LPP

 

 Linear inequality of one variable

The linear inequalities in one variable can be expressed in any one of the following forms:

         ax + b ³  0

         ax + b £ 0

         ax + b < 0

         ax + b > 0

The solution set of such inequalities give the ray, the ray may be open or it may be closed.

For example, the solution of the inequality 2x - 4 ³ 0 is x ³ 2, i.e., the solution set of this inequality consists of all real numbers greater than or equal to 2. The graph or the solution set is given in the following figure.

Linear Inequalities of Two Variables

The inequalities in two variables can be expressed  in any one of the following forms:

            ax+ by ³ c

            ax + by £ c

            ax + by < c

            ax + by > c

The solution set such inequalities is the half plane; the half plane may be open or closed. The equation corresponding to the linear inequalities of two variables is the equation of the form ax + by = c, is called the boundary line. If the solution set of the inequality consists the boundary line, then the half plane is called closed half plane and if the solution set does not consist the boundary line then the half plane is called open half.

For example, for the inequality x + y ³ 6, boundary line corresponding to this inequality is x + y = 6. This boundary line passes through the points (6, 0) and (0, 6). The solution set of this inequality opposite to that of origin and the boundary line is included in the solution set, so the solution set is the closed half plane.

Note: To find the solution set, we take a point which does not lie on the boundary line and substitute it in the given inequality. If this point satisfies the given inequality, then this point lies in the solution set and if it does not satisfies the inequality, the point does not lie on the solution and the solution set is on the opposite side of the line from that point.

The System of Linear Inequalities

Two or more than two linear inequalities taken together form the system of linear inequalities. For example,

            i) x ³ 1 and x £ 3

            ii) x + y £ 6, x ³ 0 and y ³ 0

            iii) x + y < 2 and x + 2y < 5

Any vales of the variables which satisfy the every member of the system of inequalities are called the solutions of the inequalities.

The value of variable may or may not satisfy the linear system of inequalities. If we find the value of variables or points which satisfy the system of linear inequalities, then set of all such points is called solution set. If the system of inequalities is not satisfied by the value of variable, then such steam is said to be inconsistent.

3.4    Linear Programming

Linear inequalities are used to determine the maximum or minimum value of the linear function under the given conditions. The area of linear programming deals with many practical problems in business, economics, life sciences, social sciences, etc. involving the problems of finding the maximum and minimum values of the linear function of variables which are subjected to various constraints expressed by linear inequalities of the variables. The linear programming is defined as follows:

Linear programming is a mathematical technique for finding the optimal (maximum and minimum) value of a linear function (representing cost, profit, distance, weight, quantity, etc.) called objective function of two or more variables subject to a set of linear constraints in the form of one or more linear inequalities of two or more variables.

Some Basic Terms

Objective function: The linear function z = ax + by + c which is to be optimized is called the objective function where a, b, c are real numbers and x, y are variables.

Constraints: The set of linear in-equations are known as constraints under the linear programming problem.

Decision variable: Those non-negative independent variables which are to be used in the solution of the linear programming problem are the decision variables.

Convex region: A region is said to be convex if wherever we choose any points in the region and the line segment joining them, the line segment lies in the region.

Feasible region: The closed plane region in the Cartesian plane obtained by the finite intersection of the half planes determined by a set of constraints is said to be a feasible region. The maximum and minimum values of the objective function occurs at the corner points of the feasible region.

Graphical Solution of Linear Programming Problem

The linear programming problems in two variables can be solved by graphical method. The following steps should be considered to solve the linear programming problem by graphical method.

§       Locate and identify the variables.

§       Make a chart to organize the data for the problems.

§       Restate the verbal constraints as linear inequalities.

§       Express the objective function in terms of the variables.

        §       Express the given inequalities into their corresponding equations.

§       Draw the graphs to get the feasible region and determine the vertices of feasible region.

§       Evaluate the value of the objective function at each of the vertices of feasible region.

§       Determine the optimal solution and interpret the obtained value.