LINEAR PROGRAMMING

Vikas Vasam

ID: 100-11-5919

Faculty: Prof. Dr Goran Trajkovski

CMP 561: Algorithm Analysis

VIRGINIA INTERNATIONAL UNIVERSITY

Introduction:

One of the section of mathematical programming is linear programming. Methods and linear programming models are widely used in the optimization of processes in all sectors of the economy: the development of the production program of the company, its distribution on the performers, when placing orders between the performers and the time intervals, to determine the best range of products, in problems of perspective, current and operational planning and management, traffic planning, defining a plan of trade and distribution, in the problems of development and distribution of productive forces, bases and depots of material handling systems, resources, etc. especially widely used methods and linear programming model for solving problems are savings (choice of resource-saving technologies, preparation of mixes, nesting materials), production, transportation and other tasks. Beginning of linear programming was initiated in 1939 by the Soviet mathematician and economist Kantorovich in his paper "Mathematical methods of organizing and planning production." The appearance of this work has opened a new stage in the application of mathematics in economics. Ten years later American mathematician George Dantzig developed an efficient method for solving this class of problems - the simplex method. The general idea of the simplex method to solve the LPP is as follows: ability to find initial support plan;

the presence of the optimality of the support program;

the ability to move to an improved support program.

1.1 The concept of linear programming :

Linear programming - the section of mathematical programming, applied in the development of methods for finding the extremum of linear functions of several variables by linear additional constraints imposed on the variables. According to the type of tasks its methods are divided into generic and specific. With the help of generic methods can be solved by any linear programming problem . Special methods take into account the particular model of the problem, its objective function and system constraints. Feature of linear programming problems is that the objective function reaches an extremum at the boundary of feasible solutions. The classical methods of differential calculus is associated with finding the extremum of the function at an interior point of acceptable values. Therefore it is necassary to develop new methods.

1.2 Simplex Method :

The simplex method of linear programming problems based on a reference from one plan to another, in which the objective function value increases (assuming that this problem has an optimal plan, and each of its basic plan is nondegenerate). This transition is possible, if you know some initial support plan. Consider the task for which the plan can be directly written.

1.3 The economic statement of the problem:

For fattening cattle using two types of feeds: b1, b2, in which nutrients are a1, a2, a3, a4. The content of nutrients in the units of 1 kg of each feed, the cost of 1 kg of food and nutrients in the diet of the animal are presented in Table . Make a diet, provided the minimum cost.

Nutrients Types of food Norma nutrient | | | | | |B1 | | | | |B2 | | | | | | | | | |...