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Introduction
• Linear Programming was developed by
George B Dantzing in 1947 for solving
military logistic operations.
• Meaning of Linear Programming
– The word Linear refers to linear relationship
among variables. i.e. a given change in one
variable will always cause a resulting
proportional change in another variable. For
example, doubling the investment on a
certain project will exactly double the rate of
return.
Introduction
• The word programming refers to modeling
& solving a problem mathematically that
involves the economic allocation of limited
resources by choosing a strategy among
various alternative strategies to achieve the
desired objective.
Introduction
• Linear Programming (LP) is a mathematical
modeling technique useful for the allocation of
limited resources, such as labour, material,
machine, time, warehouse space, capital,
energy etc. to several competing activities,
such as products, services, jobs, new projects
etc.
Introduction
Introduction
• Also, the general LPP calls for optimizing a
linear function of variables called the
objective function subject to a set of linear
equations and /or inequalities called the
constraints or restrictions.
General Structure of LPP
• Decision Variables: The activities that are
competing one another for sharing the
resources available. These variables are
usually interrelated in terms of utilization of
resources and need simultaneous solutions.
All these variables are considered as
continuous, controllable and non-negative.
General Structure of LPP
• The Objective Function: A LPP must have
an objective which should be clearly
identifiable and measurable in quantitative
terms. It could be of maximization of profit
(sales), minimization of cost etc. The
relationship among variables representing
objective must be linear.
General Structure of LPP
• The Constraints: There are always certain
limitations or restrictions or constraints on the use of
resources, such as labour, space, raw material,
money etc. that limit the degree to which an objective
can be achieved. Such constraints must be
expressed as linear inequalities or equations in terms
of decision variables.
Assumptions of LP Model
• Certainty
• Additivity
• Linearity (Proportionality)
• Divisibility (continuity)
Assumptions of LP Model
• Certainty: In all LLP’s, it is assumed that all the
parameters; such as availability of resources,
profit contribution of a unit or cost contribution
of a unit of decision variable and computation of
resources by a unit decision variable must be
known and fixed. Or we can say that, all the
coefficients in this objective function as well as
in the constraints are completely known with
certainty and do not change during the period
Assumptions of LP Model
of study. Thus, the profit per unit of the
product, requirements of material and labour
per unit, availability of material etc. are given
and known in the problem. The LP is
obviously deterministic in nature.
Assumptions of LP Model
• Additivity: The value of the obj. function for the
given values of decision variables and the total
sum of resources used, must be equal to the
sum of the contributions (profit or loss) earned
from each decision variable and the sum of the
resources used by each decision variable
respectively. For example, the total profit
earned by the sale of two products A & B must
Assumptions of LP Model
be equal to the sum of the profits earned
separately from A & B. Similarly, the amount
of a resource consumed by A & B must be
equal to the sum of resources used for A & B
individually.
Assumptions of LP Model
• Linearity or Proportionality: This assumption
requires the contribution of each decision
variable in both the obj function and the
constraints to be directly proportional to the
value of the decision variable. Or we can say
that, the amount of each resource used ( or
supplied) and its contribution to the profit (or
cost) in obj. fun must be proportional to the
value of each decision variable. For eg., if
Assumptions of LP Model
production of a one unit of a product uses 5
hrs of a particular resource, then making 3
units of that product uses 3*5=15 hrs of that
resource.
Assumptions of LP Model
• Divisibility or Continuity: This implies that
solution values of the decision variables and
resources can take on any non-negative values,
including fractional values of the decision
variables. For eg., it is possible to produce 8.35
quintals of wheat or 7.453 thousand gallons of a
solvent or 43.45 thousand kiloliters of milk. Such
variables are not divisible and hence are to be
assigned
Assumptions of LP Model
integer values. When it is necessary to have
integer variables, the integer programming
problem is considered to attain the desired
values.
Formulation of LPP
The term formulation referred to the process
of converting the verbal description and
numerical data into mathematical expressions
which represents the relationship among
relevant decision variable (factors), objective
& restrictions on the use of resources.
Introduction
• The term formulation referred to the process
of converting the verbal description and
numerical data into mathematical expressions
which represents the relationship among
relevant decision variable (factors), objective
& restrictions on the use of resources.
Introduction
The XYZ garment company manufactures
men's shirts and women’s t-shirts for ABC
Discount stores. ABC will accept all the
production supplied by the company. The
production process includes cutting, sewing
and packaging. XYZ employs 25 workers in the
cutting department, 35 in the sewing
department and 5 in the packaging department.
The factory works one 8-hour shift, 5 days a
week.
The following table gives the time requirements
and the profits per unit for the two garments:
Minutes per unit
Garment Cutting Sewing Packaging Unit profit (Rs.)
Shirts 20 70 12 8.00
T-shirts 60 60 4 12.00
Determine the optimal weekly production
schedule for XYZ.
Solution
Assume that XYZ produces x1 shirts and x2 t-
shirts per week.
8 x1 + 12 x2Profit got =
Time spent on cutting = 20 x1 + 60 x2 mts
Time spent on sewing = 70 x1 + 60 x2 mts
Time spent on packaging = 12 x1 + 4 x2 mts
maximize the profit z = 8 x1 + 12 x2
The objective is to find x1, x2 so as to
satisfying the constraints:
20 x1 + 60 x2 ≤ 25 40 60
70 x1 + 60 x2 ≤ 35 40 60
12 x1 + 4 x2 ≤ 5 40 60
x1, x2 ≥ 0, integers
This is a typical optimization problem.
Any values of x1, x2 that satisfy all the
constraints of the model is called a feasible
solution. We are interested in finding the
optimum feasible solution that gives the
maximum profit while satisfying all the
constraints.
More generally, an optimization problem looks
as follows:
Determine the decision variables x1, x2, …,
xn so as to optimize an objective function f
(x1, x2, …, xn) satisfying the constraints
gi (x1, x2, …, xn) ≤ bi (i=1, 2, …, m).
Linear Programming Problems(LPP)
An optimization problem is called a Linear
Programming Problem (LPP) when the
objective function and all the constraints are
linear functions of the decision variables, x1,
x2, …, xn. We also include the “non-negativity
restrictions”, namely xj ≥ 0 for all j=1, 2, …, n.
Thus a typical LPP is of the form:
Optimize (i.e. Maximize or Minimize)
z = c1 x1 + c2 x2+ …+ cn xn
subject to the constraints:
a11 x1 + a12 x2 + … + a1n xn ≤ b1
a21 x1 + a22 x2 + … + a2n xn ≤ b2
. . .
am1 x1 + am2 x2 + … + amn xn ≤ bm
x1, x2, …, xn 0
Advantages• LP helps in attaining the optimum use of
productive resources. It also indicates how a
decision maker can employ his productive
factors effectively by selecting and distributing
these resources.
• LP technique improves the quality of decisions.
• LP technique provides possible and pratical
solutions since there might be other constraints
operating operating outside the problem which
must be taken into account.
Advantages
• LP also helps in re-valuation of a basic plan for
changing conditions. If conditions change when
the plan is partly carried out, they can be
determined so as to adjust the remainder of the
plan for best results.
Limitations
• LP treats all relationship s among variables as
linear.
• While solving the an LPP, there is no guarntee
that we will get integer valued solutions.
• LP model does not take into consideration the
effect of time and uncertainnity.
Limitations
• Parameters appearing in the model are
assumed to be constant but in real-life
situations, they are frequently neither klnown
nor constant.
• It deals with single objective, whereas in real-
life situations we may come across conflicting
multi-objective problems.
Applications
• Agriculture Applications
• Military Operations
• Production Management
• Financial Management
• Marketing Managemant
• Personnel Management
General Structure of LPP
The general LPP with n decision variables and
m constraints can be stated as:
Find the values of decision variables…..