the miracle of linear program to solve problems in your business or you life

The importance of optimization and the need for a simple tool for modeling decision problems that either economic, military or other more active is made of linear programming of the search fields in the middle of the previous century. The early work (1947) is that of George B. Dantzig and associates.

Linear programming problems are usually related to limited resource allocations problems in the best possible way to maximize profit or minimize cost. The best term refers to the ability to have a set of possible decisions that achieve the same satisfaction or even profit. These decisions are usually the result of a mathematical problem.

I. The stages of formulation of PL:

Generally there are three steps to follow in order to build the model of a linear program:

1. Identify the problem variables unknown value (decision variable) and represent symbolically (exp. X1, y1).
2. Identify restrictions (constraints) of the problem and expressed by a system of linear equations.
3. Identify the objective or the selection criteria and represent linear form depending on the decision variables. Specify if the selection criterion is to maximize or minimize.

Example : agriculture Problem

A farmer wants to allocate 150 hectares of irrigable area between those tomatoes and peppers. It has 480 hours of labor and 440 m3 of water. One hectare of tomatoes request 1 hour of labor, 4 m3 of water and gives a net profit of 100 dinars. One hectare of peppers request 4 hours of labor, 2 m3 of water and gives a net profit of 200 dinars.

The office of the irrigated perimeter wants to protect the price of tomatoes and does not allow him to cultivate over 90 hectares of tomatoes. What is the best allocation of resources?

Step 1: Identify the decision variables. The two activities that the farmer must determine the surfaces are to be allocated for growing tomatoes and peppers:
• x1: the area allocated to growing tomatoes
• x2: the area allocated to the cultivation of peppers
We check that the x1 and x2 decision variables are positive.

Step 2: Identification of constraints. In this issue constraints are the availability of production factors:
• Land: the farmer has 150 hectares of land, and the stress related to the limitation of the land surface is
• Water: the cultivation of one hectare of tomatoes request 4 m3 of water and that of a hectare of peppers request 2m3 but the farmer has only 440m3. The constraint that expresses the water resource limitations is.
• Labor: The 480 hours of labor will decide (not necessarily all) shows growing tomatoes and peppers ones. Knowing that a hectare of tomatoes requires a time of labor and one hectare of peppers request 4 hours of labor then the constraint representing limitations of human resources is
• The limitations of the office of irrigated area: These limitations require that the farmer does not cultivate more than 90 hectares of tomatoes. The constraint that represents the restriction is

Step 3: Identification of the objective function. The objective function is to maximize the profit made by the tomatoes and peppers. The respective contributions 100 and 200, the two variables x1 and x2 decision are proportional to their value. The objective function is : z= (200 x1) + (200x2)