The document discusses the mathematical formulation of linear programming problems (LPP). It begins by explaining how an LPP is translated into mathematical equations with decision variables. An example problem is given involving maximizing profit with constraints on machine hours and labor hours. The objective function for the example problem is defined to maximize total profit (Z) which is expressed as a linear function of the decision variables x and y, which represent units of products A and B. The constraints define the limited available machine hours and labor hours in terms of inequalities involving x and y. Non-negativity restrictions are also included to ensure x and y are greater than or equal to 0.

1. Mathematical formulation of lpp- properties andexample
Under the guidance of
Sundar B. N.
Asst. Prof. & Course Co-ordinator
GFGCW, PG Studies in Commerce
Holenarasipura

2. MATHEMATICAL FORMULATION OF LINEAR
PROGRAMMING PROBLEM
Mathematical Formulation of LPP refers to
translating the real-world problem into the form of
mathematical equations which could be solved. It
usually requires a thorough understanding of the
problem.
Example:
2x+3y=66
4x+2y=44

3. •Objective function:
The linear programming proplems must be have a well
defined objective function for optimization. For example,
maximization of profit or minimization of cost or total elapsed time
of the system being studied. It should be expressed as linear
function of decision variables.
•Contraints:
There are always limitation on the resources which are to be
allocated among various competing activities. These resources
may be production capacity, manpower, time, space or machinery.
These must be capable of being expressed as linear wqulities or
inequaltities in terms of decision varibles.
Non-negative restriction:
Properties of mathematical
formulation of LPP

4. •Non-negative restriction:
All the variables must assume non negative
values, that is, all variables must take on values equal to or
greater than zero. Therefore, the problem should not result in
negative values for the variables.

5. PROBLEM ON MATHEMATICAL
FORMULATION OF LPP
A small scale industry manufactures two product A and
B which are processed in a machine hour and labour
hour.
Product A requires 2 hours of work in a machine hour
and 4 hours of work in the labour to manufacture
while product B requires 3 hours of work in the
machine hour and 2 hours of work in labour hour.
In one day, the industry cannot use more than 16 hours
of machine hour and 22 hours of labour hour.
It earns a profit of 3 per unit of product A and 4 per unit
of product B.
Give the mathematical formulation of the problem so as
to maximize profit.

6. MATHEMATICAL FORMULATION OF LPP
A small scale industry manufactures
two product A and B which are
processed in a machine hour and
labour hour.
Product A requires 2 hours of work in a
machine hour and 4 hours of work in
the labour to manufacture while
product B requires 3 hours of work
in the machine hour and 2 hours of
work in labour hour.
In one day, the industry cannot use
more than 16 hours of machine hour
and 22 hours of labour hour.
It earns a profit of 3 per unit of product
A and 4 per unit of product B.
Give the mathematical formulation of
the problem so as to maximise
profit.
 Formulation of objective
function
let x and y be the number of units
of product A and B, which are to
be produced. Here, x and y are
the decision variables.
Suppose Z is the profit funtion.
Since one unit of product A and
one unit of product B gives the
profit of the 3 and 4,
respectivsely, the objective
funtion is
maximize Z= 3x+4y
The requirement and availability
hours of each of the hours in
manufacturing the products are
tabulated as follow.

7. MATHEMATICAL FORMULATION OF LPP
A small scale industry
manufactures two product A and
B which are processed in a
machine hour and labour hour.
Product A requires 2 hours of work
in a machine hour and 4 hours
of work in the labour to
manufacture while product B
requires 3 hours of work in the
machine hour and 2 hours of
work in labour hour.
In one day, the industry cannot use
more than 16 hours of machine
hour and 22 hours of labour
hour.
It earns a profit of 3 per unit of
product A and 4 per unit of
product B.
Give the mathematical formulation
of the problem so as to
maximise profit.
Machine
hour
Labour
hour
Profit
Product A 2 hours 4hours 3 p/u
Product B 3 hours 2 hours 4 p/u
Available
hour p/d
16 hours 22 hours
Subject to the contraints:
Total hours of machine hours
required for both types of product
=2x+3y
Total hours of labour hours
required for both types of product
=4x+2y
hence, the constraints as per the
limited available resources are:
2x+3y +≤ 16 and
4x+2y ≤ 22

8. Since the number of units produced
for both A and B cannot be negative, the
non negative restrictions are:
thus, the mathematical formaltion of
the given problem is
maximize Z=3x+4y
Subject to the constraints
2x+3y≤16
4x+2y≤22
And non negative restrictions
x≥ 0, y≥ 0

9. Conclusion
•Identity the number of decision variables which
govern the behavior of objective function.
•Identity the set of constraints on the decision
variables and express them in the form of linear
inequation or linear equation.
•Express the objective function in the form of a
linear equation in the decision variable.
•Optimize the objective function either
graphically or mathematically.

10. Reference
Properties of mathematical formulation of lpp:
https;//ecoursesonline,iasri.res.in/mod/reso
urces/view.php?id=4943
Problems on mathematical formulation of lpp:
https;//youtu.be/LtpBMC6uzhw