Linear Programming Problem - Desklib

Linear Programming Problem
‘The mathematical experience of the student is incomplete if he never had the
opportunity to solve a problem invented by himself (Polya, 2014) ‘. In today’s world,
we come across many such situations wherein we try to assess which combinations
of two particular things are taken from varied options or in what proportion should
two things be taken so that the cost is minimum (or the profit is maximized). This
type of inequalities (or variations) form a general class of problems called
optimization problems which aim to maximize profit or minimize cost under a given
set of constraints or restrictions.
Linear Programming Problem is an important class of optimization problem. Let
us explain this with the help of an example. Suppose, a man has £1,20,000 to
invest. He has option to buy a bag worth £600 and shoes worth £1,000. He
understands that by selling one bag, he can earn a profit of £80 and £120 by selling
one shoe. He owns a storage space that can store up to 150 pieces. He now wants
to know what quantity of the items he should buy so as to maximize the profit,
assuming that he sells all the stuff that he buys.
In the situation mentioned above, the man can buy 200 bags (£1,20,000 divided by
£600) that can fetch him a profit of £16,000 or he can buy 120 shoes that can fetch
him a profit of £14,400. But in the first option, he cannot buy 200 pieces as his
storage space is limited to 150 while in the second option he can buy all 120 shoes
but the profit that he will receive is less than his first option. In this case the Linear
Programming problem method tries to mathematically formulate the problem
subject to the constraints determined by a set of linear inequalities with non-
negative variables and finds out the optimal value from a set of values that
maximizes the profit of the person.
Linear Programming Problem has some terms that are defined as follows:-
Objective Function- It is linear function, i.e. Z = ax + by which needs to be
maximized or minimized as per the case.
Constraints- These are linear inequalities or restrictions within which the problem
needs to be solved and are non-negative.
Linear Programming problem finds its huge application in the different areas such
as, management science, commerce and industry.
Let us now solve the problem given in the assignment as per the Linear
Programming Problem model discussed above.

Solution:-
Let ‘X’ be the number of units of Fusion produced per day and ‘Y’ be the number of
units of Torque produced per day.
(a) Therefore, as per the information provided, the data can be tabulated as
follows:-
Model Sales (£) Cost (£) Profit (£) Labor (in
hours)
Capital (in
hours)
Fusion 16000 10000 6000 40 40
Torque 20000 12000 8000 60 20
Constraints given:-
Total Labor Hours available per day- 60,000 hours.
Total Capital Hours available per day- 40,000 hours.
(b) We now have the following mathematical model for the given problem, where
the objective function is as follows:-
Max Z=£ 6000 X+ £ 8000 Y ,
Subject to the constraints:-
( i ) 40 X +60 Y ≤6 0 , 000
¿ , 2 X +3 Y ≤ 3 , 000
( ii ) 40 X +20 Y ≤ 4 0,000
¿ , 2 X +2Y ≤ 2 ,000
( ii i ) X , Y ≥ 0
Constraint Workings:-
( i ) 2 X +3 Y ≤ 3 ,00 0