Linear Programming Problems Formulas

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1. Linear Inequation

If a, b, c ∈ R, then the equation ax + by = c is called a linear equation in two variables x, y whereas inequalities of the form ax + by ≤ c, ax + by ≥ c, ax + by < c and ax + by > c are called Linear Inequations in two variables x & y.

2. Graphs of Linear Inequations

Consider a linear inequation ax + by ≤ c. Drawing the graph of a linear inequation means finding its solution set.
Steps to draw the graph
To draw the graph of an equation, following procedures are to be made
(i) Write the inequation ax + by ≤ c into an equation ax + by = c which represent a straight line.

(ii) Put y = 0 in ax + by = c to get point where the line meets x- axis. Similarly, put x = 0 to obtain a point where the line meets y-axis. Join these two points to obtain the graph of the line.

(iii) If the inequation is > or < , then the points lie on this line does not consider and line is drawn dotted or discontinuous.

(iv) If the inequation is ≥ or ≤, then the point lie on the line consider and line is drawn black (bold) or continuous.

To find the region that satisfies the inequation, we apply the following rules
(a) Choose a point [If possible (0, 0)] not lying on this line.

(b) Substitute its coordinates in the inequation. If the inequation is satisfied, then shade the portion of the plane which contains the chosen point, otherwise shade the portion which does not contain this point. The shaded portion represents the solution set.

3. Feasible Region

The limited (bounded) region of the graph made by two inequations are called Feasible Region. All the coordinates of the points in feasible region constitutes the solutions of system of inequations.

4. Linear Programming Problems (L.P.P.)

Linear Programming is a device to optimize the results which occurs in business under some restrictions. A general Linear Programming problem can be stated as follows:

Given a set of m linear inequalities or equations in n variables, we wish to find non- negative values of these variables which will satisfy these inequalities or equations and maximize or minimize some linear functions of the variables.

5. Some Definitions

(i) Solution:
A set of values of the decision variables which satisfy the constraints of a Linear Programming Problem (L.P.P.) is called a solution of the L.P.P.

(ii) Feasible Solution:
A solution of L.P.P. which also satisfy the non-negative restrictions of the problem is called the feasible solution.

(iii) Optimal Solution:
A feasible solution which maximize or minimize i.e. which optimize the objective function of L.P.P. called an optimal solution.

(iv) Iso- Profit Line:
The line is drawn in geometrical area of feasible region of L.P.P. for which the objective function remains constant at all the points lie on this line, is called iso- profit line.

6. Graphical method of solution of Linear Programming Problems

The graphical method for solving linear programming problems is applicable to those problems which involve only two variables. This method is based upon a theorem, called extreme point theorem, which is stated as follows-
Extreme Point Theorem:
If a L.P.P. admits an optimal solution, then at least one of the extreme (or comer) points of the feasible region gives the optimal solution.
Working Rule
(i) Find the solution set of the system of simultaneous linear inequations given by constraints and non- negativity restrictions.

(ii) Find the coordinates of each of comer points of the feasible region.

(iii) Find the values of the objective function at each of the comer points of the feasible region. By the extreme point theorem one of the comer points will provide the optimal value of the objective function. The coordinates of that comer point determine the optimal solution of the L.P.P.

7. Convex sets

In linear programming problem mostly feasible solution is a polygon in first quadrant this polygon is a convex. It means that if two points of polygon are connecting by a line then the line must be inside to polygon. For example-
(i)
(ii)
(iii)
(iv)
Figure (i) and (ii) are convex set while (iii) & (iv) are not convex set. It can be easily seen that the intersection of two convex sets is a convex set and the set of all feasible solutions of a LPP is also a convex set.

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