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Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2013 Article ID 158969 17 pageshttpdxdoiorg1011552013158969

Research ArticleA Proposal to Speed up the Computation of the Centroid ofan Interval Type-2 Fuzzy Set

Carlos E Celemin and Miguel A Melgarejo

Laboratorio de Automatica e Inteligencia Computacional Universidad Distrital Francisco Jose de Caldas Carrera 8 No 40-62Piso 7 Bogota Colombia

Correspondence should be addressed to Miguel A Melgarejo migbetgmailcom

Received 28 August 2012 Accepted 21 October 2012

Academic Editor M Onder Efe

Copyright copy 2013 C E Celemin and M A Melgarejo This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper presents two new algorithms that speed up the centroid computation of an interval type-2 fuzzy set The algorithmsinclude precomputation of the main operations and initialization based on the concept of uncertainty bounds Simulations overdifferent kinds of footprints of uncertainty reveal that the new algorithms achieve computation time reductions with respect tothe Enhanced-Karnik algorithm ranging from 40 to 70 The results suggest that the initialization used in the new algorithmseffectively reduces the number of iterations to compute the extreme points of the interval centroid while precomputation reducesthe computational cost of each iteration

1 Introduction

Type-2 fuzzy logic systems (T2-FLS) theory and its appli-cations have grown in recent years [1ndash3] One of the mainproblems related to the implementation of these systems istype reduction which computes the generalized centroid ofa type-2 fuzzy set (T2-FS) [4ndash6] This operation becomescomputationally simpler when performed over a particularclass of T2-FS namely interval type-2 fuzzy set (IT2-FS)[5 7] Basically the centroid of an IT2-FS is an interval[3 8] Therefore computing this centroid can be consideredas an optimization problem that finds the extreme points thatdefine the interval [6]

Fast computation of type reduction for IT2-FSs is anattractive problem which is critical since type-reductionprocedures for more general type-2 fuzzy sets (T2-FSs) makeuse of interval type-2 fuzzy computations [4 9ndash11] Up-to-date several iterative approaches to computing type reduc-tion for IT2-FSs have been proposed [5 12ndash18] The Karnik-Mendel (KM) algorithms are the most popular proceduresused in interval type-2 fuzzy logic systems (IT2-FLS) [17]It has been demonstrated that the KM algorithms convergemonotonically and superexponentially fast These propertiesare highly desirable in iterative algorithms [13] However

KM procedures converge in several iterations and demanda considerable amount of arithmetic and memory look-up operations [7 8 12 19] for real IT2-FLS implementa-tions

In the case of IT2 fuzzy controllers the computationalcost of type reduction is an important subject [20 21] Theoverall complexity of the controller largely depends on thetype-reduction and defuzzification stages Thus developingstrategies for reducing the computational burden and nec-essary resources for implementing these two stages is highlyconvenient In addition there are other applications of IT2-FLS in which the complexity of the hardware and softwareplatforms should be reduced in order to guarantee the fastestpossible execution of fuzzy inferences [22 23]

Noniterative type reduction of IT2-FS can be achieved bymeans of uncertainty bounds [7] however this method isapproximate and involves relatively complex computationsOther methods for computing the centroid of an IT2-FS havebeen proposed for example geometric type-reduction [4]genetically optimized type-reducers [8] interval-analysis-based type-reduction [24] and sampling defuzzification andcollapsing defuzzification [12] Although these alternativemethods exhibit interesting properties they are not as pop-ular as the KM algorithm

2 Advances in Fuzzy Systems

An enhanced version of the KM algorithm (EKM) waspresented in [16] This version includes a better initializa-tion and some computational improvements Experimentalevidence shows that the EKM algorithm saves about twoiterations which means a reduction in computation time ofmore than 39 with respect to the original algorithm Bythe time the EKM algorithm was presented and the iterativealgorithm with stop condition (IASC) was also introducedby Melgarejo et al [14 15 19] This algorithm deals withthe problem of computing type-reduction for an IT2-FSby using some properties of the centroid function [13 19]Experimental evidence showed that IASC is faster than theEKM algorithm in some cases

Recently IASC has been enhanced by Wu and Nie[18] leading to a new version called EIASC which provedto be the fastest algorithm with respect to the IASC andEKM algorithms for several type-2 fuzzy logic applicationsThe improvement introduced in EIASC mainly deals withmodifying the iterative procedure for computing the rightpoint of the interval centroid Although IASC and EIASCoutperform the EKM algorithm the initialization of theIASC-type algorithms is somewhat trivial and does notprovide an appropriate starting point which can limit theconvergence speed of these algorithms when calculating thepoints of the type-reduced set An extensive and interestingcomparison of several type-reductionmethods is provided byWu in [25] This study confirmed the convenience of EIASCover the EKMA for reducing the computational cost of anIT2-FLS The experiments that were limited to interpretedlanguage implementations also showed that EIASC and theenhanced opposite direction searching algorithm (EODS)exhibited similar performance over different applicationshowever the EODS outperformed EIASC in low-resolutioncases [26]

Regarding the aforementioned problem this work con-siders a different initialization perspective for iterative type-reduction algorithms based on the concept of inner-boundset for a type-reduced set [13] This kind of initializationhas not been tried in type-reduction computing until nowAs it will be analyzed later the inner-bound initializationreduces the distance that the algorithms need to cover inorder to find the optimal points of the interval centroidIn addition bearing in mind the computational burden ofiterative algorithms this paper also focuses on proposing analternative strategy to speed up their computation based onthe concept of precomputing Precomputation in algorithmicdesign is a powerful concept that reduces the necessaryarithmetic operations Current literature does not report theuse of precomputation in the type-reduction stage in IT2-FLSs

Using inner bound sets initialization and precomputa-tion Both the IASC and KM algorithms have been restatedleading to faster algorithms hereafter refered to IASC2 andKMA2 Our experiments considering two different imple-mentation perspectives (ie interpreted language and com-piled language) show that IASC2 and KMA2 outperformEKMA and EIASC In fact timing results suggest thatIASC2 may be the best option for implementing IT2-FLSsin dedicated hardware whereas EKMA may support the

computation of large-scale simulations of IT2-FLSs overgeneral purpose processors

Thepaper is organized as follows Section 2 provides somebackground about the centroid of an interval type-2 fuzzyset IASC2 and KMA2 are presented in Section 3 Section 4is devoted to a computational comparative study of itera-tive type-reduction methods which considers two types ofimplementation compiled-language-based and interpreted-language-based Finally we draw conclusions in Section 5

2 Background

The purpose of this section is to provide some basic infor-mation about the centroid of interval type-2 fuzzy sets Thereaderwho is not familiarwith this theory is invited to consult[3 5 13] among others

21 Type-2 Fuzzy Sets According to Mendel [5] a type-2fuzzy set denoted

119860 is characterized by a type-2membershipfunction 120583

119860

(119909 119906) where 119909 isin 119883 and 0 le 120583

119860

(119909 119906) le 1

119860 = ((119909 119906) 120583

119860

(119909 119906)) | forall119906 isin 119869

119909

sube [0 1] (1)

22 Interval Type-2 Fuzzy Sets An IT2-FS is a particular caseof a T2-FS whose values in 120583

119860

(119909 119906) are equal to one [27]An IT2-FS is fully described by its footprint of uncertainty(FOU) [3] Note that an IT2-FS set can be understood as acollection of type-1 fuzzy sets (ie traditional fuzzy sets)Themembership functions of these sets are contained within theFOU thus they are called embedded fuzzy sets [5 13]

23 Centroid of an IT2-FS Let

119860 be an IT2-FS and its cen-troid 119888(

119860) is an interval given by [5]

c (

A)= 1[119910

119897

(

119860) 119910

119903

(

119860)]

=

1

[119910

119897

119910

119903

]

(2)

where 119910

119871

and 119910

119877

are the solutions to the following optimiza-tion problems

119910

119871

=

minforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

119910

119877

=

maxforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(3)

where the universe of discourse 119883 as well as the upper andlower membership functions have been discretized in 119873

pointsA simpler way to understand the centroid of an IT2-FS

is to think about it as a set of centroids that are computedfrom all the embedded fuzzy sets [3 5] Clearly in orderto characterize an interval set it is only necessary to findits minimum and maximum The Karnik-Mendel iterativealgorithms [5 16 17] are the most popular procedures forcomputing these two points

Advances in Fuzzy Systems 3

24 Computing the Centroid of an IT2-FS It has been demon-strated that 119910

119871

and 119910

119877

can be computed in terms of theupper and lower membership functions of the footprint ofuncertainty (FOU) of

119860 [3 5] as follows

119910

119871

=

minforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

=

sum

119871

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871+1

119909

119894

119891

119894

sum

119871

119894=1

119891

119894

+ sum

119873

119894=119871+1

119891

119894

119910

119877

=

maxforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

=

sum

119877

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119877+1

119909

119894

119891

119894

sum

119877

119894=1

119891

119894

+ sum

119873

119894=119877+1

119891

119894

(4)

where 119891

119894

and 119891

119894

are the values of the lower and uppermembership functions respectively considering that theuniverse of discourse 119883 has been discretized into 119873 points119909

119894

In addition 119871 and 119877 are two switch points that satisfy thefollowing

119909

119871

le 119910

119871

le 119909

119871+1

119909

119877

le 119910

119877

le 119909

119877+1

(5)

25 Uncertainty Bounds of the Type-Reduced Set The type-reduced set of an interval type-2 fuzzy given in (2) can beapproximated by means of two interval sets called inner- andouter-bound sets [7] The end 119910

119871

and 119910

119877

points of the type-reduced set of an interval type-2 fuzzy set are bounded frombelow and above by

119910

119897

le 119910

119871

le 119910

119897

(6)

119910

119903

le 119910

119877

le 119910

119903

(7)

where [119910119897

119910

119903

] and [119910119897

119910

119903

] are the inner- and outer-bound setsrespectively For the purposes of this paper only the innerbound set is considered so it is computed as follows

119910

119860

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

119910

119861

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(8)

119910

119897

= min (119910

119860

119910

119861

)

119910

119903

= max (119910

119860

119910

119861

)

(9)

26 Properties of the Centroid Function The definition andcorresponding study of the continuous centroid functionare provided by Mendel and Liu in [13] Theoretical studiesderived from (6) can be legitimated for the discrete cen-troid function [12] Regarding this fact Mendel and Liudemonstrated several interesting properties of the continuouscentroid function which are applicable also to the discreteone Some of these properties can be summarized saying thatthe centroid function decreases or has flat spots to the left ofits minimum and it increases or has flat spots to the right of

EIASC EIASC

IASC

IASC

yL yl yR x

micro(x

)

yr

IASC2 IASC2

Figure 1 Uniform random FOU 119891

119894

(red line) and 119891

119894

(blue line)The arrows indicates the computation of minimum and maximumOrange arrows correspond to IASC2 green to EIASC and blue toIASC

this value Additionally this function has a global minimumwhen 119896 = 119871 The right-centroid function 119910

119903

has similarproperties since this function has a global maximum when119896 = 119877 Thus it increases to the left of the maximum anddecreases to the right

3 Modified Iterative Algorithms forComputing the Centroid of an IntervalType-2 Fuzzy Set

31 IASC2 The IASC-2 algorithm is presented in Table 1Thealgorithm consists of four stages sorting precomputationinitialization and recursion Sorting and precomputation arethe same for computing 119910

119871

and 119910

119877

Precomputation storesthe partial results of computing (8) only two divisions arerequired Those results will be used along the other stages ofthe algorithm

The initialization of IACS2 is characterized by its depen-dence from the inner-bound set This feature introducesa big difference with previous algorithms since they usefixed initialization points (eg EKMA IASC and EIASC)however regarding this particularity IASC2 is similar to theKMA because both algorithms include a FOU-dependentinitialization

The IASC2 works with the switch points starting insidethe centroid iterating similarly as IASC type algorithms butfrom the 119910min toward left side while the minimum 119910

119871

isreached it is in the opposite sense to IASC and EIASC Theprocedure for computing the maximum 119910

119877

begins in 119910maxand goes to the right side until it is found and the searchdirection is the same of IASC but opposite to the proposedin EIASC The advantage of this initialization for computingthe centroid is that always the switch points start close to the119910

119871

and 119910

119877

therefore less iterations are usedFigure 1 depicts a uniform random FOU with its

119910

119871

119910

119877

119910min or 119910

119897

and the 119910max or 119910

119903

and the arrows showthe direction of the search of each procedure from the initialswitch point to the minimum or maximum Also in thisexample it is possible to see the space scanned by every


Table 1 IASC2 algorithm flow

IASC2The minimum extreme point 119910

119871

of the centroid of an intervaltype-2 fuzzy set over 119909

119894

with upper membership function 119891

119894

andlower membership function 119891

119894

can be computed using thefollowing procedure

The maximum extreme point 119910

119877


119894


119894


119894

can be computed by the followingprocedure

(1) Sort 119909

119894

(119894 = 1 2 119873) in ascending order and call the sorted 119909

119894

by the same name but now 119909

1

le 119909

2

le sdot sdot sdot le 119909

119873

Match the weights 119891

119894

with their respective 119909

119894

and renumber them so that their index corresponds to the renumbered 119909

119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(15)

119867 [119899] =

119899

sum

119894=1

119891

119894

(16)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(17)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(18)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(19)

119910

119861

=

119885 [119873]

119867 [119873]

(20)

(3) Initialization Initialization119910min = min(119910

119860

119910

119861

) (21) 119910max = max(119910

119860

119910

119861

) (32)Find 119871

119894

(1 le 119871

119894

le 119873 minus 1) so that Find 119877

119894

(1 le 119877

119894

le 119873 minus 1) so that119909

119871119894le 119910min lt 119909

119871119894+1119909

119877119894le 119910max lt 119909

119877119894+1

119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

]) (22) 119864

119877

(119877

119894

) = 119885 [119877

119894

] + (119879 [119873] minus 119879 [119877

119894

]) (33)119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

]) (23) 119876

119877

(119877

119894

) = 119884 [119877

119894

] + (119883 [119873] minus 119883 [119877

119894

]) (34)

119910min =

119863

119871119894

119875

119871119894

(24) 119910max =

119864

119877

(119877

119894

)

119876

119877

(119877

119894

)

(35)

119863 = 119863

119897

(119871

119894

) (25) 119864 = 119864

119877119894(36)

119875 = 119875

119897

(119871

119894

) (26) 119876 = 119876

119877119894(37)

119896 = 119871

119894

(27) 119896 = 119877

119894

+ 1 (38)(4) Recursion Recursion

120572

119896

= 119891

119896

minus 119891

119896

(28) 120572

119896

= 119891

119896

minus 119891

119896

(39)119863 = 119863 minus 119909

119896

120572

119896

(29) 119864 = 119864 minus 119909

119896

120572

119896

(40)119875 = 119875 minus 120572

119896

(30) 119876 = 119876 minus 120572

119896

(41)

119910

119897

=

119863

119875

(31) 119910

119903

=

119864

119876

(42)

(5) If 119910

119897

le 119910min then 119910min = 119910

119897

119896 = 119896 minus 1 and go to step 4 else119910

119871

= 119910min and stopIf 119910

119877

ge 119910max then 119910max = 119888

119877

and 119896 = 119896 + 1 go to step 4 else 119910

119877

= 119910maxand stop

algorithm suggesting that this space is proportional to theamount of iterations required to converge Note that IASC2has the shortest arrows which mean few iterations

In Table 1 (21) 119910min corresponds to the minimum of theinner-bound set thus according to (6) 119910min ge 119910

119871

ThereforeTable 1 (21) provides an initialization point that is alwaysgreater than or equal to the minimum extreme point 119910

119871

Inaddition it is possible to find an integer 119871

119894

so that 119909

119871 119894le

119910min lt 119909

119871 119894+1 It is easy to show that Table 1 (22)ndash(24) is

computing the left centroid function with 119896 = 119871

119894

which fixesthe starting point of recursion from the values obtained in theprecomputation stage

Recursion Table 1 (28)ndash(31) computes the left centroidfunction with initial point in 119896 = 119871

119894

and each decrement

in 119896 leads to a decrement in 119910

119897

(119896) By doing some algebraicoperations it is easy to show that 119910

119897

(119896) = (sum

119896

119894=1

119909

119894

119891

119894

+

sum

119873

119894=119896+1

119909

119894

119891

119894

)(sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

) = (119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus

119891

119894

))(119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)) If sums are computed recursivelyit leads to 119910

119897

(119896) = (119863

119896minus1

minus 119909

119896

(119891

119896

minus 119891

119896

))(119875

119896minus1

minus (119891

119896

minus 119891

119896

))Since 119871 le 119871

119894

given that 119910

119897

le 119910min in Table 1 (21) iterationsshould decrease 119896 from 119871

119894

Thus a decrement of 119896 inducesa decrement in 119891

119896

from 119891

119896

to 119891

119896

In [5] the following wasdemonstrated

Condition 1 If 119909

119896

lt 119910(119891

1

sdot sdot sdot 119891

119873

) then 119910(119891

1


119873

) increaseswhen 119891

119896

decreases


Table 2 KMA2 algorithm flow

KMA2The KM algorithm for finding the minimum extreme point 119910

119871

ofthe centroid of an interval type-2 fuzzy set over 119909

119894

with uppermembership function 119891

119894

and lower membership function 119891

119894

canbe restated as follows

The KM algorithm for finding the maximum extreme point 119910

119877

ofthe centroid of an interval type-2 fuzzy set over xi with uppermembership function 119891

119894


119894

can bereexpressed as follows

(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(43)

119867 [119899] =

119899

sum

119894=1

119891

119894

(44)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(45)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(46)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(47)

119910

119861

=

119885 [119873]

119867 [119873]

(48)

(3) Set Set119910min = min(119910

119860

119910

119861

) (49) 119910max = max(119910

119860

119910

119861

) (56)Find 119896(1 le 119896 le 119873 minus 1) so that Find 119896(1 le 119896 le 119873 minus 1) so that

119909

119896

le 119910min lt 119909

119896+1

119909

119896

le 119910max lt 119909

119896+1

119863

119897

(119896) = 119879 [119896] + (119885 [119873] minus 119885 [119896]) (50) 119864

119877

(119896) = 119885 [119896] + (119879 [119873] minus 119879 [119896]) (57)119875

119897

(119896) = 119881 [119896] + (119884 [119873] minus 119884 [119896]) (51) 119876

119877

(119896) = 119884 [119896] + (119883 [119873] minus 119883 [119896]) (58)

119910 =

119863

119897

(119896)

119875

119897

(119896)

(52) 119910 =

119864

119877

(119896)

119876

119877

(119896)

(59)

(4) Find 119896

1015840

isin [1 119873 minus 1] such that Find 119896

1015840

isin [1 119873 minus 1] such that119909

119896

1015840 le 119910 lt 119909

119896

1015840+1

119909

119896

1015840 le 119910 lt 119909

119896

1015840+1

(5) If 119896

1015840

= 119896 stop and set 119910

119871

= 119910 else continue If 119896

1015840

= 119896 stop and set 119910

119877

= 119910 else continue(6) Set Set

119863

119897

(119896

1015840

) = 119879 [119896

1015840

] + (119885 [119873] minus 119885 [119896

1015840

]) (53) 119864

119877

(119896

1015840

) = 119885 [119896

1015840

] + (119879 [119873] minus 119879 [119896

1015840

]) (60)119875

119897

(119896

1015840

) = 119881 [119896

1015840

] + (119884 [119873] minus 119884 [119896

1015840

]) (54) 119876

119877

(119896

1015840

) = 119884 [119896

1015840

] + (119883 [119873] minus 119883 [119896

1015840

]) (61)

119910

1015840

=

119863

119897

(119896

1015840

)

119875

119897

(119896

1015840

)

(55) 119910

1015840

=

119864

119877

(119896

1015840

)

119876

119877

(119896

1015840

)

(62)

(7) Set 119910 = 119910

1015840 119896 = 119896

1015840 and go to step 4 Set 119910 = 119910

1015840 119896 = 119896

1015840 and go to step 4


119896

gt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

increases

Note that 119896 is greater than 119871 which implies that 119909

119896

gt 119910

119871

thus it is guaranteed that 119909

119896

gt 119910

119897

(119896) satisfying Condition1 Therefore it can be concluded that a decrement in 119891

119896

willinduce a decrement in 119910

119897

(119896)Finally the stop condition is stated from the properties of

the centroid function [13]They indicate that the left-centroidfunction has a global minimum in 119896 = 119871 so it is found whenthe functions start to increase

The procedure to compute 119910

119871

is only analyzed inthis section The full demonstration of this procedure isprovided in the appendix for the readers who are inter-ested in following it The computation of 119910

119877

obeys sim-ilar conceptual principles and it can be understood by

using the ideas suggested above or those provided in theappendix

32 KMA2 The KM2 algorithm is presented in Table 2 Thisalgorithm uses the same conceptual principles of KMA forfinding 119910

119871

and 119910

119877

[5] The algorithm includes precomputa-tion initialization and searching loop Precomputation andinitialization of KMA2 are the same for IASC2 having alwaysthe initial switch points near to the extreme points 119910

119871

and 119910

119877

The searching loop corresponds to steps 4ndash7 The core of theloop is Table 2 (53)ndash(55) and (A2)ndash(A4) which can be easilydemonstrated by making 119871

119894

equal to 119896 or 119896

1015840 These equationscalculate119910

119897

(119896) or119910

119903

(119896)with few sums and subtractions takingadvantage of pre-computed values Finally the stop conditionis the same for the EKMAalgorithm whichwas stated in [16]


x

1micro(x)

(a)

x

1micro(x)

(b)

x

1micro(x)

(c)

x

1micro(x)

(d)

Figure 2 FOUs cases used in the experiments (a) Uniform random FOU (b) Centered uniform random FOU (c) Right-sided uniformrandom FOU (d) Left-sided uniform random FOU

4 Computational Comparison with ExistingIterative Algorithms Implementation Basedon Compiled Language

41 Simulation Setup The purpose of these experimentsis to measure the average computing time that a type-reduction algorithm takes to compute the interval centroid ofa particular kind of FOUwhen the algorithm is implementedas a program in a compiled language like C In additioncomplementary variables like standard deviation of the com-puting time and the number of iterations required by eachalgorithm to converge are also registered The algorithmsconsidered in these experiments are IASC [14] EIASC [18]and EKMA [16] which are confronted against IASC2 andKMA2 For the sake of comparison the EKM algorithm isregarded as a reference for all the experiments

Four FOU cases were selected to evaluate the perfor-mance of the algorithms All cases corresponded to randomFOUs modified by different envelopes The first one consid-ered a constant envelope that simulated the output of a fuzzyinference engine in which most of the rules are fired at asimilar level In the second one a centered envelope was usedto simulate that rules with consequents in the middle regionof the universe of discourse are fired The last two cases wereproposed to simulate firing rules whose consequents are in

the outer regions of the universe of discourse For the sake ofclarity several examples of the FOUs considered in this workare depicted in Figure 2

The following experimental procedure was applied tocharacterize the performance of the algorithms over eachFOU case described above we increased119873 from 0 to 100withstep size of 10 and then119873was increased from 100 to 1000withstep size of 100 For each 119873 2000 Monte Carlo simulationswere used to compute 119910

119871

and 119910

119877

In order to overcome theproblem of measuring very small times 100000 simulationswere computed in each Monte Carlo trial thus the time of atype-reduction computation corresponded to the measuredtime divided by 100000

The algorithms were implemented as C programs andcompiled as SCILAB functionsThe experimentswere carriedout over SCILAB 531 numeric software running over aplatform equipped with an AMD Athlon(tm) 64 times 2 DualCore Processor 4000 + 210GHz 2GB RAM and WindowsXP operating system

411 Results Computation Time The results of computationtime for uniform random FOUs are presented in Figure 3IASC2 exhibited the best performance of all algorithms for119873 between 10 and 100 For 119873 greater than 100 KMA2 wasthe fastest algorithm followed by IASC2 The percentage


10 100 1000

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)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

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12

14

16

Rat

io

10 100 1000

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(b)

STD

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)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

0E+00

1Eminus07

2Eminus07

3Eminus07

4Eminus07

5Eminus07

6Eminus07

(c)

Figure 3 Uniform random FOU results (compiled language implementation of the algorithms) (a) Average computation time (b) ratio withrespect to the EKMA and (c) standard deviation of the computation time

of computation time reduction (PCTR) with respect to theEKMA (ie PCTR = (119905

119900

minus 119905ekm)119905ekm where 119905

119900

and 119905

119890

arethe average computation times of a particular algorithm andEKMA resp) of IASC2 was about 30 for 119873 = 10 whilein the case of KMA2 the largest PCTR was about 40 for119873 = 1000 Note that IASC and EIASC exhibited poorerperformance with respect to EKMA when 119873 grows beyond100 points however EIASC was the fastest algorithm for 119873 =

10 pointsThe standard deviation of the computation time forIASC2 andKMA2was always smaller than that of the EKMA

Figure 4 presents the results for centered random FOUsIASC2 exhibited the largest PCTR for 119873 smaller than 100points about 40 When 119873was greater than 100 KMA2 wasthe fastest algorithm This algorithm provided a maximumPCTR of about 50 for 119873 = 1000 The standard deviationof the computation time for all algorithms was often smallerthan that of EKMA

In the case of left-sided random FOUs (Figure 5) IASCwas the fastest algorithm for 119873 smaller than 100 points Thisalgorithm provided a PCTR that was between 70 and 50

Here again KMA2 was the fastest algorithm for 119873 greaterthan 100 points with a PCTR that was about 45 for thisregion In this case EIASC exhibited the worst performanceand in general the standard deviation of the computationtime was similar for all the algorithms

The results concerning right-sided randomFOUs are pre-sented in Figure 6 IASC2was the algorithm that provided thelargest PCTR for 119873 smaller than 100 points This percentagewas between 55 and 60 KMA2 accounted for the largestPCTR for 119873 greater than 100 points The reduction achievedby this algorithm was between 50 and 60 IASC exhibitedthe worst performance in this case The standard deviationof the computation time was similar in all algorithms for 119873

smaller than 100 however a big difference with respect toEKMA was observed for 119873 greater than 100 points

412 Results Iterations The amount of iterations requiredby each algorithm to find 119910

119877

was recorded in all theexperiments In Figures 7ndash10 results for the mean of the


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)

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2Eminus06

4Eminus06

6Eminus06

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1Eminus05

12Eminus05

14Eminus05

(a)

0

02

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08

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12

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10 100 1000

N

(b)

10 100 1000

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EKMIASCEIASC

IASC2KMA2

STD

of

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tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)

Figure 4 Centered random FOU results (compiled language implementation of the algorithms) (a) Average computation time (b) ratiowith respect to the EKMA and (c) standard deviation of the computation time

number of iterations regarding each FOU case are presentedA comparison of the IASC-type algorithms and also of theKM-type algorithms are provided for discussion

In the case of a uniform random FOU (Figure 7)IASC2 reduced considerably the number of iterations whencompared to the IASC and EIASC algorithmsThe percentagereductionwas about 75 for119873 = 1000with respect to EIASCOn the other hand no improvement was observed for KMA2over the EKMA in this case however the computation timewas smaller for KMA2 as presented in previous subsection

In the case of a centered random FOU the IASC2provided a significant reduction in the number of iterationscompared to the IASC and EIASC algorithms as shownin Figure 8 The percentage reduction was about 88 for119873 = 1000 with respect to EIASC Regarding the KM-typealgorithms the KMA2 reduced the number of iterations withrespect to the EKMA for 119873 smaller than 50 points Howeverthe performance of KMA2 is quite similar to that of EKMAfor 119873 greater than 100 points

Regarding the results from the left-sided random FOUcase (Figure 9) an interesting reduction in the number ofiterations was achieved by IASC2 compared to the IASC andEIASC algorithms The reduction percentage is around 90for 119873 = 1000 with respect to EIASC KMA2 displayed areduction percentage ranging from 30 for 119873 = 1000 to 83for 119873 = 10 with respect to EKMA

The IASC2 algorithm outperformed the other two IASC-type algorithms for a right-sided random FOU (Figure 10)as observed in the previous cases Considering the KM-type algorithms KMA2 provided the largest reduction in thenumber of iterations with respect to EKMA for 119873 = 10

points The reduction percentage was about 80 in this caseKMA2 also outperformed EKMA for greater resolutions

42 Discussion We believe that the four experimentsreported in this section provide enough data to claim thatIASC2 andKMA2 are faster than existing iterative algorithmslike EKMA IASC and EIASC as long as the algorithms are


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0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)

Figure 5 Left-sided random FOU results (compiled language implementation of the algorithms) (a) Average computation time (b) ratiowith respect to the EKMA and (c) standard deviation of the computation time

coded as compiled language programs The experiments arenot only limited to one type of FOU instead typical FOUcases that are expected to be obtained at the aggregated outputof an IT2-FLS were used IASC2 may be regarded as thebest solution for computing type-reduction in an IT2-FLSwhen the discretization of the output universe of discourseis smaller than 100 points On the other hand KMA2 mightbe the fastest solution when discretization is bigger than100 points Thus we believe that IASC2 should be used inreal-time applications while KMA2 should be reserved forintensive applications

Initialization based on the uncertainty bounds proved tobe more effective than previous initialization schemes Thisis evident from the reduction achieved in the number ofiterations when finding 119910

119877

(similar results were obtained inthe case of 119910

119871

) We argue that the inner-bound set is aninitial point that adapts itself to the shape of the FOU Thisis clearly different from the fixed initial points that pertainto EIASC and EKMA where the uncertainty involved in the

IT2-FS is not considered In addition a simple analysis ofthe uncertainty bounds in [7] shows that the inner-bound setlargely contributes to the computation of the outer-bound setwhich is used to estimate the interval centroid

5 Computational Comparison with ExistingIterative Algorithms Implementation Basedon Interpreted Language

51 Simulation Setup The same four FOU cases of theprevious section were used to characterize the algorithmsimplemented as interpreted programs Since an interpretedimplementation of an algorithm is slower than a compiledimplementation the experimental setup was modified Inthis case 2000 Monte Carlo trials were performed for eachFOU case however only 100 simulations were computed foreach trial so the time for executing a type-reduction is themeasured time divided by 100


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)

N

0E+00

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6Eminus06

8Eminus06

1Eminus05

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14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

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e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)

Figure 6 Right-sided random FOU results (compiled language implementation of the algorithms) (a) Average computation time (b) ratiowith respect to the EKMA and (c) standard deviation of the computation time

The algorithms were implemented as SCILAB programsThe experiments were carried out over SCILAB 531 numericsoftware running over a platform equipped with an AMDAthlon(tm) 64 times 2 Dual Core Processor 4000 + 210GHz2GB RAM and Windows XP operating system

52 Results Execution Time The results of computation timefor uniform random FOUs are presented in Figure 11 KMA2achieved the best performance among all algorithms with aPCTR of around 60 and the IASC2 PCTR was close to50 The standard deviation of the computation time for theIASC2 and KMA2 was always smaller than that of EKMAThus considering an implementation using an interpretedlanguage KMA2 should be the most appropriated solutionfor type-reduction in an IT2-FLS

Figure 12 is similar to Figure 13 These figures presentthe results for centered random FOUs and left-sided randomFOUs respectively KMA2was the fastest algorithm followedby IASC2 EIASC IASC and EKMA as the slowest The

IASC2 PCTR was almost similar to the KMA2 PCTR for119873 greater than 20 which was about 60 When 119873 = 10the IASC2 PCTR was about 46 The standard deviation ofcomputation timewas similar for all algorithms for119873 smallerthan 300 although it was significantly different with respect tothe EKMA for 119873 greater than 300 points

The results with right-sided random FOUs in Figure 14show that KMA2 was the fastest algorithm in all the casesfollowed by IASC2 with a similar performanceThe PCTR for119873 greater than 30 was about 76 with KMA2 while withIASC2 it was about 74 over the EKMA computation timeFor the other cases the reduction was around 70 and 64respectively The standard deviation of the computation timefor all algorithms was often smaller than that of EKMA

6 Conclusions

Two new algorithms for computing the centroid of an IT2-FS have been presented namely IASC2 and KMA2 IASC2follows the conceptual line of algorithms like IASC [14]


0

100

200

300

400

500

600

700

Mea

n o

f th

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10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

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of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)

Figure 7 Uniform random FOU iterations (a) IASC-type algo-rithms and (b) KM-type algorithms

and EIASC [18] KMA2 is inspired in the KM and EKMalgorithms [16] The new algorithms include an initializationthat is based on the concept of inner-bound set [7] whichreduces the number of iterations to find the centroid of anIT2-FS Moreover precomputation is considered in order toreduce the computational burden of each iteration

These features have led to motivating results over dif-ferent kinds of FOUs The new algorithms achieved inter-esting computation time reductions with respect to theEKM algorithm In the case of a compiled-language-basedimplementation IASC2 exhibited the best reduction forresolutions smaller than 100 points between 40 and 70On the other hand KMA2 exhibited the best performance forresolutions greater than 100 points resulting in a reduction

0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)

Figure 8 Centered random FOU iterations (a) IASC-type algo-rithms and (b) KM-type algorithms

ranging from 45 to 60 In the case of an interpreted-language-based implementation KMA2 was the fastest algo-rithm exhibiting a computation time reduction of around60

The implementation of IT2-FLSs can take advantage ofthe algorithms presented in this work Since IASC2 reportedthe best results for low resolutions we consider that thisalgorithm should be applied in real-time applications of IT2-FLSs Conversely KMA2 can be considered for intensiveapplications that demand high resolutions The followingstep in this paper will be focused to compare the newalgorithms IASC2 and KMA2 with the EODS algorithmsince it demonstrated to outperformEIASC in low-resolutionapplications [25]


0

100

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800

Mea

n o

f th

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10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)

Figure 9 Left-sided random FOU iterations (a) IASC-type algo-rithms and (b) KM-type algorithms

Appendix

The proof of the algorithm IASC-2 for computing 119910

119871

isdivided in four parts

(a) It will be stated that Table 1 (21) fixes a valid initializa-tion point

(b) It will be demonstrated that Table 1 (22)ndash(24) com-putes the left centroid function with 119896 = 119871

119894

(c) It will be demonstrated that recursion of step 4

computes the left centroid function with initial pointin 119896 = 119871

119894

and each decrement in 119896 leads to adecrement in 119910

119897

(119896)(d) The validity of the stop condition will be stated

0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

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of it

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IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

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s

EKMKMA2

10 100 1000

N

(b)

Figure 10 Right-sided random FOU iterations (a) IASC-typealgorithms and (b) KM-type algorithms

Proof (a) From Table 1 (15)ndash(21) it follows that

119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

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4

6

8

10

12

14

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)

10 100 1000

N

(a)

0

02

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08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

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puta

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e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)

Figure 11 Uniform random FOU results (interpreted languageimplementation of the algorithms) (a) Average computation time(b) ratio with respect to the EKMA and (c) standard deviation ofcomputation time

0

2

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)

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N

(a)

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N

(b)

0

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14

STD

of

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)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)

Figure 12 Centered random FOU results (interpreted languageimplementation of the algorithms) (a) Average computation time(b) ratio with respect to the EKMA and (c) standard deviation ofcomputation time


0

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)

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N

(a)

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N

(b)

0

005

01

015

02

025

03

035

STD

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)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)

Figure 13 Left-sided random FOU results (interpreted languageimplementation of the algorithms) (a) Average computation time(b) ratio with respect to the EKMA and (c) standard deviation ofcomputation time

0

5

10

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20

25

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)

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(a)

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)

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EKMIASCEIASC

IASC2KMA2

(c)

Figure 14 Right-sided random FOU results (interpreted languageimplementation of the algorithms) (a) Average computation time(b) ratio with respect to the EKMA and (c) standard deviation ofcomputation time


It is clear that 119910min in Table 1 (21) corresponds to theminimum of the inner-bound set [7] so according to (6)119910min ge 119910

119871

Thus Table 1 (21) provides an initialization pointthat is always greater than or equal to the minimum extremepoint 119910

119871

In addition It is possible to find an integer 119871

119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1

(b) From Table 1 (22) it follows that

119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)

From Table 1 (23) it follows that

119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)

Therefore from Table 1 (24)

119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)

(c) First we show that recursion computes the left centroidfunction with initial point 119871

119894

and decreasing 119896 Considering

119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)

Expanding it without altering the proportion

119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)

It can be observed that (A10) is a proportion composed byinitial values119863

119871 119894and119875

119871 119894and two terms that can be computed

by recursion 119863(119896) and 119875(119896) so

119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)

Sums are computed by recursion as

119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897

le 119910min in Table 1 (21) iterationsdecrease 119896 from 119871

119894

according to (A10)It has been demonstrated that (A14) is the same (A8)

computed recursively from initial point 119871

119894

So by performing


a decrement of 119896 in step 5 a decrement is introduced in 119891

119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)

In [5] it is demonstrated that

if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871

thusfrom the centroid function properties it is guaranteed that119909

119896

gt 119910

119897

(119896) satisfying (A17) Therefore it can be concludedthat a decrement in 119891

119896

as (A15) will induce a decrement in119910

119897

(119896)(d)The stop condition is stated from the properties of the

centroid function Since the centroid function has a globalminimum in 119871 once recursion has reached 119896 = 119871 119910

119897

(119896) willincrease in the next decrement of 119896 because (A17) is no longervalid So by detecting the increment in119910

119897

(119896) it can be said thatthe minimum has been found in the previous iteration

Conflict of Interests

The authors hereby declare that they do not have any directfinancial relation with the commercial identities mentionedin this paper

References

[1] J M Mendel ldquoAdvances in type-2 fuzzy sets and systemsrdquoInformation Sciences vol 177 no 1 pp 84ndash110 2007

[2] R John and S Coupland ldquoType-2 fuzzy logic a historical viewrdquoIEEE Computational Intelligence Magazine vol 2 no 1 pp 57ndash62 2007

[3] J M Mendel ldquoType-2 fuzzy sets and systems an overviewrdquoIEEE Computational Intelligence Magazine vol 2 no 1 pp 20ndash29 2007

[4] S Coupland and R John ldquoGeometric type-1 and type-2 fuzzylogic systemsrdquo IEEE Transactions on Fuzzy Systems vol 15 no1 pp 3ndash15 2007

[5] J Mendel Uncertain Rule-Based Fuzzy Logic Systems Introduc-tion and New Directions Prentice Hall Upper Saddle River NJUSA 2001

[6] J M Mendel ldquoOn a 50 savings in the computation of thecentroid of a symmetrical interval type-2 fuzzy setrdquo InformationSciences vol 172 no 3-4 pp 417ndash430 2005

[7] H Wu and J M Mendel ldquoUncertainty bounds and their usein the design of interval type-2 fuzzy logic systemsrdquo IEEETransactions on Fuzzy Systems vol 10 no 5 pp 622ndash639 2002

[8] D Wu and W W Tan ldquoComputationally efficient type-reduction strategies for a type-2 fuzzy logic controllerrdquo inProceedings of the IEEE International Conference on FuzzySystems pp 353ndash358 May 2005

[9] C Y Yeh W H R Jeng and S J Lee ldquoAn enhanced type-reduction algorithm for type-2 fuzzy setsrdquo IEEE Transactionson Fuzzy Systems vol 19 no 2 pp 227ndash240 2011

[10] C Wagner and H Hagras ldquoToward general type-2 fuzzy logicsystems based on zSlicesrdquo IEEE Transactions on Fuzzy Systemsvol 18 no 4 pp 637ndash660 2010

[11] J M Mendel F Liu and D Zhai ldquo120572-Plane representation fortype-2 fuzzy sets theory and applicationsrdquo IEEE Transactionson Fuzzy Systems vol 17 no 5 pp 1189ndash1207 2009

[12] S Coupland and R John ldquoAn investigation into alternativemethods for the defuzzification of an interval type-2 fuzzy setrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1425ndash1432 July 2006

[13] J M Mendel and F Liu ldquoSuper-exponential convergence of theKarnik-Mendel algorithms for computing the centroid of aninterval type-2 fuzzy setrdquo IEEE Transactions on Fuzzy Systemsvol 15 no 2 pp 309ndash320 2007

[14] K Duran H Bernal and M Melgarejo ldquoImproved iterativealgorithm for computing the generalized centroid of an intervaltype-2 fuzzy setrdquo in Proceedings of the Annual Meeting of theNorth American Fuzzy Information Processing Society (NAFIPSrsquo08) New York NY USA May 2008

[15] K Duran H Bernal and M Melgarejo ldquoA comparative studybetween two algorithms for computing the generalized centroidof an interval Type-2 Fuzzy Setrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems pp 954ndash959 HongKong China June 2008

[16] D Wu and J M Mendel ldquoEnhanced Karnik-Mendel algo-rithmsrdquo IEEE Transactions on Fuzzy Systems vol 17 no 4 pp923ndash934 2009

[17] X Liu J M Mendel and D Wu ldquoStudy on enhanced Karnik-Mendel algorithms Initialization explanations and computa-tion improvementsrdquo Information Sciences vol 184 no 1 pp 75ndash91 2012

[18] D Wu and M Nie ldquoComparison and practical implementationof type-reduction algorithms for type-2 fuzzy sets and systemsrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 2131ndash2138 June 2011

[19] M Melgarejo ldquoA fast recursive method to compute the gener-alized centroid of an interval type-2 fuzzy setrdquo in Proceedingsof the Annual Meeting of the North American Fuzzy InformationProcessing Society (NAFIPS rsquo07) pp 190ndash194 June 2007

[20] R Sepulveda OMontiel O Castillo and PMelin ldquoEmbeddinga high speed interval type-2 fuzzy controller for a real plant intoan FPGArdquo Applied Soft Computing vol 12 no 3 pp 988ndash9982012

[21] R Sepulveda O Montiel O Castillo and P Melin ldquoModellingand simulation of the defuzzification stage of a type-2 fuzzycontroller using VHDL coderdquo Control and Intelligent Systemsvol 39 no 1 pp 33ndash40 2011

[22] O Montiel R Sepulveda Y Maldonado and O CastilloldquoDesign and simulation of the type-2 fuzzification stage usingactive membership functionsrdquo Studies in Computational Intelli-gence vol 257 pp 273ndash293 2009

[23] YMaldonado O Castillo and PMelin ldquoOptimization ofmem-bership functions for an incremental fuzzy PD control based ongenetic algorithmsrdquo Studies in Computational Intelligence vol318 pp 195ndash211 2010


[24] C Li J Yi and D Zhao ldquoA novel type-reduction methodfor interval type-2 fuzzy logic systemsrdquo in Proceedings of the5th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo08) pp 157ndash161 October 2008

[25] D Wu ldquoApproaches for reducing the computational cost oflogic systems overview and comparisonsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 1 pp 80ndash90 2013

[26] H Hu YWang and Y Cai ldquoAdvantages of the enhanced oppo-site direction searching algorithm for computing the centroidof an interval type-2 fuzzy setrdquo Asian Journal of Control vol 14no 6 pp 1ndash9 2012

[27] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006

Submit your manuscripts athttpwwwhindawicom

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ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



An enhanced version of the KM algorithm (EKM) waspresented in [16] This version includes a better initializa-tion and some computational improvements Experimentalevidence shows that the EKM algorithm saves about twoiterations which means a reduction in computation time ofmore than 39 with respect to the original algorithm Bythe time the EKM algorithm was presented and the iterativealgorithm with stop condition (IASC) was also introducedby Melgarejo et al [14 15 19] This algorithm deals withthe problem of computing type-reduction for an IT2-FSby using some properties of the centroid function [13 19]Experimental evidence showed that IASC is faster than theEKM algorithm in some cases

Recently IASC has been enhanced by Wu and Nie[18] leading to a new version called EIASC which provedto be the fastest algorithm with respect to the IASC andEKM algorithms for several type-2 fuzzy logic applicationsThe improvement introduced in EIASC mainly deals withmodifying the iterative procedure for computing the rightpoint of the interval centroid Although IASC and EIASCoutperform the EKM algorithm the initialization of theIASC-type algorithms is somewhat trivial and does notprovide an appropriate starting point which can limit theconvergence speed of these algorithms when calculating thepoints of the type-reduced set An extensive and interestingcomparison of several type-reductionmethods is provided byWu in [25] This study confirmed the convenience of EIASCover the EKMA for reducing the computational cost of anIT2-FLS The experiments that were limited to interpretedlanguage implementations also showed that EIASC and theenhanced opposite direction searching algorithm (EODS)exhibited similar performance over different applicationshowever the EODS outperformed EIASC in low-resolutioncases [26]

Regarding the aforementioned problem this work con-siders a different initialization perspective for iterative type-reduction algorithms based on the concept of inner-boundset for a type-reduced set [13] This kind of initializationhas not been tried in type-reduction computing until nowAs it will be analyzed later the inner-bound initializationreduces the distance that the algorithms need to cover inorder to find the optimal points of the interval centroidIn addition bearing in mind the computational burden ofiterative algorithms this paper also focuses on proposing analternative strategy to speed up their computation based onthe concept of precomputing Precomputation in algorithmicdesign is a powerful concept that reduces the necessaryarithmetic operations Current literature does not report theuse of precomputation in the type-reduction stage in IT2-FLSs

Using inner bound sets initialization and precomputa-tion Both the IASC and KM algorithms have been restatedleading to faster algorithms hereafter refered to IASC2 andKMA2 Our experiments considering two different imple-mentation perspectives (ie interpreted language and com-piled language) show that IASC2 and KMA2 outperformEKMA and EIASC In fact timing results suggest thatIASC2 may be the best option for implementing IT2-FLSsin dedicated hardware whereas EKMA may support the

computation of large-scale simulations of IT2-FLSs overgeneral purpose processors

Thepaper is organized as follows Section 2 provides somebackground about the centroid of an interval type-2 fuzzyset IASC2 and KMA2 are presented in Section 3 Section 4is devoted to a computational comparative study of itera-tive type-reduction methods which considers two types ofimplementation compiled-language-based and interpreted-language-based Finally we draw conclusions in Section 5

2 Background

The purpose of this section is to provide some basic infor-mation about the centroid of interval type-2 fuzzy sets Thereaderwho is not familiarwith this theory is invited to consult[3 5 13] among others

21 Type-2 Fuzzy Sets According to Mendel [5] a type-2fuzzy set denoted

119860 is characterized by a type-2membershipfunction 120583

119860

(119909 119906) where 119909 isin 119883 and 0 le 120583

119860

(119909 119906) le 1

119860 = ((119909 119906) 120583

119860

(119909 119906)) | forall119906 isin 119869

119909

sube [0 1] (1)

22 Interval Type-2 Fuzzy Sets An IT2-FS is a particular caseof a T2-FS whose values in 120583

119860

(119909 119906) are equal to one [27]An IT2-FS is fully described by its footprint of uncertainty(FOU) [3] Note that an IT2-FS set can be understood as acollection of type-1 fuzzy sets (ie traditional fuzzy sets)Themembership functions of these sets are contained within theFOU thus they are called embedded fuzzy sets [5 13]

23 Centroid of an IT2-FS Let

119860 be an IT2-FS and its cen-troid 119888(

119860) is an interval given by [5]

c (

A)= 1[119910

119897

(

119860) 119910

119903

(

119860)]

=

1

[119910

119897

119910

119903

]

(2)

where 119910

119871

and 119910

119877

are the solutions to the following optimiza-tion problems

119910

119871

=

minforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

119910

119877

=

maxforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(3)

where the universe of discourse 119883 as well as the upper andlower membership functions have been discretized in 119873

pointsA simpler way to understand the centroid of an IT2-FS

is to think about it as a set of centroids that are computedfrom all the embedded fuzzy sets [3 5] Clearly in orderto characterize an interval set it is only necessary to findits minimum and maximum The Karnik-Mendel iterativealgorithms [5 16 17] are the most popular procedures forcomputing these two points



119871

and 119910

119877


119860 [3 5] as follows

119910

119871

=

minforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

=

sum

119871

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871+1

119909

119894

119891

119894

sum

119871

119894=1

119891

119894

+ sum

119873

119894=119871+1

119891

119894

119910

119877

=

maxforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

=

sum

119877

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119877+1

119909

119894

119891

119894

sum

119877

119894=1

119891

119894

+ sum

119873

119894=119877+1

119891

119894

(4)

where 119891

119894

and 119891

119894


119894


119909

119871

le 119910

119871

le 119909

119871+1

119909

119877

le 119910

119877

le 119909

119877+1

(5)


119871

and 119910

119877


119910

119897

le 119910

119871

le 119910

119897

(6)

119910

119903

le 119910

119877

le 119910

119903

(7)

where [119910119897

119910

119903

] and [119910119897

119910

119903


119910

119860

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

119910

119861

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(8)

119910

119897

= min (119910

119860

119910

119861

)

119910

119903

= max (119910

119860

119910

119861

)

(9)


EIASC EIASC

IASC

IASC

yL yl yR x

micro(x

)

yr

IASC2 IASC2


119894


119894



119903




119871

and 119910

119877




119871


119877


119871

and 119910

119877


119910

119871

119910

119877

119910min or 119910

119897


119903





119871


119894


119894


119894



119877


119894


119894


119894


(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(15)

119867 [119899] =

119899

sum

119894=1

119891

119894

(16)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(17)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(18)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(19)

119910

119861

=

119885 [119873]

119867 [119873]

(20)


119860

119910

119861

) (21) 119910max = max(119910

119860

119910

119861

) (32)Find 119871

119894

(1 le 119871

119894


119894

(1 le 119877

119894


119871119894le 119910min lt 119909

119871119894+1119909

119877119894le 119910max lt 119909

119877119894+1

119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

]) (22) 119864

119877

(119877

119894

) = 119885 [119877

119894

] + (119879 [119873] minus 119879 [119877

119894

]) (33)119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

]) (23) 119876

119877

(119877

119894

) = 119884 [119877

119894

] + (119883 [119873] minus 119883 [119877

119894

]) (34)

119910min =

119863

119871119894

119875

119871119894

(24) 119910max =

119864

119877

(119877

119894

)

119876

119877

(119877

119894

)

(35)

119863 = 119863

119897

(119871

119894

) (25) 119864 = 119864

119877119894(36)

119875 = 119875

119897

(119871

119894

) (26) 119876 = 119876

119877119894(37)

119896 = 119871

119894

(27) 119896 = 119877

119894


120572

119896

= 119891

119896

minus 119891

119896

(28) 120572

119896

= 119891

119896

minus 119891

119896

(39)119863 = 119863 minus 119909

119896

120572

119896

(29) 119864 = 119864 minus 119909

119896

120572

119896

(40)119875 = 119875 minus 120572

119896

(30) 119876 = 119876 minus 120572

119896

(41)

119910

119897

=

119863

119875

(31) 119910

119903

=

119864

119876

(42)

(5) If 119910

119897

le 119910min then 119910min = 119910

119897


119871


119877

ge 119910max then 119910max = 119888

119877


119877

= 119910maxand stop



119871


119871


119894

so that 119909

119871 119894le

119910min lt 119909



119894



119894

and each decrement


119897


119897

(119896) = (sum

119896

119894=1

119909

119894

119891

119894

+

sum

119873

119894=119896+1

119909

119894

119891

119894

)(sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

) = (119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus

119891

119894

))(119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894


119897

(119896) = (119863

119896minus1

minus 119909

119896

(119891

119896

minus 119891

119896

))(119875

119896minus1

minus (119891

119896

minus 119891

119896

))Since 119871 le 119871

119894

given that 119910

119897


119894


119896

from 119891

119896

to 119891

119896



119896

lt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

decreases




119871


119894


119894


119894



119877


119894


119894


(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(43)

119867 [119899] =

119899

sum

119894=1

119891

119894

(44)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(45)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(46)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(47)

119910

119861

=

119885 [119873]

119867 [119873]

(48)

(3) Set Set119910min = min(119910

119860

119910

119861

) (49) 119910max = max(119910

119860

119910

119861


119909

119896

le 119910min lt 119909

119896+1

119909

119896

le 119910max lt 119909

119896+1

119863

119897

(119896) = 119879 [119896] + (119885 [119873] minus 119885 [119896]) (50) 119864

119877

(119896) = 119885 [119896] + (119879 [119873] minus 119879 [119896]) (57)119875

119897

(119896) = 119881 [119896] + (119884 [119873] minus 119884 [119896]) (51) 119876

119877

(119896) = 119884 [119896] + (119883 [119873] minus 119883 [119896]) (58)

119910 =

119863

119897

(119896)

119875

119897

(119896)

(52) 119910 =

119864

119877

(119896)

119876

119877

(119896)

(59)

(4) Find 119896

1015840


1015840


119896

1015840 le 119910 lt 119909

119896

1015840+1

119909

119896

1015840 le 119910 lt 119909

119896

1015840+1

(5) If 119896

1015840

= 119896 stop and set 119910

119871


1015840

= 119896 stop and set 119910

119877


119863

119897

(119896

1015840

) = 119879 [119896

1015840

] + (119885 [119873] minus 119885 [119896

1015840

]) (53) 119864

119877

(119896

1015840

) = 119885 [119896

1015840

] + (119879 [119873] minus 119879 [119896

1015840

]) (60)119875

119897

(119896

1015840

) = 119881 [119896

1015840

] + (119884 [119873] minus 119884 [119896

1015840

]) (54) 119876

119877

(119896

1015840

) = 119884 [119896

1015840

] + (119883 [119873] minus 119883 [119896

1015840

]) (61)

119910

1015840

=

119863

119897

(119896

1015840

)

119875

119897

(119896

1015840

)

(55) 119910

1015840

=

119864

119877

(119896

1015840

)

119876

119877

(119896

1015840

)

(62)

(7) Set 119910 = 119910

1015840 119896 = 119896

1015840 and go to step 4 Set 119910 = 119910

1015840 119896 = 119896



119896

gt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

increases


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897




119871


119877




119871

and 119910

119877


119871

and 119910

119877


119894

equal to 119896 or 119896


119897

(119896) or119910

119903



x

1micro(x)

(a)

x

1micro(x)

(b)

x

1micro(x)

(c)

x

1micro(x)

(d)







119871

and 119910

119877





10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

16

Rat

io

10 100 1000

N

(b)

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

0E+00

1Eminus07

2Eminus07

3Eminus07

4Eminus07

5Eminus07

6Eminus07

(c)



119900


119900

and 119905

119890









119877



10 100 1000

N

Ave

rage

com

puta

tion

tim

e (s

)

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)










10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119871

and 119910

119877


119860 [3 5] as follows

119910

119871

=

minforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

=

sum

119871

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871+1

119909

119894

119891

119894

sum

119871

119894=1

119891

119894

+ sum

119873

119894=119871+1

119891

119894

119910

119877

=

maxforall119891

119894

isin [119891

119894

119891

119894

]

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

=

sum

119877

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119877+1

119909

119894

119891

119894

sum

119877

119894=1

119891

119894

+ sum

119873

119894=119877+1

119891

119894

(4)

where 119891

119894

and 119891

119894


119894


119909

119871

le 119910

119871

le 119909

119871+1

119909

119877

le 119910

119877

le 119909

119877+1

(5)


119871

and 119910

119877


119910

119897

le 119910

119871

le 119910

119897

(6)

119910

119903

le 119910

119877

le 119910

119903

(7)

where [119910119897

119910

119903

] and [119910119897

119910

119903


119910

119860

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

119910

119861

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(8)

119910

119897

= min (119910

119860

119910

119861

)

119910

119903

= max (119910

119860

119910

119861

)

(9)


EIASC EIASC

IASC

IASC

yL yl yR x

micro(x

)

yr

IASC2 IASC2


119894


119894



119903




119871

and 119910

119877




119871


119877


119871

and 119910

119877


119910

119871

119910

119877

119910min or 119910

119897


119903





119871


119894


119894


119894



119877


119894


119894


119894


(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(15)

119867 [119899] =

119899

sum

119894=1

119891

119894

(16)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(17)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(18)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(19)

119910

119861

=

119885 [119873]

119867 [119873]

(20)


119860

119910

119861

) (21) 119910max = max(119910

119860

119910

119861

) (32)Find 119871

119894

(1 le 119871

119894


119894

(1 le 119877

119894


119871119894le 119910min lt 119909

119871119894+1119909

119877119894le 119910max lt 119909

119877119894+1

119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

]) (22) 119864

119877

(119877

119894

) = 119885 [119877

119894

] + (119879 [119873] minus 119879 [119877

119894

]) (33)119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

]) (23) 119876

119877

(119877

119894

) = 119884 [119877

119894

] + (119883 [119873] minus 119883 [119877

119894

]) (34)

119910min =

119863

119871119894

119875

119871119894

(24) 119910max =

119864

119877

(119877

119894

)

119876

119877

(119877

119894

)

(35)

119863 = 119863

119897

(119871

119894

) (25) 119864 = 119864

119877119894(36)

119875 = 119875

119897

(119871

119894

) (26) 119876 = 119876

119877119894(37)

119896 = 119871

119894

(27) 119896 = 119877

119894


120572

119896

= 119891

119896

minus 119891

119896

(28) 120572

119896

= 119891

119896

minus 119891

119896

(39)119863 = 119863 minus 119909

119896

120572

119896

(29) 119864 = 119864 minus 119909

119896

120572

119896

(40)119875 = 119875 minus 120572

119896

(30) 119876 = 119876 minus 120572

119896

(41)

119910

119897

=

119863

119875

(31) 119910

119903

=

119864

119876

(42)

(5) If 119910

119897

le 119910min then 119910min = 119910

119897


119871


119877

ge 119910max then 119910max = 119888

119877


119877

= 119910maxand stop



119871


119871


119894

so that 119909

119871 119894le

119910min lt 119909



119894



119894

and each decrement


119897


119897

(119896) = (sum

119896

119894=1

119909

119894

119891

119894

+

sum

119873

119894=119896+1

119909

119894

119891

119894

)(sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

) = (119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus

119891

119894

))(119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894


119897

(119896) = (119863

119896minus1

minus 119909

119896

(119891

119896

minus 119891

119896

))(119875

119896minus1

minus (119891

119896

minus 119891

119896

))Since 119871 le 119871

119894

given that 119910

119897


119894


119896

from 119891

119896

to 119891

119896



119896

lt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

decreases




119871


119894


119894


119894



119877


119894


119894


(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(43)

119867 [119899] =

119899

sum

119894=1

119891

119894

(44)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(45)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(46)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(47)

119910

119861

=

119885 [119873]

119867 [119873]

(48)

(3) Set Set119910min = min(119910

119860

119910

119861

) (49) 119910max = max(119910

119860

119910

119861


119909

119896

le 119910min lt 119909

119896+1

119909

119896

le 119910max lt 119909

119896+1

119863

119897

(119896) = 119879 [119896] + (119885 [119873] minus 119885 [119896]) (50) 119864

119877

(119896) = 119885 [119896] + (119879 [119873] minus 119879 [119896]) (57)119875

119897

(119896) = 119881 [119896] + (119884 [119873] minus 119884 [119896]) (51) 119876

119877

(119896) = 119884 [119896] + (119883 [119873] minus 119883 [119896]) (58)

119910 =

119863

119897

(119896)

119875

119897

(119896)

(52) 119910 =

119864

119877

(119896)

119876

119877

(119896)

(59)

(4) Find 119896

1015840


1015840


119896

1015840 le 119910 lt 119909

119896

1015840+1

119909

119896

1015840 le 119910 lt 119909

119896

1015840+1

(5) If 119896

1015840

= 119896 stop and set 119910

119871


1015840

= 119896 stop and set 119910

119877


119863

119897

(119896

1015840

) = 119879 [119896

1015840

] + (119885 [119873] minus 119885 [119896

1015840

]) (53) 119864

119877

(119896

1015840

) = 119885 [119896

1015840

] + (119879 [119873] minus 119879 [119896

1015840

]) (60)119875

119897

(119896

1015840

) = 119881 [119896

1015840

] + (119884 [119873] minus 119884 [119896

1015840

]) (54) 119876

119877

(119896

1015840

) = 119884 [119896

1015840

] + (119883 [119873] minus 119883 [119896

1015840

]) (61)

119910

1015840

=

119863

119897

(119896

1015840

)

119875

119897

(119896

1015840

)

(55) 119910

1015840

=

119864

119877

(119896

1015840

)

119876

119877

(119896

1015840

)

(62)

(7) Set 119910 = 119910

1015840 119896 = 119896

1015840 and go to step 4 Set 119910 = 119910

1015840 119896 = 119896



119896

gt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

increases


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897




119871


119877




119871

and 119910

119877


119871

and 119910

119877


119894

equal to 119896 or 119896


119897

(119896) or119910

119903



x

1micro(x)

(a)

x

1micro(x)

(b)

x

1micro(x)

(c)

x

1micro(x)

(d)







119871

and 119910

119877





10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

16

Rat

io

10 100 1000

N

(b)

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

0E+00

1Eminus07

2Eminus07

3Eminus07

4Eminus07

5Eminus07

6Eminus07

(c)



119900


119900

and 119905

119890









119877



10 100 1000

N

Ave

rage

com

puta

tion

tim

e (s

)

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)










10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119871


119894


119894


119894



119877


119894


119894


119894


(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(15)

119867 [119899] =

119899

sum

119894=1

119891

119894

(16)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(17)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(18)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(19)

119910

119861

=

119885 [119873]

119867 [119873]

(20)


119860

119910

119861

) (21) 119910max = max(119910

119860

119910

119861

) (32)Find 119871

119894

(1 le 119871

119894


119894

(1 le 119877

119894


119871119894le 119910min lt 119909

119871119894+1119909

119877119894le 119910max lt 119909

119877119894+1

119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

]) (22) 119864

119877

(119877

119894

) = 119885 [119877

119894

] + (119879 [119873] minus 119879 [119877

119894

]) (33)119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

]) (23) 119876

119877

(119877

119894

) = 119884 [119877

119894

] + (119883 [119873] minus 119883 [119877

119894

]) (34)

119910min =

119863

119871119894

119875

119871119894

(24) 119910max =

119864

119877

(119877

119894

)

119876

119877

(119877

119894

)

(35)

119863 = 119863

119897

(119871

119894

) (25) 119864 = 119864

119877119894(36)

119875 = 119875

119897

(119871

119894

) (26) 119876 = 119876

119877119894(37)

119896 = 119871

119894

(27) 119896 = 119877

119894


120572

119896

= 119891

119896

minus 119891

119896

(28) 120572

119896

= 119891

119896

minus 119891

119896

(39)119863 = 119863 minus 119909

119896

120572

119896

(29) 119864 = 119864 minus 119909

119896

120572

119896

(40)119875 = 119875 minus 120572

119896

(30) 119876 = 119876 minus 120572

119896

(41)

119910

119897

=

119863

119875

(31) 119910

119903

=

119864

119876

(42)

(5) If 119910

119897

le 119910min then 119910min = 119910

119897


119871


119877

ge 119910max then 119910max = 119888

119877


119877

= 119910maxand stop



119871


119871


119894

so that 119909

119871 119894le

119910min lt 119909



119894



119894

and each decrement


119897


119897

(119896) = (sum

119896

119894=1

119909

119894

119891

119894

+

sum

119873

119894=119896+1

119909

119894

119891

119894

)(sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

) = (119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus

119891

119894

))(119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894


119897

(119896) = (119863

119896minus1

minus 119909

119896

(119891

119896

minus 119891

119896

))(119875

119896minus1

minus (119891

119896

minus 119891

119896

))Since 119871 le 119871

119894

given that 119910

119897


119894


119896

from 119891

119896

to 119891

119896



119896

lt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

decreases




119871


119894


119894


119894



119877


119894


119894


(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(43)

119867 [119899] =

119899

sum

119894=1

119891

119894

(44)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(45)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(46)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(47)

119910

119861

=

119885 [119873]

119867 [119873]

(48)

(3) Set Set119910min = min(119910

119860

119910

119861

) (49) 119910max = max(119910

119860

119910

119861


119909

119896

le 119910min lt 119909

119896+1

119909

119896

le 119910max lt 119909

119896+1

119863

119897

(119896) = 119879 [119896] + (119885 [119873] minus 119885 [119896]) (50) 119864

119877

(119896) = 119885 [119896] + (119879 [119873] minus 119879 [119896]) (57)119875

119897

(119896) = 119881 [119896] + (119884 [119873] minus 119884 [119896]) (51) 119876

119877

(119896) = 119884 [119896] + (119883 [119873] minus 119883 [119896]) (58)

119910 =

119863

119897

(119896)

119875

119897

(119896)

(52) 119910 =

119864

119877

(119896)

119876

119877

(119896)

(59)

(4) Find 119896

1015840


1015840


119896

1015840 le 119910 lt 119909

119896

1015840+1

119909

119896

1015840 le 119910 lt 119909

119896

1015840+1

(5) If 119896

1015840

= 119896 stop and set 119910

119871


1015840

= 119896 stop and set 119910

119877


119863

119897

(119896

1015840

) = 119879 [119896

1015840

] + (119885 [119873] minus 119885 [119896

1015840

]) (53) 119864

119877

(119896

1015840

) = 119885 [119896

1015840

] + (119879 [119873] minus 119879 [119896

1015840

]) (60)119875

119897

(119896

1015840

) = 119881 [119896

1015840

] + (119884 [119873] minus 119884 [119896

1015840

]) (54) 119876

119877

(119896

1015840

) = 119884 [119896

1015840

] + (119883 [119873] minus 119883 [119896

1015840

]) (61)

119910

1015840

=

119863

119897

(119896

1015840

)

119875

119897

(119896

1015840

)

(55) 119910

1015840

=

119864

119877

(119896

1015840

)

119876

119877

(119896

1015840

)

(62)

(7) Set 119910 = 119910

1015840 119896 = 119896

1015840 and go to step 4 Set 119910 = 119910

1015840 119896 = 119896



119896

gt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

increases


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897




119871


119877




119871

and 119910

119877


119871

and 119910

119877


119894

equal to 119896 or 119896


119897

(119896) or119910

119903



x

1micro(x)

(a)

x

1micro(x)

(b)

x

1micro(x)

(c)

x

1micro(x)

(d)







119871

and 119910

119877





10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

16

Rat

io

10 100 1000

N

(b)

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

0E+00

1Eminus07

2Eminus07

3Eminus07

4Eminus07

5Eminus07

6Eminus07

(c)



119900


119900

and 119905

119890









119877



10 100 1000

N

Ave

rage

com

puta

tion

tim

e (s

)

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)










10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119871


119894


119894


119894



119877


119894


119894


(1) Sort 119909

119894


119894


1

le 119909

2


119873


119894


119894


119894

(2) Precomputation

119881 [119899] =

119899

sum

119894=1

119891

119894

(43)

119867 [119899] =

119899

sum

119894=1

119891

119894

(44)

119879 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(45)

119885 [119899] =

119899

sum

119894=1

119909

119894

119891

119894

(46)

with 119899 = 1 2 119873

119910

119860

=

119879 [119873]

119881 [119873]

(47)

119910

119861

=

119885 [119873]

119867 [119873]

(48)

(3) Set Set119910min = min(119910

119860

119910

119861

) (49) 119910max = max(119910

119860

119910

119861


119909

119896

le 119910min lt 119909

119896+1

119909

119896

le 119910max lt 119909

119896+1

119863

119897

(119896) = 119879 [119896] + (119885 [119873] minus 119885 [119896]) (50) 119864

119877

(119896) = 119885 [119896] + (119879 [119873] minus 119879 [119896]) (57)119875

119897

(119896) = 119881 [119896] + (119884 [119873] minus 119884 [119896]) (51) 119876

119877

(119896) = 119884 [119896] + (119883 [119873] minus 119883 [119896]) (58)

119910 =

119863

119897

(119896)

119875

119897

(119896)

(52) 119910 =

119864

119877

(119896)

119876

119877

(119896)

(59)

(4) Find 119896

1015840


1015840


119896

1015840 le 119910 lt 119909

119896

1015840+1

119909

119896

1015840 le 119910 lt 119909

119896

1015840+1

(5) If 119896

1015840

= 119896 stop and set 119910

119871


1015840

= 119896 stop and set 119910

119877


119863

119897

(119896

1015840

) = 119879 [119896

1015840

] + (119885 [119873] minus 119885 [119896

1015840

]) (53) 119864

119877

(119896

1015840

) = 119885 [119896

1015840

] + (119879 [119873] minus 119879 [119896

1015840

]) (60)119875

119897

(119896

1015840

) = 119881 [119896

1015840

] + (119884 [119873] minus 119884 [119896

1015840

]) (54) 119876

119877

(119896

1015840

) = 119884 [119896

1015840

] + (119883 [119873] minus 119883 [119896

1015840

]) (61)

119910

1015840

=

119863

119897

(119896

1015840

)

119875

119897

(119896

1015840

)

(55) 119910

1015840

=

119864

119877

(119896

1015840

)

119876

119877

(119896

1015840

)

(62)

(7) Set 119910 = 119910

1015840 119896 = 119896

1015840 and go to step 4 Set 119910 = 119910

1015840 119896 = 119896



119896

gt 119910(119891

1


119873

) then 119910(119891

1


119873


119896

increases


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897




119871


119877




119871

and 119910

119877


119871

and 119910

119877


119894

equal to 119896 or 119896


119897

(119896) or119910

119903



x

1micro(x)

(a)

x

1micro(x)

(b)

x

1micro(x)

(c)

x

1micro(x)

(d)







119871

and 119910

119877





10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

16

Rat

io

10 100 1000

N

(b)

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

0E+00

1Eminus07

2Eminus07

3Eminus07

4Eminus07

5Eminus07

6Eminus07

(c)



119900


119900

and 119905

119890









119877



10 100 1000

N

Ave

rage

com

puta

tion

tim

e (s

)

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)










10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




x

1micro(x)

(a)

x

1micro(x)

(b)

x

1micro(x)

(c)

x

1micro(x)

(d)







119871

and 119910

119877





10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

16

Rat

io

10 100 1000

N

(b)

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

0E+00

1Eminus07

2Eminus07

3Eminus07

4Eminus07

5Eminus07

6Eminus07

(c)



119900


119900

and 119905

119890









119877



10 100 1000

N

Ave

rage

com

puta

tion

tim

e (s

)

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)










10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

16

Rat

io

10 100 1000

N

(b)

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

0E+00

1Eminus07

2Eminus07

3Eminus07

4Eminus07

5Eminus07

6Eminus07

(c)



119900


119900

and 119905

119890









119877



10 100 1000

N

Ave

rage

com

puta

tion

tim

e (s

)

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)










10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




10 100 1000

N

Ave

rage

com

puta

tion

tim

e (s

)

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

0

02

04

06

08

1

12

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

35Eminus07

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)










10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

(a)

02

0

04

06

08

1

12

16

14

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

3Eminus07

25Eminus07

2Eminus07

15Eminus07

1Eminus07

5Eminus08

(c)




119877


119871






10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

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200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

12

14

16

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




10 100 1000

Ave

rage

com

puta

tion

tim

e (s

)

N

0E+00

2Eminus06

4Eminus06

6Eminus06

8Eminus06

1Eminus05

12Eminus05

14Eminus05

16Eminus05

18Eminus05

(a)

02

0

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

STD

of

the

com

puta

tion

tim

e (s

)

0E+00

1Eminus06

9Eminus07

8Eminus07

7Eminus07

6Eminus07

5Eminus07

4Eminus07

3Eminus07

2Eminus07

1Eminus07

(c)







6 Conclusions



0

100

200

300

400

500

600

700

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)




0

100

200

300

400

500

600

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

10 100 1000

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

N

(b)





0

100

200

300

400

500

600

700

800

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

10 100 1000

N

IASCEIASC

IASC2

(a)

0

05

1

15

2

25

3

35

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)


Appendix


119871




119894



119894


119897


0

100

200

300

400

500

600

700

800

900

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

05

1

15

2

25

3

Mea

n o

f th

e n

um

ber

of it

erat

ion

s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

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1

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Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

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08

STD

of

the

com

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tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

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Ave

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tim

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)

10 100 1000

N

(a)

0

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1

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Rat

io

10 100 1000

N

(b)

0

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1

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STD

of

the

com

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tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

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Ave

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tim

e (s

)

10 100 1000

N

(a)

0

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1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

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tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

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Ave

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com

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tim

e (s

)

10 100 1000

N

(a)

0

02

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1

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Rat

io

10 100 1000

N

(b)

0

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1

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3

STD

of

the

com

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tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

100

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n o

f th

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s

10 100 1000

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IASCEIASC

IASC2

(a)

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1

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2

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Mea

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s

EKMKMA2

10 100 1000

N

(b)




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Mea

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Mea

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s

EKMKMA2

N

(b)





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800

Mea

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s

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IASCEIASC

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(a)

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35

Mea

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s

EKMKMA2

10 100 1000

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(b)


Appendix


119871




119894



119894


119897


0

100

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500

600

700

800

900

Mea

n o

f th

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s

IASCEIASC

IASC2

10 100 1000

N

(a)

0

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2

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3

Mea

n o

f th

e n

um

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of it

erat

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s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

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10 100 1000

N

(a)

0

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08

1

12

Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

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16

Ave

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com

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tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

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8

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Ave

rage

com

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tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

100

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Mea

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IASCEIASC

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(a)

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Mea

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(b)


Appendix


119871




119894



119894


119897


0

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800

900

Mea

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e n

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IASCEIASC

IASC2

10 100 1000

N

(a)

0

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3

Mea

n o

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of it

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s

EKMKMA2

10 100 1000

N

(b)



119910

119860

=

119879 [119873]

119881 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A1)

119910

119861

=

119885 [119873]

119867 [119873]

=

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

(A2)

119910min = min(

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

sum

119873

119894=1

119909

119894

119891

119894

sum

119873

119894=1

119891

119894

) (A3)

119910min = 119910

119897

(A4)


0

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10 100 1000

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1

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Rat

io

10 100 1000

N

(b)

0

01

02

03

04

05

06

07

08

STD

of

the

com

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tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

4

6

8

10

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16

Ave

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com

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tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

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1

12

Rat

io

10 100 1000

N

(b)

0

02

04

06

08

1

12

14

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

6

8

10

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Ave

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com

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tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

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tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

2

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tim

e (s

)

10 100 1000

N

(a)

0

02

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1

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Rat

io

10 100 1000

N

(b)

0

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06

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STD

of

the

com

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tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

2

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Ave

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tim

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)

10 100 1000

N

(a)

0

02

04

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1

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Rat

io

10 100 1000

N

(b)

0

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04

06

08

1

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14

STD

of

the

com

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tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)



0

2

4

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Ave

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tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

2

4

6

8

10

12

14

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

005

01

015

02

025

03

035

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)


0

5

10

15

20

25

Ave

rage

com

puta

tion

tim

e (s

)

10 100 1000

N

(a)

0

02

04

06

08

1

12

Rat

io

10 100 1000

N

(b)

0

05

1

15

2

25

3

STD

of

the

com

puta

tion

tim

e (s

)

10 100 1000

N

EKMIASCEIASC

IASC2KMA2

(c)




119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119871


119871


119894

so that119909

119871 119894le 119910min lt 119909

119871 119894+1


119863

119897

(119871

119894

) = 119879 [119871

119894

] + (119885 [119873] minus 119885 [119871

119894

])

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+ (

119873

sum

119894=1

119909

119894

119891

119894

minus

119871 119894

sum

119894=1

119909

119894

119891

119894

)

119863

119871 119894= 119863

119897

(119871

119894

) =

119871 119894

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119871 119894+1

119909

119894

119891

119894

(A5)


119875

119897

(119871

119894

) = 119881 [119871

119894

] + (119884 [119873] minus 119884 [119871

119894

])

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+ (

119873

sum

119894=1

119891

119894

minus

119871 119894

sum

119894=1

119891

119894

)

119875

119871 119894= 119875

119897

(119871

119894

) =

119871 119894

sum

119894=1

119891

119894

+

119873

sum

119894=119871 119894+1

119891

119894

(A6)


119910min =

119863

119871 119894

119875

119871 119894

=

sum

119871 119894

119894=1

119909

119894

119891

119894

+ (sum

119873

119894=1

119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ (sum

119873

119894=1

119891

119894

minus sum

119871 119894

119894=1

119891

119894

)

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=1

119891

119894

119910min =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

(A7)


119894


119910

119897

(119896) =

sum

119896

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119896+1

119909

119894

119891

119894

sum

119896

119894=1

119891

119894

+ sum

119873

119894=119896+1

119891

119894

(A8)


119910

119897

(119896) = (

119896

sum

119894=1

119909

119894

119891

119894

+

119873

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

+

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

minus

119871 119894

sum

119894=119896+1

119909

119894

119891

119894

)

times (

119896

sum

119894=1

119891

119894

+

119873

sum

119894=119896+1

119891

119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894119894

+

119871 119894

sum

119894=119896+1

119891

119894

minus

119871 119894

sum

119894=119896+1

119891

119894

)

minus1

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

+ sum

119871 119894

119894=119896+1

119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

119891

119894

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

+ sum

119871 119894

119894=119896+1

119891

119894

minus sum

119871 119894

119894=119896+1

119891

119894

119910

119897

(119896) =

sum

119871 119894

119894=1

119909

119894

119891

119894

+ sum

119873

119894=119871 119894+1119909

119894

119891

119894

minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

sum

119871 119894

119894=1

119891

119894

+ sum

119873

119894=119871 119894+1119891

119894

minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A9)

119910

119897

(119896) =

119863

119871 119894minus sum

119871 119894

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875

119871 119894minus sum

119871 119894

119894=119896+1

(119891

119894

minus 119891

119894

)

(A10)


119871 119894and119875



119910

119897

(119896) =

119863

119871 119894minus 119863 (119896)

119875

119871 119894minus 119875 (119896)

119863 (119896) =

119871 119894

sum

119894=119896+1

119909

119894

(119891

119894

minus 119891

119894

)

119875 (119896) =

119871 119894

sum

119894=119896+1

(119891

119894

minus 119891

119894

)

(A11)


119863

119896minus1

= 119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

) (A12)

119875

119896minus1

= 119875

119896

minus (119891

119896

minus 119891

119896

) (A13)

119910

119897

(119896 minus 1) =

119863

119896minus1

119875

119896minus1

=

119863

119896

minus 119909

119896

(119891

119896

minus 119891

119896

)

119875

119896

minus (119891

119896

minus 119891

119896

)

(A14)

Since 119871 le 119871

119894

given that 119910

119897


119894



119894

So by performing



119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119894

that is

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

119891 = (119891

1

119891

119873

) = (119891

1

119891

119896minus1

119891

119896

119891

119896+1

119891

119873

)

(A15)


if 119909

119896

lt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

decreases(A16)

If 119909

119896

gt 119910 (119891

1

119891

119873

) then 119910 (119891

1

119891

119873

) increases

when 119891

119896

increases(A17)


119896

gt 119910

119871


119896

gt 119910

119897


119896


119897



119897


119897




References




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Research Article A Proposal to Speed up the Computation of ...downloads.hindawi.com › journals › afs › 2013 › 158969.pdf · use of interval type- fuzzy computations [ , ]