xl , x2 ,….., xn ≥ 0 where all al1 , al2 ,…., amn ; bl , b2 ,…., bm ; cl , c2 ,…., cn are constants and xl , x2 ,…., xn are variables .
Slack and Surplus Variables
The positive variables which are added to left hand sides of the constraints to convert them into equalities are called the slack variables . The positive variables which are subtracted from the left hand sides of the constraints to convert them into equalities are called the surplus variables .
Important Definitions and Results
( i ) Solution of a LPP A set of values of the variables xl , x2 ,…., xn satisfying the constraints of a LPP is called a solution of the LPP .
( ii ) Feasible Solution of a LPP A set of values of the variables xl , x2 ,…., xn satisfying the constraints and non-negative restrictions of a LPP is called a feasible solution of the LPP .
( iii ) Optimal Solution of a LPP A feasible solution of a LPP is said to , be optimal ( or optimum ), if it also optimizes the objective function of the problem .
( iv ) Graphical Solution of a LPP The solution of a LPP obtained by graphical method i . e ., by drawing the graphs corresponding to the constraints and the non-negative restrictions is called the graphical solution of a LPP .
( v ) Unbounded Solution If the value of the objective function can be increased or decreased indefinitely , such solutions are called unbounded solutions .
( vi ) Fundamental Extreme Point Theorem An optimum solution of a LPP , if it exists , occurs at one of the extreme points ( i . e ., corner points ) of the convex Polygon of the set of all feasible solutions .
Solution of Simultaneous Linear Inequations
The graph or the solution set of a system of simultaneous linear inequations is the region containing the points ( x , y ) which satisfy all the inequations of the given system simultaneously .
To draw the graph of the simultaneous linear inequations , we find the region of the xy-plane , common to all the portions comprWng the solution sets of the given inequations . If there is no region common to all the solutions of the given inequations , we say that the solution set of the system of inequations is empty .