility Location**: Linear programming can be used to determine the optimal location for a facility. For example, a company may need to decide where to locate its new factory in order to minimize costs and maximize profits. Linear programming can be used to determine the best location for the factory.

## Advantages of Linear programming

Linear programming is an efficient and powerful method used to optimize decision-making processes. It has several advantages, including:

- It allows for the maximization of profits or minimization of costs by determining the best combination of resources and activities.
- It can be used to optimize a wide range of problems, such as scheduling, production, distribution, and resource allocation.
- It is easy to implement, as it is based on linear equations that can be solved using a variety of techniques.
- It can be used to model complex problems with multiple constraints and objectives.
- It is capable of providing optimal solutions to problems that are not possible to solve using traditional methods.
- It provides a systematic approach to decision-making, which can be used to identify optimal solutions in a time-efficient manner.

## Limitations of Linear programming

Linear programming is a powerful tool for helping decision-makers find optimal solutions; however, it has several limitations. These include:

- Linear programming assumes that objective functions and constraints are linear, which may not always be the case. For instance, non-linear objectives such as minimizing the sum of squared errors cannot be solved using linear programming.
- Linear programming assumes that all variables are continuous, which may not always be the case. For instance, discrete variables such as integer, Boolean, and categorical variables cannot be solved using linear programming.
- Linear programming assumes that all constraints are convex, which may not always be the case. For instance, a non-convex constraint may lead to multiple local optimal solutions, which may not be the desired global solution.
- Linear programming assumes that all constraints are homogeneous, which may not always be the case. For instance, a non-homogeneous constraint such as a budget constraint may lead to an infeasible solution.
- Linear programming assumes that all variables have finite values, which may not always be the case. For instance, a variable with an infinite number of possible values may lead to an infeasible solution.

Linear programming is a method used to find the optimal solutions to decision-making problems. Other approaches related to Linear programming include:

**Integer programming**: This is a type of mathematical optimization method used when the decision variables in a problem must be expressed as whole numbers.**Nonlinear programming**: This technique is used when the objective function and/or the constraint functions of a problem are nonlinear.**Dynamic programming**: This approach is used when the objective function or constraints depend on the solutions of subproblems.**Goal programming**: This is a mathematical optimization approach used to solve problems with multiple objectives.

In summary, there are several approaches related to linear programming, including integer programming, nonlinear programming, dynamic programming, and goal programming. These methods can be used to find optimal solutions to decision-making problems.

## References

- Dantzig, G. B. (1998).
*Linear programming and extensions*. Princeton university press.