Decomposition approaches to integration without a measure

0.16 MB·English

by  Salvatore Greco

Decomposition approaches to integration
SalvatoreGreco⋆12,RadkoMesiar⋆⋆34,FabioRindone⋆⋆⋆1,andLadislavSipeky†3
1 DepartmentofEconomicsandBusiness
95029Catania,Italy
5
1 2 PortsmouthBusinessSchool,Operations&SystemsManagementUniversityofPortsmouth,
0 PortsmouthPO1E,UnitedKingdom
2 3 DepartmentofMathematicsandDescriptiveGeometry,FacultyofCivilEngineering
SlovakUniversityofTechnology
n
Bratislava,Slovakia
a
J 4 InstituteofTheoryofInformationandAutomation
CzechAcademyofSciences
1
Prague,CzechRepublic

]
A
F
Abstract. FollowingtheideaofEvenandLehrer[3],wediscussageneralap-
.
h proachtointegrationbasedondecompositionoftheintegratedfunction.Wedis-
t
a tinguishsub-decompositionbasedintegrals(ineconomicslinkedwithoptimiza-
m tion problems to maximize the possible profit) and super-decomposition based
integrals (linked with costs minimization). We provide several examples (both
[
theoreticalandrealistic)tostressthatourapproachgeneralizesthatofEvenand

1 Lehrer [3] and also covers problems of linear programming and combinatorial
v optimization.Finally,weintroducesomenewtypesofintegralsrelatedtoopti-
6 mizationtasks.
2
3
0
1 Introduction
0
.
1
Theideaofdecompositionoftheintegratedfunctionf fortheintegrationpurposesisa
0
5 basicfeatureofconstructions/definitionsofintegralssinceever.Recall,e.g.,Eudoxus
1 ofCnidus(408-355BC)exhaustionprinciple,RiemannandLebesgueintegrals(lower
: andupperintegralsums), etc..Integrationalwaysmergestwo sourcesofinformation,
v
i the integrated function and weights of special functions used for decomposition pur-
X
poses(e.g.,measuresassigningweightsto sets, i.e.,to characterizefunctionsofsets),
r intoa single representativevalue.In thiscontribution,we willdealwithnon-negative
a
(measurable) functions and non-negative weights only, supposing always the mono-
tonicity of the considered weights, and vanishing of such weights for null functions.
Bothfromtransparencyofourideasaswellasfortheapplicationpurposesineconomics
and multicriteriadecisionsupport,we will always dealwith a fixed finite space N =
1,...,n ,wheren Nisafixedpositiveinteger.Thenthepowerset2N beingcon-
{ } ∈
sideredexcludesany measurabilityconstraints.Each functionf : N [0, [= R
+
→ ∞
⋆[email protected]
⋆⋆[email protected]
⋆⋆⋆[email protected]
†[email protected]
canbeseenasann-dimensionalvectorx Rn,x=(x ,...,x )=(f(1),...,f(n)).
∈ + 1 n
The aim of this contribution is a proposal of a general approach to decomposition
based integration, distinguishing sub-decompositions and super-decompositions. We
will stress several integrals known from the literature as particular instances of our
approach. Moreover, several new types of integrals related to optimization tasks will
beintroducedandexemplified.Thepaperisorganizedasfollows.InSection2wepro-
posetheideaofsub-decompositionbasedintegralsand,similarly,super-decomposition
approachto integrationis discussedinSection 3. We provideseveralexamplesofap-
plicationof decompositionintegrals,boththeoreticalaswell asrealistic. In Section4
weconfrontourapproachwithpreviousresearchinliterature,especiallywiththeidea
ofEvenandLehrer[3].ParticulardecompositionbasedintegralsarediscussedinSec-
tion5.Finally,someconcludingremarksandformalproposalforfutureresearchesare
addedinSection6.
2 Sub-decomposition based integrals
AnysystemofvectorsofRn,(xi)k =(x1,...,xk) (Rn)k withk N,iscalleda
+ i=1 ∈ + ∈
icsolalneyction,andthseucsehttohfataltlhceorelleecxtiisotnssxis=R0n==(∪0k,∈.N.(.R,0n+))wk.itAhdxecom(xpio)ksitiofnorsyssotmeme
D ⊆ Rn 6 ∈ i=1
collection(xi)k .
i=1 ∈D
Givenadecompositionsystem ,wedenote
D
˜ = x Rn x (xi)k forsomecollection(xi)n .
D ∈ +| ∈ i=1 i=1 ∈D
Conversely,f(cid:8)oranyX Rn,wedefine (cid:9)
⊆ +
= (xi)k xi X foralli=1,...,k
DX { i=1 ∈Rn| ∈ }
as the complete decomposition system generated by X, and clearly ˜ = X and,
X
moreover, istheunionofalldecompositionsystems suchthat ˜D=X.
X
D D D
Definition1 Let be a decompositionsystem. A mappingA : ˜ R is called a
+
D D →
weightingfunctionon whenever
D
– A(x) A(y)ifx,y ˜,x y(monotonicity),
– A(x)>≤0forsomex∈D˜ a≤ndA(0)=0whenever0 ˜ (boundaryconditions).
∈D ∈D
Observethatif ˜ =Rn,thenanyweightingfunctionAcanbeseenasanaggrega-
D +
tionfunction(inthesenseof[5],withrelatedboundarycondition,i.e.,sup A(x) x
{ | ∈
Rn =+ replacedbysup A(x) x Rn >0).
+} ∞ { | ∈ +}
ThefollowingexampleisinspiredtoEvenandLehrer[3,exampleinSection2].
Example1 Considertwodifferentworkagencies and .Eachagencyprovidesa
1 2
A A
coupleofworkerswithexactlythesameskills.However,eachofthefourworkerscan
work alone, or together with one or more partners. The possible teams are identified
with
= 0,1,2 2 (0,0) N2,
T { } \{ }⊆ 0
2
where (1,0), (0,1) representbasic teams formed by a single worker from agency
1
A
and respectively,while,e.g.,(2,1)istheteamformedbythetwoworkersfrom
2 1
A A
and one indifferently chosen from . Suppose we know the efficiency of each team,
2
A
measuredinsomeworkunit,givenbytheweightingfunctionE : R :
+
T →
E(1,0)=1.0
E(2,0)=2.2

E(0,1)=1.1
EEE(((012,,,211)))===223...025 (1)
E(1,2)=3.0
Clearly,wewanttomaximizeourefficEie(n2c,y2)by=ch4o.3osingthebestgroupofteamswithin
thedecompositionsystem(letusnotethat ˜ = )
T
D T
k
= xj k xj with xj (2,2) .
DT  j=1 | ∈T ≤ 
(cid:0) (cid:1) Xj=1 
Wewillreturnonthisexamplelater. 
Let be a decomposition system and let A : ˜ R be a weighting
n +
D ⊆ R D →
functionon .Fromnow,wecall(A, )abaseforintegrationonRn (shortly,abase).
D D +
Given a base (A, ), a vector x Rn is called (A, )-sub-integrable if (we use the
D ∈ + D
conventionsup =0)
∅
k k
sup A(yj) (yj)k , yj x <+ , (2)
{ | j=1 ∈D ≤ } ∞
j=1 j=1
X X
andwedefinethesetof(A, )-sub-integrablevectorsas
D
S = x Rn xis(A, )-sub-integrable .
(A,D) ∈ + | D
Let us note that S(A,D) = (cid:8), since the null vector 0 is (A, )-su(cid:9)b-integrablefor any
6 ∅ D
base.
Now,wecanintroduceoursub-decompositionbasedintegral.
Definition2 Let (A, ) bea baseforintegrationonRn, thenthe (A, )-basedsub-
D + D
decompositionintegralisthefunctionalI :S R definedby
(A,D) (A,D) +
→
k k
I (x)=sup A(yj) (yj)k , yj x . (3)
(A,D) { | j=1 ∈D ≤ }
j=1 j=1
X X
ThefollowingLemma1followsdirectlybydefinitionsofS andI .
(A,D) (A,D)
Lemma1. Forally S andx y,thenx S andI (x) I (y).
(A,D) (A,D) (A,D) (A,D)
∈ ≤ ∈ ≤
3
Remark1 Let beadecompositionsystemandletB :Rn R beanaggregation
D + → +
function which is super-additive [B(x+y) B(x)+B(y)], then for the weighting
≥
function A = B it holds S = Rn and I (x) B(x) for each x Rn.
|D˜ (A,D) + (A,D) ≤ ∈ +
Obviously,if(x) ,thenI (x)=B(x).
(A,D)
∈D
Regarding the domain S of sub-decompositionintegral I , this depends
(A,D) (A,D)
on bothA and . Supposethat(A, ) and (A′, ′) are two bases suchthat ′
D D D D ⊆ D
andA≤A′|D˜,thenS(A,D) ⊇S(A′,D′) andI(A,D) ≤I(A′,D′)onS(A′,D′).Thiswillbe
clearalsointhefollowingrelevantexamples.
Example2 Consider,e.g.,n = 2, = andA : R2 R givenbyA(x,y) = x.
D R2 + → +
Then S = Rn and I (x,y) = x. If we consider the weighting function
(A,R2) + (A,R2)
A′(x,y)=x+√y,thenS(A′,R2) ={(x,0)|x∈R+}andI(A′,R2)(x,0)=x,while
foranyx R andy >0,(x,y)isnon(A′, )-sub-integrable,indeed,beingA′sub-
+ 2
∈ R
additive, sup k A′(x ,y ) k (x ,y )=(x,y) lim nA′(x, y) =
{ i=1 i i | i=1 i i } ≥ n→+∞ n n
lim (x+√ny)=+ .
n→+∞
P ∞ P
Consider a set of objects (criteria) N = 1,...,n , and define a chain a sys-
{ }
tem (E )k such that E ... E N. Let be the set of all collections
j j=1 1 ⊆ ⊆ k ⊆ D
c 1 k , being c positive constants and (E )k a chain. Now consider the
j · Ej j=1 j j j=1
weightingfunctionsA: ˜ R ,determinedbyamonotonemeasurem:2N R
(cid:0) (cid:1) + +
D → →
(m( )=0,m(N)>0,andm(E ) m(E )wheneverE E N),bymeansof
1 2 1 2
A(c∅1 )=c m(E).InthiscaseS≤ =Rn andI i⊆stheC⊆hoquetintegral[2]
· E · (A,D) + (A,D)
withrespecttomeasurem.
OtherthatfortheChoquetintegral,inmajorityofintegralsknownsofar(Lebesque,
Choquet, Shilkret, Concave, Pan, etc. integrals), decomposition systems such that
any x ˜ can be written in the form c 1 , where c is a positive conDstant and E
E
a subse∈t oDf N (1 is the corresponding c·haracteristic function) are considered, and
E
the correspondingweightingfunctionsA : ˜ R arethendeterminedbyA(x) =
+
A(c 1 ) = c m(E), being m : 2N DR→monotone measures. Hence all these
E +
· · →
integralsarecoveredbyourapproach.FormoredetailsseeSections4and5.
Typicaleconomicalproblemsdealwithfinitenumberofgoodsg ,...,g ,andthen
1 n
weight (price) is assigned to groups of goods represented by multisets, i.e., vectors
x NnwhereN = 0,1,2,... .Notethatduetolimitationsinstoring/production, ˜
∈ 0 0 { } D
isthenmostlyfinite.Forthispurpose,thenextresultisimportantforrealapplications.
Proposition1 Foranybase(A, )suchthat ˜ isfinite,thenS =Rn.
D D (A,D) +
Proof. Let ˜ = x1,...,xm befinite,andwithoutlossofgenerality,wecanimagine
thatxj =0D,j ={1,...,m.N}ow,foranyy Rn thereexistn ,...,n Nsuchthat
6 ∈ + 1 m ∈
foreachj =1,...,mthevectorn xj exceedsyinsomecomponent.Thus,wehave
j
k k
sup A(yj) (yj)k , yj y n A(x1)+...+n A(xm)<+ .
{ | j=1 ∈D ≤ }≤ 1 k ∞
j=1 j=1
X X
(cid:3)
4
Remark2 If(A, ) isabasesuchthatS = Rn, thenI : Rn R isa
weightingfunctionDonRn.Indeed,monoton(iAc,iDty)ofI + andc(oAn,dDi)tionI+ → (0+)=0
+ (A,D) (A,D)
followbydefinitionand,moreover,sinceAisaweightingfunction,thereexistsx ˜
∈ D
suchthatA(x) > 0;supposethatxbelongstothe collection(yj)k , itfollows
j=1 ∈ D
thatI k yj k A(yj) A(x)>0.
(A,D) j=1 ≥ j=1 ≥
Letusnotet(cid:16)hPatI(A,D)(cid:17)restrPictedon ˜isnot,ingeneral,aweightingfunction,consider,
D
e.g., ((1,3,0),(3,1,0)) withA(1,3,0) = A(3,1,0) = 2,thenI (1,3,0) =
(A,D)
D{ }
I (3,1,0)= sup = 0.Nowsupposethereexistsx ˜ suchthatI (x) > 0
(A,D) (A,D)
∅ ∈ D
andthenI isaweightingfunctionon ˜andwecanconsiderI .However,
(A,D) D (I(A,D),D)
in this case, the two weighing function A and I are non comparable and also
(A,D)
consideringsub-decompositionintegrals,we havethatI isnoncomparablewith
(A,D)
I ,seeExample3.
(I(A,D),D)
A case where I and I are comparable is when the weighting function
(A,D) (I(A,D),D)
Aissuper-additive,sinceinthiscaseforanyxandanycollection(yj)k such
J=1 ∈ D
that k yj x, it follows that k A(yj) A( k yj) A(x), and then
j=1 ≤ j=1 ≤ j=1 ≤
I (x) A(x)and,consequently,I I .
(A,DP) ≤ P(I(A,D),D) ≤ (A,DP)
Finally,letusnotethatwhen = andS =Rn,thenI =I .
D Rn (A,D) + (A,D) (I(A,D),D)
Example3 Consider ((0,2,1),(2,0,0)),((2,2,1),(0,1,2)),((0,1,2)) andthe
D{ }
weightingfunctionA(0,2,1)=A(2,0,0)=A(0,1,2)=2,A(2,2,1)=3.
ItfollowsthatI (0,2,1) = I (2,0,0) = sup = 0,I (0,1,2) = 2and
(A,D) (A,D) (A,D)
∅
I (2,2,1)=4,I (2,3,3)=5.
(A,D) (A,D)
Finally,itiseasilycomputedthatI (2,2,1)=0andI (2,3,3)=6.
(I(A,D),D) (I(A,D),D)
When = , we are able to enunciate sufficient conditions for existence of
n
D R
I onallRn (fortheproofofTheorem1andsubsequentcorollaries,see[7]).
(A,Rn) +
Theorem1 S =Rn ifandonlyiftheconstantvector1=(1,...,1)is(A, )-
(A,Rn) + D
sub-integrable.
Corollary1 Let A : Rn R be a weighting function on such that for each
+ → + Rn
y Rn,A(x) c max y ,...,y ,wherecissomefixedconstantfrom]0, [.Then
∈ + ≤ · { 1 n} ∞
S =Rn.
(A,Rn) +
Due to Corollary 1, also the domination by a weighted sum W : Rn R ,
W(x) = n w x , with w = (w ,...,w ) Rn 0 , is sufficient+to→guara+n-
i=1 i i 1 n ∈ + \{ }
teethatS =Rn (i.e.,A(y) W(y)foreachy Rn isconsidered).
(PA,Rn) + ≤ ∈ +
Corollary2 Let A : Rn R be a weighting function on and let, for a fixed
+ → + Rn
ε > 0, A(y) y Rn, max y ,...,y k be bounded by a fixed constant c,
{ k | ∈ + { 1 n} ≤ }
independentlyofk ]0,ε].ThenS =Rn.
∈ (A,Rn) +
Thefollowingexampleshowsthat, in general(i.e.when ( ), Theorem1 is
n
D R
notvalid.
5
Example4 ConsiderinR2 thefollowingdecompositionsystem
+
ntimes
1 1 1 1
=(1,1), , ,..., , 
D  z(cid:18)n n(cid:19) }| (cid:18)n n(cid:19){
 
  n∈N
Now,independentlyfromtheweightingfunction,I(A,D)(1,1)= sup = 0.Onthe
∅
otherhand,ifwechooseA(x,y)=x+√y,wehave
n
1 1
I (2,2)=sup A(1,1)+ A , =sup 3+√n =+ .
(A,D) ( X1 (cid:18)n n(cid:19))n∈N (cid:8) (cid:9)n∈N ∞
Example5 LetusreconsiderExample1.Tochoosethebestgroupofteams,wehave
tocomputeefficiencyofvariouscompletegroups(i.e.wherewe useallfourworkers),
whichcanbeeasilydoneduetosmallquantityofdata.
2 E(1,0)+E(0,2)=2.0+2.0=4.0
·
2 E(1,0)+2 E(0,1)=2.0+2.2=4.2
 · ·
E(2,0)+E(0,2)=2.2+2.0=4.2
EEE(((212,,,220)))++=E24.·(31E,(00),=1)3=.02+.21+.02=.24=.04.4. (4)
E(1,1)+E(1,0)+E(0,1)=2.2+1.0+1.1=4.3
System (4)unE2d·(e2Erl,i(1n1)e,s+1)aEs=(th04e,.41b)e=st3so.5lu+tio1n.1c=orr4e.s6po=ndI(sEt,oDTI()E(2,D,T2)).(2,2). This ex-
ample can be generalized, by thinking that the two agencies and can provide
1 2
A A
any number of workers and then the possible teams are identified with elements of
= N2 (0,0) . Supposing that we know the efficiency of all possible teams, ex-
T 0 \{ }
pressed by the weighting function E : R and supposing that the first agency
+
T →
provides n workers and the second agency n , then the best group of teams corre-
1 2
spondtodecompositionsof(n ,n )allowingthecomputationofI (n ,n ).For
1 2 (E,DT) 1 2
n andn largeenoughweneedtheuseoflinearprogrammingtechniquestocompute
1 2
I (n ,n ), however I (n ,n ) is the theoretical solution to the problem,
(E,DT) 1 2 (E,DT) 1 2
inthesense thatthesub-decompositionintegraldefinitionprovidesthealgorithmicto
solvetheproblem.
LetusconsiderExamples1and5.Theoptimalsolutionwefound,I (2,2)=
(E,DT)
4.6, can be also obtainedby using the concaveintegral[9] and choosingan “ad hoc”
measure as described in the following. We identify the set of the four workers with
N = 1,2,3,4 where 1 and 2 are the two workers from the first agency and 2,3
{ }
those from the second. Consider the measure ν : 2N R given by µ( ) = 0,
+
→ ∅
µ(1) = µ(2) = E(1,0),µ(3) = µ(4) = E(0,1),µ(12) = E(2,0),µ(34) = E(0,2),
µ(13)= µ(14)= µ(23) = µ(24) = E(1,1),µ(123)= µ(124)= E(2,1),µ(134)=
6
µ(234)=E(1,2)andµ(1234)= E(2,2).Nowthebestsolutionfortheproblempro-
cav
posed in Example 1 is given by (1,1,1,1)dν = 1 ν(123)+ 1 ν(3) = 4.6.
· ·
Also the generalizationof the problem discussed at the end of Example 5 can be ob-
R cav
tainedusingtheconcaveintegral,inthesensethatI (n ,n ) = ydν where
(E,DT) 1 2
N = 1,2,...,(n1 + n2) , y = (1,1,...,1) Nn1+n2 and ν : 2N R+ is an
{ } ∈ R →
opportunecapacity.Howeverthis is possible onlybecause we havechosen an integer
componentsvector (n ,n ) and we have allowed only for decomposition of it in in-
1 2
teger components vectors. Suppose to have two numerical control machines M and
1
M andtheycanworkaloneortogether,thefirstmachinedependsonaparameterα
2 1
andthe secondona parameterα , with (α ,α ) (2√2,2). Thepossiblesetting of
2 1 2
≤
thesetwomachinesareidentifiedwith =]0,α ] ]0,α ],andweknowtheefficiency
1 2
T ×
of each combination of these machines given by E : R . finally the best set-
+
T →
ting forthe coupleof machinesis obtainedbysolvingI (α ,α ). Supposethat
(E,DT) 1 2
I (α ,α )=E(2√2,√2)+E(0,2 √2).Inthiscasenomeasurecanbespec-
(E,DT) 1 2 −
ifiedinordertosolvetheproblemusingtheconcaveintegral.
3 Super-decomposition basedintegrals
Weopenthissectionwitharealisticexample,providingmotivationstoourapproachto
super-decompositionintegral.
Example6 Consider a Fast Food (FF) which, basically, offers three goods (basic-
offers)
g =hamburger, g =chips, g =coke.
1 2 3
To increase the sales, the FF proposes also discounted compound-offers, e.g. to
buyconjointly1[hamburger+chips]islessexpansivethan1hamburgerand1chips
boughtseparately.LetussupposethattheFFsetofoffersis
= (1,0,0),(0,1,0),(0,0,1),(1,1,1),(2,0,0),(1,0,1),(0,1,1),(2,1,1) ,
S { }
where(1,0,0),(0,1,0)and(0,0,1)represent,respectively,thebasicoffershamburger,
chipsandcoke,while,e.g.,(1,1,0)representsthecompoundoffer[hamburger+chips].
Toattracttheconsumers,FFproposeapricefunctionP : R ,whichistypically
+
S →
strictlysub-additive,i.e.,
n
P(x,y,z)< P(x ,y ,z ),
i i i
i=1
X
for all (x,y,z), (x ,y ,z ) such that (x,y,z) = n (x ,y ,z ), n 2. For
i i i ∈ S i=1 i i i ≥
example,P(1,1,1) < P(1,0,1)+P(0,1,0) < P(1,0,0)+P(0,1,0)+P(0,0,1).
P
LetussupposethatFFpricesare
P(1,0,0)=2.80, P(0,1,0)=1.60, P(0,0,1)=1.80, P(1,1,1)=4.80,
P(2,0,0)=P(1,0,1)=P(0,1,1)=3, andP(2,1,1)=5.50.
7
Let us suppose also that a group of friends have to buy altogether 50 hamburgers,
30chipsand60cokes,and,obviously,theywantto payaslittle aspossiblebytaking
advantageofFFoffers.Thisisalinearprogrammingproblem,whichcanbeformalized
asfollows(x isintegerquantityof(1,0,0),P =P(1,0,0)andsoon)
a a
P (50,30,60)=min x P +x P +x P +x P +x P +x P +
G a a b b c c aa aa ac ac bc bc
{
+x P +x P
abc abc aabc aabc
 }
x +2x +x +x +2x =50
xxab++xxbca+a+xxabac+c+xxaaabb+cc =x 30aa=bc60
c ac bc abc aabc
x ,x ,...,x integer.
Butaconbsider, faoarbcexample, the necessity to buy 19 hamburgers, 10 chips and(150)
cokes.Since5.5 10<5.5 9+(2.8+1.6+1.8),weunderstandthattofindtheoptimal
· ·
solution,inequation(5)wemustreplaceequalityonconstrainswithinequality,i.e.,
P (50,30,60)=min x P +x P +x P +x P +x P +x P +
G a a b b c c aa aa ac ac bc bc
{
+x P +x P
abc abc aabc aabc
 }
x +2x +x +x +2x 50
xxab++xxbca+a+xxabac+c+xxaaabb+cc ≥x 30aabc6≥0
c ac bc abc aabc
≥
x ,x ,...,x integer.
Weawilbl returnaoabncthis example later, after introducing sub-decomposition bas(e6d)
integrals.
Sub-decompositionbased integralscan be considered as an optimizationproblem
tomaximizethepossibleprofit.Inadualwaymodelingtheminimizationofthecosts,
onecanintroducesuper-decompositionbasedintegrals.
However,thereisacrucialdifferenceconcerningthepossibleinputsx Rn tobe
∈ +
evaluatedbya super-decompositionbased integral.Indeed,for a fixeddecomposition
system , ˆ = k yj = (yj)k is the set of maximal elements of
D D { j=1 | B j=1 ∈ D}
the set of all elements x Rn covered by some collection from , i.e., a super-
decompositionbasPed integ∈ralc+an be defined only on the domBains ¯ D Rn given by
D ⊆ +
¯ = x Rn x k yj forsomecollection = [0,y].Obviously,if
D { ∈ +| ≤ j=1 B ∈ D} ∪y∈Dˆ
˜ =Rn thenalso ¯ =Rn.
GDivena+base(A, D)Pandx+ ¯,itresultsthat
D ∈D
k k
0 inf A(yj) x yj, (yj)k < .
≤ { | ≤ j=1 ∈D} ∞
j=1 j=1
X X
If there exists x ¯ such that inf k A(yj) x k yj, (yj)k > 0,
∈ D { j=1 | ≤ j=1 j=1 ∈D}
(A, ) iscalled a base forsup-integration(shortly,asup-base).For example,(A,R )
n
D P P
is not a feasible base for sup-integration when considering the product A = Π or
A=min[considerthedecompositionx=(x ,0,...,0)+...+(0,...,0,x )].
1 n
8
Definition3 Let (A, ) be a base for sup-integrationon Rn, then the (A, )-based
D + D
super-decompositionintegralisthefunctionalI(A,D) : ¯ R definedby
+
D→
k k
I(A,D)(x)=inf A(yj) x yj, (yj)k , (7)
{ | ≤ j=1 ∈D}
j=1 j=1
X X
Obviously,if = (anthen ¯ = Rn),thenI(A,D) : Rn R isanaggrega-
D Rn D + + → +
tionfunction.
Remark3 If an aggregation function B : Rn R is sub-additive [A(x +y)
+ → + ≤
A(x)+A(y)],andifconsideringtheweightingfunctionA = B ,thecouple(A, )
is a sup-base, then I(A,D)(x) B(x) for each x ¯. Obviou|sD˜ly, if (x) , thDen
≥ ∈ D ∈ D
I(A,D)(x)=B(x).
Example7 ContinuinginExample6,wecanassume
˜ = = (1,0,0),(0,1,0),(0,0,1),(1,1,1),(2,0,0),(1,0,1),(0,1,1),(2,1,1) ,
D S { }
and isthedecompositionsystem containingallcollectionsbuildingwithelements
S
D
from . It is clear that the solution of problem (6) (the minimal price that the group
S
shouldpaytosatisfytheirconstrains)isI(P,DS)(50,30,60).Usingalinearprogram-
mingsolveritresults
I(P,S)(50,30,60)=10 P +30 P +20 P =205.
aabc ac bc
· · ·
Example8 Let us consider the probabilistic sum (this is a weighting function and a
t-conorm)B : [0,1]2 R givenbyB(x,y) = x+y xy andthe decomposition
+
system = (xj)→k xj [0,1]nj = 1,...−,k .Then ˜ = [0,1]n,
D[0,1]n { j=1 ∈ Rn | ∈ } D[0,1]n
D¯[0,1]n =Rn+,andI(B,D[0,1]n) :R2+ →R+ isgivenby
if(x,y) [k,k+1]2
(k+1)(x+y k) xy ∈
I(B,D[0,1]n)(x,y)= − − forsomek ∈N,
max(x,y) otherwise.
ObservethatI(B,D[0,1]n)canbeseenasapseudo-additionon[0, ](whenextendedby
∞
monotonicityalsoforinfiniteinputs),[16],[8],I(B,D) =(<k,k+1,B > k N ),
0
| ∈
i.e.,itisassociative,commutativeaggregationfunctiononR2 withneutralelemente=
+
0.LetusnotethatI(B,D[0,1]n)(x,y)+I(A,D[0,1]n)(x,y) = x+y forallx,y ∈ [0,∞],
i.e.,ourintegralssolvesFrank’sfunctionalequation[4],[8]on[0, ].
∞
9
4 Relationwithsomeother integrals
LetN = 1,...,n beafinitesetandletm:2N R beamonotonemeasure.Even
+
{ } →
andLehrer[3]consideradecompositionset beinganon-emptysetoffinitesystems5
of subsets of N, that is = ,..., ,Hwith = Ei mi for all i = 1,...,k,
H {C1 Ck} Ci j j=1
beingEi N forallj =1,...,m .The -decompositionintegralisgivenby
j ⊆ i H (cid:0) (cid:1)
k k
I (x)=sup a m(E ) (E )k , a ,...,a 0, a 1 x . (8)
H,m { j j | j j=1 ∈H 1 k ≥ j Ej ≤ }
j=1 j=1
X X
ItisnotdifficulttocheckthatthenI =I ,wherethedecompositionsystem
H,m (Am,DH)
isdefinedby = (a 1 )k (E )k ,a ,...,a 0 ,andtheweight-
DH DH { j Ej j=1| j j=1 ∈H 1 k ≥ }
ingfunctionA : ˜ R isgivenbyA (c 1 )= c m(E).Thusourapproach
m H + m E
D → · ·
extendstheproposalofEvenandLehrer[3].Inparticular,itholds:
– if = (E)E N ,thenI istheShilkretintegral[15];
H { | ⊆ } (Am,DH)
– if = (E )k (E )k isachain ,thenI istheChoquetintegral[2];
H { j j=1 | j j=1 } (Am,DH)
– if = (E )k E ,...,E isapartitionofN ,thenI isthePAN
H { j j=1 |{ 1 k} } (Am,DH)
integral[17];ifmisadditive,thentheclassicalLebesqueintegralisrecovered;
– if = (E )k E N, j =1,...,k ,I istheconcaveintegral[9].
H { j j=1 | j ⊆ } (Am,DH)
Thecouple(A,ν) is defineda fuzzycapacity[9] if (1,...,1) A [0,1]n and
∈ ⊆
ν : A R is monotonic,continuous,and there is a positive K such that for every
+
a = (a→,...,a ) A,itholdsν(a) K n a .Theconcaveintegralofx Rn
1 n ∈ ≤ i=1 i ∈ +
withrespecttothefuzzycapacity(A,ν)[9]is
P
cav k k
xd(A,ν)=sup α ν(a ) a A,α 0,i=1,...,nand α a x .
i i i i i i
( | ∈ ≥ ≤ )
Z i=1 i=1
X X
IfweconsiderX = α a α 0anda A , = (xi)k xi X and
the weightingfunctio{nA· :|X ≥ R defi∈ned}byDAX(α {a) =i=α1 ∈ν(Ran)|then∈,itre}sults
+
I (x)= cavxd(A,ν)fo→rallx Rn. · ·
(A,DX) ∈ +
Forseveralotherintegralscoveredbyourapproachwerecommend[3][12].
R
Recently introduced superadditive integral [7] deals with a fixed decomposition
system = , and then the weighting function A defined on ˜ = Rn is just
D Rn D +
an aggregation function. The superadditive integral A∗ : Rn R is given by
+ → +
A∗(x)=sup( k A(yj) k yj x ).Obviously,A∗ =I .
{ j=1 | j=1 ≤ } (A,Rn)
Intheframeworkofsuper-decompositionbasedintegrals,werecallthat,foramonotone
P P
measurem:
– ifH={(Ej)kj=1 | (Ej)kj=1 isachain},thenI(Am,DH) istheChoquetintegral;
5Effectively,EvenandLehrer[3]speakaboutsetswhereaswespeakaboutsystems.Precisely,
theydefineacollectionDtobeasetofsubsetsofN,i.e.D⊆2N,andthentheyconsidersets
ofcollections.Howevertheirapproachcanbeequivalentlygivenusingsystemsandthisallow
ustodemonstratethatourapproachismoregeneral.
10