Linear programming problem

Linear Programming Problem (LPP) Formulation, Graphical
Method, Simplex Method, Dual LPP, Basic Concepts of Sensitivity
Analysis.
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Linear Programming Problem (LPP)
 Linear Programming is a mathematical technique useful for allocation of ‘scarce’ or ‘limited’ resources, to several
competingactivitiesonthebasisofagivencriterion ofoptimality.
 It dealswith theoptimization(maximizationorminimization)ofa functionofvariablesknownasobjective functions.
 SometimesreferredtoasLINEAROPTIMISATION
 Thewordlinearreferstolinear relationshipamongvariablesin amodel.
 Thetermprogrammingreferstotheprocessofdetermining a particularprogrammeorplanofaction.
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 Linear Programming may be defined as a method of determining an optimum programme of interdependent activities in view of
availableresources.
 The objective in a Linear Programming Problem is to maximise profit or minimise cost, as the case may be, subject toa number of
limitationsknownasconstraints.
 Forthis,anobjective functionisconstructedwhichrepresentstotalprofitortotalcostasthe casemaybe.
 The constraints are expressed in the form of inequalities or equations. Both the objective function and the constraints are linear
relationshipbetweenthevariables.
 The solution to a Linear Programming Problem shows how much should be produced (or sold or purchased) which will optimise
theobjectivefunctionandsatisfytheconstraints.
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Components of LPP:
1.Decision Variables:Thedecision variablewilldecide theo/p.
It gives theultimatesolutiontothe problem.Foranyproblem1st stepistoidentifythedecision variables.
2.Constraints:Theremustbelimitation ofResources.Theygive risetoconstraints.It maybeequalitiesorinequalities.
3.Objective function:It representshoweach decision variablewouldaffectthecost orsimplythe Valueneedstobeoptimised.
4.Data:Thesequantifytherelationshipsbetween obj:Fn andtheconstraints
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Essential ingredients/characteristics ofLPP
1. The objective function
In its general form,it is representedas:
Optimize(MaximizeorMinimize) Z=c1x1+c2x2+. . .+cnxn
wherec1,c2,c3………..cn areconstants
x1,x2,x3………..xn areDecision Variables.
2) Linear Constraints
There are a set of restrictions imposed on the variables appearing in the objective function. These restrictions are due to the
limitationsofresources.
Therestrictionsareoftheform
a₁x₁+a₂x₂+anx≥ b(or≤ b)
wherea₁,a₂,….andbareconstants.
Theremaybemorethanoneconstraint.
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3) Feasible solution:
A feasiblesolutiontoalinearprogrammingproblemisa setofvaluesforthevariablesX₁,X₂,............ Xnwhich satisfytheconstraints.
4) Optimal solution:
There can be one or more feasible solutions to a linear programming problem. A feasible solution which optimises the objective
functionisknownasoptimalsolution.
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 FormationofMathematicalmodelto a linear programmingproblem
Step1 :Identifytheobjectives asmaximisationorminimisation.
Step2:Mention theobjective quantitativelyandexpressit asa linearfunctionofvariables,knownasobjective function.
Step 3: Identify the constraints which are conditions stipulated in the problem. Constraints are those relating to availability of
resources, or conditions on quantities to be purchased or produced or sold etc. Constraints are expressed in the form of linear
inequalitiesorequations.
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PROBLEMS
Ex. 1:Amanufacturerof furnituremakestwo products,chairsandtables. Processingof theseproductsisdone on
two machines A and B. A chair requires 2 hours on machine A and 6 hours on machine B. A table requires 5 hours
on machine A andno time on machine B. There are 16 hours of time per day available on machine A and 30 hours
on machine B. Profit gained by the manufacturer from a chair is Re. I and from a table is Rs. 5 respectively.
Formulate theproblem into aL.P.P.inorder to maximizethe totalprofit.
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Ex. 2: A home resourceful decorator manufacturers two types of Lamps say A and B. Both lamps go through two
technicians first a cutter and second a finisher. Lamp A requires 2 hours of the cutter's time and 1 hour of the
finisher's time; Lamp B requires 1 hour of cutter's and 2 hours of finisher's time. The cutter has 104 hours and
finisher has 76 hours of available time each month. Profit on the Lamp A is Rs. 6.00 and on one B lamp is Rs.
11.00.Formulate amathematicalmodel.
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Solution to LPP
An LPP canbe solved by
1) Graphical Method
2) Simplex method
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Q:Draw lines corresponding to the equation
x=0,y=0,x=3,y=2, 3x+4y=12
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SIMPLEX METHOD
 If theLPPhaslargernumberofvariables,thesuitablemethod ofsolvingis Simplexmethod.
 Simplex method is a Linear Programming technique in which we start with a certain solution which is feasible. We improve this
solutionin anumberof consecutivestagesuntilwearriveatanoptimalsolution.
 For arriving at the solution of LPP by this method, the constraints and the objective function are presented in a table known as
simplextable.
 Itisan iterative(step by step)procedurein which weproceedin systematicstepsfroman initial BasicFeasible Solution toanother
Basic Feasible Solution and finally, in a finite number of steps to an optimal basic feasible solution, in such a way that the value of
theobjectivefunctionisbetter
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 Feasible solution
Itis the set of values of thevariables whichsatisfy all the constraints and non-negative restrictions of the problem.
 Optimal (Optimum) Solution
A feasible solution to a LinearProgramming Problem is said to be optimum if it optimises the objective function, Z,of the problem.
 Basic feasible solution
A feasible solution to a Linear Programming Problem in which the vectors associated to non zero variables are linearly independent is
called a basic feasible solution.
 Slack variables
If a constraint has a sign ≤ (less than or equal to) then in order to make it an equality (=) we have to add some variable to the left hand side.
Thesearecalled the slack variables. Thevalue of this variable can usually be interpreted as the amountof unused resources.
For example, consider the constraint: 2x1+ x2≤ 800
Inorder to convert the constraint into equation we add s, to L. H. S.
thenwe have 2x1+ x₂ + S₁ = 800.ThenS, is the slack variable.
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 Surplus variables
Ifa constraint has sign≥ ,thenin orderto makeit anequality have to subtract some variable from its L. H. S. Thesearecalled surplus variables.
Thevalue of this variable can be interpreted as the amountover andabove the requiredminimumlevel.
For example, consider the constraint: 2x₁ + 4x2≥ 12
Inorder to convert this into an equation we subtract s₂from the LHS of the inequality.
Then2x1+ 4x2-S2= 12.Hence s2 is the surplus variable.
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How to construct a simplex table ?
 Simplextableconsistsofrowsandcolumns.
 If there are 'm' original variables and 'n' introduced variables, then there will be 3 + m +n columns in the simplex table.
[Introducedvariablesareslack,surplusorartificialvariables].
 First column (B) contains the basic variables. Second column (C) shows the coefficient of the basic variables in the objective
function.
 Third column (XB) gives the values of basic variables. Each of next 'm+n' columns contain coefficient of variables in the
constraints,when theyareconvertedintoequations.
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 Basis (B)
Thevariables whose values are not restricted to zero in the currentbasic solution, arelisted in one column of the simplex table knownas
Basis (B).
 Basic variables
Thevariables which arelisted in the basis arecalled basic variables and others areknownas nonbasic variables.
 Vector
Anycolumn or row of a simplex table is called a vector.
Eg; X₁ - vector, X₂ -vector etc.
Ina simplex table, thereis a vector associated with every variable. Thevectors associated with the basic variables are unitvectors.
 UnitVector
A vector with one element 1and all other elements zero, is a unit vector.
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 Net Evaluation (Δj)
Δjis thenetprofitorlossif oneunitofthevariablein therespective columnis introduced.
Ie:, Δjshowswhatistheprofit(orloss)if oneunitofx;isintroduced.
TherowcontainingΔjvaluesis callednetevaluationroworindex row.
Δj =Zj-Cj
wherecj isthecoefficient ofxjvariablesin theobjective function
zjis thesum oftheproductsofcoefficientsofbasicvariables
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 MinimumRatio
Minimumratio is the lowest nonnegative ratio in the replacingratio column.
Θ( minimum ratio)= XB
Key Column Element
 KeyColumn (incomingvector)
The column whichhashighestnegativeΔjin a maximisationproblem or thehighestpositiveΔjin a minimisation problem
 Keyrow(outgoingvector)
The row whichrelatestothe minimumratio,is theOutgoingvector.
 Keyelement
Keyelementis thatelementof thesimplextable whichliesboth in thekeyrow and keycolumn.
 Iteration
Iterationmeansstepby stepprocessfollowed in simplex methodto movefrom one basic feasible solution to another
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Linear programming problem

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