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.

Board |
SCERT, Kerala |

Text Book |
NCERT Based |

Class |
Plus Two |

Subject |
Maths Notes |

Chapter |
Chapter 12 |

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.

**Convex set:**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.

**Theorem 1:**

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.

**Theorem 2:**

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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