Scale Invariance and Symmetry Relationships In Non-Extensive Statistical Mechanics

0.21 MB·English

by  Mark Fleischer

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n Scale Invariance and Symmetry Relationships
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In Non-Extensive Statistical Mechanics
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Mark Fleischer
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InformationOperationsGroup
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e JohnsHopkinsUniversity
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AppliedPhysicsLaboratory
- 11100JohnsHopkinsRoad
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Laurel, Maryland20723-6099
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s email: [email protected]
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a (cid:13)c 2004by MarkFleischer
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- December 10,2004
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n
o
c Abstract
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Thisarticleextends resultsdescribed inarecent articledetailingastructural

1 scaleinvariancepropertyofthesimulatedannealing(SA)algorithm.Theseexten-
v sionsarebasedongeneralizationsoftheSAalgorithmbasedonTsallisstatistics
3 andanon-extensiveformofentropy. Thesescaleinvariancepropertiesshowhow
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arbitraryaggregationsofenergylevelsretaincertainmathematicalcharacteristics.
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In applying the non-extensive forms of statistical mechanics to illuminate these
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scale invariance properties, an interesting energy transformation is revealed that
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hasanumberofpotentiallyusefulapplications. Thisenergytransformationfunc-
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0 tionalsorevealsanumberofsymmetryproperties. Furtherextensionsofthisre-
/ searchindicatehowthisenergytransformationfunctionrelatestopowerlawdistri-
t
a butionsandpotentialapplicationforovercomingtheso-called“brokenergodicity”
m problemprevalentinmanycomputersimulationsofcriticalphenomena.
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n 1 Introduction
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Recentadvancesinstatisticalmechanicshavehelpedtoexplainthebehavioroflarge
:
v ensemblesofinteractingsystems. SincethetimeofBoltzmann,therehasbeenagreat
Xi deal of effort in attempting to understand the behavior of such systems. In his day,
agreatdealofprogresswasmadeusingaformofentropythatseemedtoexplainthe
r
a behavioroftheso-called“idealgas”model,amodelpremisedontheverylimitedform
ofinteractionimpliedbyelasticityassumptions.Notwithstandingtheseassumptions,it
wasabletopredict,quiteaccurately,manyattributesofthebehaviorofgasesinthermal
equilibrium.
1
Despiteitsenormoussuccessfortheseidealizedsystems,theBoltzmannversionof
statisticalmechanicshashadlimitedsuccessinexplainingthebehaviorofmorecom-
plexsystemswherethecomponentsinteractinwaysthataredistinctlyinelastic—that
is to say, the particles do not obey simple Newtonian mechanics. In these models,
the particles themselves may absorb energy during collisions in forms other than by
changesintheirvelocity. Theassumptionsoftheidealgasmodelthattheseparticles
haveno internalstructurethereforecannotbe assumed. Or, it may be that these par-
ticles exhibit attractive or repulsive forces between them. In any event, their gross
behavior is not well predicted in systems governedby the Boltzmann-Gibbsentropy
formula: S = −k W π lnπ where the π represents the probabilityof the ith
BG i=1 i i i
energy partition and W represents the size (number of energy states) of the system.
P
Thesecomplexsystemsobeyvariouspowerlawsandcanexhibitphasetransitionsand
relatedcriticalphenomenawheretherearedrasticshiftsintheirenergyconfigurations.
Thesepowerlawsandcriticalitypropertiesseemtobequitediverseandubiquitous
in nature. To help explainthese behaviors, Tsallis [10, 11] developeda new entropy
expressionthatformsthebasisofanon-extensiveformofthermodynamics:
k 1− W pq
i=1 i
S = (1)
q (cid:16) qP−1 (cid:17)
wherekisaconstantandS istheentropyparameterizedbytheentropicparameterq.
q
Inclassicalstatisticalmechanics,entropyfallsintoaclassofvariablesthatarereferred
toasextensivebecausetheyscalewiththesizeofthesystem. Intensivevariables,such
astemperature,donotscale withthesize ofthesystem1. Tsallis’formofentropyis
non-extensivebecausetheentropyoftheunionoftwoindependentsystemsisnotequal
tothesumoftheentropiesofeachsystem. Thatis,forindependentsystemsAandB,
(1−q)S (A)S (B)
S (A+B)=S (A)+S (B)+ q q (2)
q q q
k
andTsallispointsoutthatq =1recoverstheextensivitypropertiesoftheBoltzmann-
Gibbs entropy formula and further, that limq→1Sq = SBG, hence can be seen as a
generalization of Boltzmann-Gibbs [10, 11]. Moreover, the stationary probabilities
p (t)ofthesystembeinginsomeifurtherillustratesthegeneralizationoftheclassic
i
entropyforminthat
limp (t)=π (t) (3)
i i
q→1
forallt>0[10,p.483].
Theentropicparameterqcontrolsimportantaspectsoftheprobabilitydistribution
oftheenergylevelsmakingitispossibletomarkedlychangethenatureofthethermo-
dynamicsystem beingmodeled. Forexample,whenq > 1thestationaryprobability
distributionshiftsfromanexponentialformtoonewithheavytailsthatgivesrisetoa
1Combinetwovesselsofgaseachwiththesamevolumeandpressureintoanothervesseloftwicethe
volume, andthe pressure andtemperature ofthe combined gas will bethe sameas before. Energyand
entropy,however,areexamplesofextensivevariablesinthatcombiningseveralsourcesofeitherenergyor
entropyandyouincreasethetotalenergyorentropy.
2
powerlawdistribution,orconversely,onecanincreasethestationaryprobabilityoflow
energystateswhenq <1,somethingquiteusefulinoptimization(seee.g.,[1,12]).
Oneofthebestwaysforstudyingtheimplicationsofthisnon-extensiveformofsta-
tisticalmechanicsisthroughtheuseofcomputersimulations.Thesimulatedannealing
(SA)algorithmprovidesthesebasicsimulationtoolsasitis,atheart,asimulationofa
thermodynamicsystemalthoughithasbeenusedprincipallyforsolvingoptimization
problems. Tsallisdevelopedageneralizedsimulatedannealing(GSA)algorithm[12]
basedonmaximizingtheentropyin(1).
Giventhesedevelopments,itseemsquiteappropriateandusefultoexaminecertain
scaleinvarianceproperties[3]oftheclassicSAalgorithminlightofthenon-extensive
formofentropyandtheGSA.Thesescaleinvariancepropertiesshedlightonanumber
ofbehaviorsandillustratesomecuriouseffectsontheconfigurationspaceitself.Anen-
ergytransformationfunctionisidentifiedandusedtoillustrateanumberofsymmetry
propertiesandpotentialapplications.Inaddition,thebehaviorofparallelformsofSA,
also exhibitscale invariancepropertiesandcan beused to illustratecertainanalogies
inthebehavioroflargeensemblesofinteractingparticlesorsystems,abasicaspectof
non-extensivestatisticalmechanicalsystems.
Thisarticleisorganizedasfollows:Section2providessomebackgroundintonon-
extensivethermodynamics,simulatedannealing,andSAsscaleinvarianceproperties.
Section 3 develops the scale invariance properties of the non-extensive form of SA.
These scale invariance properties indicate the existence of an energy transformation
function. Section5describescertainasymptoticpropertiesofthisenergytransforma-
tion function that suggest how it turns an exponential distribution into a power-law
distribution. Section 6 providessome discussion on the implicationsof the scale in-
variance properties and the energy landscape transformation describes extensions of
thisworkanddirectionsforfutureresearchinvolvingMarkovChainMonteCarlosim-
ulationofcomplexsystems.
2 Background
2.1 Non-Extensive Thermodynamics andGeneralized SA
TheGSAdevelopedbyTsallis[12],entailsadifferentformfortheacceptanceproba-
bilitiesandstationaryprobabilitiesbecauseoftheformof(1), i.e.,thestationarydis-
tributionisbasedonmaximizingthevalueofS ratherthanS . Thelatterleadsto
q BG
thewell-knownBoltzmann-Gibbsdistribution
e−fi/t
π (t)= (4)
i
Z
BG
where ZBG = Wj=1e−fj/t is the normalizing Boltzmann Partition Function. Note
thatbecausewewilloftenbereferringtotheSAalgorithmandtheMetropolisAlgo-
P
rithm,thevalueofthe“energy”functionwillbedenotedbyf whichistypicallyused
i
inoptimizationproblemstodenoteanobjectivefunctionvalue,butthiscanalsodenote
someenergyvalue.
3
Maximizingthenon-extensiveentropysubjecttocertainconstraints,describedbe-
low,givesrisetoastationaryprobabilitydistribution
1
1+ q−1 f 1−q
p (t)= t i (5)
i
Z
(cid:2) (cid:0) q(cid:1) (cid:3)
1
where Z = W 1+ q−1 f 1−q . Note that p (t) will be used throughout to
q j=1 t j i
representthestationaryprobabilityinthe non-extensivecaseandπ (t) willrepresent
i
P (cid:2) (cid:0) (cid:1) (cid:3)
thestationaryprobabilityintheclassic,extensivecase.
Thedefinitionofthestationaryprobabilityp (t)in(5)isbasedontheparticularset
i
ofconstraintsusedindefiningathermodynamicsystem.Threedistinctandnoteworthy
sets of constraints have been studied all of which employ the standard normalizing
constraint p (t) = 1. Whatdistinguishesthese constraintsets isthe relationships
i i
theydefinebetweenprobabilitiesandenergyorobjectivefunctionvalues. Tsallis[13]
P
reportsthatdifferentformsoftheseconstraintshaveanumberofimplicationsforthe
non-extensiveformoftheprobabilityp (t). Here,theformdefinedbyTsallis’Type2
i
constraint[14]forGSAisusedwhere,
W
pqf =U(2), aconstant, (6)
i i
i=1
X
whereasin his originalpaper, the constraint p˜f = U was used. This later con-
i i i
straintwasreferredtoastheType1constraintin[13,p. 537]andleadstoadifferent
P
stationaryprobabilitydenotedherebyp˜. Itbearsemphasisthatthelimitin(3)holds
i
forallthedifferentstationaryprobabilitiesthatarisefromtheuseofthedifferentcon-
straints [13]. Initially, the notions of scale invariant structures will be based on the
Type2 constraintand thenlater, in a moreusefulcontext,the scale invarianceof the
Type1constraintwillbeexamined.
2.2 ClassicalSA Scale Invariance
Among the many interesting aspects of the SA and Metropolis algorithms is a scale
invariancepropertyassociatedwiththestationaryprobabilitiesofvariousstates. This
scale invariance property is manifest in the identical mathematical forms of certain
quantitiesinvolvingindividualstates, aggregatedstates ofthesolutionspace, andthe
aggregationof states associated with multiple processorsin an expandedstate space.
See[3,4]foracompletedescription.
Oneaspectofthisscaleinvarianceinvolvestheratechangeofthestationaryprob-
abilityofastateiwithrespecttotemperaturet:
∂π (t) π (t)
i = i [f −hfi(t)] (7)
∂t t2 i
wherehfi(t)istheexpectedobjectivefunctionvalueattemperaturet. Foraggregated
statesA={i ,i ,...,i },define
1 2 m
π (t)f
π (t)≡ π (t), f (t)≡ i∈A i i, (8)
A i A
π (t)
i∈A P A
X
4
theconditionalexpectationoftheobjectivefunctionvaluegiventhatthecurrentstate
isinsetA(see[3,p.224]foramorecompletetreatment).Itthenfollowsthat
∂π (t) π (t)
A = A [f (t)−hfi(t)] (9)
∂t t2 A
wherethesimilarityof(7)and(9)indicatesaformofscaleinvariance.
Scaleinvariancealsoextendstosecondmoments.ForstandardSA,
∂hfi(t) ∂f (t) σ2(t)
= Ω = Ω (10)
∂t ∂t t2
where σ2(t) represents the variance over the entire solution space at temperature t.
Ω
Scaleinvarianceisexhibitedbythefactthat
∂f (t) σ2(t)
A = A (11)
∂t t2
andσ2(t)isthevarianceofobjectivefunctionvaluesattemperaturetconditionedon
A
thecurrentstatebeinginsetA. See[3,p.232-33]fordetails.
3 SA Scale Invariance with Non-Extensive Entropy
3.1 The BasisofScaleInvariance
Todemonstratescaleinvariancebasedonaggregatedstatesinthenon-extensivecase,
a basisformakingcomparisonsmustbe established. To thatend,the followingtem-
peraturederivativeofp (t)iscalculatedforthenon-extensiveSAcase. Fornotational
i
1
convenience,simplicityandtoensurepositivity,letN (t)≡ 1+ q−1 f 1−q (here-
i t i
inafterwewilldropthe(t)fromN (t)tofurthersimplifytheexpressions)andtaking
i
(cid:2) (cid:0) (cid:1) (cid:3)
thederivativeof(5),
∂p (t) Z ∂ N −N ∂ Z
i = q∂t i i∂t q. (12)
∂t (Z )2
q
Notethat
∂Ni = ∂ 1+ q−1 f 1−1q = fiNiq, (13)
∂t ∂t t i t2
(cid:20) (cid:18) (cid:19) (cid:21)
Z = N ,and p (t)=N /Z andhence
q j j i i q
P ∂ ∂ ∂Nj fjNjq
Z = N = = .
∂t q ∂t j ∂t t2
j j j
X X X
Substitutingthisand(13)into(12)yields
N fiNiq − Ni f Nq
∂pi(t) j j t2 t2 j j j
=
∂t (cid:16)P (cid:17)( jNj)2 P
P
5
= pi(t)fiNiq−1 − pi(t) jfjNjq
t2 t2 N
Pj j
= pi(t) f Nq−1− jPfjNjq (14)
t2 i i N
" P j j #
Tofurthersimplifyandclarify,define P
fˆ(f ,q,t)≡f Nq−1 = fi (15)
i i i i 1+ q−1 f
t i
andwilloftenbedenotedsimplybyfˆ orfˆ(t)where(cid:0)itis(cid:1)understoodtoinvolvef ,q
i i i
andt. Substituting(15)into(14)andfurthersimplifyingyields
∂p (t) p (t)
i = i fˆ(t)−hfˆi(t) (16)
∂t t2 i
h i
and which interestingly enough is analogousto (7). Notice that for q = 1, fˆ(t) =
i
fi, ∀t > 0. Tsallis[10,11]establishedthatlimq→1pi(t) = πi(t),hence,inthelimit
asq → 1,(16)→(7). Thesymbolfˆ(t)willbereferredtoastheq-transformationof
i
f . Lateron,someinterestingpropertiesofthistransformationwillbeexamined.
i
3.2 ScaleInvariancefrom theAggregationofStates
Insimilarfashionasin[3],definep (t)= p (t)forA⊂ Ω,whereΩistheset
A i∈A i
ofmicrostatesandwhereW =|Ω|and
P
p (t)fˆ(t)
fˆ (t)= i∈A i i , (17)
A
p (t)
P A
the conditional expectation of fˆ(t) conditioned on the current state being in set A.
i
Takingthederivative,
∂p (t) ∂ ∂p (t)
A = p (t)= i . (18)
i
∂t ∂t ∂t
i∈A i∈A
X X
Substituting(16)and(17)into(18)yields
∂p (t) p (t)
A = i fˆ(t)−hfˆi(t)
∂t t2 i
Xi∈A h i
p (t)fˆ(t) p (t)hfˆi(t)
i i i
= −
t2 t2
i∈A i∈A
X X
p (t) p (t)fˆ(t) p (t)hfˆi(t)
A i i A
= −
t2 p (t) t2
i∈A A
X
p (t)
= A fˆ (t)−hfˆi(t) (19)
t2 A
h i
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where for aggregatedstates, (19) has a similar mathematicalstructure as (16), hence
exhibits a scale invariance property the foundation of which is based on the energy
transformationfunctionfˆ(f ,q,t).
i i
3.2.1 ScaleInvarianceinSecondMoments
Scaleinvarianceinthe non-extensiveformofSA forsecondmomentsisindicatedin
the following where again the q-transformation of f and f (t) is used. First, the
i A
parallelstotheclassiccaseisillustrated. Thus,
∂hfˆi(t) ∂fˆ (t) ∂
= Ω = p (t)fˆ(t)
i i
∂t ∂t ∂t
"i∈Ω #
X
∂
= p (t)fˆ(t)
i i
∂t
Xi∈Ω h i
∂p (t) ∂fˆ(t)
= i fˆ(t)+p (t) i . (20)
i i
∂t ∂t
i∈Ω" #
X
Substituting(16)intothefirstpartof(20)andsimplifyingyields
∂hfˆi(t) p (t)fˆ2(t) fˆ (t)p (t)fˆ(t) ∂fˆ(t)
= i i − Ω i i + p (t) i . (21)
∂t t2 t2 i ∂t
i∈Ω i∈Ω i∈Ω
X X X
Notingtheformofthefirsttwotermsin(21)andthefactthatinthethirdterm
∂fˆ(t) 2 q−1
i = fˆ(t) (22)
∂t i t2
(cid:16) (cid:17) (cid:18) (cid:19)
and substituting into (21) and dropping the (t) for notational clarity and adding the
symbolΩtodenoteexpectationsovertheentirestatespaceyields
∂hfˆi hfˆ2i −hfˆi2 +(q−1)hfˆ2i
Ω = Ω Ω Ω
∂t t2
σˆ2 (q−1)hfˆ2i
= Ω + Ω (23)
t2 t2
whereσˆ2 representsthevarianceoftheq-transformedobjectivefunctionvaluesover
Ω
the entire objective function space (at temperature t). Obviously, this expression is
slightlydifferentfromtheclassicSAcaseasindicatedin(10). Notethatlimq→1σˆΩ2 =
σ2 sothattheexpressionsin(23)and(10)becomeequivalent.
Ω
The expression in (23) quite clearly shows the effects of the value of q—values
greater (less) than 1 increase (decrease) the rate of change of the expected objective
function values (q-transformedvalues). Eq. (23) also providesthe basis for another
form of scale invariance. Thus, after going through similar steps as in (20) through
(23)weget
∂fˆA = i∈Apifˆi2 − fˆA2 + q−1 i∈Apifˆi2. (24)
∂t t2p t2 t2 p
P A (cid:18) (cid:19)P A
7
Noting that the first and third terms indicate conditionalexpectationsconditionedon
thecurrentstatebeinginsetA,then(24)canbere-writtenintheconvenientnotation
∂fˆ hfˆ2i −hfˆi2 +(q−1)hfˆ2i
A = A A A
∂t t2
σˆ2 (q−1)hfˆ2i
= A + A (25)
t2 t2
where (25) is clearly analogous to (23) and so exhibits a form of scale invariance.
Again,inthelimitasq →1bothoftheseequationscorrespondtothescaleinvariance
ofstandard(Boltzmann-Gibbs)SA as indicatedin (10) and(11). What isinteresting
howeveristhefactthatthetermsinvolving(q−1)alsoscalewiththeaggregatedset
Ainthenon-extensivecase.
3.3 ScaleInvariancefrom theAggregationofProcessors
3.3.1 TheClassicSACase
Fleischer[5,6,7]describesanotherformofscaleinvariancebasedontheaggregation
parallel and independent processes each running the SA algorithm independently of
oneanother. Thistypeofaggregationcontrastssharplywiththeaggregationofstates
and hence directly affects the nature of the scale invariance and how the algorithm
itselffunctions. Inthisrespect,onemayviewtheSAalgorithmasasingleprocessor
orparticlethatprobabilisticallyvisitsasinglestateateachiterationofthealgorithm.
Thus,whenasetofstatesisaggregatedandfixed,theparticlevisitstheaggregatedstate
probabilisticallyandinawaythatdictateshowthestationaryprobabilityandobjective
function value should reasonably be defined, i.e., based on the sum of the stationary
probabilitiesandthe conditionalexpectationof the objectivefunctionrespectivelyas
givenin(8).
In aggregatingp independentprocessorshowever, the state space Ω becomes Ωp
andthestationaryprobabilityandobjectivefunctionvaluesassociatedwiththesetof
processors must be defined differently and requires the following definitions. First,
defineastateintheproductspacespannedbypprocessorsasi i ...i . InclassicSA
1 2 p
scaleinvariance
p
π (t)= π (t) (26)
i1i2...ip im
m=1
Y
andisbasedonthe conceptofthe jointprobabilityofindependentprocessors. Simi-
larly,theobjectivefunctionassociatedwith asetofp independentparticlesissimply
thesumofeachparticle’sobjectivefunctionvalueasopposedtotheconditionalexpec-
tation,anddefinedby
p
f ≡ f .
i1i2...ip im
m=1
X
Scaleinvarianceisindicatedby
e−fi1i2...ip/t
π (t)=
i1i2...ip ··· e−fj1j2...jp/t
jp j1
P P
8
(see[3,6])whichhasthesameformas(4)andthefactthatthetemperaturederivative
isproportionaltothedifferencebetweenafunctionoftheobjectivefunctionvalueand
itsexpectation:
∂π (t) π (t)
i1i2...ip = i1i2...ip f −hfi (t) (27)
∂t t2 i1i2...ip Ωp
(cid:2) (cid:3)
[3,p.222]wheretheexpectationisoverallthestatesoftheproductspace.
3.3.2 TheNon-ExtensiveSACase
Takingcuesfromthisearlierwork,definethestationaryprobabilityofstatei i ...i
1 2 p
spannedbypindependentprocessorsandbasedontheTsallisentropyby
p
p (t)≡ p (t).
i1i2...ip im
m=1
Y
Simplifyingusing the case of p = 2 (the following expressionsare readily extended
to the more general case) and letting a = (q −1)/t for notational clarity, the joint
probabilityofastatei i is
1 2
1 1
p (t) = [1+afi1]1−q[1+afi2]1−q
i1i2 Z2
q
1
= [1+a(fi1 +fi2 +afi1fi2)]1−q
Z2
q
= [1+af˜i1i2(t)]1−1q (28)
Z2
q
wheref˜ (t)≡f +f +af f andwhereitisalsothecasethat
i1i2 i1 i2 i1 i2
2
W
1
Zq2 = [1+afi]1−q
!
i=1
X
W W 1
= 1+af˜ (t) 1−q .
i1i2
iX1=1iX2=1h i
Thus, the form of p (t) in the productspace closely follows that of p (t) with the
i1i2 i
advantagethatthenon-extensivitypropertyisexemplifiedbyf˜ (t)whichissome-
i1i2
whatsimilartotheentropyrelationshipsin(2). Notethatforq =1, f˜ (t)=f =
i1i2 i1i2
f +f forallt.Thus,inthelimitasq →1,(28)isequivalenttotheBoltzmann-Gibbs
i1 i2
casein(26)(recall(3)).
Examiningthederivativeofthestationaryprobabilityusing(16)alsoreflectsascale
invariancepropertyandyields
∂p (t) ∂
i1i2 = [p (t)p (t)]
∂t ∂t i1 i2
9
p (t)
= p (t) i2 fˆ −hfˆi
i1 t2 i2 Ω
ph(t) i
+p (t) i1 fˆ −hfˆi
i2 t2 i1 Ω
p (t) h i
= i1i2 fˆ −hfˆi
t2 i1i2 Ω2
h i
andingeneralitfollowsthat
∂pi1∂··t·ip(t) = pi1·t··2ip(t) fˆi1···ip −hfˆiΩp (29)
h i
wherefˆi1···ip = pm=1fˆim andhfˆiΩp =phfˆiΩ andequalstheexpectationoffˆi1···ip.
Note that (29) again has a factor equal to the difference between a function of the
P
objectivefunctionvaluesanditsexpectationandhasthesamestructureasin(7),(9),
(16),(19)and(27).
Finally,letusexaminethetemperaturederivativeofhfˆi andseehowitcompares
Ωp
to(23)and(25). Thus,
∂hfˆi ∂
∂tΩp = ∂t pi1···ip(t)fˆi1···ip(t).
i1X···ip
Usingtheresultsfrom(20)through(22)andnotingthatfˆi1···ip(t)isasumofterms,
∂fˆi1···ip = q−1 p fˆ2 = q−1 fˆ2
∂t t2 im t2 i1···ip
(cid:18) (cid:19)m=1 (cid:18) (cid:19)
X
yields
∂hfˆi hfˆ2i −hfˆi2 +(q−1)hfˆ2i
Ωp = Ωp Ωp Ωp
∂t t2
σˆ2 (q−1)hfˆ2i
= Ωp + Ωp (30)
t2 t2
where σˆΩ2p represents the variance of the sums fˆi1···ip(t) (again, bear in mind that
the dependence on t is not indicated). Eq. (30) has the same form as (20) and (25)
indicatingaformofscaleinvariance.Whatisremarkablehoweveristhatinaggregating
processors,theobjectivefunctionisdefinedasthesumoftheobjectivefunctionvalues
ofeachprocessorwhereasinaggregatingstates,theobjectivefunctionisdefinedasthe
conditionalexpectationofobjectivefunctions,yettheyhavethesameformandinthe
limitasq →1areequivalenttotheaggregationofprocessorsinclassicSA.
3.4 Consistency
Fleischer [3] describes the concept of consistency in the scale invariant structure of
SAbasedonthedefinitionoftheaggregatedobjectivefunctionf . Thisconsistency
A
10