taylor introms11ge ppt 02

7/25/2019 Taylor Introms11GE Ppt 02

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2-1Copyright 2013 Pearson Education

Modeling with

LinearProgramming

Chapter 2


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Chapter Topics

Model Formulation

A Maximization Model Example

Graphical Solutions of Linear ProramminModels

A Minimization Model Example

!rreular "#pes of Linear ProramminModels

Characteristics of Linear ProramminPro$lems


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&$'ecti(es of $usiness decisions fre)uentl#in(ol(e maximizing proft or minimizingcosts*

Linear prorammin uses linear algebraicrelationships to represent a +rm,sdecisions i(en a $usiness objective andresource constraints*

Steps in application.1* !dentif# pro$lem as sol(a$le $# linear

prorammin*2* Formulate a mathematical model of the

unstructured pro$lem*

Linear Programming: AnOverview


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Decision variables - mathematical s#m$olsrepresentin le(els of acti(it# of a +rm*

Objective function - a linear mathematicalrelationship descri$in an o$'ecti(e of the +rm interms of decision (aria$les - this function is to $emaximized or minimized*

Constraints 0 re)uirements or restrictions placedon the +rm $# the operatin en(ironment stated in

linear relationships of the decision (aria$les*

Parameters - numerical coeicients and constantsused in the o$'ecti(e function and constraints*

Model Components


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ummar! of Model "ormulationteps

tep #. Clearl# de+ne the decision(aria$les

tep $. Construct the o$'ecti(efunction

tep %. Formulate the constraints


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LP Model "ormulationA Ma&imi'ation (&ample )# of %*

Product mix pro$lem - 4ea(er Cree5 Potter# Compan# 6o7 man# $o7ls and mus should $e produced to

maximize pro+ts i(en la$or and materials constraints8

Product resource re)uirements and unit pro+t.

Resource Requirements

Produ

ct

Labor

(Hr./Unit

)

Clay

(Lb./Unit

)

Proft

($/Unit)

Bowl 1 4 40

Mug 2 3 50


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LP Model "ormulationA Ma&imi'ation (&ample )$ of %*

+esource /: hrs of la$or per da#Availabilit!: 12: l$s of cla#

Decision x1; num$er of $o7ls to produce

per da#,ariables: x2; num$er of mus to produce per

da#

Objective Maximize < ; =/:x1> =:x2

"unction: ?here < ; pro+t per da#

+esource 1x1 > 2x2/: hours of la$or

Constraints: /x1> %x212: pounds of cla#

-on.-egativit! x1 :@ x2 :


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LP Model "ormulationA Ma&imi'ation (&ample )% of %*

Complete Linear Programming Model:

Maximize < ; =/:x1> =:x2

su$'ect to. 1x1> 2x2 /:/x2> %x2 12:x1 x2 :


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Aeasible solution does not (iolate anyofthe constraints.

Example. x1 ; $o7lsx2 ; 1: mus

< ; =/:x1> =:x2 ; =9::

La$or constraint chec5. 1D > 21:D ; 2 //: hours

Cla# constraint chec5. /D > %1:D ; 9: /

12: pounds

"easible olutions


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An ineasible solution(iolates at leastone of the constraints.

Example. x1; 1: $o7lsx2; 2: mus

< ; =/:x1> =:x2 ; =1/::

La$or constraint chec5. 11:D > 22:D ; : 0/: hours

1nfeasible olutions



Graphical solution is limited to linearprorammin models containin only twodecision variables can $e used 7iththree (aria$les $ut onl# 7ith reatdiicult#D*

Graphical methods pro(ide visualizationo how a solution for a linearprorammin pro$lem is o$tained*

2raphical olution of LPModels



Coordinate A&es2raphical olution of Ma&imi'ationModel )# of #$*

Fiure 2*2 Coordinates forra hical anal sis

Maximize < ; =/:x1>=:x2su$'ect to. 1x1> 2x2

/:

/x2> %x2

12: x1 x2 :

3#is bowls

3$is mugs


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Labor Constraint2raphical olution of Ma&imi'ationModel )$ of #$*

Fiure 2*% Graph of la$orconstraint


/:

/x2> %x2

12: x1 x2 :


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Labor Constraint Area2raphical olution of Ma&imi'ationModel )% of #$*

Fiure 2*/ La$or constraintarea


/:

/x2> %x2

12: x1 x2 :



Cla! Constraint Area2raphical olution of Ma&imi'ationModel )4 of #$*

Fiure 2* "heconstraint area forcla


/:

/x2> %x2

12: x1 x2 :



5oth Constraints2raphical olution of Ma&imi'ationModel )6 of #$*

Fiure 2*3 Graph of $oth modelconstraints

Maximize < ; =/:x1>

=:x2su$'ect to. 1x1> 2x2

/:

/x2> %x2

12: x1 x2 :



"easible olution Area2raphical olution of Ma&imi'ationModel )7 of #$*

Fiure 2*9 "he feasi$lesolution area

Maximize < ; =/:x1>


/:

/x2> %x2

12: x1 x2 :



Objective "unction olution 8 9;;2raphical olution of Ma&imi'ationModel )< of #$*

Fiure 2* &$'ecti(e function line for< ; =::

Maximize < ; =/:x1>


/:

/x2> %x2

12: x1 x2 :

l i bj i i l i


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Alternative Objective "unction olutionLines2raphical olution of Ma&imi'ation Model

) of #$*

Fiure 2*BAlternati(e o$'ecti(efunction lines forpro+ts < of =::

=12:: and =13::

Maximize < ; =/:x1>

=:x2

su$'ect to. 1x1> 2x2

/:

/x2> %x2

12: x

1 x

2:

i l l i


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Optimal olution2raphical olution of Ma&imi'ationModel )= of #$*

Fiure 2*1: !denti+cation of optimalsolution oint

Maximize < ; =/:x1>


/:

/x2> %x2

12: x1 x2 :

O i l l i C di



Optimal olution Coordinates2raphical olution of Ma&imi'ationModel )#; of #$*

Fiure 2*11 &ptimal solutioncoordinates

Maximize < ; =/:x1>


/:/x

2> %x

2

12: x1 x2 :

( )C * P i l i



(&treme )Corner* Point olutions2raphical olution of Ma&imi'ationModel )## of #$*

Fiure 2*12 Solutions at allcorner oints

Maximize < ; =/:x1>


/:/x

2> %x

2

12: x1 x2 :

O ti l l ti f - Obj ti


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Optimal olution for -ew Objective"unction2raphical olution of Ma&imi'ation

Model )#$ of #$*

Maximize < ; =9:x1

>

=2:x2su$'ect to. 1x1> 2x2

/:/x

2> %x

2

12: x1 x2 :

Fiure 2*1% &ptimal solution 7ith < ;

l > , i bl


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Standard form re)uires that all constraints$e in the form of e)uations e)ualitiesD*

A slac5 (aria$le is added to a constraint

7ea5 ine)ualit#D to con(ert it to ane)uation ;D*

A slac5 (aria$le t#picall# represents an

unused resource* A slac5 (aria$le contributes nothing to

the o$'ecti(e function (alue*

lac> ,ariables

Li P i M d l



Linear Programming Model:tandard "orm

Max < ; /:x1> :x2> s1

> s2su$'ect to.1x1> 2x2 > s1 ;

/:/x2> %x2 > s2 ;

12: x1 x2 s1 s2 :

?here.

x1; num$er of $o7ls

x2; num$er of mus

s1 s2are slac5 (aria$les

Fiure 2*1/ Solutions at points A 4 and

LP M d l " l ti Mi i i ti


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LP Model "ormulation ? Minimi'ation)# of


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Decision ,ariables:x1; $as of Super-ro

x2; $as of Crop-)uic5

The Objective "unction:Minimize < ; =3x1> %x2?here. =3x1; cost of $as of Super-

Gro=%x

2; cost of $as of Crop-uic5

Model Constraints:2x1> /x213 l$ nitroen constraintD

/x1

> %x2

2/ l$ phosphate constraintD

x1 x2: non-neati(it# constraintD

LP Model "ormulation ?Minimi'ation )$ of


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Minimize < ; =3x1

> =%x2

su$'ect to. 2x1> /x2 13

/x2> %x2 2/

x1 x2 :

Fiure 2*13 Constraint lines for

fertilizer model

Constraint 2raph ? Minimi'ation)% of


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2-2BCopyright 2013 Pearson EducationFiure 2*19 Feasi$le solutionarea

"easible +egion? Minimi'ation)4 of =%x2

su$'ect to. 2x1> /x2 13

/x2> %x2 2/

x1 x2 :

O ti l l ti P i t


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2-%:Copyright 2013 Pearson EducationFiure 2*1 "he optimalsolution oint

Optimal olution Point ?Minimi'ation )6 of =%x2su$'ect to. 2x1> /x2 13

/x2> %x2 2/

x1 x2 :

"he optimalsolution of aminimizationpro$lem is at theextreme pointclosest to the

oriin*

urplus ,ariables Minimi'ation )7


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A surplus (aria$le is subtracted rom aconstraint to con(ert it to an e)uation ;D*

A surplus (aria$le represents an excessa$o(e a constraint re)uirement le(el*

A surplus (aria$le contributes nothing tothe calculated (alue of the o$'ecti(efunction*

Su$tractin surplus (aria$les in the farmerpro$lem constraints.

2x1> /x2- s1; 13

nitroenD /x > %x - s ; 2/

urplus ,ariables ? Minimi'ation )7of


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Fiure 2*1B Graph of the fertilizer

example

2raphical olutions ? Minimi'ation)< of =%x2> :s1

> :s2su$'ect to. 2x1> /x2 0 s1; 13

/x2> %x20 s2 ; 2/

x1 x2 s1 s2:

1rregular T!pes of Linear


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For some linear prorammin models theeneral rules do not appl#*

Special t#pes of pro$lems include those7ith.

Multiple optimal solutions

!nfeasi$le solutions

n$ounded solutions

1rregular T!pes of LinearProgramming Problems

Multiple Optimal olutions 5eaver


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2-%/Copyright 2013 Pearson EducationFiure 2*2: Example 7ith multipleo timal solutions

Multiple Optimal olutions 5eaverCree> Potter!

"he o$'ecti(e function isparallel to a constraintline*

Maximize %:x2su$'ect to. 1x1> 2x2 /:

/x2> %x2 12:

x1 x2 :

?here.x1; num$er of $o7ls

x2; num$er of mus

An 1nfeasible Problem


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An 1nfeasible Problem

Fiure 2*21 Graph of an infeasi$lepro$lem

E(er# possi$le solutionviolatesat least oneconstraint.

Maximize < ; x1> %x2su$'ect to. /x1> 2x2

x1/

x23

x1 x2:

An @nbounded Problem


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An @nbounded Problem

Fiure 2*22 Graph of an un$oundedpro$lem

alue of the o$'ecti(efunction increases

inde+nitel#.Maximize < ; /x1> 2x2su$'ect to. x1/

x22

x1 x2:

Characteristics of Linear


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Characteristics of LinearProgramming Problems

A decision amonst alternati(e courses of action isre)uired*

"he decision is represented in the model $#decision variables*

"he pro$lem encompasses a oal expressed as anobjective function that the decision ma5er7ants to achie(e*

Hestrictions represented $# constraints*exist

that limit the extent of achie(ement of theo$'ecti(e*

"he o$'ecti(e and constraints must $e de+na$le $#linearmathematical functional relationships*

Properties of Linear


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Proportionalit!- "he rate of chane slopeD ofthe o$'ecti(e function and constraint e)uations isconstant*

Additivit!- "erms in the o$'ecti(e function andconstraint e)uations must $e additi(e*

Divisibilit!- Iecision (aria$les can ta5e on an#fractional (alue and are therefore continuous asopposed to inteer in nature*

Certaint!- alues of all the model parametersare assumed to $e 5no7n 7ith certaint# non-pro$a$ilisticD*

Properties of LinearProgramming Models

Problem tatement


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Problem tatement(&ample Problem -o # )# of %*

J 6ot do mixture in 1:::-pound $atches*

J "7o inredients chic5en =%Kl$D and $eef=Kl$D*

J Hecipe re)uirements.

at least :: pounds ofchic5en

at least 2:: pounds of$eef

J Hatio of chic5en to $eef must $e at least 2

to 1*

olution


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tep #:!dentif# decision (aria$les*

x1; l$ of chic5en in mixture

x2; l$ of $eef in mixture

tep $:

Formulate the o$'ecti(e function*

Minimize < ; =%x1> =x27here < ; cost per 1:::-l$ $atch

=%x1; cost of chic5en

=x2; cost of $eef

olution(&ample Problem -o # )$ of %*

olution


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tep %:

Esta$lish Model Constraints x1> x2; 1::: l$

x1:: l$ of chic5en

x22:: l$ of $eef

x1Kx22K1 or x1- 2x2:

x1 x2:

The Model: Minimize < ; =%x1> x2 su$'ect to. x1> x2; 1::: l$

x1:

x22::

x1- 2x2:

olution(&ample Problem -o # )% of %*

(&ample Problem -o $ )# of %*


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Sol(e the follo7inmodel raphicall#.Maximize < ; /x1> x2su$'ect to. x1> 2x2

1: 3x1> 3x2

%3 x1/

x1 x2:

Step 1. Plot theconstraints as e)uations

(&ample Problem -o $ )# of %*

Fiure 2*2% Constrainte uations

(&ample Problem -o $ )$ of %*


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(&ample Problem -o $ )$ of %*

Maximize < ; /x1> x2su$'ect to. x1> 2x2

1:

3x1> 3x2%3 x1/

x1 x2:

Step 2. Ietermine thefeasi$le solution space

Fiure 2*2/ Feasi$le solution space andextreme points

(&ample Problem -o $ )% of %*


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(&ample Problem -o $ )% of %*

Maximize < ; /x1> x2su$'ect to. x1> 2x2

1: 3x1> 3x2

%3 x1/

x1 x2:

Step % and /.Ietermine the solutionpoints and optimalsolution

Fiure 2*2 &ptimal solutionoint


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taylor introms11ge ppt 02

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