Application of linear programming

Object of the interest of the given paper is the area of linear programming and its application in economic practice. It is possible to find basic characteristics, definitions, possibilities of records as well as description of selected task solution of linear programming in the paper. The authors focused on solution of a certain issue and therefore showed overall approach within solution of this kind of issues we come across each day in economic practice. The main goal of the paper is application of steps and algorithms focused on solution of issues connected with minimalization of costs created within purchase of materials used in a production company.


INTRODUCTION
Theory of mathematical programming as Brezina et al. state [1] "was elaborated in accordance with task solution of effective exploitation of bounded disposable resources necessary for reaching the given goals".Moreover, the authors state that "each task of mathematical programming is constructed with the aim to display some economic situation in which we try to find the best possible solution within specific bounded prerequisites".Numeric optimization methods are a part of various quantitative economic investigations, whereas many of them can be expressed by linear functions.Linear programming as a part of operational research has significantly rich history and nowadays represents scientific discipline of which the issues, tasks and questions are described in a great details.A linear programming (sometimes known as linear optimization) problem may be defined as the problem of maximizing or minimizing a linear function subject to linear constraints.The constraints may be equalities or inequalities.Simplistically, linear programming is the optimization of an outcome based on some set of constraints using a linear mathematical model.

MATERIAL AND METHODS
According to Fábry [2] basically we distinguish three basic forms of linear programming tasks (LP) 1 .

I. General form of LP tasks, it is a task to find points
, in which the linear form of variables reaches: (1a) maximum, resp.minimum in set of all the points suitable for equalities and inequalities in the form: is indication of so called Kronecker delta2 .

Newly given variables
, where are called additional variables.3. Canonical task (2) about maximum can be equivalently converted to a task about minimum by changing objective function to .

III. Standardized form of LP tasks defining general form (1) can be formulated as follows:
To find points , in which linear form of variables reaches: (3a) maximum, resp.minimum in set of all points suitable for inequalities in form: One of the possible methods used for solution of LP tasks is a graphical method which, as Klvaňa [3] states, is valuable especially because it is graphic.In spite of the fact that optional LP task can be interpreted geometrically there are limitations for graphic solution and representation of these tasks.
1. Set of possible solutions is a part of maximally three-dimensional space .LP task can be given in general form (1).

Dimension of space
, in which the task is solved, can be more than 3, although LP task must be given in canonical form (2) and must be stated , where .
Specific procedure of using graphical method for LP tasks solution which consists of four steps is given in the Fig. 1.It shows a case for two variables , where , solution of which can be drawn in double-dimensional graph, i.e. in a plane.
We display objective function in graph (1a).The easiest images are contour lines called as well isolines which create a set of parallels.

Coordinative axis is indicated as and coordinative axis is indicated as
Step I.
We draw set of possible solutions by displaying all the half-planes suitable inequalities and all the straight lines suitable for equalities in limited conditions (1b).Set of suitable solutions is intersection of all the half-planes and straight lines.If this intersection is not empty it is therefore convex polyedric set in plane.
Step II.
Step III.
We find point or points from the set of possible solutions (if they exist) in which objective function reaches required extreme.
Step IV.

Solution of the task by graphical method:
To solve the given nutrition task we use geometrical method and algorithm which is given in the previous part.Specific solution is in the Fig. 2. According to the fact that and variables can reach negative values, the set of possible solutions is always given in the first quadrant.The Fig. 2 shows that such LP task does not have a solution because the intersection of all half-planes, i.e. a set of possible solutions is an empty set.In other words, a set of limited conditions of the LP task is not consistent.
In such case the company has more possibilities how to solve the given task.They could for example not to accept the order for a sale of component in the amount of 9 units or sell smaller amount.Another alternative could be increasing of stock capacity and so forth.

Solution of modified task by graphical method:
In this case we voted for the first out of above mentioned possibilities as a solution, i.e. the company denies the sale of component.It means that from the mathematical model of the original task about nutrition issue, inequality is omitted and we add the second condition of non-negativity .
Graphical solution of modified task is shown in the Fig. 3 which shows a set of possible solutions given in light blue color.It creates a pentagon with its points and E. Each point3 of this set is a solution of the given LP task.To look for extreme in objective function is essential only on the edges of possible set .Therefore table 2 shows values of objective function for these five edges.Table 2 as well as Fig. 3 show that minimum value is reached by objective function on the edge , i.e. in a point where set of possible solutions are touched by isoline in the smallest value (in the Fig. 3 we can see line in green color).From the above mentioned findings we can state that within given limitations it is more advantageous for the company to purchase optimally 2 units of component and 2 units of components with overall costs 64 euro.

CONCLUSIONS
As we already mentioned linear programming is a part of operational research known as well as managerial science.Their meaning is to provide managers with exact quantitative solutions for decision-making they come across in everyday economic practice.Specific goal of the

1 .
For values , where from (1b) there are no limitations.These can reach positive, negative and zero values.2. Objective function is not identically equal zero, i.e. exists , where .3. There are cases when general formulation does not include the condition of nonnegativity for all the variables , where .II.Canonical form of LP tasks representing general form (1) which can be formulated as follows: To find points , in which the linear form of variables reaches:

Fig. 1 .
Fig. 1.Algorithm of graphical method of LP task solution for two variables and

Fig. 2
Fig. 2 Geometrical solution of the nutrition task

Fig. 3
Fig. 3 Geometric solution of modified nutrition task

Tab 2
Values of objective function on the edges of possible set of tasks