Topic 10 Linear Progrmming

7/29/2019 Topic 10 Linear Progrmming

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Topic 10

Linear programming

Formulation and Graphical solution


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Introdcution

Decision problems are faced by managers which

involve allocation of resources to various

activities with some objective.

All managers have limited resources , so he has to

take a decision as to how best the resources be

allocated among the various activities.

The decision problems can be formulated andsolved as mathematical programming problems.


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Linear Programming

A technique for choosing the best alternative

from a set of feasible alternatives, in

situations in which the objective function as

well as the constraint can be expressed as

linear mathematical functions.

Requirements of LPP


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Assumptions of LPP

The values of the variables are non-negative

Constraints are linear

The number of constraints is not very large The objective function is also linear


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Methods of solving LPP

2 Methods used for solving LPP

GRAPHICAL METHOD

SIMPLEX METHOD


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GRAPHICAL METHOD

In this method, the given set of inequalities

are plotted on a graph paper, and the region

satisfying all the linear inequalities

simultaneously is obtained.

Generally the region is a convex polygon. This

region is the solution of the linear inequalities.


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Important terms in LPP

Limitations

Objective function

Solution Feasible solution

Optimal feasible solutions


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Uses of Linear Programming

In industry

In agriculture

In diet problems In war

In Transportation problems

In allocation of work


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example

Draw the graphs of following linear

inequalities and obtain the maximum value of

the objective function Z=5x + 7y.

x0, y0, x + y 4, 3x + 8y 24 and x and y are

integers.


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Example

Two types of hens are kept in a poultry farm. A

type of hen costs Rs 20 each and B type of hen

costs Rs 30 each. A type of hen lays 4 eggs per

week and B type of hen lays 6 eggs per week.

At the most 40 hens can be kept in the

poultry. Not more than Rs 1050 is to be spent

on the hens. How many hens of each typeshould be purchased to get maximum eggs?


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Example A firm is engaged in producing two products, A and B.

Each unit of product A requires 2 kg of raw materialand 4 labour hours of processing, whereas each unit of

product B requires 3 kg of raw material and 3 hours oflabour, of the same type. Every week , the firm has an

availability of 60 kgs of raw materials and 96 labourhours. One unit of product A sold yields Rs 40 and oneunit of product B sold gives Rs 35 as profit.

Formulate this problem as LPP to determine as to howmany units of each of the products should be produced

per week so that the firm can earn the maximum

profit. Assume that there is no marketing constraint sothat all that is produced can be sold.


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Example

The agricultural research institute suggested to afarmer to spread out at least 4800 kg of a specialphosphate fertilizer and not less than 7200 kg of aspecial nitrogen fertilizer to raise productivity of crops

in his fields. there are two sources of obtaining these-mixtures A and B. Both of these are available in bagsweighing 100 kg each and they cost Rs 40 and Rs 24respectively. Mixture A contains phosphate andnitrogen equivalent of 20 kg and 80 kg respectively,

while mixture B contains these ingredients equivalentof 50 kg each. Formulate LPP and determine how manybags of each type should the farmer buy in order toobtain the required fertilizer at minimum cost.

Topic 10 Linear Progrmming

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