Fuzzy Covering and Partitioning Problems Based on the ... · by some probability distribution and an a priori certain information on the knowledge competitions is given by some weights

Fuzzy Covering and Partitioning Problems Based on the ExpertValuations: Application in Optimal Choice of Candidates

GIA SIRBILADZEI. Javakhishvili Tbilisi State University

Department of Exact and Natural Sciences2, University St., 0143 Tbilisi

[email protected]

BEZHAN GHVABERIDZEI. Javakhishvili Tbilisi State University


[email protected]

ANNA SIKHARULIDZEI. Javakhishvili Tbilisi State University


[email protected]

BIDZINA MATSABERIDZEI. Javakhishvili Tbilisi State University


[email protected]

DAVID DEVADZERustaveli Batumii State UniversityDepartment of computer science35, Ninoshvili St., 6010 Batumi

[email protected]

Abstract: A new approach in the discrete optimization (partitioning and covering) problems is presentedbased on the expert knowledge presentations. An a priori uncertain information on the alternatives is givenby some probability distribution and an a priori certain information on the knowledge competitions is givenby some weights. A new criterion is introduced for a minimal fuzzy covering or partitioning problem whichis a minimal value of average misbelief in possible alternatives. A bicriterial problem is obtained usingthe new criterion and the criterion of minimization of average price of the covering or partitioning. Theproposed approach is illustrated by an example for the partitioning problem.

Key–Words: Minimal fuzzy covering or partitioning, Minimal compatibility level, Positive and negativediscrimination measures, Average misbelief criterion, Bicriterial problem

1 Introduction

Optimization and decision-making problems are tra-ditionally handled by either the deterministic or theprobabilistic approach. The former provides an ap-proximate solution, completely ignoring uncertainty,while the latter assumes that any uncertainty can berepresented as a probability distribution. Of course,both approaches only partially capture reality uncer-tainty that indeed exists but not in the form of known

probability distributions.The existing literature clearly supports the notion

of using the fuzzy set theory and soft-computing tech-niques to further expand the human capability in mak-ing optimal decisions involving non-probabilistic un-certainty [3], [4], [9], [14]–[20], [23]–[37], [40]–[44].

In the Preface of the Journal of Fuzzy Optimiza-tion and Decision Making (vol. I, 2002, pp. 11–12)Professor L. A. Zadeh had said: “My 1970 paper withR.E. Bellman, ‘Decision-Making in a Fuzzy Environ-

WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS

Gia Sirbiladze, Bezhan Ghvaberidze, Anna Sikharulidze, Bidzina Matsaberidze, David Devadze

ISSN: 1790-0832 1840 Issue 11, Volume 6, November 2009

ment’ was intended to suggest a framework based onthe theory of fuzzy sets for dealing with imprecisionand partial truth in optimization and decision analy-sis. In the intervening years, a voluminous literatureon applications of fuzzy logic to decision analysis hascome into existence.”

In particular, when constructing decision-makingsystems [3], [9], [14]–[25], [29]–[36], [42]–[44]. theuse of fuzzy set theory is rather effective, since in-formation fuzziness is a typical property of any sys-tem of this kind. Frequently, the main material forthe construction of such systems is expert knowledgeand representations. The use of such systems contain-ing subjective, fuzzy uncertainty [11] leads to naturalgeneralizations of the above-mentioned problems inthe form of fuzzy optimization problems.

Fuzzy programming problems has been discussedwidely in literature [3], [9], [14], [19], [20], [24],[25], [28], [31], [32], [33], [35], [36], [38] and ap-plied in such various disciplines as operations re-search, economic management, business administra-tion, engineering and so on. Liu B. (Liu, [19] 2002)presents a brief review on fuzzy programming models,and classifies them into three broad classes: expectedvalue models, chance-constrained programming andchance-dependent programming.

Our further study belongs to the first class, wherewe used the instrument of fuzzy statistics and fuzzyset theory for our investigation.

In this paper we continue the investigation ofa discrete fuzzy optimization problem, presented ini.e. a minimal set covering and partitioning problemswith expert data. The obtained bicriterial optimizationproblem is a specific compromised approach betweenexpert and objective methods of optimization.

2 Preliminary Concepts

2.1 Classical set covering problem

Partitioning, covering and packing problems serve asa mathematical model for many theoretical and ap-plied problems such as the coloring of graphs, con-struction of perfect codes and minimal disjunctivenormal forms, drawing up of block-diagrams, infor-mation search, drawing up of traffic schedules, ad-ministrative division into zones and so on [2], [7], [8],[12].

Let us introduce some basic notions [2], [12].Suppose that we are given the finite set R ={r1, . . . , rm} and the family of its subsets S ={S1, . . . , Sn}. Let S′ = {Sj1 , . . . , Sjp}, 1 ≤ p ≤ n,be some subfamily of the family S. If each elementri is contained in at most (at least) one of the setsSj belong to S′, then S′ is called a packing (cover-ing) of the set R. A covering which is simultaneouslya packing is called a partitioning of the set R. LetA = ‖aij‖m×n be an incidence matrix of elements Rand subsets Sj : aij = 1 if ri ∈ Sj , and aij = 0 ifri 6∈ Sj . Each subfamily S′ of the family S is givenby means of the characteristic vector which has thecomponent xj = 1 if the subset Sj is contained in S′,and xj = 0 otherwise. If to each Sj ∈ S we assigna (positive) price cj , then partitioning, covering andpacking problems take the form

1) minAx=e

(c, x); 2) minAx≥e

(c, x); 3) maxAx≤e

(c, x);

here c = (c1, . . . , cn) is the price vector, x =(x1, . . . , xn) is the vector with components 0 and 1,and e is the vector consisting of 1’s. Note that in manyinteresting problems cj = 1, j = 1, . . . , n (such is, forinstance, the problem on finding a minimal dominat-ing set in the graph), but this does not simplify thesolution process of these problems.

2.2 On the most typical value (MTV) of acompatibility function with respect to aprobabilistic or fuzzy measure

If data are represented in intervals, their distributionis obscure, they overlap and are described or obtainedby an individual expert (insufficient expert data), thenthey are considered to be of combined nature. In thatcase, along with probabilistic-statistical uncertainty,there arises the so-called possibilistic uncertainty pro-duced by an individual (expert) and demanding theapplication of fuzzy analysis methods. In such situa-tions only probabilistic-possibilistic analysis can pro-vide satisfactory results by using the fuzzy methods tobe discussed below.

In describing such data functionally, in many realsituations the property of additivity remains unre-vealed for a measurable representation of a set and thiscreates an additional restriction. Hence, to study sub-jective insufficient expert data it is frequently better touse monotone estimators instead of additive ones.




We introduce the definition of a fuzzy measure(Sugeno, [39]) adapted to the case of a finite referen-tial.

Definition 1 LetX = {x1, x2, . . . , xn} be a finite setand g a set function

g : P(X)→ [0, 1],

where P is the power set of X . We will say g is afuzzy measure on X if it satisfies:

(i) g(∅) = 0; g(X) = 1.(ii) ∀A,B ⊆ X , if A ⊆ B, then g(A) ≤ g(B).

A fuzzy measure is a normalized and monotoneset function. it can be considered as as extension ofthe probability concept, where additivity is replacedby the weaker condition of monotonicity.

Let us, for example, consider three typical symp-toms x1, x2 ,x3, which indicate some illness y. Let anexpert (physician) provide objective-subjective datausing his/her wide experience and medical records ofpatients (another expert would certainly provide dif-ferent data).

Assume that we have the following information:80% of patients with illness y exhibit the symptomsx1 and x2, 20% of them have the symptoms x1 andx3. This information can be written using the mono-tone instead of the additive, measure g defined on thesubsets of the set X = {x1, x2, x3} (Table 1).

Table 1: Distribution table showing the dual measuresg and g∗

A ⊆ X g g∗

{x1} 0 1{x2} 0 0.8{x3} 0 0.2{x1, x2} 0.8 1{x1, x3} 0.2 1{x2, x3} 0 1{x1, x2, x3} 0 1

g∗ is called the dual measure of g defined byg∗(A) = 1 − g(A). Note that g∗ contains the sameinformation as g but is written in a different way.

Non-additive but monotone measures were firstused in fuzzy analysis in the 80s by M. Sugeno [39].

The fuzzy integral is a functional which assignssome number or a compatibility value to each fuzzysubset when the fuzzy measure is already fixed. Asknown [39], the concept of a fuzzy integral makesit possible to condense information provided by acompatibility function and a fuzzy measure. Hav-ing the fuzzy measure determined, we can estimatea fuzzy subset by the most typical compatibility value(MTV ). The MTV is essentially different in con-tent and significance from a probabilistic average evenwhen a probabilistic measure is used instead of afuzzy measure. The pre-image of the MTV with re-spect to a compatibility function distinguishes fromthe universe the most typical representative values ofthe considered fuzzy subset.

As already known, fuzzy averages differ both inform and content from probabilistic-statistical aver-ages and other numerical characteristics such as modeand median. Nevertheless, in some cases “nonfuzzy”(objective) and “fuzzy” (subjective) averages coincide[34], [39]. For a given set of fuzzy subsets with com-patibility function values from the interval [0; 1], thefuzzy average determines the most typical representa-tive compatibility value ME – Monotone Expectation.

The following fuzzy integral (based on the Cho-quet operator [6]) is the monotone expectation, whichwas defined by Bolanos et.al. [5]:

2.2.1 Fuzzy measure and a monotone expectation

Definition 2 ([5]) Let g be a fuzzy measure onX andh : X → R+

0 a non-negative function. The monotoneexpectation of h with respect to g is

Eg(h) =∫ +∞

0g(Hα) dα,

where Hα = {x ∈ X | h(x) ≥ α}.

The monotone expectation always exists and it isfinite for each g and h. It is obvious that Eg(·) is ageneralization of the mathematical expectation: thatis what it becomes when the used fuzzy measure is aprobability measure, that is,

EP (h) =∫Xh dP,

where P denotes a probability measure, EP – mathe-matical expectation.




Some of the most important properties of themonotone expectation are see in [24]-[31], [39], [44].

Since the monotone expectation is a generaliza-tion of the mathematical expectation, it can be ques-tioned whether the former possesses some weakerproperty in relation to additivity than the latter. Thefollowing proposition gives an expression of themonotone expectation that permits us to analyse thatquestion.

Proposition 1 ([5]) If the values of a non-negativefunction h are ordered as

h(x1) ≤ h(x2) ≤ · · · ≤ h(xn),

then the monotone expectation of h with respect to afuzzy measure g can be written as

Eg(h) =n∑i=1

h(xi)(g(Ai)− g(Ai+1)),

where Ai = {xi, xi+1, . . . , xn}, i = 1, . . . , n,g(An+1) = 0.

Thus, the monotone expectation is an additivefunctional for functions ordered equally.

We can also notice thatEg(h) is an average of theh function values weighted by

pi = g(Ai)− g(Ai+1), i = 1, . . . , n, pn = g(An).

Asn∑i=1

pi = g(A1) = g(X) = 1 and pi ≥ 0,

i = 1, . . . , n,

the values pi can be interpreted as the values of aprobability function. Then Eg(h) is equivalent to themathematical expectation of h with respect to thatprobability distribution.

The values pi depend on the fuzzy measure g andthe sets Ai, which depend on h only in the order de-termined by its values. So we can say:

Proposition 2 The monotone expectation of a non-negative function h with respect to a fuzzy measure gcoincides with the mathematical expectation of hwithrespect to a probability that depends only on g and theordering of the values of h.

2.2.2 Fussy measure and the fuzzy expected value(FEV)

In this section, we discuss the main estimators offuzzy statistics: the fuzzy expected value (FEV) ofthe population. The FEV determines MTV for a com-patibility function.

Let h be a compatibility function of some fuzzysubset of X , h : X → [0, 1] be an F -measurablefunction.

Definition 3 ([16]) The FEV of the compatibilityfunction h with respect to the fuzzy measure g isSugeno’s integral over X:

FEV (h) =6∫X

h ◦ g(·) ≡ supα∈[0,1]

{α ∧ g(Hα)} ,

where ∧ denotes a minimum of two arguments.

It clearly follows that the FEV somehow “aver-ages” the values of the compatibility function h not inthe sense of a statistical average but by cutting subsetsof the α level, whose values of a fuzzy measure g areeither sufficiently “high” or sufficiently “low”.

Thus the FEV gives a concrete value of the com-patibility function h, this value being the most typicalcharacteristic of all possible values with respect to thefuzzy measure g, obtained by cutting off the “upper”and “lower” strips on the graph of g(Hα).

Thus the information carried by h and g gets con-densed in the FEV which is the most typical value ofall compatibility values.

Consider the situation where X ={x1, x2, . . . , xn} is a finite set.

Proposition 3 ([39]) If the values of a compatibilityfunction h are ordered as

h(x1) ≤ h(x2) ≤ · · · ≤ h(xn),

then the FEV of h with respect to a fuzzy measure gcan be written as

FEV = maxi{h(xi) ∧ g(Ai)} = min

i{h(xi) ∨ g(Ai)} ,

where Ai = {xi, xi+1, . . . , xn}, i = 1, . . . , n, andwhere ∨ is a maximum of two arguments.




3 Aggregation by the monotone ex-pectation in the set Covering andPartitioning Problems

Our further consideration concerns minimal fuzzycovering and partitioning problems. Let S ={S1, S2, . . . , Sn} be some family of fuzzy subsets onR. Denote the compatibility level µ

Sj(ri) ≡ bij for

ri ∈ R, j = 1, 2, . . . , n. In constructing Sj , weuse certain subjective expert estimates: µ

Sj(ri) > 0

means that element ri will be covered by a fuzzy setSj with some positive level, even if this level is small.

Definition 4 Any subfamily S′ = {Sjk} ⊂ S, k =1, . . . , p, 1 ≤ p ≤ n, of fuzzy subsets is called a fuzzycovering (fuzzy partitioning) of the setR if for each rithere exists (there exists only one) fuzzy subset Sjk ⊂S′ such that µ

Sjk(ri) > α, where 0 ≤ α < 1 is called

the minimal compatibility level, by which an elementri is covered by a fuzzy set Sjk .

So we have defined the values of elements of anincidence matrix:

aik = 1 if µSjk

(ri) > α and aik = 0 otherwise.

It is clear that if α = 0, then we receive a classicalcase.

If to each Sj ∈ S we assign a (positive) price cj ,then the fuzzy covering (fuzzy partitioning)problem isformulated as follows: find a fuzzy covering (fuzzypartitioning) S′ of the set R having the least pricewith the least misbelief in subjective data. Thus un-der an optimal fuzzy covering (fuzzy partitioning) weunderstand a covering (partitioning)defined by meansof two criteria: 1) minimization of a covering (par-titioning) average price with probability uncertaintyproduced by a priori probability distribution on prices;2) minimization of average misbelief in fuzzy uncer-tainty produced by the assumption of fuzzy covering(fuzzy partitioning). We obtain a bicriterion discreteoptimization problem. Note that if under S we un-derstand the classical covering (partitioning), then thisproblem can be reduced to the well known covering(partitioning) problem [7].

Suppose we are given some fuzzy set on R+0 with

the definition: “a large ratio” := (L−R) with a nonde-creasing compatibility function µL−R : R+

0 → [0, 1].Analogously [23], we introduce the notation of

positive and negative discriminations:

pij =1

n− 1

n∑k=1k 6=j

µL−R

( bijbik

),

nij =1

n− 1

n∑k=1k 6=j

µL−R

(bikbij

),

{i = 1, . . . ,m,j = 1, . . . , n,

(1)

where a heuristic explanation of the positive (pij) andthe negative (nij) discrimination measure is that pijrepresents the accumulated belief that the element Sjis more indicative (in the sense of covering or parti-tioning) of an element ri than anyone of the remain-ing elements rl (l = 1, . . . ,m, l 6= i), while nijrepresents the belief that an element Sj is more in-dicative of not an element ri, but of other elementsrl (l = 1, . . . ,m, l 6= i) with respect to belief cov-ering (partitioning) levels bij of the fuzzy subsets Sj ,j = 1, . . . , n.

Suppose we are given some weights’ distribution(r1 r1 . . . rmw1 w2 . . . wm

),M∑i=1

wj = 1, on the set R,

which indicates an a priori information about the pref-erences on choice of elements ri.

Let two fuzzy sets be given on [0; 1]: one de-fined as “large” with some nondecreasing compatibil-ity functions µlarge : [0; 1] → [0; 1], and the otherdefined as “small” with some nonincreasing compati-bility functions µsmall : [0; 1] → [0; 1]. We introducethe following values

πj =m∑i=1

pijwi, vj =m∑i=1

nijwi, (2)

j = 1, . . . , n,

where πj and vj are called weighted average posi-tive and the negative discrimination measures of thecovering (partitioning), respectively, for elements Sj ,j = 1, . . . , n.




Now, on the set {S1, . . . , Sn} we construct a mis-belief distribution of a covering (partitioning), whereboth the positive and the negative discrimination mea-sure (πj , vj) are taken into account;

δj = νµsmall(πj) + (1− ν)µlarge(vj), (3)

j = 1, . . . , n,

where ν, 0 < ν < 1, is a weighted parameter whichindicates preference of positive or negative discrimi-nations.

The information content of δj is as follows: a cov-ering (partitioning) level of misbelief in “the accep-tance” of an element Sj .

Let S′ = {Sjk}, k = 1, 2, . . . , p; 1 ≤ p ≤ n,be some fuzzy covering (partitioning). It can be char-acterized by the binary vector x

S′ = (x1, . . . , xn),where

xi =

1, if the fuzzy subset Si is

contained in S′,0, otherwise.

Let us consider the misbelief distribution on xS′

S ′ ⇔(x1, . . . , xnδ1, . . . , δn

).

We say that the values xj are chosen with somea priori information and we can consider some proba-bility distribution on x

S′ :

PS′ =

(x1, . . . , xn

p1, . . . , pn

).

Thus for each fuzzy covering (partitioning) S ′we have constructed the fuzzy misbelief distribution(δ1, . . . , δn) on x

S′ and the probability distributionPS′ . Applying the method of fuzzy statistics which

was presented in Subsection 2.2.1 [5], [34], [37], [30],[39], we define a fuzzy average value of S ′ as a mono-tone expectation [6] [34], (which here coincides withmathematical expectation)

Ep(S ′)def≡∫ 1

0PS ′(µS ′ ≥ α)dα =

=n∑j=1

xj δjpj . (4)

Note that the value Ep(S ′) is an average measure ofmisbelief in a fuzzy covering (partitioning). Minimiz-ing the average misbelief in the fuzzy covering (parti-tioning) S ′, we obtain the criterion

n∑j=1

xj δj′ → min, (5)

where δ′ = (δ1p1, . . . , δnpn).Finally, the minimal fuzzy covering problem (par-

titioning) is reduced to a bicriterial problem of thetype (minsum-minsum) [12] for an ordinary covering(partitioning) with the target functions

f1 =n∑j=1

c′j xj → min

(minimization of an average price)

(c′j = cjpj),

f2 =n∑j=1

δj′ xj → min

(minimization of an average misbelief).

(6)

If X is the set of all Boolean vectors satisfyingthe conditions of the fuzzy covering problem, then byconsidering the scalar optimization problem

λf1 + (1− λ)f2 → min, (7)

(x1, . . . , xn) ∈ X, λ ∈ (0, 1),

where

X = {xS′ ∈ {0, 1}

n | S′ ⊂ S, S′ is the covering} ≡≡ {x ∈ {0, 1}n | Ax ≥ e}

and λ is a weighted parameter, we can find, in thegeneral case, some Pareto optima [12].

An aggregation by the FEV in Minimal FuzzyCovering Problem is the problem of our future inves-tigation.

4 Example: Application in the Opti-mal Choice of Candidates

As an application let us consider an example based onthe problem from [7] called the problem of a choice




of translators. Suppose some company needs tohire translators from Polish, French, German, Greek,Italian, Spanish, Russian, Chinese, Portuguese andJapanese into English and there are sixteen applicantsA, B, C, D, . . . , P. It is assumed that each candidateknows only some subset from the above-mentionedset of languages and demands the definite salary. Weintroduce, in the problem, a new element of estima-tion of their knowledge of languages and admit thatthey should be examined in the above-mentioned lan-guages. The results of examinations (which usuallyincludes expert valuations) are normalized and repre-sented in the form of numbers bij which determinethe level of knowledge of the j-th candidate with re-spect to the i-th language. Information on the candi-dates, estimates of language knowledge and salariesdemanded by the candidates are represented in theform of Table 2, where numbers aij are given in theupper part of each cell, and numbers bij in the lowerpart (aij = 1 if µ

Sj(ri) = bij > α = 0, 5 and aij = 0

otherwise; that means that the minimal compatibilitylevel or determined minimal level of language knowl-edge competitions is equal to 0,5).

Table 2: Data base of language knowledge and demanded salaries of the candidates A, B, C, D, . . . , P

To construct a target function which guaranteesminimal misbelief, we introduce the following con-crete compatibility functions (any other choice of ac-ceptable functions does not influence the final deci-sions [10]):

µL−R(x) : =x

x+ 2,

µlarge(x) : = x, 0 ≤ x ≤ 1,µsmall(x) : = 1− x, 0 ≤ x ≤ 1.

Positive and negative discrimination measures (1)and their average values are calculated. On the set{x1, . . . , xn} the uniform probability distribution wastaken in the role of a priori information measure, be-cause we did not have any initial information about thecandidates. The parameter ν is equal to 0,5. The uni-form weights wi = 1

n were taken in the role of a prioriinformation measure, because there was no preferencefor any language.




Finally, a fuzzy distribution of misbelief for thecandidates is written in the form

δ1 = 0, 484, δ2 = 0, 550, δ3 = 0, 452,

δ4 = 0, 483, δ5 = 0, 448, δ6 = 0, 483,

δ7 = 0, 549, δ8 = 0, 548, δ9 = 0, 548,

δ10 = 0, 548, δ11 = 0, 549, δ12 = 0, 483,

δ13 = 0, 483, δ14 = 0, 416, δ15 = 0, 482,

δ16 = 0, 484.

After passing to dimensionless values of targetfunctions we obtain

f1 = 0, 620x1 + 0, 200x2 + 0, 700x3+

+ 0, 800x4 + 0, 800x5 + 0, 900x6+

+ 0, 180x7 + 0, 320x8 + 0, 160x9+

+ 0, 220x10 + 0, 240x11 + 0, 700x12+

+ 0, 900x13 + 0, 920x14 + 0, 500x15+

+ 0, 680x16,

f2 = 0, 269x1 + 0, 306x2 + 0, 251x3+

+ 0, 268x4 + 0, 249x5 + 0, 269x6+

+ 0, 305x7 + 0, 305x8 + 0, 305x9+

+ 0, 305x10 + 0, 305x11 + 0, 269x12+

+ 0, 268x13 + 0, 231x14 + 0, 268x15+

+ 0, 269x16.

Note that for f1 an optimal solution is the can-didates B, G, I, J, K should be hired, and for f2 thecandidates M, N. For λ = 0, 5, by scalarized lin-ear convolution, criterion (7) (if both criteria are as-sumed equivalent) gives the solution M, N. To solvethe scalar problem on a minimal covering we use analgorithm of search tree type from [7].

During the research a software has been devel-oped. The software consists of two basic modules: thefirst module is responsible for reducing minimal fuzzycovering to classical covering problem and the sec-ond module is responsible for solving classical cover-ing problem using both exact (search tree type algo-rithm [7]) and approximate (greedy algorithm) meth-ods. The software is in the processes of developmentand is considered to be the basis for decision makingsupport systems software.

5 Conclusion

We apply the methods of fuzzy statistics [23], [34],[30] to the considered discrete optimization problemswith fuzzy data. In an appropriate manner we intro-duce the definitions of positive and negative discrim-inations of expert knowledge of optimization prob-lem parameters, i.e. parameters of possible solutionsand alternatives (candidates). We thereby determinea fuzzy distribution of misbelief on the set of alter-natives. As a result we obtain a bicriterial discreteoptimization problem which is solved by the methodof linear convolution of criteria. The scalar problemis solved by the search tree algorithm from [7]. Theobtained bicriterial optimization problem is a specificcompromised approach between the expert (fuzzy)and the objective (probabilistic) method of optimiza-tion of decision-making, where both the minimizationof average misbelief in alternatives and the minimiza-tion of an average price for alternatives are taken intoaccount. The constructed approach (minimal fuzzycovering or partitioning) to the solution of the discreteoptimization problem with data of combined (expert-objective) nature can be regarded as more trustworthyfrom the standpoint of application than the classicaloptimization methods.

6 Acknowledgement

The designated project has been fulfilled by financialsupport of the Georgian National Science Foundation(Grant #GNSF/ST08/1-361). Any idea in this publica-tion is possessed by the author and may not representthe opinion of the Georgian National Science Founda-tion itself.

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