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## Kerala Plus Two Maths Notes Chapter 12 Linear Programming

Introduction

A special class of optmisation problems such as finding maximum profit, minimum cost, or minimum use of resources, etc, is Linear Programming Problems. In this chapter we study some linear programming problems and their solutions graphically.

A. Basic Concepts

A linear Programming Problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called the objective function) subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative.

A few important LPP are;

- Diet Problem.
- Manufacturing Problem.
- Transportation Problem.

1. The common region determined by the constraints including the non-negative constraints x ≥ 0, y ≥ 0 of a LPP is called the feasible region.

2. Points within and on the boundary of the feasible region represents feasible solution of the constraints. Any point outside the feasible region is an infeasible solution.

3. Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.

I. Corner point Method

- Find the feasible region of the LPP and determine its corner points (vertices).
- Evaluate the objective function Z = ax + by at each corner points. Let M and m be the maxjmum and minimum values at these points.
- If the feasible region is bounded, M, and m respectively are the maximum and minimum values of the objective function.
- If the feasible region is unbounded, then
- M is the maximum value of the objective function, if the open half-plane determined by ax + by > M has no points in common with the feasible region. Otherwise the objective function no maximum value.
- m is the minimum value of the objective function, if the open half-plane determined by ax + by < m has no point in common with the feasible region. Otherwise, the objective function has no minimum value.