Plus Two Maths Notes Chapter 12 Linear Programming is part of Plus Two Maths Notes. Here we have given Plus Two Maths Notes Chapter 12 Linear Programming.
|Text Book||NCERT Based|
|Chapter Name||Linear Programming|
|Category||Plus Two Kerala|
Kerala Plus Two Maths Notes Chapter 12 Linear Programming
- A Linear Programming Problem (LPP) is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several rabies (called objective function) subject to the conditions that the variables (called decision variables) are non-negative and satisfy a set of linear inequalities (called linear constraints).
- The linear inequalities or equations or restrictions on the variables of a linear programming problem are called constraints.
- Optimization problem:
A problem which seeks to maximize or minimize a linear function.
- A few important linear programming problems are:
- Manufacturing problems
- Diet problems
- Transportation problems
- A solution of the LPP:
A set of values of the decision variables which salis& the constraints of an LPP.
- Feasible solution (feasible region):
A solution of an LPP which also satisfies the non-negativity restrictions of the problem.
The set of all feasible solutions oían LPP.
- Optimal solution: A feasible solution which optimizes (maximize or minimize) the objective function of an L.PP.
- If an LPP admits an optimal solution, then at least one of the extreme (or corner) points of the feasible region give the optimal solution.
- There are two methods to solve an LPP graphically:
- Corner-point method
- ISO-profit or ISO-cost method
- The following theorems are fundamental in solving linear programming problems.
Let R be the feasible region for an LLP and let z = ax+by be the objective function. When z has an optimal value, where the variables x and y are subject to constraints described by linear inequalities. This optimal value must occur at a corner point of the feasible region.
Let R be the feasible region for a linear programming problem and let z = ax+by be the objective function. Let R is bounded, then the objective function z has both maximum and minimum value on R and each of these occurs at a corner. point of R.
- Corner point method:
This is a method for solving LLP. The method comprises of following steps.
- Find the feasible region of LIP and determine its comer points.
- Evaluate the objective function Z = ax+by at each corner point. Let M and m, respectively denote the largest and smallest values of these points.
- When a feasible region is bounded. M and m are the maximum and minimum values of z.
- When a feasible region is unbounded. we have N1 is the maximum value of Z, If the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, z has no maximum value. Similarly. m is the minimum values of z. If the open half plane determined by ax + by < m has no point in common with a feasible region. Otherwise, z has no minimum value.
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