Linear Programming
Department of General Sciences
College of Basic Education / Haditha - University of Anbar
The sciences in general are intertwined and dependent on each other. Mathematics can be considered as one of the most important sciences. Mathematics is even called the mother of sciences, which is included in all other sciences to solve or clarify problems, etc. In this article, we will present the contribution of mathematicians along with economists in creating a modern scientific concept that helps the higher management of establishments to take a number of critical decisions that they face. Hence, both economists and mathematicians contributed to the development of this method, which began to appear in 1920 at the hands of the famous economist (Leontiev) in Its development for the analysis of inputs and outputs, then continued its development at the hands of the English mathematician (Dantizg) who discovered the Simplex method, one of the methods of solving linear programming, and linear programming is one of the modern scientific methods of operations research.
First: Define Linear Programming:
It is a modern mathematical method used to find the best uses for the available limited resources. This method has two aspects: Programming, which means the possibility of using the computer to find different programs to use the limited resources available to the facility and in a manner consistent with the restrictions imposed on the facility and in line with the restrictions imposed on these resources. As for linearity, it means the relationships between all the specific variables of the problem under study. Linear relationships, that is, the response of all variables is one and is consistent with the response of the objective function.
Second: The Linear Programming Model: It is a set of equations and inequalities in addition to the objective function. We always strive to find the optimal solution, which achieves all constraints with the existence of the Objection Function.
Third: The objective of linear programming: The main objective of using linear programming models is to solve a problem facing the administration. Therefore, linear programming is used, and here it is necessary to transfer the problem from its initial state (the state of speech or the constructive state, represented by verbal narration of all the details of the problem) to The case of equations and inequalities expressing the problem is under study, and here the linear programming model should explain the dimensions of the original problem in all its details. Finally, the mathematical solution to the linear programming model, which originally represents a solution to the research problem, is possible to obtain the optimal solution. Therefore, all matters of experience and know-how must be used in the formulation of linear programming models.
Fourth: Types of solutions in linear programming: The solutions in linear programming are of three types:
1- Solution: It is a solution that can be reached in any set of equations.
2- The feasible solution, which is the solution that can be found after reaching the solution in the first case, and this solution fulfills all constraints in general.
3- The optimal solution, which is the solution that can be found after reaching the possible solution, and this solution fulfills all constraints with the presence of the objective function.
Fifth: Methods of solving linear programming models: (Method of Solution of L.P Models)
There are two basic ways to solve linear programming models:
1- Graphical Method: This method is suitable for solving linear programming problems that contain only two variables and is used if the variables are restricted or unrestricted by the sign. Sin in practical life.
2- The Simplex Method This method was developed by the British mathematician (Dantizg G.) in 1947. This method begins with finding a possible basic initial solution, then moving to a basic solution that may be better than the previous one, by replacing the variables with one of the non-essential variables. the basic .
Keywords: Linear programming, linear programming model, linear programming solutions, graphic method, simplex method.