Friday, November 27, 2009

How to Formulate the Problem as A Linear Programming Problem

It is very necessary to formulate the given business problem as a linear programming problem  because after this , we can calculate the solution of LPP . In other words if we want to know the quantity of our business
product or any other things , so that our cost will minimize or our profit maximize , then first of we will write two quantity as X and y and try to find our objective function . It may possible that we have to minimize it or it may possible it has to maximize .
After this try to make different conditions , it may possible that several conditions of time , price , production or sale capacity will apply and last condition x,y will always > o r equal to Zero .

Let us take one example

A firm manufactures two types of helmets , each helmet of type I requires twice as much labour time as type II . If all helmets are of type II only , the firm can manufacture a total of 500 helmets daily. The market limits daily sales of type one and type two to 150 and 250 helmets . Assuming that the profits per helmet are Rs. 8 for type I and Rs. 5 for type II , formulate the problem as a linear programming problem to determine the number of helmets to be produced of each type so as to maximise the profit .


In this example manufacturer is producing two types of helmets type I and Type II

Write type I helmet = X

Type II helmet = Y

Now find the objective of LPP

In last line is give the objective . Objective is to maximize the profit by producing and selling the helmet

Now we know that profit means = No. of unit of X produce  multiply with its  profit per unit + No. of unit of Y produce multiply with its profit per unit

So miximize

Z =  X x 8 + Y x 5

or we can say

Z = 8 X + 5Y

Now next step is to fine different conditions

In above question two condition is given .

First condition of Time

 Company has limited labourer who works in limited time . The total capacity  to produce is 500 helmets daily by labours . Now it is we simple if we multiply the labour time with unit produce and both will add , it must be below or equal to total 500 helmet daily.

or we can say

for producing X labour time is double for producing Y

it  means we multiply 2 times with X and add 1 time multiply with Y unit

2X + 1 Y it is daily product but condition is that it is = or < 500

or we can say

2X + Y = or < 500

Second condition of  Sale capacity

company can sell only 150 units of X or less

so we can say in math language

X < or = 150

Company can sell only 250 units of Y or less , then we can say

Y < or = 250

Now we can formulate all LPP in following way


Z = 8 X + 5 Y

subject to the constraints or conditions of LPP

2 X + Y < or = 500

X < or = 150

Y < or = 250

always X , Y > or = 0
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