Linear programming deals with optimizing a linear objective function subject to linear constraints. It involves determining the values of decision variables to maximize or minimize the objective function. The general linear programming model involves maximizing or minimizing a linear combination of n decision variables subject to m linear constraints, along with non-negativity restrictions on the decision variables. Formulating a linear programming problem involves identifying decision variables, expressing constraints and the objective function linearly in terms of the variables, and adding non-negativity restrictions.

1. Linear Programming
Shubhagata Roy
School of Management Sciences
Varanasi

2. Introduction:
Linear Programming deals with the optimization (max. or min.) of a function of
variables, known as ‘objective function’, subject to a set of linear equations and/or
inequations, known as ‘constraints’. The objective function may be profit, cost,
production capacity or any other measure of effectiveness, which is to be obtained in
the best possible or optimal manner.
There are two words to be understood – ‘Linear’ and ‘Programming’. The word
‘linear’ refers to linear relationship among the variables in a model. Hence, a given
change in one variable will always result in a proportional change in the other
variable(s). For example, doubling the investment on a certain project will also
double the rate of return.
The word ‘programming’ refers to the mathematical modeling and solving
a problem that involves the economic allocation of limited resources, by choosing a
particular course of action or strategy among the various alternatives in order to
achieve the desired result.

3. Structure of Linear Programming Model:
A linear programming problem may be defined as the problem of ‘maximizing or
minimizing a linear function subject to linear constraints’.
The constraints may be equalities or inequalities. Here is a simple example.
Find numbers X1 and X2 that maximizes the sum (X1 + X2) subject to the constraints
X1 + 2X2 ≤ 4
4X1 + 2X2 ≤ 12
−X1 + X2 ≤ 1
and, X1 ≥ 0, X2 ≥ 0
In this problem there are two unknowns, and five constraints. All the constraints are
inequalities and they are all linear in the sense that each involves an inequality in
some linear function of the variables. The last two constraints X1 ≥ 0 and X2 ≥ 0, are
special. These are called ‘non-negativity constraints’ and are often found in linear
programming problems. The other constraints are then called the ‘main
constraints’. The function to be maximized (or minimized) is called the ‘objective
function’ and is denoted by ‘Z’. Hence, the objective function is ‘maximize Z = X1
+ X2’.

4. General Form of LPP:
The general linear programming model with ‘n’ decision variables and ‘m’ constraints
can be stated in the following form:
Find the values of X1,X2,……, Xn so as to
Maximize (or Minimize) Z = C1X1+C2X2+…….+CnXn
subject to the constraints
a11X1+ a12X2 +….+ a1nXn (≤ or = or ≥) b1
a21X1+ a22X2 +….+ a2nXn (≤ or = or ≥) b2
. . . .
. . . .
am1X1+ am2X2 +….+ amnXn (≤ or = or ≥) bm
and X1,X2,……, Xn ≥ 0
where aij’s are the ‘technological coefficients’ or ‘input-output coefficients’;
bi represents the ‘total availability of ith resource’ and Cij’s are the ‘coefficients representing
per unit contribution of decision variable Xj’.
i = 1,2,3,…., m and j = 1,2,3,……, n.

5. Steps in Formulation of LPP:
Step1: Identify the decision variables and assign symbols x1,x2,x3…. to them.
These decision variables are those quantities whose values we wish to
determine.
Step2: Identify the set of constraints and express them as linear equations/
inequations in terms of the decision variables. These constraints are the given
conditions.
Step3: Identify the objective function and express it as a linear function of
decision variables. It might take the form of maximizing profit or production
or minimizing cost.
Step4: Add the non-negativity restrictions on the decision variables, as in the
physical problems, negative values of decision variables have no valid
interpretation.

6. Steps in Formulation of LPP:
Step1: Identify the decision variables and assign symbols x1,x2,x3…. to them.
These decision variables are those quantities whose values we wish to
determine.
Step2: Identify the set of constraints and express them as linear equations/
inequations in terms of the decision variables. These constraints are the given
conditions.
Step3: Identify the objective function and express it as a linear function of
decision variables. It might take the form of maximizing profit or production
or minimizing cost.
Step4: Add the non-negativity restrictions on the decision variables, as in the
physical problems, negative values of decision variables have no valid
interpretation.