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 .
Solution
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
Maximize
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
