ds linear programmimg

7/24/2019 DS Linear Programmimg

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Linear

Programming:Model Formulation

and GraphicalSolution

KOMAL GANGI Asst. Professor

!IAS

Prepared"#:


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!opics

Linear Programming An overview

Model Formulation

Characteristics of Linear ProgrammingProblems

Assumptions of a Linear ProgrammingModel

Advantages and Limitations of a Linear

Programming A Ma!imi"ation Model #!ample

$raphical %olutions of Linear Programming

Models


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*b+ectives of business decisions fre,uentl(involve maximizing proft or minimizingcosts

Linear programming uses linear algebraicrelationships to represent a rm.sdecisions/ given a business objective/ andresource constraints

%teps in application01 &dentif( problem as solvable b( linear

programming2 Formulate a mathematical model of the

unstructured problem

Linear Programming: AnO$er$ie%


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&ecision $aria"les - mathematical s(mbolsrepresenting levels of activit( of a rm

O"'ecti$e function - a linear mathematicalrelationship describing an ob+ective of the rm/

in terms of decision variables - this function is tobe ma!imi"ed or minimi"ed

(onstraints re,uirements or restrictionsplaced on the rm b( the operating environment/

stated in linear relationships of the decisionvariables

Parameters - numerical coeicients andconstants used in the ob+ective function and

constraints

Model (omponents


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Decisionvariables) Activities) Mathematical symbols representing levels of activity of a

firm.

Denoted by x1,x2,.xn

If values are under the control of decision maker, such

variables are said to be controllable

ther!ise, they are uncontrollable

In an "# Model, all Decision $ariables are continuous,controllable and %on& negative

'1(), '2(),..'n()

Model (omponents


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Decisionvariables) Activities) In a product&mix manufacturing, management may use

an "# model to determine ho! many units of each

products to be manufactured by using limited resources

such as personnel, machinery, money , material , etc

Model (omponents


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Objective Function * linear mathematical relationship describing an

ob+ective of the firm, in terms of decision variables

his function is to be maximi-ed or minimi-ed.

he ob+ectivegoal/ function of each "# problem is

expressed in terms of decision variables to optimi-e the

criterion of optimality also called measure of

performance/ such as profit, cost, revenue, distance, etc.

Model (omponents


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Objective Function ptimi-e Maximi-e or Minimi-e/

0 c1x1 c2x2 . cnxn

3here0 is the measure of performance variable3hichis a function of x1, x2,. xn.

4uantities c1,

c2,

. 5n

are parameters that represent

the contribution of a unit of the respective variable x1, x2,

. 'n to the measure of performance 0.

Model (omponents


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The Constraints 6e7uirements or restrictions placed on the firm by the

operating environment, stated in linear relationships of

the decision variables.

5ertain limitations on the use of resources e.g. labour,machine, ra! material, space, money, etc that limit to the

degree to !hich an ob+ective can be achieved

8uch constraints must be expressed as linear e7ualities

or ine7ualities in terms of decision variables he solution of an "# model must satisfy these

constraints

Model (omponents


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Summar# of Model FormulationSteps

Step *0 Clearl( dene the decisionvariables

Step +0 Construct the ob+ectivefunction

Step ,0 Formulate the constraints


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(haracteristics of LinearProgramming Pro"lems

A decision amongst alternative courses of actionis re,uired

'he decision is represented in the model b(decision $aria"les

'he problem encompasses a goal/ e!pressed asan o"'ecti$e function/ that the decision ma9erwants to achieve

:estrictions ;represented b( constraints-e!ist

that limit the e!tent of achievement of theob+ective

'he ob+ective and constraints must be denableb( linearmathematical functional relationships


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Certainty

$alues of all the model parameters are assumed to be

kno!n !ith certainty non&probabilistic/.

*vailability of resources, profitor cost/ contribution of aunit of decision variable and consumption of resources

by a unit of decision variable

Divisibility(or continuity) 9he solution values of

Decision variables can take on any fractional value andare therefore continuous as opposed to integer in

nature. :g& ;.2


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Proportionality(Linearity) & he rate of change slope/

of the ob+ective function and constraint e7uations is

constant.

Additivity & erms in the ob+ective function and

constraint e7uations must be additive.

Assumptions of LinearProgramming Model


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It helps decision & makers to use their productiveresource effectively.

he decision&making approach of the user becomes

more ob+ective and less sub+ective.

In a production process, bottle necks may occur. >or

example, in a factory some machines may be in great

demand !hile others may lie idle for some time. *

significant advantage of linear programming is

highlighting of such bottle necks.

Ad$antages of LinearProgramming Model


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"inear programming is applicable only to problems !here the

constraints and ob+ective function are linear i.e., !here they

can be expressed as e7uations !hich represent straight lines.In real life situations, !hen constraints or ob+ective functions

are not linear, this techni7ue cannot be used.

>actors such as uncertainty, and time are not taken into

consideration.

#arameters in the model are assumed to be constant but in

real life situations they are not constants.

"inear programming deals !ith only single ob+ective , !hereas

in real life situations may have multiple and conflicting

ob+ectives. In solving a "## there is no guarantee that !e get an integer

value. In some cases of no of men?machine a non&integer

value is meaningless.

Limitations of LinearProgramming Model


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LP Model FormulationA Maimi/ation 0ample )* of 1-

Product mi! problem -


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LP Model FormulationA Maimi/ation 0ample )+ of 1-


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LP Model FormulationA Maimi/ation 0ample ), of 1-

2esource 8 hrs of labor per da(A$aila"ilit#: 128 lbs of cla(

&ecision !1? number of bowls to produce per

da(3aria"les: !2? number of mugs to produce per

da(

O"'ecti$e Ma!imi"e @ ? 8!1B 38!2

Function: here @ ? prot per da(

2esource 1!1 B 2!28 hours of labor

(onstraints: !1B )!2128 pounds of cla(

Non4Negati$it# !1 8D !2 8


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LP Model FormulationA Maimi/ation 0ample )1 of 1-

(omplete Linear Programming Model:

Ma!imi"e @ ? 8!1B 38!2

sub+ect to0 1!1B 2!2 8!2B )!2 128!1/ !2 8


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Aeasible solution does not violate anyofthe constraints0

#!ample0!1 ? 3 bowls!2 ? 18 mugs

@ ? 8!1B 38!2 ? 588

Labor constraint chec90 1;3E B 2;18E ? 23 58 hours

Cla( constraint chec90 ;3E B );18E ? 58 5

128 pounds

Feasi"le Solutions

f i"l l i


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An ineasible solutionviolates at leastone of the constraints0

#!ample0!1? 18 bowls!2? 28 mugs

@ ? 8!1B 38!2 ? 188

Labor constraint chec90 1;18E B 2;28E ? 38 68 hours

Infeasi"le Solutions


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$raphical solution is limited to linearprogramming models containing only twodecision variables ;can be used with three

variables but onl( with great diicult(E

$raphical methods provide visualization o howa solution for a linear programming problem isobtained

$raphical methods can be classied under two

categories0 1 &so-Prot;CostE Line Method

2 #!treme-point evaluation Method

Graphical Solution of LPModels

( di t A


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(oordinate AesGraphical Solution of Maimi/ationModel )* of *+-

Figure +.+ Coordinates for$ra hical Anal sis

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2B )!

2

128 !1/ !2 8

7* is "o%ls

7+ is mugs

L " ( t i t


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La"or (onstraintGraphical Solution of Maimi/ationModel )+ of *+-

Figure +., $raph of LaborConstraint

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2B )!

2

128 !1/ !2 8

L " ( t i t A


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La"or (onstraint AreaGraphical Solution of Maimi/ationModel ), of *+-

Figure +.1 Labor ConstraintArea

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2B )!

2

128 !1/ !2 8

(l ( t i t A


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(la# (onstraint AreaGraphical Solution of Maimi/ationModel )1 of *+-

Figure +.8 Cla( ConstraintArea

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

9 th ( t i t


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9oth (onstraintsGraphical Solution of Maimi/ationModel )8 of *+-

Figure +. $raph of


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Feasi"le Solution AreaGraphical Solution of Maimi/ationModel ) of *+-

Figure +.; Feasible%olution Area

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

O"' ti F ti S l ti =>??


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O"'ecti$e Function Solution < =>??Graphical Solution of Maimi/ationModel ); of *+-

Figure +.> *b+ection Function Linefor @ ? 688

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

Alternati$e O"'ecti$e Function Solution


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Alternati$e O"'ecti$e Function SolutionLinesGraphical Solution of Maimi/ation Model

)> of *+-

Figure +.@Alternative *b+ectiveFunction Lines

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

Optimal Solution


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Optimal SolutionGraphical Solution of Maimi/ationModel )@ of *+-

Figure +.*? &dentication of *ptimal%olution Point

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

Optimal Solution (oordinates


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Optimal Solution (oordinatesGraphical Solution of Maimi/ationModel )*? of *+-

Figure +.** *ptimal %olutionCoordinates

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

0treme )(orner- Point Solutions


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0treme )(orner- Point SolutionsGraphical Solution of Maimi/ationModel )** of *+-

Figure +.*+ %olutions at AllCorner Points

Ma!imi"e @ ? 8!1

B

38!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

Optimal Solution for Ne% O"'ecti$e


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Optimal Solution for Ne% O"'ecti$eFunctionGraphical Solution of Maimi/ation

Model )*+ of *+-

Ma!imi"e @ ? 58!1

B

28!2sub+ect to0 1!1B 2!2

8!

2

B )!2

128 !1/ !2 8

Figure +.*, *ptimal %olution with @ ?

Slac 3aria"les


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%tandard form re,uires that all constraintsbe in the form of e,uations ;e,ualitiesE

A slac9 variable is added to a constraint

;wea9 ine,ualit(E to convert it to ane,uation ;?E

A slac9 variable t(picall( represents an

unused resource A slac9 variable contributes nothing to

the ob+ective function value

Slac 3aria"les

Linear Programming Model:


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Linear Programming Model:Standard Form

Ma! @ ? 8!1B 38!2B s1

B s2sub+ect to01!1B 2!2 B s1 ?

8!2B )!2 B s2 ?

128 !1/ !2/ s1/ s2 8

here0 !1? number of bowls

!2? number of mugs

s1/ s2are slac9 variables

Figure +.*1 %olution Points A/


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LP Model Formulation B Minimi/ation)* of >-

'wo brands of fertili"er available - %uper-gro/ Crop-,uic9

Field re,uires at least 14 pounds of nitrogen and2 pounds of phosphate

%uper-gro costs 4 per bag/ Crop-,uic9 ) per bag Problem0 =ow much of each brand to purchase to

minimi"e total cost of fertili"er given following data>

LP Model Formulation Minimi/ation


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LP Model Formulation B Minimi/ation)+ of >-

Figure +.*8Fertili"ing farmer.seld

LP Model Formulation


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&ecision 3aria"les:!1? bags of %uper-gro

!2? bags of Crop-,uic9

!he O"'ecti$e Function:Minimi"e @ ? 4!1B )!2here0 4!1? cost of bags of %uper-$ro

)!2? cost of bags of Crop-uic9

Model (onstraints:2!1B !214 lb ;nitrogen constraintE

!1B )!22 lb ;phosphate constraintE

!1

/ !2

8 ;non-negativit( constraintE

LP Model Formulation BMinimi/ation ), of >-

(onstraint Graph Minimi/ation


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Minimi"e @ ? 4!1B )!2sub+ect to0 2!1B !2 14

!2B )!2 2

!1/ !2 8

Figure +.* $raph of -

Feasi"le 2egion Minimi/ation


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2-1Figure +.*; Feasible %olutionArea

Feasi"le 2egionB Minimi/ation)8 of >-


!2B )!2 2

!1/ !2 8

Optimal Solution Point B


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2-2Figure +.*> *ptimum%olution Point

Optimal Solution Point BMinimi/ation ) of >-


!2B )!2 2

!1/ !2 8

Surplus 3aria"les B Minimi/ation );


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A surplus variable is subtracted rom aconstraint to convert it to an e,uation ;?E

A surplus variable represents an excessabove a constraint re,uirement level

A surplus variable contributes nothing tothe calculated value of the ob+ectivefunction

%ubtracting surplus variables in the farmerproblem constraints0

2!1B !2- s1? 14

;nitrogenE !1B )!2- s2? 2

Surplus 3aria"les B Minimi/ation );of >-

Graphical Solutions B Minimi/ation


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2-Figure +.*@ $raph of Fertili"er#!am le

Graphical Solutions B Minimi/ation)> of >-

Minimi"e @ ? 4!1B )!2B 8s1

B 8s2sub+ect to0 2!1B !2 s1? 14

!2B )!2 s2 ? 2

!1/ !2/ s1/ s28

Irregular !#pes of Linear


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For some linear programming models/ thegeneral rules do not appl(

%pecial t(pes of problems include those

with0 Multiple optimal solutions

&nfeasible solutions

Gnbounded solutions

Irregular !#pes of LinearProgramming Pro"lems

Multiple Optimal Solutions 9ea$er


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2-4Figure +.+? #!ample with Multiple* timal %olutions

Multiple Optimal Solutions 9ea$er(ree Potter#

'he ob+ective function isparallel to a constraintline

Ma!imi"e @?8!1B )8!2sub+ect to0 1!1B 2!2 8

!2B )!2 128

!1/ !2 8

here0!1? number of bowls

!2? number of mugs

An Infeasi"le Pro"lem


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An Infeasi"le Pro"lem

Figure +.+* $raph of an &nfeasibleProblem

#ver( possible solution$iolatesat least oneconstraint0

Ma!imi"e @ ? 3!1B )!2sub+ect to0 !1B 2!26

!1

!24

!1/ !28

An Cn"ounded Pro"lem


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An Cn"ounded Pro"lem

Figure +.++ $raph of an GnboundedProblem

Halue of the ob+ectivefunction increases

indenitel(0Ma!imi"e @ ? !1B 2!2sub+ect to0 !1

!22

!1/ !28

Pro"lem Statement


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Pro"lem Statement0ample Pro"lem No. * )* of ,-

I =ot dog mi!ture in 1888-pound batches

I 'wo ingredients/ chic9en ;)JlbE and beef;3JlbE

I :ecipe re,uirements0 at least 388 pounds of

Kchic9en

at least 288 pounds ofKbeef

I :atio of chic9en to beef must be at least 2to 1

Solution


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Step *:&dentif( decision variables

!1? lb of chic9en in mi!ture

!2? lb of beef in mi!ture

Step +:

Formulate the ob+ective function

Minimi"e @ ? )!1B 3!2where @ ? cost per 1/888-lb batch

)!1? cost of chic9en

3!2? cost of beef

Solution0ample Pro"lem No. * )+ of ,-

Solution


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Step ,:

#stablish Model Constraints !1B !2? 1/888 lb

!1388 lb of chic9en

!2288 lb of beef

!1J!22J1 or !1- 2!28

!1/ !28

!he Model: Minimi"e @ ? )!1B 3!2 sub+ect to0 !1B !2? 1/888 lb

!138

!2288

!1- 2!28

Solution0ample Pro"lem No. * ), of ,-

0ample Pro"lem No. + )* of ,-


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%olve the followingmodel graphicall(0Ma!imi"e @ ? !1B 3!2sub+ect to0 !1B 2!2

18 4!1B 4!2

)4 !1

!1/ !28

%tep 10 Plot theconstraints as e,uations

0ample Pro"lem No. + )* of ,-

Figure +.+, Constraint# uations

0ample Pro"lem No. + )+ of ,-


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0ample Pro"lem No. + )+ of ,-

Ma!imi"e @ ? !1B 3!2sub+ect to0 !1B 2!2

18

4!1B 4!2

)4 !1

!1/ !28

%tep 20 etermine thefeasible solution space

Figure +.+1 Feasible %olution %pace and#!treme Points

0ample Pro"lem No. + ), of ,-


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0ample Pro"lem No. + ), of ,-

Ma!imi"e @ ? !1B 3!2sub+ect to0 !1B 2!2

18 4!1B 4!2

)4 !1

!1/ !28

%tep ) and 0etermine the solutionpoints and optimalsolution

Figure +.+8 *ptimal %olution

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