Linear Programming —CONSTRAINTS

Before we start discussing Constraints in Linear programming, we need to understand its meaning and use.

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Studying the concepts of Linear Programming

Briefly said, Linear Programming is a mathematical technique to solve the operations management problems. It is also called a method for optimization of Linear Objective functions that are always subject to certain constraints. In simple terms, we can say that Linear Programming is to find out the ways to achieve the maximum performance or out come (called optimization) in a given Mathematical model. The constraints could be a list of requirements, which are represented by linear equations.

  • Types of constraints

In every situation, there are certain limitations and restrictions that affect the variable values There can be less severe limitations and such constraints are considered “non-binding” constraints as they may not affect the ultimate solution for finding optimal values. A “binding constraint” is the one that changes optimal solution also, if there is a change in value of such constraint.
The constraints can be mainly termed as “equality and inequality constraints” As well as “duality constraints”.

In view of the above, Linear Programming is the most useful technique for getting the optimum results in any Operation Research studies.

  • Linear Programming (LP) Structure

An LP model will normally consist of an objective function and constraints on that function.

Here is simple example to explain this
Z = a1X1 + a2X2 + a3X3 + . . .        and so on    + anXn

Here the Constraints could be:
b11X1 + b12X2 + b13X3 + . . . + b1nXn      is  <  c1
and the equation goes ……..

bm1X1 + bm2X2 + bm3X3 + . . . + bmnXn <  cm

In this system of linear equations, we have to optimize the objective function value of Z where as the ‘a’ , ‘b’, ‘c’ are the constraints that are derived from the specifics of the problem. The values of ‘x’ will be the decision variables for which we have to find the optimum values.

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Linear Programming Constraints

  • Certain LP assumptions

First linear programming requires a linearity in the equations as is evident from the above. However, the linearity further requires that;

  • There should be proportionality. This means that a change in a variable should result in a proportionate change in that variable’s contribution to the total value of the function.
  • The decision variables can be divided into non –integral values, while taking on fractional values.
  • The function value will always be the sum of contribution of each term in the equation.

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