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7/30/2019 3. Linear Programming- Formulation
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Operations Research
2
Linear ProgrammingProblem LPP
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Introduction
A model consisting of linear relationships representing a
firms objective which is to be either maximized (in the
case of profit, production) or minimized (in the case of
men, materials and so on) and has to be expressed as
a linear function of the decision variables and resourceconstraint.
Linear programming is a method of mathematical
programming that involves optimisation of a certainfunction, called objective function, subject to certain
constraints and restrictions.
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Introduction
Linear means that the relationships handled are those
represented in straight lines. i.e. the relationships in
form of
y = a+bx
Programming means taking decisions systematically.
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Basic Concepts
Linear assumption- straight line or proportional
relationships among the relevant variables.
Process and its level- combination of particular inputs to
produce a particular output.
Criterion function- objective function which states the
determinants of the quantity either to be maximized orminimized.
Constraints or inequalities- restrictions imposed upon
decision variables.
Feasible solution- possible solutions which can beworked upon given constraints.
Optimum solution- best of the feasible solutions.
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Applications
Personal Assignment problem
Transportation problem
Efficiency on operation of system of Dams
Optimum Estimation of Executive compensation
Agricultural applications
Military applications
Production management
Marketing management
Manpower management
Physical distributions
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Components of LPP
Objective Function To maximise or minimise
Constraints
Involving , =, or sign
Usually, a maximisation problem has
type of constraints and a minimisation problem has
type. But a given problem can have constraints involving
any of the signs.
Non-negativity Condition
Variables to be non-negative
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Components of LPP
Maximize Z = 50x1 + 75x2 + 60x3
Subject to
5x1 + 8x2 + 7x3 480
4x1 + 2x2 + 3x3 240x1 2x2 + x3 20
x1 , x2 , x3 0
Non-negativity
condition
Objective
FunctionConstraints
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Assumptions underlying LinearProgramming
Proportionality-assumed to have constant return toscale
Additivity- total of all the activities is given by thesum total of each activity conducted separately.
Continuity-the decision variables are continuous.
Certainty- prior knowledge of all the coefficients inthe objective function, constraints and resource values
Finite Choices- finite number of choices areavailable to decision maker.
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Linear Programming Problem
LP is a mathematical modeling technique used to determinea level of operational activity in order to achieve an objective,subject to restrictions called constraints
Another necessary requirement is that decision variablesshould be interrelated and non-negative.
The resources must be limited.
In most linear programming problems, the decision variablesare permitted to take any non-negative values that satisfy theconstraints.
There are some problems in which the variables haveintegral values only. These problems are not LPP, but veryoften they can be solved by LPP techniques.
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Mathematical Formulation of LPP
The procedure for mathematical formulation of LPP consistsof the following steps:
Step 1 Identify the decision variables of the problem. Step 2 Formulate the objective function to be optimised
(maximised or minimised) as a linear function of the decisionvariables.
Step 3 Formulate the constraints of the problem such as resourcelimitations, market conditions, interrelation between variables andothers as linear equation or in-equations in terms of the decisionvariables.
Step 4 Add the non-negativity constraint so that negative values of
the decisions variables do not have any valid physicalinterpretation. The objective function, the set of constraint and thenon-negative constraint together form a linear programming
problem.
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Statements of Basic TheoremsAnd Properties
GENERAL FORMULATION OF LPP
In order to find the values ofn decision variablesx1,x2,,xn tomaximise or minimise the objective function
z= c1x1 + c2x2+ + cn xn (2.3.1)and also satisfy m-constraints
Max/min z = c1x1 + c2x2 + ... + cnxnsubject to:
a11x1 + a12x2 + ... + a1nxn(, =, ) b1a21x1 + a22x2 + ... + a2nxn(, =, ) b2
:a
m1
x1
+ am2
x2
+ ... + amn
xn
(, =, ) bm
xj = decision variablesbi = constraint levelscj = objective function coefficientsaij = constraint coefficients
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Statements of Basic Theoremsand Properties
The constraints may be in the form of an inequality ( or )
or even in the form of an equation (=) and finally satisfying
the non-negativity restrictions
x1 0, x2 0, , xj 0, xn 0 (2.3.3)
By convention, the values of right hand side parameters bi
(i = 1, 2, 3, , m) are restricted to non-negative values only.
Any negative can be changed to positive by multiplying 1
on both sides (in this case direction of inequality will change).
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Standard Form of LPP
Step 1All the constraints should be converted to equations excepts
for the non-negativity restrictions which remain as inequalities (0).Constraints of the inequality type can be changed to equations by(adding or subtracting) the left side of each such constraint by non-negative variablesslack variables / surplus variables.
Step 2 The right side element of each constraint should be madenon-negative, if required.
Step 3 All variables must have non-negative values
Step 4 The objective function should be maximization form
The minimization of a functionf(x) is equivalent to themaximization of the negative expression of this functionf(x), that is,Minf(x) =Max {f(x)}
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General Form of LPP
The general form of LPP with constraints is:
Maximisez= c1x1 + c2x2+ + cn xn+ 0xn+ 1+ + 0xn+ msubject to
a11x1+ + a1n xn+xn+ 1 = b1
a21x1+ + a2n xn+xn + 2 = b2 =
am1x1+ + amn xn+xn+ m= bm
x1 0,x20, , xn + m 0.
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General Form of LPP
Definition If the constraints of a general LPP ben
aij xj bi i = 1, 2, , k (2.3.4)j = 1
then non-negative variablessiwhich are introduced to convert the
inequalities (2.3.4) to the equalities nn
aij xj +si = bi i = 1, 2, , kj = 1
are calledslack variables
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Example of LPP
Example: A manufacturer makes Products A and B and sells
at the profit of Rs. 2 and 3. Each product is processed of
machines G and H. Type A requires 1 minute processing
time on G and 2 min on H. Type B requires 1 min on G
and 1 min on H. The machine G is available for not morethan 6 hours and 40 min . While H is available for 10
hours during any working day. Formulate problem as
linear programming problem.
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Solution of LPP
Machine
Time of ProductsAvailable
time (Min.)Type A (X1) Type B (X2)
G 1 1 400
H 2 1 600
Profit per
unitRs. 2 Rs. 3
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Solution of LPP
Find X1 and X2 such that profit ;Maximize Z= 2X1 + 3 X2
Subject to ConstraintsX1 + X2 400 (Available time on G)
2X1 + X2 600 (Available time of H)
X1, X2 0 (Non Negative constraint)
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Example 2 of LPP
Example: An animal feed company must produce 200 kgs of a
mixture consisting of ingredients X1 and X2 daily. X1
costs Rs 3 / kg and X2 Rs 8 /kg. Not more than 80 kg of
X1 can be used and at least 60 kg of X2 must be used.
Find how much of each ingredient should be used if thecompany wants to minimize cost?
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Solution of LPP 2
Find X1 and X2 such that profit ;Minimize Z= 3X1 + 8 X2
Subject to ConstraintsX1 + X2 = 200 (Total mix to be produced)
X1 80 (Max use of X1)
X2 60 (Min Use of X2)
X1 0 (NonNegative Constraint)
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General Form of LPP
Definition If the constraints of a general LPP be
n
aij xj bi i = k, k+1, k +2 (2.3.5)j = 1
then non-negative variablessiwhich are introduced to convert theinequalities (2.3.) to the equalities n
n
aij xj - si = bi i = k, k+1, k +2 j = 1
are calledsurplus variables
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Matrix Form of LPP
The linear programming problem in the standard form (2.3.1), (2.3.2),(2.3.3) can be expressed in matrix form as follows:
Maximisez= CX(objective function)
subject toAX= b, b 0 (constraint equation)
X 0 (non-negativity restriction)
whereX= (x1,x2, ,xn,xn +1, , xn+ m)
C= ( c1, c2, , cn, 0, 0, , 0)
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Matrix Form of LPP
DefinitionLetXB=B
1bbe a basic feasible solution to the LPP.
Maximisez= CXwhereAX= b, andX= 0, let CBbe the cost vector
corresponding toXB. For each column vectorajinA, which is not a
column vector of B, let
m
aj=aij bi.j = 1
Then the numberzj =CBi aijis called the evaluationi = 1
corresponding toajand the number (zjcj) is called the net evaluationcorresponding to ajwhere cjis thej
th component ofC.
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Definition
Solution to a LPP A setX= {x1,x2, ,xn+ m} of variables is called
a solution to a LPP, if satisfies the set of constraints (2.3.2) only.Feasible solution Any setX= {x1,x2, ,xn+ m} of variables iscalled a feasible solution of the LPP, if it satisfies the set ofconstraints (2.3.2) and non-negativity restrictions (2.3.3).Basic solution A basic solution to (2.3.2) is a solution obtained bysetting any n variables (among m + n variables) equal to zero and
solving for remaining m variables provided the determinant of thecoefficients of these m variables is non-zero. Such m variables (anyof them may be zero) are called basic variables, and the remainingvariables which are set as zero are called non-basic variables.
The number of basic solutions must be atmost m + n Cm.
Basic feasible solution A basic solution to a LPP is called as abasic feasible solution if it satisfies the non-negative restriction.
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Solution to LPP
There are two types of basic feasible solutions.
(i) Non-degenerate All mbasic variables are positive, and
remaining n variables will be zero.
(ii) Degenerate A basic feasible solution is degenerate, if one or
more basic variables are zero.Optimum basic feasible solution A basic feasible solution to a
LPP is said to be its optimum solution if it optimises (maximises or
minimises) the objective function.
Unbounded solution If the value of the objective functionzcan be
increased or decreased indefinitely, such solutions are called
unbounded solutions.
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Fundamental Theorem of LPP
THEOREM
The collection of all feasible solutions to linear programmingproblems constitute a convex set whose extreme pointscorrespond to the basic feasible solutions.
THEOREMIf an LPP has a feasible solution, then it also has a basic feasible
solution.
THEOREM(Replacement of a Basis vector)Let an LPP have a basic feasible solution. If we drop one of the basisvectors and introduce a non-basis vector in the basis set, then the new
solution obtained is also a basic feasible solution.
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Fundamental Theorem of LPP
THEOREM (Unbounded Solution)
Let there exist a basic feasible solution to a given LPP. If for at
least onej, for whichyij < 0 (i = 1, 2, , m) andzjcjis negative,
then there does not exist any optimum solution to this LPP.
THEOREM(condition of optimality)
A sufficient condition for the initial basic feasible solution to an LPP
to be optimum (maximum) is thatzjcj0 for allj for which the
column vector aj A is not in the basis B.