respected teacher can you define the concept of linear programing in a simplified manner ?
The Objective Function is a linear function of variables which is to be optimised i.e., maximised or minimised. e.g., profit function, cost function etc. The objective function may be expressed as a linear expression.
Constraints
A linear equation represents a straight line. Limited time, labour etc. may be expressed as linear inequations or equations and are called constraints.
e.g., If 2 tables and 3 chairs are to be made in not more than 10 hours and 1 table is made in x hours while 1 chair is made in y hours, then this constraint may be written as
A linear equation is also called a linear constraint as it restricts the freedom of choice of the values of x and y.
Optimisation
A decision which is considered the best one, taking into consideration all the circumstances is called an optimal decision. The process of getting the best possible outcome is called optimisation. The best profit is the maximum profit. Hence optizmizing the profit would mean maximising the profit. Optimising the cost would mean minimising the cost as this would be most favourable.
Solution of a LPP
A set of values of the variables x1, x2,….xn which satisfy all the constraints is called the solution of the LPP.
Feasible Solution
A set of values of the variables x1, x2, x3,….,xn which satisfy all the constraints and also the non-negativity conditions is called the feasible solution of the LPP.
Optimal Solution
The feasible solution, which optimises (i.e., maximizes or minimizes as the case may be) the objective function is called the optimal solution.
Convex Region
A region is said to be a convex region, if for any two points of the region say (x1, y1) and (x2, y2) the line segment joining these points, is also in the region.
Non-convex Sets
We know that the feasible solution of an LPP is a set. This set (If it is finite non-empty) is convex for any L.P.P.
There are four types of feasible region normally we get
Type 1:
The region is totally empty, then there is no solution to the problem.
Type 2:
The region contains exactly one point, the point is optimal solution
Type 3:
The region may be bounded convex set consisting of more than and point
Type 4:
The region may be unbounded.
