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welcome to the course introduction to operations
research operations research is a field of
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study where we optimize performance under
given constraints operations research also
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helps us develop models for decision making
among the several topics in operations research
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linear programming is the most important and
the most popular one linear programming problems
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have applications in the various fields some
of which include manufacturing systems publics
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systems business supply chain management and
analytics in linear programming we try to
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optimize a linear objective function subject
to linear constraints and with non negativity
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restrictions on the variables every activity
is carried out with an objective in mind and
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it is important to optimize the objective
these activities are also carried out under
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some constraints or conditions and linear
programming helps us understand these and
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helps us to formulate and solve problems which
come out of this practical situations where
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several activities are modeled we start linear
programming with the simple example to understand
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the notation and also to understand the aspects
of formulating a problem into a linear programming
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problem the first formulation that we will
be looking at is called a product mix problem
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where we try to find out what is the mix of
products that an organization should produce
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we start with a very simple example you can
see the text of the example and then we convert
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this into a formulation and then we define
the terms and notation related to linear programming
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now let's consider a shop that makes two types
of sweets or two types of products this are
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called a and b the shop uses two resources
which are indicated here as flour and sugar
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they make this sweets in packets so to make
one packet of a they need three kg of flour
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and three kg of sugar to make one packet of
b they need three kg of flour and four kg
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of sugar they have with them twenty one kg
of flour and twenty eight kg of sugar now
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this sweets when made are sold at rupees thousand
and rupees nine hundred per packet respectively
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find the best product mix to maximize the
revenue generated by the shop
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now we will look at this example and we will
try to formulate this as a linear programming
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problem the first thing that the shop has
to decide is how many packets of the two sweets
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that are going to be made so we start defining
this variables to this problem as x one and
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x two which are two variables where x one
represents the number of packets of sweet
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a made and x two represents the number of
packets sweet b made and in general we could
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say x one x two would represent the number
or the quantity of the products that are being
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made now by making sweets and by selling them
the revenue generated is rupees thousand per
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packet for a and nine hundred per packet of
b so if we make x one number of packets of
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a then the revenue generated is thousand into
x one and if we make x two packets of sweet
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b the revenue generated is nine hundred into
x two
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therefore the total revenue generated is thousand
x one plus nine hundred x two which is shown
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here thousand x one plus nine hundred x two
is the revenue that is generated by the production
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and sale of these products now we writ this
revenue function as thousand x one plus nine
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hundred x two now obviously we want to maximize
the revenue that is generated and therefore
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we wish to maximize the function thousand
x one plus nine hundred x two now the availability
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of these resources which are flour and sugar
they a limit or restrict the values that x
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one x two can take or in general restrict
the values that the quantity of production
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can take now these restriction are written
as constrains and they are shown here now
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we know that twenty one kg of flour is available
to make one packet of a we need three kg of
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flour to make one packet of b we also need
three kg of flour
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so to make x one packets of a we need three
x one kg of flour to make x two packets of
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b we need three x two kg of flour and therefore
the total quantity of flour required is three
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x one plus three x two which is shown here
the quantity of flour available is twenty
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one which is shown in right hand side and
now this twenty one is going to limit the
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values that x one and x two can take therefore
three x one plus three x two which is the
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quantity of flour required cannot exceed twenty
one which is the quantity of flour available
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this is modeled as three x one plus three
x two is less than or equal to twenty one
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which means three x one plus three x two cannot
exceed twenty one
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similarly we have twenty eight kg of sugar
which is going to restrict the values that
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x one and x two can take in a certain way
to make one packet of a we need three kg of
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sugar to make x one packets of a v require
three into x one kg of sugar to make one packet
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of b we need four kg of sugar to make x two
packets of b b required four into x two kg
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of sugar and three x one plus four x two which
is the requirement cannot exceed twenty eight
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which is the availability so the two resources
availability of the two resources which is
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twenty one kg of flour and twenty eight kg
of sugar are going to restrict or limit the
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values that x one and x two can take and that
is represented by three x one plus three x
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two less than or equal to twenty one and three
x one plus four x two less than or equal to
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twenty eight
we also have this condition x one and x two
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greater than or equal to zero which means
we would like to make non negative quantities
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of both the products which also implies that
we cannot make negative quantities of any
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of the products so the formulation of the
product mix problem for the given situation
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is that is let x one be the number of packets
of sweet a made let x to be the number of
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packets of sweet b made maximize thousand
x one plus nine hundred x two subject to three
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x one plus three x two less than or equal
to twenty one and three x one plus four x
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two less than or equal to twenty eight x one
and x two greater than or equal to zero so
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this completes the linear programming formulation
of the given product mix problem we will now
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go on to define what is a linear programming
problem and y this problem that we have formulated
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comes under the category of linear programming
so we start by saying let x one be the number
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of packets of sweet a made let x two be the
number of packets of sweet b made so the first
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wrote this variables and this variable that
we wrote are called decision variables these
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variables represent the decisions that have
to be made in the product mix problem so this
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variable that we have defined are called decision
variables we then wrote the revenue function
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in terms of the decision variables as thousand
x one b plus nine hundred x two if x one packets
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of sweet a are made and x two packets of sweet
b are now this function that we wrote is called
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the objective function and represents the
objective of the problem which is to generate
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the revenue and to maximize or try to have
as higher value of revenue as possible
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so the revenue function is the objective function
which is to be maximized and therefore maximize
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thousand x one plus nine hundred x two is
the objective function associated with the
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problem we then wrote three x one plus three
x two less than or equal to twenty one and
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three x one plus x four x two less than or
equal to twenty eight now these two conditions
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or constrain as they are called limit the
value that the decision variables can take
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and these constraints are shown here there
are two constraints in this problem last we
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wrote x one x two greater than or equal to
zero and said that we are not going to produce
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negative quantities of the products therefore
we have explicitly stated a non negativity
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restriction on the variables
so the same formulation that we made has now
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been described in terms of decision variable
in terms of the objective function in terms
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of constraints and non negative restriction
on the decision variables so first part of
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a linear programming formulation is to define
the decision variable then we define the objective
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function as a function of the decision variable
and in this case we have a linear function
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of the decision variable the objective function
is thousand into x one plus nine hundred into
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x two which is a linear function of x one
and x two the the function is not known linear
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for example this function is not thousand
x one square plus nine hundred x two square
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and so on it is thousand into x one plus nine
hundred into x two which is a linear function
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if we draw the the cover associated with this
we would get a line so its linear function
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of the decision variables the constraint have
left inside right a right hand side and a
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relationship in this case the relationships
are inequalities and both the inequalities
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are of the less than or equal to time the
constraints left hand side we have linear
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functions of the decision variables three
x one plus three x two is a linear function
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of x two three x one plus x one and x two
three x one plus four x two is also a linear
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function of x one and x two so the constraints
or linear functions of the decisions variables
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and we have an explicit non negativity restriction
on the decision variables
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so the linear programming problems is one
where the objective function is a linear function
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of the decision variables the constraints
are all linear inequalities or equations as
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the case may be in this example we have only
inequalities we do not have equations and
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there is an explicit non negativity restriction
on the decision variable so if we are able
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to model a write formulation which has an
objective function which is a linear function
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of the decision variables constraints which
are linear inequalities or equation and explicit
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non negativity restriction on the decision
variables we have formulated in linear programming
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therefore the problem that we just know formulated
the product mix problem is a linear programming
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problem that has an objective function which
is to maximize the revenue which is a linear
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function the constraints are linear inequalities
and there is an explicit non negativity restriction
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we also take a look at some of the assumptions
in this problem three important assumptions
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are proportionality linearity and deterministic
nature of the problem let me explain all of
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this proportionality comes from saying that
if i make one packet of sweet a i get thousands
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rupees by selling it if i make x one packets
i would get thousand into x one if i make
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half a packet i would get five hundred if
i make two packets i would get two thousand
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and so on
here for every packet i need three kg of the
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resource so if i make two packets i would
require require six kg if i make half a packets
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i would require one and half kg this is the
personality assumption the addictively or
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linearity of assumption in this case is the
revenue made by selling x one kg is thousand
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x one and the revenue to made by selling x
two kg of sweet b is nine hundred x two the
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total revenue is thousand x one plus nine
hundred x two so there is the so that is the
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additive it or linearity assumption in a similar
manner if three x one kg of the resources
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required to make sweet a and three x two is
required to make sweet be then the total requirement
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is some of these two requirements and we have
the linearity are the activity of assumption
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that way deterministic in the sense the values
are known and values are known with certainty
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when we actually formulate the problem
the availability is is a constant which is
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twenty one kg of this resource and is known
well in advance and deterministic so the three
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important assumptions are proportionality
linearity and deterministic nature of the
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numbers or the or the resources that are available
for the product mix problem in a next unit
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of this module on linear programming we will
continue to look at some more formulations
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where we try to understand minimization problems
that require greater then or equal to constrains