Introduction to Linear Programming Problems (LPPs)A PresentationByProf. Manidatta Ray

What is LPP ?

In mathematics, linear programming (LP) problems involve the optimization of a linear objective function, subject to linear equality and inequality constraints.

Put very informally, LP is about trying to get the best outcome (e.g. maximum profit, least effort, etc) given some list of constraints using a linear mathematical model.

Linear programming can be applied to various fields of study. Most extensively it is used in business and economic situations, but can also be utilized for some engineering problems. Some industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.

Components of a LP ModelThe linear programming model consists of an objective function and a set of constraints. The quantifiable goal that is to be maximized or minimized is the "objective function." This function measures the effectiveness of a particular solution for a linear programming problem . The constraints are mathematical statements that specify elements of the problem such as the limitations of available resources or a demand level that must be met.

The conditions for a mathematical model to be a linear program (LP) are:

all variables continuous (i.e. can take fractional values) a single objective (minimize or maximise) the objective and constraints are linear i.e. any term is either a constant or a constant multiplied by an unknown.

Importance of Linear Programming LP's are important - because: Many practical problems can be formulated as LP's There exists an algorithm (called the simplex algorithm) which enables us to solve LP's numerically relatively easily. The common objectives for LP's are minimise cost/ maximise profit .

Application of Linear Programming inBusiness and Industries Linear programming is a mathematical programming and optimization technique that has been widely used by business and industry for many years. It involves allocating scarce resources on the basis of a given criterion of optimality. Most often, this criterion is either maximum margin or minimum cost.

A classic application of linear programming in manufacturing is the "product mix" problem. In this problem, the objective is to determine the kinds and quantities of products to be produced that would maximize profits. The case-mix decision is similar to this manufacturing problem.

Some of the major application areas to which LP can be applied are:

Production planning Oil refinery management Distribution Financial and economic planning Manpower planning Blast furnace burdening Farm planning

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