# Linear programming

**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:**

## References

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