1. Applied Operations Research
Linear Programming
Definition: LP is a mathematical modelling technique that helps in resource allocation
decisions, and composed of linear equations and inequalities.
Uses:
 Advertising Budget allocation
 Resource allocation in Production Processes
 Manpower Planning etc.
Example for Reference: A company produces Tables and chairs, each table requires 4 hours
of carpentry work and 2 hours of painting. And each chair requires 3 hours of carpentry
work and 1 hour of painting.
Total carpentry hours available are 240 hours and 100 hours of painting time are available.
Profit contribution of each table is $ 7 and $ 5 for each chair.
Formulate a LPP keeping in consideration profit maximization.
Requirements:
Problems seek to maximize or minimize an objective, subject to some constraints:
Following are the requirements of the linear system in Operations Research:
1. Objective / Purpose –an objective or purpose defines “why” we are going to solve
the problem.(it can be maximization of profit, minimization of cost etc)
 It should be understandable
 Mathematically expressible , and
 Linear in nature
2. Constraints –constraints are the conditions that stop us from undertaking the activity
after a certain point. (it can be labor availability constraint, raw material constraint
and power constraint etc ex: if raw material is available to make only 100 cars then
availability of raw material stops you from producing more then 100 cars)
 It should be understandable
 Mathematically expressible , and
 Linear in nature
Hemant Sharma, Assistant Professor, RIMT-SMS Page 1

2. 3. There must be alternatives available
4. As expressed in points (1) and (2) all the expressions must be LINEAR in nature
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Steve asked- “What holds a Car together and prevents it from breaking?”
John said –“The SCREWS”
Assumptions of a Linear System:
There are few Properties that we assume to hold true in case of a linear system:
These properties are as follows:
1) Certainty
2) Proportionality
3) Additive nature
4) Divisibility
5) Non-negativity
Question: Why there is a need to assume the above mentioned properties?
Answer: To solve a linear system, we need to perform some mathematical operations
(addition, subtraction, multiplication and division), these assumptions are made to:
Make sure that the System remains linear while these operations are being performed (to
be continued)
Example for reference: A Company manufactures Tables and Chairs, Each table requires
4 hours of carpentry time and 2 hours of painting time and each chair requires 3 hours
of carpentry and 1 hour of painting time.
During the present production cycle, 240 hours of carpentry time and 100 hours of
painting time are available.
Profit contribution of each table is $ 7 and $ 5 for each chair.
Formulate a LPP keeping in consideration profit maximization.
Hemant Sharma, Assistant Professor, RIMT-SMS Page 2

3. Explanation of Assumptions
1) Certainty : - It means that the coefficients of all the equations and inequalities
are known and remain constant throughout.
i.e.- if in the beginning we were told that to make a table we need 4 hours of
carpentry and 2 hours of painting , then certainty means that no matter how
many tables we make, we make them in the beginning of the production cycle or
in the end of the cycle each table will take 4 hours of carpentry time and 2 hours
of painting time.
Certainty says that we need to maintain the process of production as decided
in the beginning
2) Proportionality: It means that all the resources and outcomes vary in a constant
proportion (as decided earlier), and there is NO ECONOMIES of scale operating
here.
i.e. if we need 4 hours of carpentry and 2 hours of painting time to make one
table then
No of tables Carpentry hours required Painting hours required
2 4 hours/table X 2 tables = 2 hours/table X 2 tables =
8 hours 4 hours
3 4 hours/table X 3 tables = 2 hours/table X 3 tables =
12 hours 6 hours
6 4 hours/table X6 tables 2 hours/table X 6 tables =
=24 hours 12 hours
Proportionality makes sure that all the equations and inequalities together act
as a linear system and there is dependence established (which is linear in
nature)
3) Additive Nature:- (2+2=4) - it means that if 1 table gives profit of $ 7 then 2
tables will give profit of $ 14 and 3 tables will give a profit of $ 21 and so on..
i.e. there is NO DISCOUNTING allowed. The resources add up in a linear manner
to give away outputs
OUTPUT = EXACT sum of INPUTS
(4 = 2 + 2)
Hemant Sharma, Assistant Professor, RIMT-SMS Page 3

4. 4) Divisibility :- it means that the coefficients can assume fractional values,
Why?
Suppose we have a system with 2 linear equations
2x + 3y = 15 5/2 X (2x + 3y = 15) we are trying to eliminate “x” from these
5x + 2y = 21 -- 5x + 2y = 21 equations, so as to solve for “y”
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11/2 y = 33/2  “y” = 3
This operation can be performed only, if we allow the “coefficients of “x” and “y”” to assume
fractional values, i.e. allow for division
5) Non-negativity :- implies that the Decision variables (here the number of tables and
chairs) CANNOT take negative values
i.e. Variables >= 0
(since the number of tables and chairs a company can produce CANNOT be negative)
Please consult, in case of doubts
You can mail me at sharma.hemant@ymail.com, or clarify in the class
Hemant Sharma, Assistant Professor, RIMT-SMS Page 4