taylor introms11ge ppt 02
TRANSCRIPT
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2-1Copyright 2013 Pearson Education
Modeling with
LinearProgramming
Chapter 2
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2-2Copyright 2013 Pearson Education
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 (&le )# 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
($/Unit)
Bowl 1 4 40
Mug 2 3 50
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LP Model "ormulationA Ma&imi'ation (&le )$ 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$or
Constraints: /x1> %x212: pounds of cla#
-on.-egativit! x1 :@ x2 :
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LP Model "ormulationA Ma&imi'ation (&le )% of %*
Complete Linear Programming Model:
Maximize < ; =/:x1> =:x2
su$'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
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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$tained*
2raphical olution of LPModels
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Coordinate A&es2raphical olution of Ma&imi'ationModel )# of #$*
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
Maximize < ; =/:x1>=:x2su$'ect to. 1x1> 2x2
/:
/x2> %x2
12: x1 x2 :
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Labor Constraint Area2raphical olution of Ma&imi'ationModel )% of #$*
Fiure 2*/ La$or constraintarea
Maximize < ; =/:x1>=:x2su$'ect to. 1x1> 2x2
/:
/x2> %x2
12: x1 x2 :
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Cla! Constraint Area2raphical olution of Ma&imi'ationModel )4 of #$*
Fiure 2* "heconstraint area forcla
Maximize < ; =/:x1>=:x2su$'ect to. 1x1> 2x2
/:
/x2> %x2
12: x1 x2 :
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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 :
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"easible olution Area2raphical olution of Ma&imi'ationModel )7 of #$*
Fiure 2*9 "he feasi$lesolution area
Maximize < ; =/:x1>
=:x2su$'ect to. 1x1> 2x2
/:
/x2> %x2
12: x1 x2 :
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Objective "unction olution 8 9;;2raphical olution of Ma&imi'ationModel )< of #$*
Fiure 2* &$'ecti(e function line for< ; =::
Maximize < ; =/:x1>
=:x2su$'ect to. 1x1> 2x2
/:
/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>
=:x2su$'ect to. 1x1> 2x2
/:
/x2> %x2
12: x1 x2 :
O i l l i C di
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Optimal olution Coordinates2raphical olution of Ma&imi'ationModel )#; of #$*
Fiure 2*11 &ptimal solutioncoordinates
Maximize < ; =/:x1>
=:x2su$'ect to. 1x1> 2x2
/:/x
2> %x
2
12: x1 x2 :
( )C * P i l i
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(&treme )Corner* Point olutions2raphical olution of Ma&imi'ationModel )## of #$*
Fiure 2*12 Solutions at allcorner oints
Maximize < ; =/:x1>
=:x2su$'ect to. 1x1> 2x2
/:/x
2> %x
2
12: x1 x2 :
O ti l l ti f - Obj ti
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Optimal olution for -ew Objective"unction2raphical olution of Ma&imi'ation
Model )#$ 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$'ecti(e function (alue*
lac> ,ariables
Li P i M d l
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Linear Programming Model:tandard "orm
Max < ; /:x1> :x2> s1
> s2su$'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(&le 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 $eef
olution(&le Problem -o # )$ 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(&le Problem -o # )% of %*
(&le 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
(&le Problem -o $ )# of %*
Fiure 2*2% Constrainte uations
(&le Problem -o $ )$ of %*
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(&le 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
(&le Problem -o $ )% of %*
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(&le 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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