research article ship electric propulsion simulation system...

Research ArticleShip Electric Propulsion Simulation SystemReliability Evaluation Based on Improved D-S Expert WeightCalculation Method

Bing Li Guoliang Gu Bowen Xing and Lihong Li

College of Automation Harbin Engineering University Harbin Heilongjiang 150001 China

Correspondence should be addressed to Bowen Xing bwheuoutlookcom

Received 12 August 2014 Revised 21 September 2014 Accepted 5 October 2014

Academic Editor Shaofan Li

Copyright copy 2015 Bing Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to have a better evaluation process to determine the experts weight in the evaluation process this paper proposes a newexpert weight calculation method First of all to establish electric propulsion simulation evaluation system use AHP method tocalculate the initial weight principle of index Then use the D-S to fuse the experts evaluation information combined with theweight vector structure of the expert weight objective function and through the genetic algorithm to solve the expert weight sizeAccording to the expert weight vector calculate the final weight vector Not only can it greatly make use of the experts informationand analyze the similarity of information effectively but also it calculates the weight of each expert objectively At the same time theevaluation subjective factors have been reduced by the adoption of this new method

1 Introduction

Regarding ship electric propulsion system as a modernship career development direction its safety and reliabilityare more and more concerned about [1] The simulationtechnology is one of the important means for people tostudy ship electric propulsion system [2] But the results ofsimulation credibility are worth considering and incorrectresults could lead tomajor events So analyzing the credibilityin the system simulation results determining the relativeweight of each subsystem has great significance for studyingthe mechanism of ship electric propulsion system

In the system simulation credibility analysis we need tomake sure of the mutual importance of the systems namelythe weight Analytic hierarchy process (AHP) is a compre-hensive evaluation method used in research of complicatedsystem [3] The basic idea of using analytic hierarchy processto determine weight is to invite more related experts tocompare each subsystem and to identify and analyse thejudgment matrix As a result of difference between eachexpert in knowledge experience ability and level differentexperts have different result toweight evaluation systemHow

to make better use of the evaluation experts has always beenabout the topic of comparison

The D-S evidence theory is a method widely used ininformation fusion technology [4] Chen et al proposemaking use of the Markov random fields (MRFs) and D-S evidence theory to interactive color image segmentationmethod [5] Si et al proposed a novel prediction approachthrough information fusion of improvedD-S evidence theoryand neural network to forecast the distribution of coal seamterrain [6] Li and Pang use D-S evidence theory to solvevessel collision risk assessment [7] Experimental resultsshow that the proposed approach confirms the validity andis reasonable for real application But D-S evidence theorycannot solve the conflict evidence problem

To solve this problem according to the multiple expertsjudgment matrix by using analytic hierarchy process methodand D-S evidence theory avoid the conflict in informationon the expert information synthesis from the actual caseAccording to the weight of the fusion establish an expertweight target function and determine the weight of expertsusing genetic algorithms Finally determine the weight offinal system by weighting This algorithm is effective to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 314058 5 pageshttpdxdoiorg1011552015314058

2 Mathematical Problems in Engineering

Ship electric propulsionsystem simulation credibility

A

System simulation modelB

System simulation resultC

Powerdistri-butionsystem

Propul-sion

system

Powertransfor-mationsystem

Powersupplysystem

D1


Propul-sion

system


Powersupplysystem

D2 D3D4 D5 D6 D7

D8

Figure 1 Structure diagram for ship electric propulsion simulation system reliability evaluation index system

solve the synthesis expert information conflict problems andimprove the system of the objectivity of the judgment

2 The Simulation System ReliabilityEvaluation Index System

For the ship electric propulsion simulation system reliabilityevaluation study according to user requirements and thecharacteristics of the system itself first of all we need toestablish the ship electric propulsion simulation system relia-bility evaluation index system Based on the key indicatorsin research process investigate subsystem one by one andestablish the system equivalence evaluation index systemIn the process of building evaluation system according tothe target layer criterion layer and measures layer sequencedecompose the system step by step hierarchically and makea complicated problem decomposed into several elementsFor ship electric propulsion simulation system is concernedabout target layer is ship electric propulsion simulationsystem reliability Criterion layer from the system simulationmodel to system based on the simulation results determinesthe level of index measures step by step [8] For system sim-ulation model and system simulation results are concernedabout the reliability is analyzed from the power supplysystem power distribution system power transformationsystem and propulsion system respectively Finally shipelectric propulsion simulation system reliability evaluationsystem is shown in Figure 1

3 Expert Evaluation Weight Analysis

31 The Steps to Calculate the Index Weight

Step 1 According to many expertsrsquo relative judgment matrixuse the principle of AHP to calculate the index of the initialweight

Step 2 Determine the index weight fusion through themodified D-S fusion initial index weight and determine theabsolute judgment matrix through the expert judgment

Step 3 According to each index fusion weight and absolutejudgment matrix determine the expert weight the objectivefunction and a genetic algorithm is adopted to calculate theoptimal solution to determine the expert weight

Step 4 Weigh the initial weight and expert weight anddetermine the weights of the index

32 Based on the AHP Analysis of Initial Weights Initialweight can be calculated by AHP to carry out Comparedwith the previous expert scoring method fuzzy evalua-tion method the grey correlation method Pressure-State-Response method (PSR) and artificial neural network algo-rithm AHP is a kind of qualitative analysis and quantitativeanalysis and systematic and hierarchical multiple factors ofdecision analysis method this method will be the decisionmakerrsquos experience quantitative judgment It is very con-venient in the condition of the multiobjective and lack ofnecessary data [9]

Ship electric propulsion simulation system reliabilityevaluation index system calculation generally can be dividedinto the following four steps [10]

Step 1 Each element value in judgment matrix is relativeto a certain element in a previous level associated with theeach elements in the layer pairwise comparison judgmentimportance In the judge process use 1ndash9 scale method toshow specific as is shown in Table 1

Step 2 The elementrsquos relative weight for the criterion iscalculated by judgment matrix

Step 3Compute synthetic weight of each of the layer elementsto system target

Step 4 Consistency check consistency includes absoluteconsistency (or complete consistency) and order consistencyThe so-called absolute consistency means that the judgmentmatrix 119860 If matrix A meet

119886119894119895

= 119886119894119896119886119895119896

119894 119895 119896 = 1 2 119899 (1)

We called matrix 119860 meet absolute consistency

Mathematical Problems in Engineering 3

Table 1 The scale method of 1sim9

Scale Meaning1 Two factors have the same importance

3 A factor relative to another factor a littleimportant

5 A factor relative to another obvious importantfactors

7 A factor relative to another important factorstrongly

9 A factor relative to another extremely importantfactor

2 4 6 8 Median in two adjacent judgments

ReciprocalFactors 119894 and 119895 are to judge 119887

119894119895

the factors 119895 and 119894 compare judgment119887119895119894

= 1119887119894119895

It says 119860 is absolute consistency matrix (or completeconsistency matrix) at the same time there is

119886119894119895

=119882119894

119882119895

119894 119895 119896 = 1 2 119899

119860119882 = 119899119882

(2)

Sort consistency is to point to the following if factor a isimportant than factor 119887 and factor 119887 is important than factor119888 then a factor is important than factor 119888 And the consistencycheck index CI is as follows

CI =120582max minus 119899

119899 minus 1 (3)

The 119899 is the order number of judgment matrix 119860 and 120582max isthe biggest characteristic root of judgment matrix 119860

Calculation consistency ratio CR is as follows

CR =CIRI

(4)

When CR lt 01 consider the consistency of judgementmatrix is acceptable [11] (Table 2)

33 The Construction of Expert Weight Objective FunctionBased onD-SMethod For the ship electric propulsion systemconcerned about setting the index set 119861 = 119887

1 1198872 119887

119899

experts set119863 = 1198891 1198892 119889

119898 andmatrix119860 = (119886

119894119895)119898times119899

(0 lt

119886119894119895

lt 1) is absolute judgment matrix coming from experts tomarking index weight Set index fusion weight vector 119882 =

1199081 1199082 119908

119899 expert weight vector is 119877 = 119903

1 1199032 119903

119898

and meet sum119899119895=1

119908119895= 1 sum119898

119894=1119903119894= 1

For absolute judgment matrix if evaluation from expert119889119894has no difference with other expertsrsquo evaluation the expert

119889119894has a higher similarity with other experts That is to

say expert 119889119894has higher credibility 119865

119894119895(119903) show the index

judgment deviation from experts 119889119894and other experts

119865119894119895 (119903) =

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895 (5)

Table 2 The average random consistency targets RI

Matrix order number RI1 02 03 0524 0895 1126 1267 1368 1419 14610 119

119865119895(119903) show the total index judgment deviation from 119898

experts

119865119895 (119903) =

119898

sum

119894=1

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895 (6)

For index 119887119895 deviation value 119865

119895(119903) is smaller credibility of

the judge from experts is higher For the system in the indextotal deviation value is smaller credibility of the judge fromexperts is higher According to the calculation of the systemintegration indicators weight structural expert weight targetoptimization function

min 119865 (119903) =

119899

sum

119895=1

119898

sum

119894=1

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895

st119898

sum

119894=1

119903119894= 1

0 lt 119903119894lt 1 (119894 = 1 119898)

(7)

34 Solve theObjective Function Formula (7) belongs to non-linear optimization problem the genetic algorithm suitablefor processing this kind of problem With the increase ofmatrix dimension parameters in formula (7) will increasesharply in order to find the optimal solution undermultivari-ate conditions introducing the concept of niche to enhancethe diversity of population [12 13] Algorithm flow chart isshown in Figure 2

4 Ship Electric PropulsionSimulation System Expert EvaluationWeight Calculation Conclusion

Based on the analysis of the weight of each subsystem invitethree authoritative experts to score evaluation Relative to


Start

Set running parameters

Random generation

Calculating individual fitness

Terminationconditions meet

Output

Selection crossover mutationand then get individual

Merge the individual and start nichecalculation

Fitness sortingtake individual

Yes

NoCounter t = t + 1

Individual counter t = 0

Figure 2 Flow chart for the niche genetic algorithm of AHP calcu-lation

the subsystem119861 three experts are given respectively119863111986321198633 and 1198634 judgment matrix specific as follows

1198611=

[[[[[[[[

[

11

6

1

33

6 1 2 7

31

21 5

1

3

1

7

1

51

]]]]]]]]

]

1198612=

[[[[[[

[

11

2

1

23

2 1 1 4

2 1 1 4

1

3

1

4

1

41

]]]]]]

]

1198613=

[[[[[[[[

[

1 11

53

1 11

53

5 1 1 7

1

3

1

3

1

71

]]]]]]]]

]

(8)

According to the principle of AHP calculate the correspond-ing initial weights of each subsystem and check the con-sistency of judge matrix If they do not meet the require-ments for consistency give the judge matrix again Getthree experts to 1198631 1198632 1198633 and 1198634 judgment weight1198821198871

= 01176 05345 02904005781198821198872

= 020040358703583 00816 and 119882

1198873= 01953 01953 05324 00771

According to the calculation of the initial weight usingthe improvedD-S to fuse evaluation information get119863111986321198633 and 1198634 fusion weight to subsystem 119861119882

1198871015840 = 00633

03941 05334 00091For ship electric propulsion system simulation model

credibility analysis the purpose is to determine whether themodel is accurate In this criterion the three experts give1198631 1198632 1198633 and 1198634 weight fuzzy evaluation matrix 119861 =

[01 05 03 01 02 035 035 01 02 02 05 01] Accord-ing to each subsystem fusion weight and expert fuzzy eval-uation matrix by the algorithm to determine the expertweight objective function and through the genetic algorithmoptimization determine the expert weight optimal solutionGet three subsystems1198631119863211986331198634weight evaluation of itsown weight 119877

119861= 0401 036 0238

According to the expert weight and the initial weight ofthe subsystem using the weighted method to calculate theweight of each subsystem get 1198631 1198632 1198633 1198634 weight tosubsystem 119861119882

119887= 01654 03886 03732 00706

Similarly the three experts give the judgment matrix ofeach subsystem for objectives 119860 and 119862 and calculate theweight of each subsystem according to the above method Inview of the space reasons give only the expert weight and theweight of each subsystem

For the three experts grade evaluation for1198635119863611986371198638get the weight in the evaluation process of evaluation 119877

119862=

048 0218 03 and the subsystem 1198635 1198636 1198637 1198638 weight119882119862

= 01666 01758 02974 03582Analyzing subsystem 119861 and 119862 against overall goal 119860 get

evaluation weight 119877119860

= 044 0259 03 and the weight sub-system 119861 119862 to 119860 119882

119860= 05108 04882

According to the calculation of the 119882119860 119882119861 119882119862weight

vector use the weighted method to calculate the total target

119882 = [1198821015840

1198611198821015840

119862] sdot 1198821015840

119860

=[[[

[

01653 01666

03887 01758

03731 02974

00708 03582

]]]

]

sdot [05108

04882]

= [01658 02844 03358 02110]1015840

(9)

From the calculation results above it is known that in theship electric propulsion system the simulation credibilityis the greatest impacted by power conversion subsystemsecondly they were distribution subsystem propulsion sub-system and power subsystem Power transformation simula-tion subsystem in energy conversion and harmonic aspectsaffect the credibility of the system Distribution systemsimulation subsystem produces certain effect to managementand distribution of electricity Propulsion system simulationsubsystem is aimed at mutation load For power subsystem


main consideration of its power quality the other modulesproduce small amount of influence

5 Conclusions

By using D-S theory and AHP study the ship electric propul-sion simulation system for the credibility evaluation expertweight Through the inspection of the similarity betweenthe experts determine the expert weight objective functionand the genetic algorithm was used to calculate the expertweight optimal solution This method not only can fuse theadvantages of other methods but also make better use ofthe expert advice in the evaluate process greatly It can makesubjective judgments of experts more united and avoid theone-sidedness when considering only one expert and variousjudgments from different experts on reliability of ship electricpropulsion simulation system can be treated particularlyevidence of conflict is no longer blindly negated At the sametime this method can optimize the indexes and enhance theveracity and reliability for scientific decision-making whichhas better comprehensive assessment evaluation and is moremeaningful

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Natural Science Founda-tion (NNSF) of China under Grant 51307026 Natural ScienceFoundation of Heilongjiang Province under Grant E201347Fundamental Research Funds for the Central Universitiesunder Grant HEUCFX41305 and the China PostdoctoralFunds (no 2012M510924)

References

[1] L Sheng and L Lihong ldquoApplication of FCE for credibility ofsimulation of ship electric propulsion systemrdquo Techniques ofAutomation amp Applications vol 32 no 4 pp 52ndash77 2012

[2] S Ya-feng W Chao-yang and H Zhi-ping ldquoResearch onsimulation credibilityrdquo Electronic Measurement Technology vol32 no 11 pp 8ndash11 2009

[3] H Chen andW Yong ldquoCredibility research of the simulation ofthe integrative avionics electronic system which based on cloudfocus judgementrdquo Electronic Measurement Technology vol 32no 10 pp 69ndash72 2009

[4] B Chen and J Feng ldquoMultisensor information fusion of pulsedGTAW based on improved D-S evidence theoryrdquo InternationalJournal of Advanced Manufacturing Technology vol 71 no 1ndash4pp 91ndash99 2014

[5] Y Chen A B Cremers and Z Cao ldquoInteractive color imagesegmentation via iterative evidential labelingrdquo InformationFusion vol 20 no 1 pp 292ndash304 2014

[6] L Si Z Wang C Tan and X Liu ldquoA novel approach forcoal seam terrain prediction through information fusion of

improved D-S evidence theory and neural networkrdquo Measure-ment Journal of the International Measurement Confederationvol 54 pp 140ndash151 2014

[7] B Li and F-W Pang ldquoAn approach of vessel collision risk assess-ment based on theD-S evidence theoryrdquoOceanEngineering vol74 pp 16ndash21 2013

[8] I-S Jung and C-S Lee ldquoFuzzy inference and AHP-basedalternative evaluation tool in the development of sustainableresidential landrdquo KSCE Journal of Civil Engineering vol 16 no3 pp 273ndash282 2012

[9] C-C Sun ldquoA performance evaluation model by integratingfuzzy AHP and fuzzy TOPSIS methodsrdquo Expert Systems withApplications vol 37 no 12 pp 7745ndash7754 2010

[10] G-Z Li N-L Tan and J-B Zhang ldquoCriticality analysis of sub-way train equipment based on improved analytical hierarchyprocessrdquo Journal of ElectronicMeasurement and Instrument vol26 no 6 pp 503ndash507 2012

[11] W Shi Y Li C Deng M Fan and Z Cai ldquoDesign andimplementation of water environment safety risk assessmentmodel based on AHPrdquo Chinese Journal of Scientific Instrumentvol 30 no 5 pp 1009ndash1013 2009

[12] Z-Y Yang J-B Qu and C-K Huang ldquoBridge safety evaluationbased on fuzzy synthetic evaluation method and analytichierarchy processrdquo Journal of Tianjin University Science andTechnology vol 38 no 12 pp 1063ndash1067 2005

[13] S Liu Y Zhang and D Yu ldquoCorrectinn consistency of fuzzyjudgment matrix using niche genetic algorithmrdquo Journal ofHuazhong University of Science and Technology vol 38 no 7pp 126ndash129 2010

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Function Spaces

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


Ship electric propulsionsystem simulation credibility

A

System simulation modelB

System simulation resultC


Propul-sion

system


Powersupplysystem

D1


Propul-sion

system


Powersupplysystem

D2 D3D4 D5 D6 D7

D8

Figure 1 Structure diagram for ship electric propulsion simulation system reliability evaluation index system

solve the synthesis expert information conflict problems andimprove the system of the objectivity of the judgment

2 The Simulation System ReliabilityEvaluation Index System

For the ship electric propulsion simulation system reliabilityevaluation study according to user requirements and thecharacteristics of the system itself first of all we need toestablish the ship electric propulsion simulation system relia-bility evaluation index system Based on the key indicatorsin research process investigate subsystem one by one andestablish the system equivalence evaluation index systemIn the process of building evaluation system according tothe target layer criterion layer and measures layer sequencedecompose the system step by step hierarchically and makea complicated problem decomposed into several elementsFor ship electric propulsion simulation system is concernedabout target layer is ship electric propulsion simulationsystem reliability Criterion layer from the system simulationmodel to system based on the simulation results determinesthe level of index measures step by step [8] For system sim-ulation model and system simulation results are concernedabout the reliability is analyzed from the power supplysystem power distribution system power transformationsystem and propulsion system respectively Finally shipelectric propulsion simulation system reliability evaluationsystem is shown in Figure 1

3 Expert Evaluation Weight Analysis

31 The Steps to Calculate the Index Weight

Step 1 According to many expertsrsquo relative judgment matrixuse the principle of AHP to calculate the index of the initialweight

Step 2 Determine the index weight fusion through themodified D-S fusion initial index weight and determine theabsolute judgment matrix through the expert judgment

Step 3 According to each index fusion weight and absolutejudgment matrix determine the expert weight the objectivefunction and a genetic algorithm is adopted to calculate theoptimal solution to determine the expert weight

Step 4 Weigh the initial weight and expert weight anddetermine the weights of the index

32 Based on the AHP Analysis of Initial Weights Initialweight can be calculated by AHP to carry out Comparedwith the previous expert scoring method fuzzy evalua-tion method the grey correlation method Pressure-State-Response method (PSR) and artificial neural network algo-rithm AHP is a kind of qualitative analysis and quantitativeanalysis and systematic and hierarchical multiple factors ofdecision analysis method this method will be the decisionmakerrsquos experience quantitative judgment It is very con-venient in the condition of the multiobjective and lack ofnecessary data [9]

Ship electric propulsion simulation system reliabilityevaluation index system calculation generally can be dividedinto the following four steps [10]

Step 1 Each element value in judgment matrix is relativeto a certain element in a previous level associated with theeach elements in the layer pairwise comparison judgmentimportance In the judge process use 1ndash9 scale method toshow specific as is shown in Table 1

Step 2 The elementrsquos relative weight for the criterion iscalculated by judgment matrix

Step 3Compute synthetic weight of each of the layer elementsto system target

Step 4 Consistency check consistency includes absoluteconsistency (or complete consistency) and order consistencyThe so-called absolute consistency means that the judgmentmatrix 119860 If matrix A meet

119886119894119895

= 119886119894119896119886119895119896

119894 119895 119896 = 1 2 119899 (1)

We called matrix 119860 meet absolute consistency










119894119895


= 1119887119894119895


119886119894119895

=119882119894

119882119895

119894 119895 119896 = 1 2 119899

119860119882 = 119899119882

(2)


CI =120582max minus 119899

119899 minus 1 (3)



CR =CIRI

(4)



1 1198872 119887

119899

experts set119863 = 1198891 1198892 119889

119898 andmatrix119860 = (119886

119894119895)119898times119899

(0 lt

119886119894119895


1199081 1199082 119908


1 1199032 119903

119898

and meet sum119899119895=1

119908119895= 1 sum119898

119894=1119903119894= 1




119894119895(119903) show the index


119865119894119895 (119903) =

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895 (5)




experts

119865119895 (119903) =

119898

sum

119894=1

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895 (6)




min 119865 (119903) =

119899

sum

119895=1

119898

sum

119894=1

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895

st119898

sum

119894=1

119903119894= 1

0 lt 119903119894lt 1 (119894 = 1 119898)

(7)





Start


Random generation



Output




Yes

NoCounter t = t + 1




1198611=

[[[[[[[[

[

11

6

1

33

6 1 2 7

31

21 5

1

3

1

7

1

51

]]]]]]]]

]

1198612=

[[[[[[

[

11

2

1

23

2 1 1 4

2 1 1 4

1

3

1

4

1

41

]]]]]]

]

1198613=

[[[[[[[[

[

1 11

53

1 11

53

5 1 1 7

1

3

1

3

1

71

]]]]]]]]

]

(8)


= 01176 05345 02904005781198821198872

= 020040358703583 00816 and 119882

1198873= 01953 01953 05324 00771


1198871015840 = 00633




119861= 0401 036 0238


119887= 01654 03886 03732 00706



119862=





119860= 05108 04882



119882 = [1198821015840

1198611198821015840

119862] sdot 1198821015840

119860

=[[[

[

01653 01666

03887 01758

03731 02974

00708 03582

]]]

]

sdot [05108

04882]

= [01658 02844 03358 02110]1015840

(9)




5 Conclusions




Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119894119895


= 1119887119894119895


119886119894119895

=119882119894

119882119895

119894 119895 119896 = 1 2 119899

119860119882 = 119899119882

(2)


CI =120582max minus 119899

119899 minus 1 (3)



CR =CIRI

(4)



1 1198872 119887

119899

experts set119863 = 1198891 1198892 119889

119898 andmatrix119860 = (119886

119894119895)119898times119899

(0 lt

119886119894119895


1199081 1199082 119908


1 1199032 119903

119898

and meet sum119899119895=1

119908119895= 1 sum119898

119894=1119903119894= 1




119894119895(119903) show the index


119865119894119895 (119903) =

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895 (5)




experts

119865119895 (119903) =

119898

sum

119894=1

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895 (6)




min 119865 (119903) =

119899

sum

119895=1

119898

sum

119894=1

119898

sum

119896=1

10038161003816100381610038161003816119886119894119895119903119894minus 119886119896119895119903119896

10038161003816100381610038161003816lowast 119908119895

st119898

sum

119894=1

119903119894= 1

0 lt 119903119894lt 1 (119894 = 1 119898)

(7)





Start


Random generation



Output




Yes

NoCounter t = t + 1




1198611=

[[[[[[[[

[

11

6

1

33

6 1 2 7

31

21 5

1

3

1

7

1

51

]]]]]]]]

]

1198612=

[[[[[[

[

11

2

1

23

2 1 1 4

2 1 1 4

1

3

1

4

1

41

]]]]]]

]

1198613=

[[[[[[[[

[

1 11

53

1 11

53

5 1 1 7

1

3

1

3

1

71

]]]]]]]]

]

(8)


= 01176 05345 02904005781198821198872

= 020040358703583 00816 and 119882

1198873= 01953 01953 05324 00771


1198871015840 = 00633




119861= 0401 036 0238


119887= 01654 03886 03732 00706



119862=





119860= 05108 04882



119882 = [1198821015840

1198611198821015840

119862] sdot 1198821015840

119860

=[[[

[

01653 01666

03887 01758

03731 02974

00708 03582

]]]

]

sdot [05108

04882]

= [01658 02844 03358 02110]1015840

(9)




5 Conclusions




Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start


Random generation



Output




Yes

NoCounter t = t + 1




1198611=

[[[[[[[[

[

11

6

1

33

6 1 2 7

31

21 5

1

3

1

7

1

51

]]]]]]]]

]

1198612=

[[[[[[

[

11

2

1

23

2 1 1 4

2 1 1 4

1

3

1

4

1

41

]]]]]]

]

1198613=

[[[[[[[[

[

1 11

53

1 11

53

5 1 1 7

1

3

1

3

1

71

]]]]]]]]

]

(8)


= 01176 05345 02904005781198821198872

= 020040358703583 00816 and 119882

1198873= 01953 01953 05324 00771


1198871015840 = 00633




119861= 0401 036 0238


119887= 01654 03886 03732 00706



119862=





119860= 05108 04882



119882 = [1198821015840

1198611198821015840

119862] sdot 1198821015840

119860

=[[[

[

01653 01666

03887 01758

03731 02974

00708 03582

]]]

]

sdot [05108

04882]

= [01658 02844 03358 02110]1015840

(9)




5 Conclusions




Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusions




Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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