3. Linear Programming- Formulation

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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



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.



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.



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.



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



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).



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.

3. Linear Programming- Formulation

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