Fuzzy Analytical Hierarchy Process Method to Determine the ...

Research ArticleFuzzy Analytical Hierarchy Process Method to Determine theQuality of Gemstones

Mochammad Sobandi Dwi Putra Septi Andryana Fauziah and Aris Gunaryati

Department of Informatics Universitas Nasional Jakarta 12520 Indonesia

Correspondence should be addressed to Septi Andryana septiandryanacivitasunasacid

Received 25 May 2018 Accepted 13 September 2018 Published 1 October 2018

Guest Editor Qi Zeng

Copyright copy 2018 Mochammad Sobandi Dwi Putra et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The selection of quality gemstones requires a special ability to select and assess the quality of gemstones to be tradedThe diversityof types of gemstones and consumers becomes an obstacle in itself when the knowledge and ability of individuals to analyze thequality of gemstones is minimalThe decision-making method used is Fuzzy Analytical Hierarchy Process (F-AHP)method whichis widely used in various sectors F-AHP is easy to adapt to many decision issues the research proposes a decision-making systemusing the F-AHP algorithm to analyze the quality of gemstones The results obtained with the use of F-AHPmodel in the selectionof quality gemstones show the highest quality of gemstones of all stones compared Rubi 1 with a weight value of 0152942

1 Introduction

Many times we are always faced with several options for theright decision-making It is difficult to determine an accuratechoice according to predetermined criteria Decision-makingissues are also experienced when selecting quality gemstonesFor the gemstones entrepreneur special skills are requiredto select or assess the quality of gemstones to be tradedThe diversity of types of gemstones and consumer types inchoosing gemstones is certainly a constraint when the data isincomplete and there is a lack of individual knowledge aboutanalyzing the quality of gemstones

To maintain the consistency of product quality and inaccordance with the demands of the market it is necessaryto have quality control on eligible products so that theerror does not happen again The system to be created is asolution that can assist decision-making for decision-makersin assessing and selecting quality gemstones accurately andeffectively

In previous research several studies using the F-AHPmethod have been proven from several previous studieswith the conclusion that the F-AHP method can be appliedand effective for many problems in real life Chien-ChangChou and Ker-Wei Yu [1] propose a hybrid fuzzy AHP to

deal with the decision-making problems in an uncertain andmultiple-criteria environment choiceThe F-AHP adopted bythe research [2] which combines the AHP with fuzzy settheory can not only capture the thinking logic of humanbeings but also focus on the relative importance of theevaluation criteria In journal [3] the result obtained showsthe best balance of performance for criteria from differentcategories such as physicochemical properties as well assafety environmental and health aspects The assumptionmade in F-AHP approach is that all the criteria involvedare independent of each other However in practice therelationship among criteria is usually complex and theremight be interdependencies

To control the quality we need a relevant element andmethod [4] fuzzy model can be used with various mcdmmethods [5] F-AHP model is a good referral for decision-makers [6] The fuzzy AHP method is applicable as a controlfor the quality and is useful for multicriteria decision-makingproblems [7] The criteria people think of are the size thatmakes the quality of the gemstone better but with othercriteria as a comparison can make the quality of smallerstones better than larger ones [8]

By using F-AHP method we can help a decision-makerto make more efficient flexible and realistic decisions based

HindawiAdvances in Fuzzy SystemsVolume 2018 Article ID 9094380 6 pageshttpsdoiorg10115520189094380

2 Advances in Fuzzy Systems

Table 1 Data of gemstones containing specific gravity color hardness cutting and clarity as the main criteria The data can be seen fromthe gemstone certificate

Specific gravity Color Hardness Cutting ClarityRubi 1 450 ct AAA 9 mohs Excellent I2Rubi 2 160 ct AA 9 mohs Excellent SI2Rubi 3 065 ct A 9 mohs Average I1Rubi 4 225 ct A 9 mohs Average SI1Rubi 5 105 ct B 9 mohs Poor VS

Define theProblem

Create acomparison matrix

Set up Triangular FuzzyNumber (TFN)

Checking forconsistency

Ranking and selection ofdecisions

Calculate the weightvalue of the fuzzy vector

Figure 1 Block diagram has six steps of F-AHP phase process

upon the available criteria and alternatives [9]Therefore theauthors wish to apply the F-AHP method to determine thequality of gemstones

2 Data Availability

The data and criteria in Table 1 were collected by consultingthe director of Kantor Promosi Batu Mulia Indonesia Thedata can be seen from the stone certificate issued by KantorPromosi BatuMulia Indonesia Every time we buy gemstoneswewill get certificate of authenticity of stone in the certificatethere are criteria of stone

3 Materials and Methods

Analytic Hierarchy Process (AHP) is a decision supportmethod developed to complete problem by breaking thesolution problems grouping them and then arranging theminto a hierarchical structure To obtain priority criteria thismethod uses a comparison of criteria paired with a measure-ment scale that has been determined The main input of theAHPmethod is the perception of experts or experts so thereis a factor of subjectivity in retrieval decision This methodalso takes into account data validity with inconsistency limits[10] However considerable uncertainty and doubt in givingan assessment will have an impact on the accuracy of the dataand the results obtained Based on this further theory wasdeveloped namely the method of Fuzzy Analytic HierarchyProcess Fuzzy Analytic Hierarchy Process is a method ofAnalyticHierarchy Process (AHP) developedwith fuzzy logictheory Fuzzy AHP method is used similar to the method ofAHP It is just that the Fuzzy AHPmethod sets the AHP scaleinto the fuzzy triangle scale to be accessed priority

In this section the F-AHP method was developed Theprocedure used in the proposed method is described asfollows

Step 1 (define the problem and determine the desired solution(see Figure 1)) We need to define the problem accordingto the criteria used to determine the gemstones of qualitySpecific gravity color hardness cutting and clarity are usedas the main criteria for determining gemstone quality This issome data from the gemstone certificate (see Table 1)

The weight of the stone has a unit of carat (ct) the greaterthe weight of a gemstone the greater its size Stone hardnessunit is called Mohs because the name of the first person to doresearch on the hardness of a gemstone was FriedrichMohs ageologist and mineralogist fromGermany in 1812The claritylevel of a gemstone is divided into IF VVS1 VVS2 VS1 VS2I1 I2 etc The gemstone color level is seen from the level ofclarity of color seen by the eye given the levels of B A AAAAA and so on The level of cutting quality is seen from itsproportional and symmetrical shape of gemstones pieces

Step 2 (create a comparison matrix (see Figure 1)) Afterwe know the data and criteria stone in Table 1 we need tocreate a comparison matrix The matrix used is simple hasa strong position for the consistency framework obtainsother information that may be required with all possiblecomparisons and is able to analyze the overall prioritysensitivity for changes in consideration

Here are the equations used to define pairwise compar-isons

119886119894119895 = 119908119894119908119895 119894 119895 = 1 2 119899 (1)

where n denotes the number of criteria compared Wi areweights for the i criterion and aij is the ratio of the weightof the i criterion and j

After knowing the comparison of its criteria in Table 2the next thing done is to normalize each column into the

Advances in Fuzzy Systems 3

Table 2 Comparison of criteria as the weighted value of each criterion

Criteria Specific Gravity Color Hardness Cutting ClaritySpecific Gravity 1 3 2 3 3Color 13 1 3 2 3Hardness 12 13 1 3 3Cutting 13 12 13 1 3Clarity 13 13 13 13 1

Table 3 Ratio index

n 1 2 3 4 5 6 7 8 9 10RI 000 000 058 090 112 124 132 141 145 149

matrix form by dividing each value in the column i and row jwith the largest value in column i

119886119894119895 =119886119894119895

max 119886119894119895 forall119894 119895 (2)

Then the results of the matrix normalization from Table 2 areobtained as follows

[[[[[[[[[

0 4 0 58 0 3 0 321 0 2310 133 0 193 0 45 0 214 0 2310 2 0 064 0 15 0 3214 0 2310 133 0 096 0 05 0 107 0 2310 133 0 064 0 05 0 035 0 072

]]]]]]]]]

(3)

Step 3 (checking for consistency (see Figure 1)) The compar-ison of the consistency index with a random generator (RI)value is listed in Table 3 set by Saaty [10] This value dependson the matrix order n

Consistency is expected to be near perfect to produce adecision that is close to valid

Here is the equation used to calculate the value ofconsistency

First we must recognize the value of the eigenvectorwhich is the weighted value of the criterion To calculate theeigenvector we use the following equation

119908119894 = 119886119894119899 forall119894 (4)

119908119894 is the eigen vector where 119886119894 is the sum of the matrix nor-malization values and is divided by the number of criterion(119899)

The largest eigenvalue is the number of times multiplyingthe number of columns with the main eigenvector (seeTable 4) So it can be obtained by the equation

120582119898119886119896119904 = (sum11986611987211minus1198991 times 1198831) + + (sum1198661198721119899minus119899119894 times 119883119899)

(5)

Table 4 Eigenvector on criteria

Criteria eigenvectorSpecific Gravity 03665Color 02443Hardness 01933Cutting 01236Clarity 00720

120582119898119886119896119904 = (0 3665 X 2 5) + (0 2443 X 5 667)+ (0 1933 X 6 6667)+ (0 1236 X 9 333) + (0 0720 X 13)

120582119898119886119896119904 = 5 474

(6)

After obtaining maximum lambda value the value of CI canbe determined

119862119868 = 120582119898119886119896119904 minus 119899119899 minus 1 (7)

where CI is the consistency index and maximum lambda isthe largest eigenvalue of the n-order matrix

119862119868 = 5 474 minus 55 minus 1119862119868 = 0 1185

(8)

If the value of CI is zero (0) this means the matrix isconsistent If the value of CI obtained is greater than 0 (CIgt0) then the limit of inconsistency applied by Saaty is testedTesting is measured using Consistency Ratio (CR) ie indexvalue (Table 3) or comparison between CI and RI

119862119877 = 119862119868119877119868 (9)

The RI value used is in accordance with the order n matrixIf the CR of a smaller matrix is 10 (01) this means that theinconsistency of each opinion is considered acceptable

119862119877 = 0 11851 12119862119877 = 0 1058

(10)


Table 5 TFN scale

TFN Scale L M U1 1 1 12 05 1 153 1 15 24 15 2 255 2 25 36 25 3 357 3 35 48 35 4 459 4 45 4505 0666667 1 20333333 05 0666667 1025 04 05 066666702 0333333 04 050166667 0285714 0333333 040142857 025 0285714 03333330125 0222222 025 02857140111111 0222222 0222222 025

Intensity of interest(Saaty1980)

TFN Scale

()Member degree

2

05

3 4 5 6 7 8 9

2 3 4 5 6 7 8 9

Figure 2 Graph of Fuzzy Triangle Set

The consistency value of 01058 is equivalent to 10 inconsis-tency this value can still be tolerated because the consistencyvalue index must be less than 01

Step 4 (set up Triangular Fuzzy Number (TFN) (seeFigure 1)) The F-AHP scale has three values namely thelowest value (lower L) middle value (median M) and high-est value (upper U) So each fuzzy set will be divided into 2(see Figure 2) except for the same comparison set or can beseen on the TFN scale (see Table 5)

Based on the index (see Table 5) the comparison value inTable 2 will be made into a TFN set (see Table 6)

Step 5 (calculate the weight value of the fuzzy vector (seeFigure 1)) After the AHP comparison value is transformedto F-AHP scale value fuzzy synthesis value is calculated Theprocess to get fuzzy synthesis value is shown using equationof the following formula

119878119894 =119898

sum119895=1

119872119895119892119894119883 1[sum119899119894=1sum119898119895=1119872119895119892119894] (11)

Information

119878119894 = fuzzy synthesis valuesum119898119895=1119872119895119892119894 = summing the cell value in that column

starting from column 1 in each row matrixi = rowj = columnAfter the comparison of fuzzy synthesis values (see

Table 7) we will get the defuzzification ordinate value (d1015840)From the above calculation we can calculate the values of vand d1015840 To calculate V1015840 we use the equation of the followingformula

119881 (1198722 ge 1198721)

=

1 119894119891 1198982 ge 11989810 119894119891 1198971 ge 1199062

1198971 minus 1199062(1198982 minus 1199062) minus (1198981 minus 1198971)1015840

119890119905119888

(12)

Calculating the value of the fuzzy vector weight (W1015840) calcu-lation of the fuzzy weight value is shown using the equationof the following formula

1198891015840(119860119894) = min119881 (119878119894 gt 119878119896) (13)


Table 6 TFN set of criteria each value in the criteria comparison (see Table 2) is changed to TFN referring to the TFN scale

Specific Gravity Color Hardness Cutting ClarityL M U L M U L M U L M U L M U

Specific Gravity 1 1 1 1 15 2 05 1 15 1 15 2 1 15 2Color 05 0666667 1 1 1 1 1 15 2 05 1 15 1 15 2Hardness 0666667 1 2 05 0666667 1 1 1 1 1 15 2 1 15 2Cutting 05 0666667 1 0666667 1 2 05 0666667 1 1 1 1 1 15 2Clarity 05 0666667 1 05 0666667 1 05 0666667 1 05 0666667 1 1 1 1

Table 7 Synthesis value

Synthesis ValueL M U0125 0246 04390111 0215 03870115 0215 04130101 0183 03620083 0139 0258

Table 8 Criteria weight value the result of calculation whichcontains the weight value of each criterion

criteria WeightSpecific Gravity 0279088Color 0249106Hardness 0251526Cutting 0220279Clarity 01546

collecting ordinate values that have been previously obtainedas below

Σ1198821015840 = (V1199041198961 V1199041198962 V119904119896119899) (14)

Normalization of vector weight values is obtained by theequation of the following formula

1198821015840 = (1198891015840 (1198601) 1198891015840 (1198602) 1198891015840 (119860119899))T (15)

Step 6 (ranking and selection of decisions (see Figure 1))Next is to do an alternative value calculation where the alter-native settlement measures are the same as the completionsteps on the criteria Each alternative elementrsquos weight value(see Table 8) will be calculated by the weight of the criteriaelement and will be directed to get the decision result

4 Result and Discussion

The built system consists of several menus that are the stagesin running the decision support systemThefirst thing to do islogin first To be able to use this systemwe need to login Afterlogin we will enter into the main menu On the main pagethe F-AHP algorithm and any data needed to start the systemprocess are explained After that alternative data and criteriaare entered into the systemThe gemstone data we have needto be input into the alternate data input page according to

0002004006008

01012014016018

1 2 3 4 5

Ranking

Figure 3 Ranking graph ranking is derived from the result ofalternate weight matrix by weight of criterion

the criteria along with the criteria data we input into theinput data page criteria The next thing to do is to providea comparison of the criteria and the value of alternativecomparison to each criterion After all has been done nextwe can do the following process In the process page we cansee the value of criteria comparison and TFN set of criteriaWhen the process is completed this will result in the rankingof each alternative the decision-maker can determine whichgemstones are qualified from the gemstones being comparedFrom the ranking results in Figure 3 and Table 9 it can beconcluded that alternative 1 has the most optimum weightvalue compared with other alternatives Therefore a decisioncan be made that Rubi 1 is the highest-quality gemstone of allstones compared

5 Conclusion

The conclusion of this research is as follows we created a sys-tem that can assist decision-making in assessing and choosingquality gemstones accurately and effectively by using F-AHPalgorithm

The focus on the decision of the system is more on thedecision of stones based on the same type of stone name thisis because for the decision system to be more appropriateand relevant for use as a consideration in decision-makingit is impossible to compare one stone with stones of differenttypes not in the same class quality so the end result of thesystem is based on the classification of the type of stone name

As shown in Figure 3 we obtained the result by using theF-AHP model in the selection of quality gemstones Rubi 1with a weight value of 0152942 Rubi 2 of 0075731 Ruby 3of 0050075 and Ruby 4 and Rubi 5 of 0This weighting value


Table 9 Value of alternative calculation result on criteria

Specific Gravity Color Hardness Cutting Clarity TotalRubi 1 0548005 0389588 025 042711 0427683 0152942Rubi 2 0270062 0531613 025 057289 0491956 0075371Rubi 3 0181933 0028674 025 0 0080361 0050775Rubi 4 0 0050125 025 0 0 0Rubi 5 0 0 025 0 0 0

indicates that a gemstone of the highest quality is Rubi 1 witha weight value of 0152942 of all stones compared

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Chien-Chang Chou and Ker-Wei Yu ldquoApplication of a NewHybrid Fuzzy AHP Model to the Location ChoicerdquoMathemat-ical Problems in Engineering vol 2013 Article ID 592138 12pages 2013

[2] H-Y Wu G-H Tzeng and Y-H Chen ldquoA fuzzy MCDMapproach for evaluating banking performance based on Bal-anced Scorecardrdquo Expert Systems with Applications vol 36 no6 pp 10135ndash10147 2009

[3] J Ooi M A B Promentilla R R Tan D K S Ng and NG Chemmangattuvalappil ldquoIntegration of FuzzyAnalyticHier-archy Process into multi-objective Computer Aided MolecularDesignrdquo Computers amp Chemical Engineering vol 109 pp 191ndash202 2018

[4] J-C Tu and C-L Hu ldquoApplying the Fuzzy Analytic HierarchyProcess to Construct the Product Innovative Service Systemof Wedding Photography Apparelrdquo Mathematical Problems inEngineering vol 2015 2015

[5] M Shaverdi ldquoApplication of Fuzzy AHP Approach for Finan-cial Performance Evaluation of Iranian Petrochemical SectorrdquoProcedia Computer Science pp 995ndash1004

[6] R-H Chiu L-H Lin and S-C Ting ldquoEvaluation of green portfactors and performance a fuzzy AHP analysisrdquo MathematicalProblems in Engineering vol 2014 Article ID 802976 12 pages2014

[7] L Wang J Chu and JWu ldquoSelection of optimummaintenancestrategies based on a fuzzy analytic hierarchy processrdquo Interna-tional Journal of Production Economics vol 107 no 1 pp 151ndash163 2007

[8] S-H Tsaura T-Y Chang and C-H Yen ldquoThe evaluation ofairline service quality by fuzzy MCDMrdquo Tourism Managementvol 23 no 2 pp 107ndash115 2002

[9] Kaur ldquoSelection of Vendor Based on Intuitionistic FuzzyAnalytical HierarchyProcessrdquoAdvances in Operations Researchvol 2014 Article ID 987690 10 pages 2014

[10] T L Saaty and K P Kearns Analytical Planninge Organiza-tion of Systems Pergamon Press 1985

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Table 1 Data of gemstones containing specific gravity color hardness cutting and clarity as the main criteria The data can be seen fromthe gemstone certificate

Specific gravity Color Hardness Cutting ClarityRubi 1 450 ct AAA 9 mohs Excellent I2Rubi 2 160 ct AA 9 mohs Excellent SI2Rubi 3 065 ct A 9 mohs Average I1Rubi 4 225 ct A 9 mohs Average SI1Rubi 5 105 ct B 9 mohs Poor VS

Define theProblem

Create acomparison matrix

Set up Triangular FuzzyNumber (TFN)

Checking forconsistency

Ranking and selection ofdecisions

Calculate the weightvalue of the fuzzy vector

Figure 1 Block diagram has six steps of F-AHP phase process

upon the available criteria and alternatives [9]Therefore theauthors wish to apply the F-AHP method to determine thequality of gemstones

2 Data Availability

The data and criteria in Table 1 were collected by consultingthe director of Kantor Promosi Batu Mulia Indonesia Thedata can be seen from the stone certificate issued by KantorPromosi BatuMulia Indonesia Every time we buy gemstoneswewill get certificate of authenticity of stone in the certificatethere are criteria of stone

3 Materials and Methods

Analytic Hierarchy Process (AHP) is a decision supportmethod developed to complete problem by breaking thesolution problems grouping them and then arranging theminto a hierarchical structure To obtain priority criteria thismethod uses a comparison of criteria paired with a measure-ment scale that has been determined The main input of theAHPmethod is the perception of experts or experts so thereis a factor of subjectivity in retrieval decision This methodalso takes into account data validity with inconsistency limits[10] However considerable uncertainty and doubt in givingan assessment will have an impact on the accuracy of the dataand the results obtained Based on this further theory wasdeveloped namely the method of Fuzzy Analytic HierarchyProcess Fuzzy Analytic Hierarchy Process is a method ofAnalyticHierarchy Process (AHP) developedwith fuzzy logictheory Fuzzy AHP method is used similar to the method ofAHP It is just that the Fuzzy AHPmethod sets the AHP scaleinto the fuzzy triangle scale to be accessed priority

In this section the F-AHP method was developed Theprocedure used in the proposed method is described asfollows

Step 1 (define the problem and determine the desired solution(see Figure 1)) We need to define the problem accordingto the criteria used to determine the gemstones of qualitySpecific gravity color hardness cutting and clarity are usedas the main criteria for determining gemstone quality This issome data from the gemstone certificate (see Table 1)

The weight of the stone has a unit of carat (ct) the greaterthe weight of a gemstone the greater its size Stone hardnessunit is called Mohs because the name of the first person to doresearch on the hardness of a gemstone was FriedrichMohs ageologist and mineralogist fromGermany in 1812The claritylevel of a gemstone is divided into IF VVS1 VVS2 VS1 VS2I1 I2 etc The gemstone color level is seen from the level ofclarity of color seen by the eye given the levels of B A AAAAA and so on The level of cutting quality is seen from itsproportional and symmetrical shape of gemstones pieces

Step 2 (create a comparison matrix (see Figure 1)) Afterwe know the data and criteria stone in Table 1 we need tocreate a comparison matrix The matrix used is simple hasa strong position for the consistency framework obtainsother information that may be required with all possiblecomparisons and is able to analyze the overall prioritysensitivity for changes in consideration

Here are the equations used to define pairwise compar-isons

119886119894119895 = 119908119894119908119895 119894 119895 = 1 2 119899 (1)

where n denotes the number of criteria compared Wi areweights for the i criterion and aij is the ratio of the weightof the i criterion and j

After knowing the comparison of its criteria in Table 2the next thing done is to normalize each column into the




Table 3 Ratio index

n 1 2 3 4 5 6 7 8 9 10RI 000 000 058 090 112 124 132 141 145 149


119886119894119895 =119886119894119895

max 119886119894119895 forall119894 119895 (2)


[[[[[[[[[

0 4 0 58 0 3 0 321 0 2310 133 0 193 0 45 0 214 0 2310 2 0 064 0 15 0 3214 0 2310 133 0 096 0 05 0 107 0 2310 133 0 064 0 05 0 035 0 072

]]]]]]]]]

(3)





119908119894 = 119886119894119899 forall119894 (4)




(5)



120582119898119886119896119904 = (0 3665 X 2 5) + (0 2443 X 5 667)+ (0 1933 X 6 6667)+ (0 1236 X 9 333) + (0 0720 X 13)

120582119898119886119896119904 = 5 474

(6)


119862119868 = 120582119898119886119896119904 minus 119899119899 minus 1 (7)


119862119868 = 5 474 minus 55 minus 1119862119868 = 0 1185

(8)


119862119877 = 119862119868119877119868 (9)


119862119877 = 0 11851 12119862119877 = 0 1058

(10)


Table 5 TFN scale

TFN Scale L M U1 1 1 12 05 1 153 1 15 24 15 2 255 2 25 36 25 3 357 3 35 48 35 4 459 4 45 4505 0666667 1 20333333 05 0666667 1025 04 05 066666702 0333333 04 050166667 0285714 0333333 040142857 025 0285714 03333330125 0222222 025 02857140111111 0222222 0222222 025


TFN Scale

()Member degree

2

05

3 4 5 6 7 8 9

2 3 4 5 6 7 8 9






119878119894 =119898

sum119895=1

119872119895119892119894119883 1[sum119899119894=1sum119898119895=1119872119895119892119894] (11)

Information




119881 (1198722 ge 1198721)

=

1 119894119891 1198982 ge 11989810 119894119891 1198971 ge 1199062


119890119905119888

(12)


1198891015840(119860119894) = min119881 (119878119894 gt 119878119896) (13)










Σ1198821015840 = (V1199041198961 V1199041198962 V119904119896119899) (14)


1198821015840 = (1198891015840 (1198601) 1198891015840 (1198602) 1198891015840 (119860119899))T (15)




0002004006008

01012014016018

1 2 3 4 5

Ranking



5 Conclusion








Data Availability




References
















Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in







Table 3 Ratio index

n 1 2 3 4 5 6 7 8 9 10RI 000 000 058 090 112 124 132 141 145 149


119886119894119895 =119886119894119895

max 119886119894119895 forall119894 119895 (2)


[[[[[[[[[

0 4 0 58 0 3 0 321 0 2310 133 0 193 0 45 0 214 0 2310 2 0 064 0 15 0 3214 0 2310 133 0 096 0 05 0 107 0 2310 133 0 064 0 05 0 035 0 072

]]]]]]]]]

(3)





119908119894 = 119886119894119899 forall119894 (4)




(5)



120582119898119886119896119904 = (0 3665 X 2 5) + (0 2443 X 5 667)+ (0 1933 X 6 6667)+ (0 1236 X 9 333) + (0 0720 X 13)

120582119898119886119896119904 = 5 474

(6)


119862119868 = 120582119898119886119896119904 minus 119899119899 minus 1 (7)


119862119868 = 5 474 minus 55 minus 1119862119868 = 0 1185

(8)


119862119877 = 119862119868119877119868 (9)


119862119877 = 0 11851 12119862119877 = 0 1058

(10)


Table 5 TFN scale

TFN Scale L M U1 1 1 12 05 1 153 1 15 24 15 2 255 2 25 36 25 3 357 3 35 48 35 4 459 4 45 4505 0666667 1 20333333 05 0666667 1025 04 05 066666702 0333333 04 050166667 0285714 0333333 040142857 025 0285714 03333330125 0222222 025 02857140111111 0222222 0222222 025


TFN Scale

()Member degree

2

05

3 4 5 6 7 8 9

2 3 4 5 6 7 8 9






119878119894 =119898

sum119895=1

119872119895119892119894119883 1[sum119899119894=1sum119898119895=1119872119895119892119894] (11)

Information




119881 (1198722 ge 1198721)

=

1 119894119891 1198982 ge 11989810 119894119891 1198971 ge 1199062


119890119905119888

(12)


1198891015840(119860119894) = min119881 (119878119894 gt 119878119896) (13)










Σ1198821015840 = (V1199041198961 V1199041198962 V119904119896119899) (14)


1198821015840 = (1198891015840 (1198601) 1198891015840 (1198602) 1198891015840 (119860119899))T (15)




0002004006008

01012014016018

1 2 3 4 5

Ranking



5 Conclusion








Data Availability




References
















Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table 5 TFN scale

TFN Scale L M U1 1 1 12 05 1 153 1 15 24 15 2 255 2 25 36 25 3 357 3 35 48 35 4 459 4 45 4505 0666667 1 20333333 05 0666667 1025 04 05 066666702 0333333 04 050166667 0285714 0333333 040142857 025 0285714 03333330125 0222222 025 02857140111111 0222222 0222222 025


TFN Scale

()Member degree

2

05

3 4 5 6 7 8 9

2 3 4 5 6 7 8 9






119878119894 =119898

sum119895=1

119872119895119892119894119883 1[sum119899119894=1sum119898119895=1119872119895119892119894] (11)

Information




119881 (1198722 ge 1198721)

=

1 119894119891 1198982 ge 11989810 119894119891 1198971 ge 1199062


119890119905119888

(12)


1198891015840(119860119894) = min119881 (119878119894 gt 119878119896) (13)










Σ1198821015840 = (V1199041198961 V1199041198962 V119904119896119899) (14)


1198821015840 = (1198891015840 (1198601) 1198891015840 (1198602) 1198891015840 (119860119899))T (15)




0002004006008

01012014016018

1 2 3 4 5

Ranking



5 Conclusion








Data Availability




References
















Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in













Σ1198821015840 = (V1199041198961 V1199041198962 V119904119896119899) (14)


1198821015840 = (1198891015840 (1198601) 1198891015840 (1198602) 1198891015840 (119860119899))T (15)




0002004006008

01012014016018

1 2 3 4 5

Ranking



5 Conclusion








Data Availability




References
















Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








Data Availability




References
















Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




Fuzzy Analytical Hierarchy Process Method to Determine the ...

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