Linear Programming
The word Linear Programming consist of two words
Linear This word is used to show the relationship between
decision variables which are directly proportional. For example if we increase
the production of product, it will proportionate increase the profit, this is
called a linear relationship.
Programming It implies the planning of activities in such a
way that the activities with available resources yield optimal results .
This Linear Programming states the planning of decision variables which are
directly proportional to achieve desired optimal results.
Or
Linear Programming involves the planning of activities to
obtain an optimum result.
“Linear Programming is the analysis of problems in which a
linear function of a number of variables
is to be maximized (or minimized) when those variables are subject to a number
of restraints in the form of linear inequalities”
Assumptions of Linear Programming
Linearity or
proportionality A basic assumption of Linear Programming is that there
exists proportionality in the objective function and the constraints. It
implies that if a product gives a profit of Rs 25 the profit earned from the
sale of 10 such products will be Rs 250.
Finite Choice Another
assumption of a Linear Programming is that a limited number of choices are
available to decision maker and the decision variable do not have negative
values. This assumption is a realistic one, as it is not possible to produce or
use negative quantities
Additivity
It means that if m1 minutes are required to produce a product p1 on machine A
and m2 minutes are required to produce a product p2, the total time required to
make product p1 and p2 on machine A is m1+m2 minutes. This however, may not
happen because of the change over time from product p1 to p2. Similarly the
total profit is determined y the sum of profit is calculated by each of the
products separately
Certainty
A Linear Programming model also assumes that the various parameters, such as
coefficients of the constraints, objective function coefficients and resources
values are known certainty and they do not change with the passage of time. So,
availability of materials, labour etc. the cost or profit per unit of the
product, market demand of the product are assumed to be known certainly
Continuity
One more assumption Linear programming is that the decision variables are
continuous. It implies, combinations output with fractional values, in the
context of production problems are possible.
Applications
Product Mix Linear programming helps in determining the
quantity of different products to be manufactured knowing the marginal
contribution of each product and amount of available resources used by each
product
Transportation problem Transportation n problems are faced
by business organization which have number of availability centres (plants,
warehouse etc.) with gives capacities which feed number of requirements cemtres
(warehouse, markers etc.) with given requirements.
Blending problem This type of problem is faced by the
chemical, food and petroleum industry etc. while deciding production of a
product which can be made from a variety of available raw materials of
different composition and prices.
Media selection LP has also been used in advertising firld
as a decision aid in selecting the effective media. Media helps the marketing
managers in allocation a fixed budget across various advertising media like newspapers,
magazines, radio and television etc.
Diet problem it involves determination of combination of
different nutrients such as proteins, vitamins, carbohydrates etc. for
different foods to satisfy the minimum daily nutritional requirements at the
minimum cost and minimizes the cost of raising live-stock.
Travelling salesman’s problem The problem of finding the
shortest route for a salesman starting from a given city, visiting each of the
specified cities, and returning to the original point of departure can be
handled by the assignment technique of linear
programming.
Capital investment This type of problem arises because of
the different ways in which a fixed amount of capital can be allocated to a number
of activities. The total return depends upon the manner in which the allocation
is made. This , the objective may be to
find that allocation which maximizes the total return
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