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.