research article algorithm for target recognition based on interval-valued...

Research ArticleAlgorithm for Target Recognition Based on Interval-ValuedIntuitionistic Fuzzy Sets with Grey Correlation

Xuan Huang12 Lihong Guo1 Jiang Li1 and Yang Yu1

1Changchun Institute of Optics Fine Mechanics and Physics Chinese Academy of Sciences Changchun 130033 China2University of Chinese Academy of Sciences Beijing 130039 China

Correspondence should be addressed to Lihong Guo guolhciompaccn

Received 2 January 2016 Revised 30 March 2016 Accepted 21 April 2016

Academic Editor Anna M Gil-Lafuente

Copyright copy 2016 Xuan Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to improve exact recognition ratios for aerial targets this paper presents a novel algorithm for target recognition basedon interval-valued intuitionistic fuzzy sets with grey correlation Drawbacks of some previously proposed methods are analyzedand then a novel algorithm is presented Recognition matrix of an aerial target is established first Every entry associated with thematrix is an interval-valued intuitionistic fuzzy number which is composed of interval-valued membership and nonmembershiprepresenting the relation of the target to one category in terms of one characteristic parameter Then grey correlation theory isused to analyze the recognition matrix to obtain the grey correlation degree of this unknown target to every category 200 setsof target recognition data are used to compare the proposed algorithm with traditional methods Experimental results verify thatthe correct recognition ratio can be up to 995 that satisfies the expectations which shows the proposed algorithm can solve thetarget recognition problems better The proposed algorithm can be used to solve the uncertain inference problems such as targetrecognition threat assessment and decision making

1 Introduction

Target recognition is the recognition and classification for theidentity and characteristics of one target Typically it is in thefirst class level of JDL information fusionmodelWhether theground defense systems can recognize aerial targets quicklyreliably and accurately would directly affect the situationevaluation and threat assessment and finally has an influenceon the allocation of defense power

Target recognition is to confirm the category and identityof an unknown target according to its feature parametersDue to the uncertainties in feature values most of targetrecognition methods were developed typically based on theuncertainty theory for example D-S evidence [1 2] Bayesianinference [3 4] attributemeasure [5] and fuzzy set [6] How-ever these methods describe uncertainty information withdrawbacks that resulted in the errors Also they may highlydepend on the a priori knowledge Therefore these methodsmay lead to low recognition accuracy in inference results

Correlation analysis is a method that typically has a lowrequirement in the size of samples It does not need any

classical distribution laws in computation Its results canmatch quite well the qualitative analysis Therefore applyinggrey correlation theory in target recognition can guaranteethe accuracy and real-time characteristics In [7 8] greycorrelation analysis is used for target recognition by adoptingthe normalization of feature parameters to construct greycorrelation coefficients and then giving a grey correlationsequence The results show that the fairly accurate recogni-tion ratios are obtained But this method needs to comparethe unknown target with the data chosen from the databaseIf some data in the database is not accurate enough therecognition results would be incorrect

In recent times intuitionistic fuzzy set (IFS) [9 10]is widely used in various areas such as multiple featuredecisions [11] information fusion [12] and image processing[13] On the basis of Zadehrsquos fuzzy set (ZFS) nonmembershipfunction is added in IFS such that the utility of membershipnonmembership and intuitionistic exponential can respec-tively signify the supportive objective and neutral character-isticsThus IFS is able to describe the uncertainty information

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3408191 9 pageshttpdxdoiorg10115520163408191

2 Mathematical Problems in Engineering

more flexibly and superiorly In [14] IFS is introduced toaccomplish target recognition through establishing a seriesof intuitionistic fuzzy inference rules While IFS has quitehigh confidence the issue becomes significantly difficult ifthere are numerous feature parameters The likelihood ofldquocombination explosionrdquo enables the process complicated andimpractical on real-world problems

Due to the complexity and uncertainty in practice thevalues of membership and nonmembership mostly could notbe represented by the real values Consequently Atanassovextended the intuitionistic fuzzy sets to the interval-valuedintuitionistic fuzzy sets [15 16] using an interval to expressthe membership and nonmembership The nature of uncer-tainty and fuzzy sets in objective things is described in moredetail by using this new method Interval-valued intuition-istic fuzzy set is also applied to various areas for examplecharacteristic decision [17] logistic programming [18] andpattern recognition [19]

Therefore in order to improve exact recognition ratios foraerial targets and avoid the ldquocombination explosionrdquo problemcaused by intuitionistic fuzzy inference this paper proposes anewmethod based on the interval-valued intuitionistic fuzzysets with grey correlation

Then the rest of paper is set out as follows In Section 2target recognition problem is described and limitations ofknown approaches to target recognition are analyzed InSection 3 a novel algorithm based on interval-valued intu-itionistic fuzzy sets with grey correlation is proposed whichis free of these limitations Recognition matrix of an aerialtarget is established first and then grey correlation theoryis used to analyze the recognition matrix In Section 4 theproposed algorithm is adopted in different target recognitionexperimentally and compared with other previous algo-rithms Finally the concluding section summarizes the paper

2 Description of Target Recognition

Assuming that there are 119899 categories of recognized targetsfor each of which 119896 feature parameters are gained by usingmultiple sensors All 119896 feature parameterswould be combinedto form a feature vector that describes the target In thispaper U

1= 1199061 1199062 119906

119899 and U

2= 1198861 1198862 119886

119896 are

used to represent the target category set and target featureparameter set respectively Viameasuring unknown targetXthe measurements of X = 119909

1 1199092 119909

119896 can be obtained

The so-called target recognition is actually that the featureparameter vector formed by measurements can be clusteredto the most associated similar target category Hence thetarget recognition issue can be seen as a decision issue Bycomputing and analyzing the feature parameter X the mostmatching target category 119906

119894is directly chosen from target

category set as the decisive resultNote that the targets investigated in this paper are aerial

targets In order to make recognizing targets typicality andto enhance the recognition efficiency target categories arerespectively bomber (B as 119906

1) gunship (G as 119906

2) tactical

ballistic missile (TBM as 1199063) cruise missile (CM as 119906

4)

and air-to-ground guided missile (AGM as 1199065) In these five

target categories the bomber has a large volume of missilesand significantly high attacking capability while gunship isof low penetration and speed even if with high attackingcapability The tactical ballistic missile can have a very fastspeed and should be intercepted in a very short time Also thecruise missile has a low speed during the time of cruise whilethe air-to-ground guidedmissile can be significantly fast witha small radar cross section area

As every target selected in this paper has its own promi-nent characteristics the target feature parameter set can beset as follows

U2= 119878119867119881

119867 119881119881 119860 (1)

where 119878 represents the radiation cross section area of radar119867 is the height of target 119881

119867indicates the cruise speed 119881

119881

indicates the vertical speed and119860 represents the accelerationof target

Recognizing an aerial target means the access to thefeature parameters such as radiation cross section area heightof target speed and acceleration from sensors and via theanalyses and comparisons of these parameters the member-ship of this target to each category is identified resulting insuggesting the specific category the target should belong to

Grey correlation analysis method is applied to solvethe target recognition problems Guan and He [7] use thismethod to solve the radar emitter recognition problems andLin et al [8] solve the radiation source recognition problemsbased on its improved method

The observing vector of one unknown target in termsof each characteristic parameter can be denoted by 119883

0=

1198830(119895) | 119895 = 1 2 5 Let 119872 data sets that are known

in the target recognition database be 119883119898= 119883119898(119895) | 119895 =

1 2 5 (119898 = 1 2 119872) Using grey correlation methodto recognize target the grey correlation degree betweenunknown target 119883

0and each known data 119883

119898should be

calculated as follows

120585119898(119895)

=

min119898min119895

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816+ 120588max

119898max119895

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816+ 120588max

119898max119895

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816

(2)

where 120588 is the resolution coefficientThen the known data with highest grey correlation degree

can be correlated with the unknown target1198830

This approach using grey correlation is simple andeffective to resolve the issue of some default characteristicparameters This method however needs a lot of data in thetarget recognition database such that the final recognitionresults heavily depend on the existing knowledge Thereforeif some data in the database is not accurate enough therecognition results would be incorrect

Another common method applied in target recognitionproblems is intuitionistic fuzzy inference which is proposedby Lei et al [14]They first applies IFS to describe each charac-teristic parameter The membership function and nonmem-bership function are chosen based on which the inferencerules can be established Then combining all inference rules

Mathematical Problems in Engineering 3

terminally yields the target recognition results The inferencerules are established as follows

If and Then (3)

Using the membership function and nonmembershipfunction IFS can describe uncertain information quite wellAnd the credible inference rules can be used to recognizetarget accurately Thus more credible inference rules lead tomore accurate recognition results However large numbersof categories of target characteristic parameters and intuitionfuzzy sets result in the multiple increasing on the fuzzyinference rules such that the issue of ldquocombinatorial explo-sionrdquo correspondingly comes out Moreover in practice themembership or nonmembership may not be denoted by realvalues which is a constraint of the proposed method Forsolving this drawback the interval-valued intuitionistic fuzzysets can be applied to describe the uncertain information

Therefore in order to combine the advantages of thesetwo methods while improving their drawbacks a novelalgorithm or method should be proposed The new methodshould describe uncertain feature and uncertain informationjust like the IFS method Furthermore interval-valued intu-itionistic fuzzy sets can describe uncertain information betterOn the other hand the new method should have the abilityto analyze uncertain information as grey correlation analysismethod So the algorithm for target recognition based oninterval-valued intuitionistic fuzzy sets with grey correlationis proposed

3 Algorithm of Interval-Valued IntuitionisticFuzzy Sets with Grey Correlation

This section presents an algorithm for target recognitionby combining interval-valued intuitionistic fuzzy sets withgrey correlation When using the proposed algorithm torecognize an unknown target one should first establish atarget recognition matrix composed of interval-valued intu-itionistic fuzzy numbers Each interval-valued intuitionisticfuzzy number represents the interval-valued membershipand nonmembership of the unknown target to one categoryin terms of one characteristic parameter In order to realizethe recognition of this target grey correlation method isused to analyze the target recognition matrix Then greycorrelation sequence is subsequently obtained Figure 1 showsthe flow chart of the proposed algorithm

31 Interval-Valued Intuitionistic Fuzzy Numbers Assumingthat X is a constant domain we say that

A = ⟨119909 120583119860(119909) 120574

119860(119909)⟩ | 119909 isin X (4)

is an intuitionistic fuzzy set on X In this set 120583119860 X rarr [0 1]

and 120574119860 X rarr [0 1] respectively represent the membership

and nonmembership functions from 119909 toA Moreover for all119909 isin X in A 0 ⩽ 120583

119860(119909) + 120574

119860(119909) ⩽ 1 holds true

Due to the uncertainty of objective things it is reallydifficult that 120583

119860(119909) and 120574

119860(119909) can be expressed by exact real

values Therefore the interval values are introduced which

Target feature parameters

Target recognition matrix

Grey correlation analysis

Recognition result

Figure 1 Flowchart of algorithm for target recognition

means that 120583119860(119909) and 120574

119860(119909) are given by the interval values

in the interval of [0 1] Furthermore for any 119909 isin X 0 ⩽

sup 120583119860(119909) + sup 120574

119860(119909) ⩽ 1 holds true

Therefore the sequential intervals formed by member-ship and nonmembership intervals are called interval-valuedintuitionistic fuzzy number which is ([119886 119887] [119888 119889]) where[119886 119887] sub [0 1] and [119888 119889] sub [0 1] 119887 + 119889 ⩽ 1

32 Interval-Valued Intuitionistic Fuzzy Recognition MatrixThe target recognition matrix of an unknown target isconstructed as follows

R = (120572119894119895)119899times119896

(119894 = 1 2 119899 119895 = 1 2 119896) (5)

where 120572119894119895is an interval-valued intuitionistic fuzzy number

which is composed of the interval-valued membership andthe interval-valued nonmembership of the unknown targetto category 119906

119894in terms of parameter 119886

119895 It is shown that

120572119894119895= ([inf 120583

119894119895 sup 120583

119894119895] [inf 120574

119894119895 sup 120574

119894119895]) (6)

where inf 120583119894119895and sup 120583

119894119895 respectively represent the lower

bound and upper bound of membership interval of anunknown target to category 119906


119895and

inf 120574119894119895and sup 120574

119894119895 respectively represent the lower bound

and upper bound of nonmembership interval of the target tocategory 119906


119895

For instance 120572119894119895

= ([06 08] [01 02]) indicates viathe feature parameter 119886

119895the likelihood of recognizing an

unknown target as 119906119894is between 60 and 80 while the

likelihood that the target does not belong to 119906119894is between 10

and 20The selection of upper and lower bounds is a significantly

key part for the construction of target matrix For example120583TBM(119881119867) represents the membership of an unknown targetto TBM in terms of the cruise speed 119881

119867 and 120574TBM(119881119867) rep-

resents the nonmembership of the unknown target to TBM


in terms of the cruise speed 119881119867 Let 119881

119867lie in [0 3000ms]

as the speed of TBM is very fast which is typically 4ndash6times the speed of sound By considering the nonlinearity ofmembership and nonmembership functions the piecewisefunctions are applied to solve the upper bound and lowerbound of 120583TBM(119881119867) and 120574TBM(119881119867) as follows

inf 120583TBM (119881119867)

=

119881119867

5000

119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 02 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 06 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 08 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 05 2000ms lt 119881119867⩽ 3000ms

sup 120583TBM (119881119867)

=

119881119867

10000

+ 02 119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 03 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 07 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 09 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 06 2000ms lt 119881119867⩽ 3000ms

inf 120574TBM (119881119867)

=

minus

119881119867

10000

+ 07 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 06 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 02 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 03 2000ms lt 119881119867⩽ 3000ms

sup 120574TBM (119881119867)

=

minus

119881119867

5000

+ 09 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 07 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 03 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

+ 01 1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 04 2000ms lt 119881119867⩽ 3000ms

(7)

When the cruise speed of one aerial target is knownbased on it the membership and nonmembership interval

can be calculated according to (7) For example when 119881119867=

2000ms (8) is obtained as follows

120572TBM-119881119867 = ([05 06] [03 04]) (8)

Therefore when the cruise speed is 2000ms accordingto (8) the likelihood of recognizing this target as TBM via119881

119867

is between 50 and 60 while the impossibility is between30 and 40

Besides that the TBM has the significant characteristicsof small radiation cross section area high vertical speedand large height and acceleration According to those char-acteristics the upper and lower bounds of memberships andnonmemberships of 120572TBM-119878 120572TBM-119867 120572TBM-119881119881 and 120572TBM-119860 canbe found typically

Also the piecewise functions are applied to establish theupper bound and lower bound of 120583AGM(119867) and 120574AGM(119867) as

inf 120583AGM (119867)

=

0 119867 ⩽ 3000m(119867 minus 3000)

10000

3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 08 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 06 119867 ⩾ 17000m

sup 120583AGM (119867)

=

01 119867 ⩽ 3000m(119867 minus 3000)

10000

+ 01 3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 09 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 07 119867 ⩾ 17000m

inf 120574AGM (119867)

=

08 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 08 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 015 119867 gt 17000m

sup 120574AGM (119867)

=

09 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 09 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

+ 01 11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 025 119867 gt 17000m

(9)

Similarly the memberships and nonmemberships ofother categories can be established by the same method


This section would not present the details due to the paperspace Hence all of the obtained interval-valued intuitionisticfuzzy numbers form the target recognition matrices of theunknown target

33 Grey Correlation In this part grey correlation theory isused to analyze the recognition matrix established above

First positive desired recognition strategy and negativedesired recognition strategy of an unknown target arerespectively built up The positive desired recognition strat-egy is composed of the maximal number of each column ofthe recognition matrix and the negative desired recognitionstrategy is composed of the minimal number of everycolumn of the recognition matrix So the score function andaccuracy function are used to compare the interval-valuedintuitionistic fuzzy numbers

Second grey correlation coefficient matrix of theunknown target is built up Each grey correlation coefficientis the relationship between each entry in the recognitionmatrix and each entry of desired recognition strategyAccording to grey correlation analysis method the greycorrelation coefficient of every entry of the recognitionmatrix is calculated

Finally grey correlation sequence of the unknown targetis built upGrey correlation sequence is composed of grey cor-relation degrees Via the grey correlation coefficient matrixthe grey correlation degree of the unknown target to eachcategory is calculated The category which has the maximalgrey correlation degree should be selected as the optimalrecognition category

331 Desired Recognition Strategy

Definition 1 (score function and accuracy function) Let120572 = ([119886 119887] [119888 119889]) be an interval-valued intuitionistic fuzzynumber Its score function is defined asΔ(120572) = (119886minus119888+119887minus119889)2while the so-called accuracy function is119867(120572) = (119886 + 119888 + 119887 +

119889)2

Definition 2 Assume that 1205721and 120572

2are two interval-valued

intuitionistic fuzzy numbers If Δ(1205721) lt Δ(120572

2) then 120572

1lt 1205722

In particular when Δ(1205721) = Δ(120572

2) if 119867(120572

1) lt 119867(120572

2) then

1205721lt 1205722holds

Definition 3 (positive and negative desired recognition strate-gies) The interval-valued intuitionistic fuzzy vectors are asfollows

119909+= 120572+

1 120572+

2 120572

+

119896

119909minus= 120572minus

1 120572minus

2 120572

minus

119896

(10)

which are respectively positive and negative desired recogni-tion strategies of the target recognition matrix (120572

119894119895)119899times119896

if andonly if

120572+

119895= max119894

120572119894119895 (119895 = 1 2 119896)

120572minus

119895= min119894

120572119894119895 (119895 = 1 2 119896)

(11)

hold true

From the definitions above the scales of two interval-valued intuitionistic fuzzy numbers can be determineddirectly by the score and accuracy functions It is typicalthat the associated fuzzy number would be larger with thelarger score function while if the score functions of twofuzzy numbers are equal then the larger fuzzy sets are is onlydependent on the larger accuracy function

Based on the determined rules described above to solvethe positive or negative desired strategy is necessarily toextract themaximal orminimal interval-valued intuitionisticfuzzy number in each column from the target recognitionmatrix

332 Grey Correlation Coefficient Matrix After obtainingthe positive and negative desired recognition strategies thispart calculates the grey correlation coefficient between eachentry in the target recognition matrix and each entry of thestrategy

Each grey correlation coefficient represents the proximitybetween each entry in the target recognition matrix and eachentry of the strategy The closer 120572

119894119895and 120572+

119895are the farther 120572

119894119895

and 120572minus119895are and the better will be

As |1198830(119895) minus 119883

119898(119895)| in (2) the proximity between 120572

119894119895and

120572+

119895(120572minus119895) should be calculated first Let119863+

119894119895(119863minus

119894119895) represent the

distance Use the norms of interval values to calculate thedistance as follows

119863+

119894119895= 119863+

119894119895(120572119894119895 120572+

119895) =

1

4

(

10038161003816100381610038161003816inf V minus inf 120583+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120583119894119895minus sup 120583+

119895

10038161003816100381610038161003816+

10038161003816100381610038161003816inf 120574119894119895minus inf 120574+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120574119894119895minus sup 120574+

119895

10038161003816100381610038161003816)

119863minus

119894119895= 119863minus

119894119895(120572119894119895 120572minus

119895) =

1

4

(

10038161003816100381610038161003816inf 120583119894119895minus inf 120583minus

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120583119894119895minus sup 120583minus

119895

10038161003816100381610038161003816+


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816)

(12)

Then distance matrix (119863+

119894119895)119899times119896

consists of all of thedistance119863+

119894119895between 120572

119894119895and 120572+

119895 (119863minus119894119895)119899times119896

consists of all of thedistance119863minus

119894119895between 120572

119894119895and 120572minus

119895

And then 120585+119894119895is defined as grey correlation coefficient

between 120572119894119895and 120572

+

119895 And 120585

minus

119894119895is coefficient between 120572

119894119895and

120572minus

119895 Using grey system theory and the distance matrices the

formula of grey correlation coefficients can be obtained asfollows

120585+

119894119895=

min119894119895119863+

119894119895+ 120588max

119894119895119863+

119894119895

119863+

119894119895+ 120588max

119894119895119863+

119894119895

(13)

120585minus

119894119895=

min119894119895119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

(14)

where 120588 is the recognition coefficient that is typically 05


In (13) max119894119895119863+

119894119895and min

119894119895119863+

119894119895 respectively represent

the maximal and minimal element of the distance matrix(119863+

119894119895)119899times119896

In (14) max119894119895119863minus

119894119895and min

119894119895119863minus

119894119895 respectively repre-

sent themaximal andminimal element of the distancematrix(119863minus

119894119895)119899times119896

All of grey correlation coefficients would form the grey

correlation coefficient matrices namely that are (120585+119894119895)119899times119896

and(120585minus

119894119895)119899times119896

333 Grey Correlation Sequence In this part the grey corre-lation degree between each target category 119906

119894and the positive

or the negative desired strategy is calculated By using the greycorrelation coefficient matrices the grey correlation degrees120582+

119894and 120582minus

119894 are calculated respectively as follows

120582+

119894=

119896

sum

119895=1

120585+

119894119895119886119895

120582minus

119894=

119896

sum

119895=1

120585minus

119894119895119886119895

(15)

where 119886119895represents the weight of the feature parameter and

it follows the properties of 119886119895⩾ 0 and sum119896

119895=1119886119895= 1

The special case is that 119886119895is 1119896 when the importance of

each feature parameter is treated to be equal which is natu-rally the averaged case In practice the selection of weightshowever enables the sumof errors to beminimal between thegrey correlation degree of each target category and the greycorrelation degree of desired recognition strategy namelywhich follows the optimization model

min 119865 (119886) =

119899

sum

119894=1

119896

sum

119895=1

[(1 minus 120585+

119894119895) 119886119895]

2

st119896

sum

119895=1

119886119895= 1

119886119895⩾ 0

(16)

By solving this model we have

119886119895=

sum119896

119895=1[sum119899

119894=1(1 minus 120585

+

119894119895)

2

]

minus1

minus1

sum119899

119894=1(1 minus 120585

+

119894119895)

2

(17)

Via (15)sim(17) the grey correlation degree of each targetcategory can be obtained While 120582+

119894becomes larger and 120582minus

119894

becomes smaller the target category119906119894is closer to the positive

desired recognition strategy andmore keeping away from thenegative desired recognition strategy

Introduce another definition associated with comprehen-sive grey correlation degree as follows

119889119894=

(120582+

119894)

2

(120582+

119894)2

+ (120582minus

119894)2 (18)

Target featureparameters

Target recognitionmatrix

Grey correlationcoefficient matrix

Grey correlation degrees and grey correlation sequences

Positive and negative desired recognition strategies

Figure 2 Flowchart of software for target recognition

which is typically used to represent the level that 119906119894belongs to

the positive and negative desired recognition strategies Thiscomprehensive grey correlation degree also indicates that thetarget category 119906

119894belongs to the positive desired recognition

strategy with the level of 119889119894 while belonging to the negative

desired recognition strategy with the level of 1 minus 119889119894 It is

furthermore seen that the confidence increases that the targetis recognized as 119906

119894 when 119889

119894increases

Then the grey correlation sequence of the unknowntarget can be obtained by descending the comprehensive greycorrelation degrees According to the maximal correlationrecognition the category which has the maximal compre-hensive grey correlation would be recognized as the targetcategory associated with the unknown target

4 Application of the Proposed Algorithm

41 Target Recognition Processing Software Based on theproposed algorithm presented before a target recognitionprocessing software is established by plugging the mem-bership and nonmembership functions into the softwareThen defining the feature parameters of targets as inputsthe target recognition matrices positive and negative desiredrecognition strategies grey correlation coefficient matricesand grey correlation degrees can be subsequently calculatedAnd after that the final output is the grey correlation sequenceof each target category The final recognition strategy isthe maximal grey correlation sequence that is the tool todetermine the target category Figure 2 shows the flow chartof the proposed software

In order to validate the efficacy and stability of theproposed algorithm this paper uses 200 sets of featureparameters selected randomly according to the uniform dis-tribution from the database as inputs to testify the processingcapability of the softwareThese 200 sets of feature parameterscorrespond respectively to the proposed 5 target categorieswhich means every 40 sets of data would be for each targetcategory Part of measured raw data is shown in Table 1


Consider the 1st feature parameter vector as an examplein Table 1 as follows

05 35000 1800 2000 350 (19)

which represents that the radiation cross section area ofthis target is 05m2 the height of the target is 35000mits cruise speed is 1800ms its vertical speed is 2000msand acceleration is 350ms2 This section uses this vector as

an example to illustrate the algorithm process of the targetrecognition processing software

(1) Target Recognition Matrix Using the mentioned mem-bership and nonmembership interval-valued intuitionisticfuzzy functions the target recognition matrix of relevance isconstructed as follows

R

=

[

[

[

[

[

[

[

[

[

([005 015] [06 07]) ([035 045] [04 05]) ([011 021] [069 079]) ([01 02] [07 08]) ([01 02] [07 08])

([0 01] [065 075]) ([01 02] [07 08]) ([005 015] [07 08]) ([002 012] [078 088]) ([01 02] [065 075])

([055 065] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02])

([065 075] [0 01]) ([0 01] [07 08]) ([017 027] [058 068]) ([0 01] [075 085]) ([0 01] [07 08])

([06 07] [005 015]) ([03 04] [045 055]) ([05 06] [03 04]) ([06 07] [02 03]) ([08 09] [0 01])

]

]

]

]

]

]

]

]

]

(20)

(2) Desired Recognition Strategies ([065 075] [0 01]) is themaximal entries of the first column of the target recogni-tion matrix And ([07 08] [01 02]) ([07 08] [01 02])([07 08] [01 02]) and ([08 09] [0 01]) are respectivelythe maximal entries of other four columns

([0 01] [065 075]) is the minimal entries of thefirst column of the target recognition matrix And([0 01] [07 08]) ([005 015] [07 08]) ([002 012][078 088]) and ([0 01] [07 08]) are respectively theminimal entries of other four columns

Therefore the positive and negative desired recognitionstrategies can be gained by adopting score function andaccuracy function as follows

119909+= ([065 075] [0 01]) ([07 08] [01 02])

([07 08] [01 02]) ([07 08] [01 02])

([08 09] [0 01])

119909minus= ([0 01] [065 075]) ([0 01] [07 08])

([005 015] [07 08]) ([002 012] [078 088])

([0 01] [07 08])

(21)

(3) Grey Correlation Coefficient MatrixThen (12) are appliedto compute the distance matrices

D+ = (119863+119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

06 0325 059 06 07

065 06 0625 068 0675

01 0 0 0 01

0 065 0505 0675 07

005 0375 02 01 0

]

]

]

]

]

]

]

]

]

Dminus = (119863minus119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

005 0325 0035 008 005

0 005 0 0 0075

055 065 0625 068 065

065 0 012 0025 0

06 0275 0425 058 075

]

]

]

]

]

]

]

]

]

(22)

From (22) the maximal entry of the distance matrix D+is 07 and the minimal entry is 0 So max

119894119895119863+

119894119895= 07 and

min119894119895119863+

119894119895= 0 On the other hand max

119894119895119863minus

119894119895= 075 and

min119894119895119863minus

119894119895= 0

The distance matrices and grey correlation coefficientsyield the grey correlation coefficient matrices of the target asfollows

120585+= (120585+

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

03684 05185 03723 03684 03333

035 03684 03590 03398 03415

07778 1 1 1 07778

1 035 04094 03415 03333

0875 04828 06364 07778 1

]

]

]

]

]

]

]

]

]

120585minus= (120585minus

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

08824 05357 09146 08242 08824

1 08824 1 1 08333

04054 03659 0375 03555 03659

03659 1 07576 09375 1

03846 05769 04688 03927 03333

]

]

]

]

]

]

]

]

]

(23)


Table 1 Part of original data

119878 (m2) 119867 (m) 119881119881(ms) 119881

119867(ms) 119860 (ms2) Recognition

result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065

(4) Grey Correlation Sequence By substituting the correlationcoefficient matrix (23) into (17) the weights of featureparameters are represented by the following vector

a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)

By substituting (24) into (15) the grey correlation degreebetween each target category and the positive desired recog-nition strategy is

120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)

and the grey correlation degree between each target categoryand the negative desired recognition strategy is


119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)

Using (18) the comprehensive correlation degree of eachtarget category can be calculated as follows

119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)

Hence the sequence is 1198893gt 1198895gt 1198894gt 1198891gt 1198892

The comprehensive correlation degree 1198893corresponding to

the target category 1199063is maximal It means that according to

the five feature parameters the unknown target is closest tothe feature parameters of TBM It also demonstrates that thestrategy of recognizing the unknown target as the TBM is thebest

The radiation reflection cross section area of this targetis 1 to 2 magnitudes smaller than that of the airplanewhile the cruise and vertical speeds are comparatively biggerMoreover the height is in a very high degree and acceler-ation is approximately 1 mach Each feature parameter of

Table 2 Comparison of recognition results

Number Algorithm Correct recognitionratio ()

Algorithm 1 Grey correlation 925

Algorithm 2 Intuitionistic fuzzyinference 965

Algorithm 3 The proposedalgorithm 995

the unknown targetmatches quite well the feature parametersof TBM The category of this target in the target recognitiondatabase is consistent with the recognition result by theinterval-valued intuitionistic fuzzy sets with grey correlationalgorithm Such result also signifies that the proposed algo-rithm in this paper has good accuracy and high adaptation

Table 1 lists each recognition result for each target Allof recognition results are consistent with the categories inthe target recognition database Therefore this proposedalgorithm can be of stability in multiple target recognitionproblems

42 Comparisons with Other Algorithms Using the selected200 data sets the proposed algorithm in this paper isrespectively compared to target recognition algorithms basedon grey correlation [7] and intuitionistic fuzzy inference [14]

For intuitionistic fuzzy inference algorithm the fiveparameters in (1) are similarly selected as the feature parame-ters The numbers of membership functions are respectively3 5 6 4 and 4 Therefore in this algorithm the number ofinference rules is up to 3 times 5 times 6 times 4 times 4 = 1440

Table 2 displays the experimental results by using threealgorithms

According to Table 2 for the target recognition issue theproposed algorithm in this paper based on interval-valuedintuitionistic fuzzy sets with grey correlation has a higherrecognition ratio than that of algorithm 1 and algorithm 2This experiment also typically verifies that the algorithm iscapable of dealing with the target recognition problems withuncertain information The recognition results are closer tothe practical cases

Moreover due to the numerous inference rules in algo-rithm 2 while the number of feature parameters increasesthe number of inference rules will correspondingly increasein times In such a case the issue of combination explosioneasily comes up However for the proposed algorithm in thispaper only 25membership and 25 nonmembership functionsare neededThese functions are used for constructing a targetrecognition matrix and the rest of algorithm processingis just element extractions and matrices computation forthe target recognition matrix Therefore comparing withalgorithm 2 algorithm 3 is more concise and simpler inalgorithm construction and software compiling and as wellcan effectively avoid the issue of combination explosion

According to the examples shown above the novelmethod can describe the uncertain feature or uncertain infor-mation quite well On the other hand it has a good ability toanalyze the uncertain information Therefore the proposed


method can be used to deal with the uncertain inferenceproblems such as the target recognition problems the threatassessment problems and the decision making problems Itcan construct a relation between the feature parameters andthe decision results based on the interval-valued intuitionisticfuzzy sets with grey correlation method

5 Conclusion

Thepaper introduces a target recognition algorithm based oninterval-valued intuitionistic fuzzy sets with grey correlationUsing interval-valued intuitionistic fuzzy numbers the targetrecognition matrix of an unknown target is establishedfrom which the positive and negative desired recognitionstrategies are extracted According to grey system theorythe grey correlation degree between the unknown target andeach category is built up to obtain the optimal recognitionstrategies In order to validate this proposed algorithm thetarget recognition software is compiled in which 5 typicaltarget categories and 200 data sets are combined to testifyThe results show that by this proposed algorithm the correctrecognition ratio is up to 995 which is higher than ratiosattained by only using grey correlation algorithmor intuition-istic fuzzy inference algorithm namely that are 925 and965 respectively It means that by adopting this proposedalgorithm the anticipated expectation can be reached Andfurthermore the combination explosion issue which possiblyexists in fuzzy inference algorithm can be effectively avoidedduring the time of target recognition

Therefore the target recognition algorithm based oninterval-valued intuitionistic fuzzy sets with grey correla-tion has comparatively better recognition results It canaccomplish the target recognition quite well and makes upthe drawbacks caused by only using grey correlation orfuzzy inference method Because of the abilities to describeand analyze the uncertain information this method canbe applied in the target recognition problems the threatassessment problems and the decision making problems

In future research the selection of upper and lowerbounds of the membership and nonmembership functionshould be considered deeply It is the significantly key partfor the construction of target matrix The membership andnonmembership function should be more close to the actualsituation If the target matrix is established quite well therecognition results will be more accuracy

Competing Interests

The authors declare that they have no competing interests

References

[1] Y Zhou X Yu M Cui and XWang ldquoRadar target recognitionbased onmultiple features fusionwithDempster-Shafer theoryrdquoin Proceedings of the IEEE 10th International Conference onElectronic Measurement and Instruments (ICEMI rsquo11) vol 1 pp243ndash247 Chengdu China August 2011

[2] T Tian DMing F Jie and B Lei ldquoFusion of multi-measures ininfrared target recognition based on Dempster-Shafer evidence

theoryrdquo in MIPPR Automatic Target Recognition and ImageAnalysis vol 8003 of Proceedings of SPIE pp 1ndash8 2011

[3] A Forman D B Brown and J H Hughen ldquoMUltiSensortarget recognition system (MUSTRS)rdquo inProceedings of the 27thAsilomar Conference on Signals Systems and Computers vol 1pp 263ndash267 IEEE Pacific Grove Calif USA November 1993

[4] WMei G Shan and CWang ldquoModel the uncertainty in targetrecognition using possiblized bayesrsquo theoremrdquo in Proceedings ofthe IEEE International Conference on Fuzzy Systems (FUZZ rsquo12)pp 1ndash3 Brisbane Australia June 2012

[5] X Guan Y He and X Yi ldquoAttribute measure recognitionapproach and its applications to emitter recognitionrdquo Science inChina Series F Information Sciences vol 48 no 2 pp 225ndash2332005

[6] N Li and F Liu ldquoA SAR target recognition method based onview-aspects partitioned by fuzzy clusteringrdquo Acta ElectronicaSinica vol 40 no 2 pp 394ndash399 2012

[7] X Guan and Y He ldquoA novel radar emitter recognition approachbased on grey correlation analysisrdquo Journal of System Simula-tion vol 16 no 11 pp 2601ndash2603 2004

[8] Y Lin X-C Si R-L Zhou and H Yang ldquoApplication ofimproved grey correlation algorithm on radiation source recog-nitionrdquo Journal on Communications vol 31 no 8 pp 166ndash1712010

[9] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[10] X-HYuanH-X Li andC Zhang ldquoThe theory of intuitionisticfuzzy sets based on the intuitionistic fuzzy special setsrdquo Infor-mation Sciences vol 277 no 9 pp 284ndash298 2014

[11] Y Wang Y-J Lei and Y-L Lu ldquoMultiple attribute decisionmaking method based on intuitionistic fuzzy setsrdquo SystemsEngineering and Electronics vol 29 no 12 pp 2060ndash2063 2007

[12] K Zhang X Wang C K Zhang D-Y Zhou and Q FengldquoEvaluating and sequencing of air target threat based on IFEand dynamic intuitionistic fuzzy setsrdquo Systems Engineering andElectronics vol 36 no 4 pp 697ndash701 2014

[13] S Lu Y-J Lei W-W Kong and Y Lei ldquoImage registrationalgorithm based on intuitionistic fuzzy distancerdquo Control andDecision vol 26 no 11 pp 1670ndash1674 2011

[14] Y Lei Y J Lei Y Q Feng and W W Kong ldquoTechniquesfor target recognition based on intuitionistic fuzzy reasoningrdquoControl and Decision vol 26 no 8 pp 1163ndash1168 2011

[15] K T Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[16] Z S Xu ldquoOn correlation measures of intuitionistic fuzzysetsrdquo in Intelligent Data Engineering and Automated LearningmdashIDEAL 2006 vol 4224 of Lecture Notes in Computer Science pp16ndash34 Springer Berlin Germany 2006

[17] Y J Zhang P J Ma X H Su and C P Zhang ldquoMulti-attributedecisionmaking with uncertain attribute weight information inthe framework of interval-valued intuitionistic fuzzy setrdquo ActaAutomatica Sinica vol 38 no 2 pp 220ndash228 2012

[18] H Y Zhang W X Zhang and C L Mei ldquoEntropy of interval-valued fuzzy sets based on distance and its relationship withsimilaritymeasurerdquoKnowledge-Based Systems vol 22 no 6 pp449ndash454 2009

[19] Z S Xu ldquoOn similarity measures of interval-valued intuition-istic fuzzy sets and their application to pattern recognitionsrdquoJournal of Southeast University (English Edition) vol 23 no 1pp 139ndash143 2007

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


more flexibly and superiorly In [14] IFS is introduced toaccomplish target recognition through establishing a seriesof intuitionistic fuzzy inference rules While IFS has quitehigh confidence the issue becomes significantly difficult ifthere are numerous feature parameters The likelihood ofldquocombination explosionrdquo enables the process complicated andimpractical on real-world problems

Due to the complexity and uncertainty in practice thevalues of membership and nonmembership mostly could notbe represented by the real values Consequently Atanassovextended the intuitionistic fuzzy sets to the interval-valuedintuitionistic fuzzy sets [15 16] using an interval to expressthe membership and nonmembership The nature of uncer-tainty and fuzzy sets in objective things is described in moredetail by using this new method Interval-valued intuition-istic fuzzy set is also applied to various areas for examplecharacteristic decision [17] logistic programming [18] andpattern recognition [19]

Therefore in order to improve exact recognition ratios foraerial targets and avoid the ldquocombination explosionrdquo problemcaused by intuitionistic fuzzy inference this paper proposes anewmethod based on the interval-valued intuitionistic fuzzysets with grey correlation

Then the rest of paper is set out as follows In Section 2target recognition problem is described and limitations ofknown approaches to target recognition are analyzed InSection 3 a novel algorithm based on interval-valued intu-itionistic fuzzy sets with grey correlation is proposed whichis free of these limitations Recognition matrix of an aerialtarget is established first and then grey correlation theoryis used to analyze the recognition matrix In Section 4 theproposed algorithm is adopted in different target recognitionexperimentally and compared with other previous algo-rithms Finally the concluding section summarizes the paper

2 Description of Target Recognition

Assuming that there are 119899 categories of recognized targetsfor each of which 119896 feature parameters are gained by usingmultiple sensors All 119896 feature parameterswould be combinedto form a feature vector that describes the target In thispaper U

1= 1199061 1199062 119906

119899 and U

2= 1198861 1198862 119886

119896 are

used to represent the target category set and target featureparameter set respectively Viameasuring unknown targetXthe measurements of X = 119909

1 1199092 119909

119896 can be obtained

The so-called target recognition is actually that the featureparameter vector formed by measurements can be clusteredto the most associated similar target category Hence thetarget recognition issue can be seen as a decision issue Bycomputing and analyzing the feature parameter X the mostmatching target category 119906

119894is directly chosen from target

category set as the decisive resultNote that the targets investigated in this paper are aerial

targets In order to make recognizing targets typicality andto enhance the recognition efficiency target categories arerespectively bomber (B as 119906

1) gunship (G as 119906

2) tactical

ballistic missile (TBM as 1199063) cruise missile (CM as 119906

4)

and air-to-ground guided missile (AGM as 1199065) In these five

target categories the bomber has a large volume of missilesand significantly high attacking capability while gunship isof low penetration and speed even if with high attackingcapability The tactical ballistic missile can have a very fastspeed and should be intercepted in a very short time Also thecruise missile has a low speed during the time of cruise whilethe air-to-ground guidedmissile can be significantly fast witha small radar cross section area

As every target selected in this paper has its own promi-nent characteristics the target feature parameter set can beset as follows

U2= 119878119867119881

119867 119881119881 119860 (1)

where 119878 represents the radiation cross section area of radar119867 is the height of target 119881

119867indicates the cruise speed 119881

119881

indicates the vertical speed and119860 represents the accelerationof target

Recognizing an aerial target means the access to thefeature parameters such as radiation cross section area heightof target speed and acceleration from sensors and via theanalyses and comparisons of these parameters the member-ship of this target to each category is identified resulting insuggesting the specific category the target should belong to

Grey correlation analysis method is applied to solvethe target recognition problems Guan and He [7] use thismethod to solve the radar emitter recognition problems andLin et al [8] solve the radiation source recognition problemsbased on its improved method

The observing vector of one unknown target in termsof each characteristic parameter can be denoted by 119883

0=

1198830(119895) | 119895 = 1 2 5 Let 119872 data sets that are known

in the target recognition database be 119883119898= 119883119898(119895) | 119895 =

1 2 5 (119898 = 1 2 119872) Using grey correlation methodto recognize target the grey correlation degree betweenunknown target 119883

0and each known data 119883

119898should be

calculated as follows

120585119898(119895)

=

min119898min119895

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816+ 120588max

119898max119895

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816+ 120588max

119898max119895

10038161003816100381610038161198830(119895) minus 119883

119898(119895)

1003816100381610038161003816

(2)

where 120588 is the resolution coefficientThen the known data with highest grey correlation degree

can be correlated with the unknown target1198830

This approach using grey correlation is simple andeffective to resolve the issue of some default characteristicparameters This method however needs a lot of data in thetarget recognition database such that the final recognitionresults heavily depend on the existing knowledge Thereforeif some data in the database is not accurate enough therecognition results would be incorrect

Another common method applied in target recognitionproblems is intuitionistic fuzzy inference which is proposedby Lei et al [14]They first applies IFS to describe each charac-teristic parameter The membership function and nonmem-bership function are chosen based on which the inferencerules can be established Then combining all inference rules



If and Then (3)






A = ⟨119909 120583119860(119909) 120574

119860(119909)⟩ | 119909 isin X (4)




119860(119909) + 120574

119860(119909) ⩽ 1 holds true


119860(119909) and 120574






Recognition result


means that 120583119860(119909) and 120574



sup 120583119860(119909) + sup 120574

119860(119909) ⩽ 1 holds true



R = (120572119894119895)119899times119896

(119894 = 1 2 119899 119895 = 1 2 119896) (5)





120572119894119895= ([inf 120583

119894119895 sup 120583

119894119895] [inf 120574

119894119895 sup 120574

119894119895]) (6)

where inf 120583119894119895and sup 120583




119895and

inf 120574119894119895and sup 120574




119895

For instance 120572119894119895







119867 and 120574TBM(119881119867) rep-




119867lie in [0 3000ms]


inf 120583TBM (119881119867)

=

119881119867

5000

119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 02 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 06 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 08 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 05 2000ms lt 119881119867⩽ 3000ms

sup 120583TBM (119881119867)

=

119881119867

10000

+ 02 119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 03 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 07 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 09 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 06 2000ms lt 119881119867⩽ 3000ms

inf 120574TBM (119881119867)

=

minus

119881119867

10000

+ 07 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 06 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 02 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 03 2000ms lt 119881119867⩽ 3000ms

sup 120574TBM (119881119867)

=

minus

119881119867

5000

+ 09 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 07 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 03 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

+ 01 1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 04 2000ms lt 119881119867⩽ 3000ms

(7)




120572TBM-119881119867 = ([05 06] [03 04]) (8)


119867




inf 120583AGM (119867)

=

0 119867 ⩽ 3000m(119867 minus 3000)

10000

3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 08 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 06 119867 ⩾ 17000m

sup 120583AGM (119867)

=

01 119867 ⩽ 3000m(119867 minus 3000)

10000

+ 01 3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 09 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 07 119867 ⩾ 17000m

inf 120574AGM (119867)

=

08 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 08 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 015 119867 gt 17000m

sup 120574AGM (119867)

=

09 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 09 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

+ 01 11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 025 119867 gt 17000m

(9)










119889)2




2) then 120572

1lt 1205722


2) if 119867(120572

1) lt 119867(120572

2) then

1205721lt 1205722holds


119909+= 120572+

1 120572+

2 120572

+

119896


1 120572minus

2 120572

minus

119896

(10)


119894119895)119899times119896

if andonly if

120572+

119895= max119894

120572119894119895 (119895 = 1 2 119896)

120572minus

119895= min119894

120572119894119895 (119895 = 1 2 119896)

(11)

hold true





119894119895and 120572+


119894119895


As |1198830(119895) minus 119883


119894119895and

120572+


119894119895(119863minus



119863+

119894119895= 119863+

119894119895(120572119894119895 120572+

119895) =

1

4

(

10038161003816100381610038161003816inf V minus inf 120583+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120583119894119895minus sup 120583+

119895

10038161003816100381610038161003816+

10038161003816100381610038161003816inf 120574119894119895minus inf 120574+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120574119894119895minus sup 120574+

119895

10038161003816100381610038161003816)

119863minus

119894119895= 119863minus

119894119895(120572119894119895 120572minus

119895) =

1

4

(


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816+


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816)

(12)


119894119895)119899times119896


119894119895between 120572

119894119895and 120572+

119895 (119863minus119894119895)119899times119896


119894119895between 120572

119894119895and 120572minus

119895


between 120572119894119895and 120572

+

119895 And 120585

minus


119894119895and

120572minus



120585+

119894119895=

min119894119895119863+

119894119895+ 120588max

119894119895119863+

119894119895

119863+

119894119895+ 120588max

119894119895119863+

119894119895

(13)

120585minus

119894119895=

min119894119895119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

(14)



In (13) max119894119895119863+

119894119895and min

119894119895119863+



119894119895)119899times119896

In (14) max119894119895119863minus

119894119895and min

119894119895119863minus



119894119895)119899times119896



and(120585minus

119894119895)119899times119896




119894and 120582minus


120582+

119894=

119896

sum

119895=1

120585+

119894119895119886119895

120582minus

119894=

119896

sum

119895=1

120585minus

119894119895119886119895

(15)



119895=1119886119895= 1



min 119865 (119886) =

119899

sum

119894=1

119896

sum

119895=1

[(1 minus 120585+

119894119895) 119886119895]

2

st119896

sum

119895=1

119886119895= 1

119886119895⩾ 0

(16)


119886119895=

sum119896

119895=1[sum119899

119894=1(1 minus 120585

+

119894119895)

2

]

minus1

minus1

sum119899

119894=1(1 minus 120585

+

119894119895)

2

(17)



119894




119889119894=

(120582+

119894)

2

(120582+

119894)2

+ (120582minus

119894)2 (18)













119894 when 119889

119894increases







05 35000 1800 2000 350 (19)




R

=

[

[

[

[

[

[

[

[

[

([005 015] [06 07]) ([035 045] [04 05]) ([011 021] [069 079]) ([01 02] [07 08]) ([01 02] [07 08])

([0 01] [065 075]) ([01 02] [07 08]) ([005 015] [07 08]) ([002 012] [078 088]) ([01 02] [065 075])

([055 065] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02])

([065 075] [0 01]) ([0 01] [07 08]) ([017 027] [058 068]) ([0 01] [075 085]) ([0 01] [07 08])

([06 07] [005 015]) ([03 04] [045 055]) ([05 06] [03 04]) ([06 07] [02 03]) ([08 09] [0 01])

]

]

]

]

]

]

]

]

]

(20)




119909+= ([065 075] [0 01]) ([07 08] [01 02])

([07 08] [01 02]) ([07 08] [01 02])

([08 09] [0 01])

119909minus= ([0 01] [065 075]) ([0 01] [07 08])

([005 015] [07 08]) ([002 012] [078 088])

([0 01] [07 08])

(21)


D+ = (119863+119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

06 0325 059 06 07

065 06 0625 068 0675

01 0 0 0 01

0 065 0505 0675 07

005 0375 02 01 0

]

]

]

]

]

]

]

]

]


=

[

[

[

[

[

[

[

[

[

005 0325 0035 008 005

0 005 0 0 0075

055 065 0625 068 065

065 0 012 0025 0

06 0275 0425 058 075

]

]

]

]

]

]

]

]

]

(22)


119894119895119863+

119894119895= 07 and

min119894119895119863+


119894119895119863minus

119894119895= 075 and

min119894119895119863minus

119894119895= 0


120585+= (120585+

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

03684 05185 03723 03684 03333

035 03684 03590 03398 03415

07778 1 1 1 07778

1 035 04094 03415 03333

0875 04828 06364 07778 1

]

]

]

]

]

]

]

]

]


119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

08824 05357 09146 08242 08824

1 08824 1 1 08333

04054 03659 0375 03555 03659

03659 1 07576 09375 1

03846 05769 04688 03927 03333

]

]

]

]

]

]

]

]

]

(23)



119878 (m2) 119867 (m) 119881119881(ms) 119881


result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065


a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)


120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)



119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)


119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)





















5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











If and Then (3)






A = ⟨119909 120583119860(119909) 120574

119860(119909)⟩ | 119909 isin X (4)




119860(119909) + 120574

119860(119909) ⩽ 1 holds true


119860(119909) and 120574






Recognition result


means that 120583119860(119909) and 120574



sup 120583119860(119909) + sup 120574

119860(119909) ⩽ 1 holds true



R = (120572119894119895)119899times119896

(119894 = 1 2 119899 119895 = 1 2 119896) (5)





120572119894119895= ([inf 120583

119894119895 sup 120583

119894119895] [inf 120574

119894119895 sup 120574

119894119895]) (6)

where inf 120583119894119895and sup 120583




119895and

inf 120574119894119895and sup 120574




119895

For instance 120572119894119895







119867 and 120574TBM(119881119867) rep-




119867lie in [0 3000ms]


inf 120583TBM (119881119867)

=

119881119867

5000

119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 02 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 06 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 08 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 05 2000ms lt 119881119867⩽ 3000ms

sup 120583TBM (119881119867)

=

119881119867

10000

+ 02 119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 03 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 07 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 09 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 06 2000ms lt 119881119867⩽ 3000ms

inf 120574TBM (119881119867)

=

minus

119881119867

10000

+ 07 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 06 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 02 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 03 2000ms lt 119881119867⩽ 3000ms

sup 120574TBM (119881119867)

=

minus

119881119867

5000

+ 09 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 07 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 03 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

+ 01 1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 04 2000ms lt 119881119867⩽ 3000ms

(7)




120572TBM-119881119867 = ([05 06] [03 04]) (8)


119867




inf 120583AGM (119867)

=

0 119867 ⩽ 3000m(119867 minus 3000)

10000

3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 08 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 06 119867 ⩾ 17000m

sup 120583AGM (119867)

=

01 119867 ⩽ 3000m(119867 minus 3000)

10000

+ 01 3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 09 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 07 119867 ⩾ 17000m

inf 120574AGM (119867)

=

08 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 08 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 015 119867 gt 17000m

sup 120574AGM (119867)

=

09 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 09 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

+ 01 11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 025 119867 gt 17000m

(9)










119889)2




2) then 120572

1lt 1205722


2) if 119867(120572

1) lt 119867(120572

2) then

1205721lt 1205722holds


119909+= 120572+

1 120572+

2 120572

+

119896


1 120572minus

2 120572

minus

119896

(10)


119894119895)119899times119896

if andonly if

120572+

119895= max119894

120572119894119895 (119895 = 1 2 119896)

120572minus

119895= min119894

120572119894119895 (119895 = 1 2 119896)

(11)

hold true





119894119895and 120572+


119894119895


As |1198830(119895) minus 119883


119894119895and

120572+


119894119895(119863minus



119863+

119894119895= 119863+

119894119895(120572119894119895 120572+

119895) =

1

4

(

10038161003816100381610038161003816inf V minus inf 120583+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120583119894119895minus sup 120583+

119895

10038161003816100381610038161003816+

10038161003816100381610038161003816inf 120574119894119895minus inf 120574+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120574119894119895minus sup 120574+

119895

10038161003816100381610038161003816)

119863minus

119894119895= 119863minus

119894119895(120572119894119895 120572minus

119895) =

1

4

(


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816+


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816)

(12)


119894119895)119899times119896


119894119895between 120572

119894119895and 120572+

119895 (119863minus119894119895)119899times119896


119894119895between 120572

119894119895and 120572minus

119895


between 120572119894119895and 120572

+

119895 And 120585

minus


119894119895and

120572minus



120585+

119894119895=

min119894119895119863+

119894119895+ 120588max

119894119895119863+

119894119895

119863+

119894119895+ 120588max

119894119895119863+

119894119895

(13)

120585minus

119894119895=

min119894119895119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

(14)



In (13) max119894119895119863+

119894119895and min

119894119895119863+



119894119895)119899times119896

In (14) max119894119895119863minus

119894119895and min

119894119895119863minus



119894119895)119899times119896



and(120585minus

119894119895)119899times119896




119894and 120582minus


120582+

119894=

119896

sum

119895=1

120585+

119894119895119886119895

120582minus

119894=

119896

sum

119895=1

120585minus

119894119895119886119895

(15)



119895=1119886119895= 1



min 119865 (119886) =

119899

sum

119894=1

119896

sum

119895=1

[(1 minus 120585+

119894119895) 119886119895]

2

st119896

sum

119895=1

119886119895= 1

119886119895⩾ 0

(16)


119886119895=

sum119896

119895=1[sum119899

119894=1(1 minus 120585

+

119894119895)

2

]

minus1

minus1

sum119899

119894=1(1 minus 120585

+

119894119895)

2

(17)



119894




119889119894=

(120582+

119894)

2

(120582+

119894)2

+ (120582minus

119894)2 (18)













119894 when 119889

119894increases







05 35000 1800 2000 350 (19)




R

=

[

[

[

[

[

[

[

[

[

([005 015] [06 07]) ([035 045] [04 05]) ([011 021] [069 079]) ([01 02] [07 08]) ([01 02] [07 08])

([0 01] [065 075]) ([01 02] [07 08]) ([005 015] [07 08]) ([002 012] [078 088]) ([01 02] [065 075])

([055 065] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02])

([065 075] [0 01]) ([0 01] [07 08]) ([017 027] [058 068]) ([0 01] [075 085]) ([0 01] [07 08])

([06 07] [005 015]) ([03 04] [045 055]) ([05 06] [03 04]) ([06 07] [02 03]) ([08 09] [0 01])

]

]

]

]

]

]

]

]

]

(20)




119909+= ([065 075] [0 01]) ([07 08] [01 02])

([07 08] [01 02]) ([07 08] [01 02])

([08 09] [0 01])

119909minus= ([0 01] [065 075]) ([0 01] [07 08])

([005 015] [07 08]) ([002 012] [078 088])

([0 01] [07 08])

(21)


D+ = (119863+119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

06 0325 059 06 07

065 06 0625 068 0675

01 0 0 0 01

0 065 0505 0675 07

005 0375 02 01 0

]

]

]

]

]

]

]

]

]


=

[

[

[

[

[

[

[

[

[

005 0325 0035 008 005

0 005 0 0 0075

055 065 0625 068 065

065 0 012 0025 0

06 0275 0425 058 075

]

]

]

]

]

]

]

]

]

(22)


119894119895119863+

119894119895= 07 and

min119894119895119863+


119894119895119863minus

119894119895= 075 and

min119894119895119863minus

119894119895= 0


120585+= (120585+

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

03684 05185 03723 03684 03333

035 03684 03590 03398 03415

07778 1 1 1 07778

1 035 04094 03415 03333

0875 04828 06364 07778 1

]

]

]

]

]

]

]

]

]


119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

08824 05357 09146 08242 08824

1 08824 1 1 08333

04054 03659 0375 03555 03659

03659 1 07576 09375 1

03846 05769 04688 03927 03333

]

]

]

]

]

]

]

]

]

(23)



119878 (m2) 119867 (m) 119881119881(ms) 119881


result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065


a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)


120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)



119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)


119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)





















5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867lie in [0 3000ms]


inf 120583TBM (119881119867)

=

119881119867

5000

119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 02 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 06 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 08 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 05 2000ms lt 119881119867⩽ 3000ms

sup 120583TBM (119881119867)

=

119881119867

10000

+ 02 119881119867⩽ 1000ms

(119881119867minus 1000)

1000

+ 03 1000ms lt 119881119867⩽ 1400ms

(119881119867minus 1400)

1500

+ 07 1400ms lt 119881119867⩽ 1700ms

(1700 minus 119881119867)

1000

+ 09 1700ms lt 119881119867⩽ 2000ms

(2000 minus 119881119867)

2500

+ 06 2000ms lt 119881119867⩽ 3000ms

inf 120574TBM (119881119867)

=

minus

119881119867

10000

+ 07 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 06 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 02 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 03 2000ms lt 119881119867⩽ 3000ms

sup 120574TBM (119881119867)

=

minus

119881119867

5000

+ 09 119881119867⩽ 1000ms

(1000 minus 119881119867)

1000

+ 07 1000ms lt 119881119867⩽ 1400ms

(1400 minus 119881119867)

1500

+ 03 1400ms lt 119881119867⩽ 1700ms

(119881119867minus 1700)

1000

+ 01 1700ms lt 119881119867⩽ 2000ms

(119881119867minus 2000)

2500

+ 04 2000ms lt 119881119867⩽ 3000ms

(7)




120572TBM-119881119867 = ([05 06] [03 04]) (8)


119867




inf 120583AGM (119867)

=

0 119867 ⩽ 3000m(119867 minus 3000)

10000

3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 08 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 06 119867 ⩾ 17000m

sup 120583AGM (119867)

=

01 119867 ⩽ 3000m(119867 minus 3000)

10000

+ 01 3000m lt 119867 ⩽ 11000m(11000 minus 119867)

30000

+ 09 11000m lt 119867 ⩽ 17000m(17000 minus 119867)

60000

+ 07 119867 ⩾ 17000m

inf 120574AGM (119867)

=

08 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 08 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 015 119867 gt 17000m

sup 120574AGM (119867)

=

09 119867 ⩽ 3000m(3000 minus 119867)

10000

+ 09 3000m lt 119867 ⩽ 11000m(119867 minus 11000)

40000

+ 01 11000m lt 119867 ⩽ 17000m(119867 minus 17000)

60000

+ 025 119867 gt 17000m

(9)










119889)2




2) then 120572

1lt 1205722


2) if 119867(120572

1) lt 119867(120572

2) then

1205721lt 1205722holds


119909+= 120572+

1 120572+

2 120572

+

119896


1 120572minus

2 120572

minus

119896

(10)


119894119895)119899times119896

if andonly if

120572+

119895= max119894

120572119894119895 (119895 = 1 2 119896)

120572minus

119895= min119894

120572119894119895 (119895 = 1 2 119896)

(11)

hold true





119894119895and 120572+


119894119895


As |1198830(119895) minus 119883


119894119895and

120572+


119894119895(119863minus



119863+

119894119895= 119863+

119894119895(120572119894119895 120572+

119895) =

1

4

(

10038161003816100381610038161003816inf V minus inf 120583+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120583119894119895minus sup 120583+

119895

10038161003816100381610038161003816+

10038161003816100381610038161003816inf 120574119894119895minus inf 120574+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120574119894119895minus sup 120574+

119895

10038161003816100381610038161003816)

119863minus

119894119895= 119863minus

119894119895(120572119894119895 120572minus

119895) =

1

4

(


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816+


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816)

(12)


119894119895)119899times119896


119894119895between 120572

119894119895and 120572+

119895 (119863minus119894119895)119899times119896


119894119895between 120572

119894119895and 120572minus

119895


between 120572119894119895and 120572

+

119895 And 120585

minus


119894119895and

120572minus



120585+

119894119895=

min119894119895119863+

119894119895+ 120588max

119894119895119863+

119894119895

119863+

119894119895+ 120588max

119894119895119863+

119894119895

(13)

120585minus

119894119895=

min119894119895119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

(14)



In (13) max119894119895119863+

119894119895and min

119894119895119863+



119894119895)119899times119896

In (14) max119894119895119863minus

119894119895and min

119894119895119863minus



119894119895)119899times119896



and(120585minus

119894119895)119899times119896




119894and 120582minus


120582+

119894=

119896

sum

119895=1

120585+

119894119895119886119895

120582minus

119894=

119896

sum

119895=1

120585minus

119894119895119886119895

(15)



119895=1119886119895= 1



min 119865 (119886) =

119899

sum

119894=1

119896

sum

119895=1

[(1 minus 120585+

119894119895) 119886119895]

2

st119896

sum

119895=1

119886119895= 1

119886119895⩾ 0

(16)


119886119895=

sum119896

119895=1[sum119899

119894=1(1 minus 120585

+

119894119895)

2

]

minus1

minus1

sum119899

119894=1(1 minus 120585

+

119894119895)

2

(17)



119894




119889119894=

(120582+

119894)

2

(120582+

119894)2

+ (120582minus

119894)2 (18)













119894 when 119889

119894increases







05 35000 1800 2000 350 (19)




R

=

[

[

[

[

[

[

[

[

[

([005 015] [06 07]) ([035 045] [04 05]) ([011 021] [069 079]) ([01 02] [07 08]) ([01 02] [07 08])

([0 01] [065 075]) ([01 02] [07 08]) ([005 015] [07 08]) ([002 012] [078 088]) ([01 02] [065 075])

([055 065] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02])

([065 075] [0 01]) ([0 01] [07 08]) ([017 027] [058 068]) ([0 01] [075 085]) ([0 01] [07 08])

([06 07] [005 015]) ([03 04] [045 055]) ([05 06] [03 04]) ([06 07] [02 03]) ([08 09] [0 01])

]

]

]

]

]

]

]

]

]

(20)




119909+= ([065 075] [0 01]) ([07 08] [01 02])

([07 08] [01 02]) ([07 08] [01 02])

([08 09] [0 01])

119909minus= ([0 01] [065 075]) ([0 01] [07 08])

([005 015] [07 08]) ([002 012] [078 088])

([0 01] [07 08])

(21)


D+ = (119863+119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

06 0325 059 06 07

065 06 0625 068 0675

01 0 0 0 01

0 065 0505 0675 07

005 0375 02 01 0

]

]

]

]

]

]

]

]

]


=

[

[

[

[

[

[

[

[

[

005 0325 0035 008 005

0 005 0 0 0075

055 065 0625 068 065

065 0 012 0025 0

06 0275 0425 058 075

]

]

]

]

]

]

]

]

]

(22)


119894119895119863+

119894119895= 07 and

min119894119895119863+


119894119895119863minus

119894119895= 075 and

min119894119895119863minus

119894119895= 0


120585+= (120585+

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

03684 05185 03723 03684 03333

035 03684 03590 03398 03415

07778 1 1 1 07778

1 035 04094 03415 03333

0875 04828 06364 07778 1

]

]

]

]

]

]

]

]

]


119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

08824 05357 09146 08242 08824

1 08824 1 1 08333

04054 03659 0375 03555 03659

03659 1 07576 09375 1

03846 05769 04688 03927 03333

]

]

]

]

]

]

]

]

]

(23)



119878 (m2) 119867 (m) 119881119881(ms) 119881


result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065


a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)


120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)



119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)


119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)





















5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119889)2




2) then 120572

1lt 1205722


2) if 119867(120572

1) lt 119867(120572

2) then

1205721lt 1205722holds


119909+= 120572+

1 120572+

2 120572

+

119896


1 120572minus

2 120572

minus

119896

(10)


119894119895)119899times119896

if andonly if

120572+

119895= max119894

120572119894119895 (119895 = 1 2 119896)

120572minus

119895= min119894

120572119894119895 (119895 = 1 2 119896)

(11)

hold true





119894119895and 120572+


119894119895


As |1198830(119895) minus 119883


119894119895and

120572+


119894119895(119863minus



119863+

119894119895= 119863+

119894119895(120572119894119895 120572+

119895) =

1

4

(

10038161003816100381610038161003816inf V minus inf 120583+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120583119894119895minus sup 120583+

119895

10038161003816100381610038161003816+

10038161003816100381610038161003816inf 120574119894119895minus inf 120574+

119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816sup 120574119894119895minus sup 120574+

119895

10038161003816100381610038161003816)

119863minus

119894119895= 119863minus

119894119895(120572119894119895 120572minus

119895) =

1

4

(


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816+


119895

10038161003816100381610038161003816

+


119895

10038161003816100381610038161003816)

(12)


119894119895)119899times119896


119894119895between 120572

119894119895and 120572+

119895 (119863minus119894119895)119899times119896


119894119895between 120572

119894119895and 120572minus

119895


between 120572119894119895and 120572

+

119895 And 120585

minus


119894119895and

120572minus



120585+

119894119895=

min119894119895119863+

119894119895+ 120588max

119894119895119863+

119894119895

119863+

119894119895+ 120588max

119894119895119863+

119894119895

(13)

120585minus

119894119895=

min119894119895119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

119863minus

119894119895+ 120588max

119894119895119863minus

119894119895

(14)



In (13) max119894119895119863+

119894119895and min

119894119895119863+



119894119895)119899times119896

In (14) max119894119895119863minus

119894119895and min

119894119895119863minus



119894119895)119899times119896



and(120585minus

119894119895)119899times119896




119894and 120582minus


120582+

119894=

119896

sum

119895=1

120585+

119894119895119886119895

120582minus

119894=

119896

sum

119895=1

120585minus

119894119895119886119895

(15)



119895=1119886119895= 1



min 119865 (119886) =

119899

sum

119894=1

119896

sum

119895=1

[(1 minus 120585+

119894119895) 119886119895]

2

st119896

sum

119895=1

119886119895= 1

119886119895⩾ 0

(16)


119886119895=

sum119896

119895=1[sum119899

119894=1(1 minus 120585

+

119894119895)

2

]

minus1

minus1

sum119899

119894=1(1 minus 120585

+

119894119895)

2

(17)



119894




119889119894=

(120582+

119894)

2

(120582+

119894)2

+ (120582minus

119894)2 (18)













119894 when 119889

119894increases







05 35000 1800 2000 350 (19)




R

=

[

[

[

[

[

[

[

[

[

([005 015] [06 07]) ([035 045] [04 05]) ([011 021] [069 079]) ([01 02] [07 08]) ([01 02] [07 08])

([0 01] [065 075]) ([01 02] [07 08]) ([005 015] [07 08]) ([002 012] [078 088]) ([01 02] [065 075])

([055 065] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02])

([065 075] [0 01]) ([0 01] [07 08]) ([017 027] [058 068]) ([0 01] [075 085]) ([0 01] [07 08])

([06 07] [005 015]) ([03 04] [045 055]) ([05 06] [03 04]) ([06 07] [02 03]) ([08 09] [0 01])

]

]

]

]

]

]

]

]

]

(20)




119909+= ([065 075] [0 01]) ([07 08] [01 02])

([07 08] [01 02]) ([07 08] [01 02])

([08 09] [0 01])

119909minus= ([0 01] [065 075]) ([0 01] [07 08])

([005 015] [07 08]) ([002 012] [078 088])

([0 01] [07 08])

(21)


D+ = (119863+119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

06 0325 059 06 07

065 06 0625 068 0675

01 0 0 0 01

0 065 0505 0675 07

005 0375 02 01 0

]

]

]

]

]

]

]

]

]


=

[

[

[

[

[

[

[

[

[

005 0325 0035 008 005

0 005 0 0 0075

055 065 0625 068 065

065 0 012 0025 0

06 0275 0425 058 075

]

]

]

]

]

]

]

]

]

(22)


119894119895119863+

119894119895= 07 and

min119894119895119863+


119894119895119863minus

119894119895= 075 and

min119894119895119863minus

119894119895= 0


120585+= (120585+

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

03684 05185 03723 03684 03333

035 03684 03590 03398 03415

07778 1 1 1 07778

1 035 04094 03415 03333

0875 04828 06364 07778 1

]

]

]

]

]

]

]

]

]


119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

08824 05357 09146 08242 08824

1 08824 1 1 08333

04054 03659 0375 03555 03659

03659 1 07576 09375 1

03846 05769 04688 03927 03333

]

]

]

]

]

]

]

]

]

(23)



119878 (m2) 119867 (m) 119881119881(ms) 119881


result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065


a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)


120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)



119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)


119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)





















5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










In (13) max119894119895119863+

119894119895and min

119894119895119863+



119894119895)119899times119896

In (14) max119894119895119863minus

119894119895and min

119894119895119863minus



119894119895)119899times119896



and(120585minus

119894119895)119899times119896




119894and 120582minus


120582+

119894=

119896

sum

119895=1

120585+

119894119895119886119895

120582minus

119894=

119896

sum

119895=1

120585minus

119894119895119886119895

(15)



119895=1119886119895= 1



min 119865 (119886) =

119899

sum

119894=1

119896

sum

119895=1

[(1 minus 120585+

119894119895) 119886119895]

2

st119896

sum

119895=1

119886119895= 1

119886119895⩾ 0

(16)


119886119895=

sum119896

119895=1[sum119899

119894=1(1 minus 120585

+

119894119895)

2

]

minus1

minus1

sum119899

119894=1(1 minus 120585

+

119894119895)

2

(17)



119894




119889119894=

(120582+

119894)

2

(120582+

119894)2

+ (120582minus

119894)2 (18)













119894 when 119889

119894increases







05 35000 1800 2000 350 (19)




R

=

[

[

[

[

[

[

[

[

[

([005 015] [06 07]) ([035 045] [04 05]) ([011 021] [069 079]) ([01 02] [07 08]) ([01 02] [07 08])

([0 01] [065 075]) ([01 02] [07 08]) ([005 015] [07 08]) ([002 012] [078 088]) ([01 02] [065 075])

([055 065] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02])

([065 075] [0 01]) ([0 01] [07 08]) ([017 027] [058 068]) ([0 01] [075 085]) ([0 01] [07 08])

([06 07] [005 015]) ([03 04] [045 055]) ([05 06] [03 04]) ([06 07] [02 03]) ([08 09] [0 01])

]

]

]

]

]

]

]

]

]

(20)




119909+= ([065 075] [0 01]) ([07 08] [01 02])

([07 08] [01 02]) ([07 08] [01 02])

([08 09] [0 01])

119909minus= ([0 01] [065 075]) ([0 01] [07 08])

([005 015] [07 08]) ([002 012] [078 088])

([0 01] [07 08])

(21)


D+ = (119863+119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

06 0325 059 06 07

065 06 0625 068 0675

01 0 0 0 01

0 065 0505 0675 07

005 0375 02 01 0

]

]

]

]

]

]

]

]

]


=

[

[

[

[

[

[

[

[

[

005 0325 0035 008 005

0 005 0 0 0075

055 065 0625 068 065

065 0 012 0025 0

06 0275 0425 058 075

]

]

]

]

]

]

]

]

]

(22)


119894119895119863+

119894119895= 07 and

min119894119895119863+


119894119895119863minus

119894119895= 075 and

min119894119895119863minus

119894119895= 0


120585+= (120585+

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

03684 05185 03723 03684 03333

035 03684 03590 03398 03415

07778 1 1 1 07778

1 035 04094 03415 03333

0875 04828 06364 07778 1

]

]

]

]

]

]

]

]

]


119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

08824 05357 09146 08242 08824

1 08824 1 1 08333

04054 03659 0375 03555 03659

03659 1 07576 09375 1

03846 05769 04688 03927 03333

]

]

]

]

]

]

]

]

]

(23)



119878 (m2) 119867 (m) 119881119881(ms) 119881


result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065


a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)


120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)



119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)


119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)





















5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











05 35000 1800 2000 350 (19)




R

=

[

[

[

[

[

[

[

[

[

([005 015] [06 07]) ([035 045] [04 05]) ([011 021] [069 079]) ([01 02] [07 08]) ([01 02] [07 08])

([0 01] [065 075]) ([01 02] [07 08]) ([005 015] [07 08]) ([002 012] [078 088]) ([01 02] [065 075])

([055 065] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02]) ([07 08] [01 02])

([065 075] [0 01]) ([0 01] [07 08]) ([017 027] [058 068]) ([0 01] [075 085]) ([0 01] [07 08])

([06 07] [005 015]) ([03 04] [045 055]) ([05 06] [03 04]) ([06 07] [02 03]) ([08 09] [0 01])

]

]

]

]

]

]

]

]

]

(20)




119909+= ([065 075] [0 01]) ([07 08] [01 02])

([07 08] [01 02]) ([07 08] [01 02])

([08 09] [0 01])

119909minus= ([0 01] [065 075]) ([0 01] [07 08])

([005 015] [07 08]) ([002 012] [078 088])

([0 01] [07 08])

(21)


D+ = (119863+119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

06 0325 059 06 07

065 06 0625 068 0675

01 0 0 0 01

0 065 0505 0675 07

005 0375 02 01 0

]

]

]

]

]

]

]

]

]


=

[

[

[

[

[

[

[

[

[

005 0325 0035 008 005

0 005 0 0 0075

055 065 0625 068 065

065 0 012 0025 0

06 0275 0425 058 075

]

]

]

]

]

]

]

]

]

(22)


119894119895119863+

119894119895= 07 and

min119894119895119863+


119894119895119863minus

119894119895= 075 and

min119894119895119863minus

119894119895= 0


120585+= (120585+

119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

03684 05185 03723 03684 03333

035 03684 03590 03398 03415

07778 1 1 1 07778

1 035 04094 03415 03333

0875 04828 06364 07778 1

]

]

]

]

]

]

]

]

]


119894119895)119899times119896

=

[

[

[

[

[

[

[

[

[

08824 05357 09146 08242 08824

1 08824 1 1 08333

04054 03659 0375 03555 03659

03659 1 07576 09375 1

03846 05769 04688 03927 03333

]

]

]

]

]

]

]

]

]

(23)



119878 (m2) 119867 (m) 119881119881(ms) 119881


result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065


a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)


120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)



119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)


119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)





















5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878 (m2) 119867 (m) 119881119881(ms) 119881


result1 05 35000 1800 2000 350 119906

3

2 06 40000 2000 1950 420 1199063

3 10 15000 250 8 05 1199061

4 15 12000 270 10 07 1199061

5 044 90 340 0 02 1199064

6 067 75 300 0 015 1199064

7 5 3000 70 10 05 1199062

8 3 2000 60 12 04 1199062

9 07 10000 1400 1500 400 1199065

10 08 11000 1500 1300 390 1199065


a = (1198861 1198862 1198863 1198864 1198865)119879

= (02718 01824 01874 01828 01756)

119879

(24)


120582+= (120582+

119894)119899times1

= (03903 03517 09006 05333 07629)

119879

(25)



119894)119899times1

= (08146 09493 03764 07708 04279)

119879

(26)


119889 = (119889119894)119899times1

= (01867 01206 08513 03237 08180)

119879

(27)





















5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusion




Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article algorithm for target recognition based on interval-valued...

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