research article fuzzy multicriteria abc supplier

Research ArticleFuzzy Multicriteria ABC Supplier Classification inGlobal Supply Chain

Petar Kefer1 Dragan D Milanovic2 Mirjana Misita2 and Aleksandar Zunjic2

1Omni Surfaces 40 Kodiak Cres Toronto ON Canada M3J 3G52Faculty of Mechanical Engineering Kraljice Marije 16 11000 Belgrade Serbia

Correspondence should be addressed to Mirjana Misita mmisitamasbgacrs

Received 13 April 2016 Revised 5 July 2016 Accepted 19 July 2016

Academic Editor Monica A Lopez-Campos

Copyright copy 2016 Petar Kefer 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

The determination of the optimal purchasing strategy in enterprise that is a part of global supply chain could be performed in twosteps In step one a classification of potential suppliers is performed in order to determine the optimal portfolio of suppliers Thisis delivered by using the fuzzy multicriteria proposed ABC classification method Uncertainties in relative importance of criteriaand their values are described by linguistic expressions Modelling of linguistic expressions is based on the fuzzy sets theory In thesecond step ranking of optimal portfolio of suppliers is performed by using the modified ELECTRE method The obtained resultsrepresent valuable input for determining the long time purchasing strategy and building partnership with the best suppliers Thedeveloped two-stepmodel is verified on real life dataThe obtained results indicate good compliance with the opinionsmanagementin this type of industry It is worth to mention that the proposed model can be easily extended and adopted to the analysis of otherissues of management which could be applicable in different research areas

1 Introduction

During the usual activities in enterprises many products andservices originated out of enterprise are used In that mannerthe organization should ensure that purchased and usedproducts and services conform to specified requirements [1]In compliance with that an organization shall establish andapply criteria for evaluation selection and monitoring ofperformance and reevaluate suppliers (external providers)On the other hand distribution and procurement in thesame time are usually based on complex procedures involv-ing many management and control functions at variouslevels Strategies for better supplymanagementmust promotean effective and appropriate products and services supplyproviding products and services in requested quantities justin time and yet the inventory costs have to be as lowas possible Development of supply policy and appropriatestrategy have to be defined on the enterprise level or on thelevel of the whole supply chain which is a complex problem

Recent work majorly focuses on analytical and decisionmodelling [2] for supplier selection as well as on supplierdevelopment [3] Development of sophisticated models for

optimal supplier selection strategy [4] for different kind ofproducts requires time effort and resources which lead toan increase in inventory control costs and to an increasein total costs too In order to decrease inventory controlcosts in practice the first important step is to perform theclassification of potential suppliers into classes in compliancewith their ability to provide processes or products andservices in accordance with specified requirements Basedon the suppliers classification results management definesappropriate ways to manage and control specific suppliersand rank them for cooperation and building partner con-nections Building partnership should be based on obtainingrespect and trust with mutual benefit [5]

In practice one of the widely used techniques for classifi-cation of different items into classes (applicable to the classi-fication of suppliers too) is the ABC method which is basedon Pareto analysis [6]This method is easy to understand anduse in practice since in the conventional ABC classificationmethod inventory items are divided into three classes AB and C The values of deterministic classification criteriaare classified into descending row As it is known 5ndash10

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9139483 11 pageshttpdxdoiorg10115520169139483

2 Mathematical Problems in Engineering

of analysed inventory items ranked at first place belong togroup A next 15 correspond to group B and the rest ofinventory items correspond to group C Selection of theclassification criterion depends on the kind of problem beingconsidered The issue of selection of suppliers may be statedas a multicriteria optimization task In literature there isa lot of papers that employ different methods for rankingand assessment of suppliers The most used methods areAnalytical Hierarchy Process (AHP) [7] Technique forOrderof Preference by Similarity to Ideal Solution (TOPSIS) [8]ELimination Et Choix Traduisant la REalite (ELECTRE) [9]or combination of two or more methods [10] In compliancewith the rank of suppliers decision makers may choose themost suitable suppliers work building partnership relations

The conventional ABC methodology may sometimes notbe appropriate to provide a good classification of inventoryitems in practice [11] If there is a need to make the clas-sification more sophisticated more than one classificationcriterion and imprecise data about items have to be usedThe classification task becomes a multicriteria classificationproblem in the presence of uncertainty

In this paper the authors focus on the supplier classifica-tion in global supply chain where number of suppliers is largedue to demands of global market The number and types ofthe selection criteria are not clearly defined in the literatureand there are no specific guidelines that treat this issue Thespecificity of construction industry products their variabledemand over time and in particular high cost indicatethe importance of suppliersrsquo classification problem A newfuzzy multicriteria ABC model for classification of suppliersin construction industry is proposed Three eliminatorycriteria are selected to make a base of potential suppliers (1)customer care (2) quality (ratio between price and specificperformances of products) and (3) delivery method (ratiobetween costs and delivery rate)

The objective of paper is to define the appropriate pur-chasing strategy in the global supply chain through two steps(1) classification of a large number of potential suppliers byapplying proposed fuzzymulticriteriaABC (2) the ranking ofclassifiedA group of suppliers (optimal portfolio of suppliers)by using the proposed model consisted from fuzzy AHP andfuzzy ELECTREmethods For the classification problem theproposed fuzzy multicriteria ABC methodology is suitablesince it treats uncertain criteria in appropriate way andit is easy and suitable for usage leading to determinationof optimal portfolio of suppliers This optimal portfolio ofsuppliers consisted of those who are suitable for furtherselection of the group that will be used for building partner-shipThemethods of multicriteria decisionmakingmethodsfuzzy AHP and fuzzy ELECTRE are used for ranking ofselected suppliers In this manner resources such as time andcosts of defining appropriate supply strategy may be reducedsignificantly and in the same time effectiveness of supplyprocess is increased In the final consequence this should leadto increase effectiveness of all business processes in globalsupply chain

Each of three considered criteria is described by using lin-guistic expressions specified by enterprisemanagement teamThese linguistic expressions are modelled using triangular

fuzzy numbers by analogy to Aleksic et al [12] Uncertaintyin criteria values used in supplier classification are mod-elled by fuzzy sets [13ndash15] The motivation for using thismethodology came from its suitability for handling impreciseand ambiguous data and it supports usage of decisionmaking methods which operate with such data It may besaid that fuzzy sets theory resembles the human reasoningin its use of approximate information and uncertainty togenerate decisions [16] In thismanner fuzzy sets have severaladvantages over theories that treat similar problems (a) theyare based on a natural language (b) they are conceptually easyto understand (c) they could be combined with conventionalmethods and techniques for dealing and reasoning withuncertain data and (d) they can capture most nonlinearrelations in problems of arbitrary complexity [17]

The paper is organized in the followingwayThe literaturereview is given in Section 2 Section 3 presents evaluationframework and modelling of uncertainties In Section 4the proposed algorithm is presented Section 5 is used forthe verification of the model using an illustrative exampleConclusions are presented in Section 6

2 Literature Review

A number of papers presented in the literature were dealingwith the problem of items classification in uncertain envi-ronments for inventory control purposes and proposed fuzzymulticriteria ABC classification approaches after realizingthe importance of considering multiple criteria in the ABCanalysis [18] In addition to the overall cost some othercriteria such as lead time inventory holding cost limitationof the warehouse space and order cost were recognized asbeing important for items classification

In the literature as well as in the practice determinationof the optimal portfolio of suppliers is based onmathematicalmodels Possibly there may be a lot of potential suppliers soin the first step the eliminatory criteria should be appliedBuilding partnersrsquo relations with suppliers should be per-formed after the portfolio of suitable suppliers is generatedHowever there is no wide literature which considers supplierselection when there is large number of potential suppliers

Techniques of artificial intelligence such as backprop-agation networks support vector machines and 119896-nearestneighbors may be used for ABC inventory classificationtaking into account several criteria [19] Also ABC classifi-cation may be supported by using different methods such asAnalytic Hierarchy Process and Data Envelopment Analysis[20] In some cases existing models may be analysed andimproved such as extension of theNg-model formulticriteriainventory ABC classification [20] In this way classificationmay be performed by respect to multicriteria and theirweights simultaneouslyThe criteria weightsmay be obtainedby using DEA [20] If criteria are uncertain such as thefact that demand frequency and costs are in focus fuzzyABC model for classification of items according to theirvalue may be applied [21] As the values of classificationcriteria are uncertain they may be described by triangularfuzzy numbers [15] and may be aggregated into one bymultiplying two corresponding criteria values In this way

Mathematical Problems in Engineering 3

value of aggregated classification criteria is described by fuzzynumber which may be defuzzified and classification may beperformed by application of conventional ABC In a case ofexisting variables with either nominal or nonnominal valuesand incorporated management experience and judgmenta fuzzy rule based approach to ABC classification may beapplied [22]

The values of the treated classification items should beranked according to the value of classification criterion inmonotonically decreasing string Typically items of classA represent about 5ndash10 of the total number of itemsapproximately next 15 of items correspond to the groupB and the rest of the items belong to the group C Theitems of class A are the most significant in the treatedissue respecting the classification criteria In treated problemof suppliersrsquo selection ABC method may be deployed fordetermining optimal portfolio of suppliers (classifiedAgroupof suppliers) Optimal portfolio of suppliers should be rankedin order to propose optimal purchasing strategy throughbuilding potential partnership with the best ranked suppliersOptimal portfolio of suppliers may be ranked by applyingdifferent supplier selection models [23] Amongst manymethodologies ELECTRE may be seen as a proven asset formulticriteria decision analysis finding applications in widescientific fields [24] When the issue of supplier selection isin focus the determination of preference of alternatives maybe set as a group decision making problem [25] ELECTREmethod may be used for dealing with multicriteria decisionproblems such as determination of master contractor whenthere are several subcontractors [26] In this case index ofpreference is modified and it is calculated as a product offuzzy triangular numbers ELECTREmay bemodified in waythat the value of criteria for each supplier may be assessedby three decision makers whose assessments are modelledby fuzzy numbers [27] In this proposed fuzzy ELECTREHamming distance is used for comparing the suppliers on thetreated criteria

In compliance with the results of good practice it maybe noticed that the evaluation criteria used for the supplierselection often do not have the same relative importanceTherelative importance does not depend on supplier and it is notsubordinated to change over time Usually decision makersdeliver better opinions by using linguistic expressions thanprecise numbers Itmay be suggested that it is closer to humanthinking to compare relative importance of each pair ofcriteria than to perform direct assessment Respecting thesefacts many authors determine relative importance throughthe fuzzy AHP framework [28] Handling of uncertaintiesmay be performed by using extent analysis [29]

3 Evaluation Framework andModelling of Uncertainties

Step 1 Potential suppliers may be formally presented as a setof indices 120580 = 1 119894 119868 where 119868 is the total number ofsuppliers and 119894 is the index of the possible supplier In thiscase a set of suppliers is defined according to results of goodpractice

Step 2 Evaluation criteria are presented by set of indices 120581 =1 119896 119870 where 119870 is the total number of criteria and119896 is index of criterion The number and type of criteria aredefined by the management team based on the experiencethe results of benchmarking and current information aboutsuppliers which are presented in reports

Step 3 Management team of the enterprise is formallypresented by set of indices 120576 = 1 119890 119864 where 119864 isthe total number of the decision makers and 119890 is the index ofdecision maker In the considered problem the managementteam of treated enterprise which exists within the global sup-ply chain consisted of purchasing manager main managerplantmanager and financialmanager It may be assumed thatthe decision makers have different importance for evaluationand selection of suppliersrsquo problem The decision makerrsquosweight is denoted as 120596

119890 119890 = 1 119864 These weights are given

with respect to the results of good practice For consideredproblem the weights of decision makers are 03 03 02 and02 respectively

Step 4 The relative importance of each pair of criteria isassessed by each decision maker The decision maker usespredefined linguistic expressions which are modelled bytriangular fuzzy numbers (TFNs) The aggregated values ofthe fuzzy pairwisematrix of the relative importance of criteriaare calculated by using Fuzzy Averaging Ordered Method(FOWA) [30] The weights vector is given by extent analysesmethod [29] The criteria weights are described by precisevalues

Step 5 The possible suppliers should be evaluated accordingto the predefined period of time Usually it is a period ofone year In general the time period is divided into smalltime intervals In other words the assessment of suppliers isperformed in discreet time periods Time period is presentedby set of indices 120591 = 1 119905 119879 where 119879 is the totalnumber of discretized intervals and 119905 is the index of timeinterval

Step 6 The criterion value for supplier is assessed by eachdecision maker for each time period 119905 Decision makersuse predefined linguistic expressions which are modelled byTFNs The aggregated values of criteria values for suppliersover time period are given by using fuzzy averaging method

Step 7 The crisp values of decision matrix are given byapplying moment method [15]

Step 8 Classification of possible suppliers with respect to allcriteria and their weights is performed by the proposed ABCmodel

Step 9 Fuzzy decision matrix of suppliers which belong togroup A is stated The rank of these suppliers is determinedby fuzzy ELECTRE method

31 Modelling of Uncertainties Rating of the relative impor-tance of evaluation criteria and their values are based on


uncertain and imprecise knowledge of decision makersModelling of these uncertainties is based on fuzzy set theory[13 15] which is a suitable mathematical tool for presentinguncertain numbers in quantitativeway Fuzzy set is defined byits membership function which can be obtained in differentways [14] The uncertainties which exist in real problemsare often modelled by TFNs because they offer a goodcompromise between descriptive power and computationalsimplicityThe number of TFNs assigned to the uncertaintiesinto the relative importance of criteria and criteria valuesis defined by management team of global supply chain Thedomain of defined TFNs is defined on the real line whichbelong to different intervals In the literature there are norules or suggestions on how to determine the domain andgranularity of fuzzy numbers

311 Modelling of Criteria Relative Importance The relativeimportance of evaluation criteria is unchangeable during theconsidered period of time The assessment of relative impor-tance of each pair of identified criteria is performed by eachdecision maker They use predefined linguistic expressionswhich are modelled by TFNs 119890

1198961198961015840 = (119909 119897

119890

1198961198961015840 119898119890

1198961198961015840 119906119890

1198961198961015840)

which are defined in the following way

Low importance (L) (119909 1 1 5)Medium importance (M) (119909 1 3 5)High importance (H) (119909 1 5 5)

The domains of these TFNs are defined real line into interval[1 5]The value 1 denotes that criterion 119896 over criterion 1198961015840 hasequal importance Value 5means that the relative importanceof criterion 119896 over criterion 1198961015840 has the most importance

If the strong relative importance of criterion 1198961015840 overcriterion 119896 holds then the pairwise comparison scale canbe represented by the fuzzy number 119890

1198961198961015840 = (

119890

1198961015840119896)minus1

=

(1119906119890

1198961015840119896 1119898119890

1198961015840119896 1119897119890

1198961015840119896)

32 Modelling of Criteria Values In the practice uncertaincriteria values are assessed bymanagement team at the globalsupply chain level for each time intervalTheir judgments arebased on evidence data results of good practice experienceand so forth It could be assumed that management teammakes decision by consensus In this paper fuzzy rating ofuncertain criteria values at the time interval level is describedby linguistic expressions which can be represented as TFNsV119905119894119896= (119910 119871

119905

119894119896119872119905

119894119896 119880119905

119894119896) Values in the domain of these TFNs

are defined on measurement scale which belong to areal setwithin the interval [0 1] Value 0 and 1 denote that criterion119896 for each supplier at the time interval level is at the lowestvalue and the highest value respectively

Specifically seven linguistic expressions which are mod-elled by TFNs are used

Very low (119910 0 0 025)Low (119910 01 02 03)Fairly medium moderate (119910 015 03 045)Moderate (119910 035 05 065)

Fairly high (119910 055 07 085)High (119910 07 08 09)Very high (119910 075 1 1)

4 The Proposed Algorithm

The algorithm of the proposed model is presented as follows

Step 1 Fuzzy assessment of the relative importance of eachcriteria pair is presented in matrix form

[119890

1198961198961015840]119870times119870

(1)

Step 2 The aggregated value of each pair of criteria iscalculated

1198961198961015840 =

119864

sum

119890=1

120596119890sdot 119890

1198961198961015840 (2)

The fuzzy pairwise comparison matrix of the relative impor-tance of criteria is constructed

[1198961198961015840]119870times119870

(3)

The weights vector is determined by using the concept ofextent analysis [29] which is presented in the followingmanner The value of fuzzy synthetic extent 119878

119896 with respect

to the 119896th criterion is defined as follows

119896= (

119870

sum

1198961015840=1

1198971198961198961015840

119870

sum

1198961015840=1

1198981198961198961015840

119870

sum

1198961015840=1

1198971198961198961015840)

sdot (

119870

sum

119896=1

119870

sum

1198961015840=1

1198971198961198961015840

119870

sum

119896=1

119870

sum

1198961015840=1

1198981198961198961015840

119870

sum

119896=1

119870

sum

1198961015840=1

1199061198961198961015840)

(4)

The weights vector is represented as follows

119882119901= ((Bel (

1)) (Bel (

119896)) (Bel (

119870))) (5)

The measure of belief according to which TFN 119896 is bigger

than all other TFNs 1198961015840 is denoted as Bel(

119896) This value

is obtained by applying the method for fuzzy numberscomparison [31 32] These values are crisp

The normalized weights vector is given by using linearnormalization procedure for benefit type criteria [33]

Step 3 The fuzzy rating of criterion 119896 for each supplier atthe time interval level 119905 is performed by each decisionmakerThese values are presented by TFNs V119905

119894119896

Step 4 Thevalue of criterion 119896 for each supplier for the wholetime period V

119894119896 is obtained by using the fuzzy averaging

method

V119894119896=1

119879sdot

119879

sum

119905=1

V119905119894119896 (6)


Step 5 The weighted fuzzy criterion value 119894119896 is calculated

as follows

119894119896= 119908119896sdot V119894119896 (7)

Step 6 The class A of items is determined as followsClass A contains suppliers that have high rated criteria

customer care quality and delivery method The highestvalues of identified criteria are represented by crisp values1199081 1199082 1199083 respectively According to fuzzy algebra rules

these crisp values should be represented by TFNs (1199081 1)

(1199082 1) and (119908

3 1) respectively In order to determine

whether a supplier belongs to class A the Euclidian distanceof supplier 119894 119894 = 1 119868 represented by (

1198941 1198942 1198943) from the

highest criteria values is calculated as follows

dist (119894 ref119860) = defuzz

radic

3

sum

119896=1

(119908119896minus 119894119896)2

(8)

As 119894119896are fuzzy numbers their distance to ref119860 is also a fuzzy

number The supports of fuzzy numbers dist(119894 ref119860) can bedescribed in discrete forms by discrete with membershipdegree min

119896=1119870(120583119894119896

(119910))Once the distances of all possible suppliers from ref119860 are

calculated they are ranked in the ascending orderThefirst 5ndash10 of the corresponding ranked suppliers are classified intoclass A

Step 7 The class C of items is determined as followsSuppliers that have low criteria values belong to class C

The reference point ref119862 is defined as the arranged tripletcrisp values (0 0 0) for care about clients quality anddelivery method

The weighted ref119862 is also represented as arranged triplet(0 0 0) Similarly as in the previous step a fuzzy distancebetween supplier 119894 119894 = 1 119868 and weighted ref119862 iscalculated as follows


radic

3

sum

119896=1

(0 minus 119894119896)2

(9)

The distances have to be ranked in the ascending order Thefirst 80 distances correspond to suppliers which belong toclass C

Step 8 Suppliers that are not classified either into class A orinto class C form class B

Step 9 Theweighted decision matrix for suppliers from classof A is constructed

Step 10 The sets of concordance 1198781198941198941015840 and the sets of discor-

dance 1198731198781198941198941015840 are determined in the following way If

1198941015840119896ge

119894119896 (119894 1198941015840

= 1 1198681) then criteria 119896 belongs to the set of

concordance The total number of suppliers of class A isdenoted as 119868

1 1198681le 119868 If

1198941015840119896lt 119894119896(119894 1198941015840

= 1 1198681) then

criterion 119896 belongs to the set of disordinance The fuzzynumbers

1198941015840119896 119894119896 are compared by using method which is

developed in [31 32]

Step 11 Construction of concordance matrix 119862 = [1198881198941198941015840]119868times119868

and discordance matrix119873 = [119899

1198941198941015840]119868times119868

is performed where

1198881198941198941015840 =

0 1198941015840119896lt 119894119896if 1198781198941198941015840 = 0

sum

119896isin1198781198941198941015840

1199081198961198941015840119896ge 119894119896if 1198781198941198941015840 = 0

(10)

1198991198941198941015840 =

0 1198941015840119896gt 119894119896

max119896isin119873119878

1198941198941015840(119889 (1198941015840119896 119894119896))

max119896=1119870

119889 (1198941015840119896 119894119896)

otherwise(11)

where 119889(119894119896 1198941015840119896) is distance between two fuzzy numbers

which is obtained by applying vertex method [34]

119889 (119894119896 1198941015840119896)

= radic1

3sdot [(119897119894119896minus 1198971198941015840119896)2

+ (119898119894119896minus 1198981198941015840119896)2

+ (119906119894119896minus 1199061198941015840119896)2

]

(12)

Mean value of concordance coefficient 119888119904119903 and coefficient

of discordance 119899119904119903 is obtained according to the following

expressions

119888119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198881198941198941015840

119899119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840

(13)

Step 12 The concordance matrix should be generated119872 =

[1198981198941198941015840]119868times119868

Elements of this matrix are determined by thefollowing rules

(i) Elements on the main diagonal are not defined

(ii) 1198981198941198941015840 = 0 for those pair of alternatives (119894 1198941015840) where 119888

1198941198941015840 lt

119888119904119903or 1198991198941198941015840 gt 119899119904119903

(iii) 1198981198941198941015840 = 1 for those pair of alternatives (119894 1198941015840) where 119888

1198941198941015840 ge

119888119904119903and 1198991198941198941015840 le 119899119904119903

Step 13 Rank of suppliers of group A with respect to allcriteria and their weights is determined according to theconcordance matrix The best supplier is one with greatestnumber in the concordance matrix

5 Illustrative Example

Illustrative example is presented as a verification for theproposed model The large group of potential suppliers isanalysed for the ABC classification since all of supplierscould deliver needed products and they are geographicallydistributed all over the world It is worth to mention thatanalysed enterprise is performing suppliersrsquo selection inbuilding and civil engineering industry


Fuzzy assessment according to the Step 1 should beperformed

[[[[[

[

1 1 1 11

1198721

1198671

1198721

119871

1

1198711

1198721

1198721

119867

1 1 1 1 1119872 1 119871

1 1 1 1

]]]]]

]

(14)

Aggregated values of judgements of decision makers areobtained by FOWA This procedure is presented on criteria2 and 3 (Step 2)

23=

4

sum

119890=1

120596119890sdot 119890

1198961198961015840

= 03 sdot (1 1 1) + 03 sdot (1 3 5) + 02 sdot (1 1 1)

+ 02 sdot (1 1 5) = (1 16 3)

(15)

Fuzzy pairwise comparison matrix of the aggregated relativeimportance of criteria is

[[

[

(1 1 1) (015 0425 1) (02 0505 1)

(1 235 6667) (1 1 1) (1 16 3)

(1 1980 5) (0333 0625 1) (1 1 1)

]]

]

(16)

By applying the procedure developed by Chang [29] thenormalized weights vector is119882 = (022 047 031)

By applying the proposed algorithm (Steps 3ndash5) theresults are obtained and presented in Table 1

According to Step 6 of the proposed algorithm the rankof suppliers is obtained and presented in Table 2

Ranked suppliers classified into class A are (119894 = 2) (119894 = 1)(119894 = 41) and (119894 = 36)


Ranked suppliers are classified into class C are (119894 = 27)(119894 = 26) (119894 = 40) (119894 = 34) (119894 = 18) (119894 = 39) (119894 = 17) (119894 = 21)(119894 = 25) (119894 = 38) (119894 = 15) (119894 = 16) (119894 = 14) (119894 = 20) (119894 = 31)(119894 = 37) (119894 = 42) (119894 = 11) (119894 = 19) (119894 = 22) (119894 = 10) (119894 = 23)(119894 = 30) (119894 = 28) (119894 = 29) (119894 = 6) (119894 = 4) (119894 = 8) (119894 = 24)(119894 = 12) (119894 = 7) (119894 = 13) and (119894 = 3)

The rest of suppliers belong to class B (Step 8 of theproposed algorithm) These suppliers are (119894 = 5) (119894 = 9)(119894 = 32) and (119894 = 25)

The fuzzy weighted decision matrix is constructed (Step10 of the proposed algorithm) and presented in Table 4

The sets of concordance and sets of discordance are (Step10 of the proposed algorithm)

11987821= 1198961 1198962

11987311987821= 1198963

119878241

= 1198962

119873119878241

= 1198961 1198963

119878236

= 1198962

119873119878236

= 1198961 1198963

11987812= 1198963

11987311987821= 1198961 1198962

119878141

= 1198962

119873119878141

= 1198961 1198963

119878136

= 1198962

119873119878136

= 1198961 1198963

119878412

= 1198961 1198963

119873119878412

= 1198962

119878411

= 1198961 1198963

119873119878411

= 1198962

1198784136

= 1198962

1198731198784136

= 1198961 1198963

119878362

= 1198961 1198963

119873119878362

= 1198962

119878361

= 1198961 1198963

119873119878361

= 1198962

1198783641

= 1198961 1198963

1198731198783641

= 1198962

(17)

According to Step 11 of the proposed algorithm the matrix ofconcordance and thematrix of discordancemay be presentedin the following way

119862 =

[[[[[

[

mdash 069 047 047

031 mdash 047 047

053 053 mdash 047

053 053 053 mdash

]]]]]

]

(18)

Distance between TFNs (see (12)) 2119896 41119896

119896 = 1 2 3 are

119889 (21 411)

= radic1

3sdot [(0151 minus 0118)

2

+ (0192 minus 0148)2

+ (0201 minus 0179)2

]

= 00342

119889 (22 412)

= radic1

3sdot [(0294 minus 030)

2

+ (0352 minus 0376)2

+ (0411 minus 0423)2

]

= 00159


Table 1 The weighted aggregated criteria values

Country Supplier Seaport 119896 = 1 119896 = 2 119896 = 3

Brazil i = 1 Vitoria (0162 0209 0214) (0294 0352 411) (0182 0232 0263)Brazil i = 2 Vitoria (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)Brazil i = 3 Vitoria (0154 0204 0212) (0276 0341 0406) (0109 0155 0202)Brazil i = 4 Vitoria (0151 0192 0206) (0270 0329 0388) (0073 0116 0159)Brazil i = 5 Vitoria (0154 0204 0212) (0346 0447 0458) (0089 0132 0174)Brazil i = 6 Vitoria (0151 0193 0207) (0253 0317 0382) (0089 0132 0175)Brazil i = 7 Vitoria (0160 020 0209) (0294 0364 0406) (0051 0085 0116)Brazil i = 8 Saupe (0066 0099 0132) (0294 0352 0411) (0139 0186 0232)Brazil i = 9 Vitoria (0162 0209 0215) (0311 0364 0417) (0078 0124 0171)Brazil i = 10 Vitoria (0160 0198 0209) (0164 0235 0306) (0139 0186 0232)Brazil i = 11 Vitoria (0162 0209 0215) (0117 0188 0258) (0167 0209 0252)Brazil i = 12 Vitoria (0151 0192 0206) (0341 0352 0411) (0078 0124 0171)Brazil i = 13 Vitoria (0129 0195 0190) (0311 0364 0417) (0078 0124 0171)Brazil i = 14 Vitoria (0110 0143 0176) (0111 0188 0259) (0139 0171 0194)Brazil i = 15 Santos (0063 0094 0124) (0182 0115 0146) (0050 0085 0139)Brazil i = 16 Vitoria (0110 0143 0176) (0159 0223 0288) (0054 0093 0132)Brazil i = 17 Vitoria (0063 0094 0124) (0094 0164 0235) (0101 0139 0178)Brazil i = 18 Saupe (0066 0099 0132) (0118 0188 0258) (0027 0054 0101)Brazil i = 19 Vitoria (0099 0132 0165) (0235 0317 0370) (0066 0101 0155)Brazil i = 20 Saupe (0041 0072 0102) (0205 0270 0335) (0031 0062 0112)China i = 21 Xiamen (0118 0154 0176) (0053 0106 0188) (0140 0171 0221)India i = 22 Chennai (0129 0165 0187) (0229 0294 0358) (0089 0132 0174)India i = 23 Chennai (0126 0165 0176) (0247 0305 0364) (0089 0132 0174)India i = 24 Mumbai (0129 0165 0187) (030 0376 0423) (0031 0062 0113)India i = 25 Chennai (0129 0165 0187) (0317 0388 0429) (0074 0116 0159)India i = 26 Mumbai (0041 0072 0102) (0071 0117 0194) (0078 0109 0159)India i = 27 Chennai (0019 0038 0072) (0035 0070 0135) (0054 0093 0132)Italy i = 28 Carrara (0118 0148 0179) (0253 0317 0382) (0093 0139 0186)Italy i = 29 Carrara (0052 0082 0113) (0235 0388 0429) (0050 0085 0104)Italy i = 30 Verona (0096 0126 0157) (0235 0306 0376) (0132 0171 0209)Italy i = 31 Carrara (0140 0176 0198) (0229 0294 0358) (0132 0170 0209)Italy i = 32 Carrara (0074 0104 0135) (0305 040 0435) (0151 0201 0232)Italy i = 33 Verona (0074 0104 0135) (0141 0211 0282) (0046 0077 0128)Italy i = 34 Verona (0071 0099 0126) (0088 0153 0217) (0062 0093 0143)Portugal i = 35 Lisboa (0060 0066 0115) (0165 0235 0306) (0050 0085 0140)Singapore i = 36 Singapore (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)Spain i = 37 Vigo (0079 0104 0115) (0223 0282 0341) (0116 0155 0194)Spain i = 38 Porbido (0601 0088 0115) (0182 0247 0288) (0035 0070 0105)Spain i = 39 Novelda (0074 0104 0135) (0088 0153 0217) (0120 0163 0205)Spain i = 40 Valencia (0066 0099 0132) (0065 0129 0091) (0054 0078 014)Taiwan i = 41 Keelung (0118 0148 0179) (030 0376 0423) (0163 0209 0236)Taiwan i = 42 Keelung (0069 0094 0118) (0229 0294 0358) (0097 0140 0182)

119889 (23 413)

= radic1

3sdot [(0209 minus 0163)

2

+ (0256 minus 0209)2

+ (0283 minus 0236)2

]

= 00467

(19)

The value of discordance matrix element (see (11)) is

119899241

max119896=13

(00342 00467)

max119896=123

(00342 00159 00467)=00467

00467

= 1

(20)


Table 2 Rank suppliers from refA

Supplier dist(119894 ref119860) = defuzz

radic

3

sum

119896=1

(119908119896minus 119894119896)2


radic

3

sum

119896=1

(119908119896minus 119894119896)2

i = 2 01354 i = 42 02716i = 1 0144 i = 19 02737i = 41 01594 i = 29 02864i = 36 01655 i = 11 02867i = 32 01785 i = 31 03129i = 5 01824 i = 14 03265i = 3 02027 i = 15 03376i = 8 02096 i = 16 03378i = 9 02147 i = 20 03503i = 25 02197 i = 38 03532i = 12 02224 i = 35 03581i = 13 02291 i = 33 0366i = 6 02351 i = 39 03682i = 30 02352 i = 17 03712i = 28 02405 i = 27 03904i = 4 02418 i = 21 03908i = 7 02494 i = 18 03983i = 23 02498 i = 34 04017i = 22 02567 i = 40 04271i = 10 02598 i = 26 04285i = 37 02699i = 24 02699

Table 3 Rank suppliers from refC


radic

3

sum

119896=1

(0 minus 119894119896)2


radic

3

sum

119896=1

(0 minus 119894119896)2

i = 27 01254 i = 19 03558i = 26 01789 i = 22 03611i = 40 01826 i = 10 03661i = 34 02055 i = 23 03693i = 18 02197 i = 30 03720i = 39 02231 i = 28 03764i = 17 02349 i = 29 03813i = 33 02481 i = 6 03931i = 21 02559 i = 4 03968i = 35 02595 i = 8 04111i = 38 02714 i = 24 04113i = 15 02792 i = 12 04178i = 16 02840 i = 7 04199i = 14 02864 i = 13 04202i = 20 02864 i = 3 04242i = 31 03111 i = 25 04330i = 37 03382 i = 9 04353i = 42 03414 i = 32 04518i = 11 03481 i = 5 04971


Table 4 The weighted decision matrix

119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)

Table 5 Rank of suppliers of group A

Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2

The values of discordance matrix are calculated in same wayThe matrix of discordance is as follows

119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)

The mean value of concordance coefficient 119888119904119903and coefficient

of disconcordance 119899119904119903are calculated in the following way (see

(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)

Thematrix of consistent domination is obtained according toStep 12 of the proposed algorithm such as

119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)

The rank of suppliers of group A is given by using procedure(Step 13 of the proposed algorithm) The obtained rank ispresented in Table 5

51 Discussion There are no specific guidelines for determin-ing the classification criteria in suppliersrsquo selection problemso these criteria vary in different economy branches In thispaper three eliminatory criteria are chosen based on theresults of good practice in order to make a base of potentialsuppliers The analysed criteria are (1) customer care (2)quality (ratio between price and specific performances ofproducts) and (3) delivery method (ratio between costs anddelivery rate) In global supply chains there is large number

of potential suppliers so determining the optimal portfolioof suppliers is very important task since it may save timedecrease costs and provide input for defining optimal supplystrategy Comparing this model to application of developedABC models [19 21] it may be stated that main implicationof this model is using the new approach in classification todetermine A and C class The proposed model takes intoaccount the type of criteria for suppliers selection so it maybe assumed that it represents its main advantage

Based on the obtained ranking results by applying themodified ELECTRE method (see Table 5) it may be con-cluded that suppliers (119894 = 41) and (119894 = 36) have the sameimportance for the treated enterprise as a part of global supplychain With respect to the obtained result the managementteam may define different supply strategy for short-timeperiod (one year) such as exclusive purchasing form one ofthe selected suppliers The selection of adequate purchasingstrategy has a crucial influence on profit of enterprise withinglobal supply chain as well as on its market position

In the same time the obtained results present inputdata for development of the methodology for selectingsuppliers in different environment supply conditions andexternal circumstances This methodology should includevarious aspects of technical economic social organizationalmarket-oriented and environmental character By using thementioned methodology management team can choose thebest supplier that is suitable for building long-term and stablecooperationThis cooperation can potentially include provid-ing capabilities for innovation and development reliability inother partnerships preparedness to share risk and profit withthe company

The proposed model is focused on the real-world sit-uation in domain determining short-time and long-timepurchasing strategy With regard to paper which treats theproblemof supplier assessment andwhich can be found in theliterature this paper pioneers the application of classificationfor building optimal portfolio of suppliers and their rankingand selection within the supply chain management

Research implications of this paper may be presented asa comparison with similar papers that have used ELECTREmethod for solving similar problems as well as with papersthat have used ABC classification The weight of criteriaand preference rating can be stated as fuzzy group decisionmaking problems

Aggregation of relative weights of criteria importance isperformed by using FOWA operator comparing it to pro-cedure proposed by Marbini and Tavana [27] or procedureproposed by Alencar et al [25] In practice it is reasonableto expect that decision makers have different weights soFOWA operator seems to be more suitable The time interval


for assessment of preference rating is divided in sub-timeintervals compared to other models where an assessment isperformedduring thewhole time interval [25 27] It is knownthat it is more precise to make assessment in shorter timeinterval

6 Conclusion

Quick and continuous changes occurring in the businessenvironment lead to opening issues in the organization andadaptation in different type of industries One of the mostimportant management tasks is determining of purchasingstrategy for the short-time and long-time period respec-tively This is because purchasing strategy has significantinfluence on successful establishment of global supply chainAs it is known in the global supply chain there are numerouspotential suppliers so that the considered problem is verycomplex

The theoretical contributions of this paper are presentedas follows In the first place conventional assessment ofsuppliers is performed with respect to quality and priceSometimes the other influencing criteria are disregardedThe evaluation criteria are selected according to the literaturereview which is conducted

Secondly it is appropriate to use linguistic terms insteadof numerical values for describing uncertainties into (1) therelative importance of each pair of criteria and (2) criteriavalues which exist in the considered problem Modelling oflinguistic variables is based on the fuzzy set theory Weightsvector of criteria is obtained by applying extent analysisapproach The fuzzy AHP may be considered as suitable forcapturing the vagueness of human thinking style and in thesame time it may be effectively employed for solving theissue of determining criteria weights in the supplier selectionproblem

The large number of potential suppliers in global supplychain enhances the complexity of the process of supplierselection In that manner it is necessary before delivery ofthe process of selection to deliver the process of suppliersrsquoclassification This may be denoted as the third contributionof the paper since a new fuzzymulticriteriaABC classificationof suppliers is proposed The effectiveness of the proposedalgorithm is tested using real-world data of 42 suppliers thatoperate in different geographical locations all over the worldThe classification obtained using the algorithm is in goodagreement with the judgment of the management team of theconsidered global supply chain

The fourth contribution makes the ranking of the suppli-ers denoted as group A The ranking is conducted by usingthe fuzzy ELECTRE proposed in this paper Modification ofELECTRE may be presented as (1) determination of sets onconcordance and sets of discordance [31] and (2) calculationof coefficient of discordance

Beside abovementioned various advantages the proposedmodel has some constraints It may be seen that there is thelack of research foundation in the literature which relatesto selection of criteria that are used for suppliersrsquo selectionin building and civil engineering industry Those criteriamay vary from constrains of supplierrsquos capacity aggregated

quality duty taxes and risk factors to political stability andso forth These extensions could undoubtedly increase thecomputational complexities The second constraint relates toABC method since it may be deployed only if 119896 is less orequal to 3 ELECTRE should be expanded with more criteriain order to determine purchasing strategy for long time andin the same time to establish partnership with supplier

This paper focuses on the complex circumstances alarge number of suppliers from different countries multipledecision makers involved in decision process and multipleuncertainties The established mathematical model can helpboth practitioners and researchers to further utilize anddeploy the purchasing strategy in global supply chain

Competing Interests

The authors declare that they have no competing interests

References

[1] ISO 90012015 Quality management systemsmdashRequirements[2] G Bruno E Esposito A Genovese andM Simpson ldquoApplying

supplier selection methodologies in a multi-stakeholder envi-ronment a case study and a critical assessmentrdquo Expert Systemswith Applications vol 43 pp 271ndash285 2016

[3] A C Trapp and J Sarkis ldquoIdentifying robust portfolios of sup-pliers a sustainability selection and development perspectiverdquoJournal of Cleaner Production vol 112 part 3 pp 2088ndash21002016

[4] P Amorim E Curcio B Almada-Lobo A P Barbosa-Povoaand I E Grossmann ldquoSupplier selection in the processed foodindustry under uncertaintyrdquo European Journal of OperationalResearch vol 252 no 3 pp 801ndash814 2016

[5] P H Andersen C Ellegaard andH Kragh ldquoIrsquom yourman howsuppliers gain strategic status in buying companiesrdquo Journal ofPurchasing and Supply Management vol 22 no 2 pp 72ndash812014

[6] B Du S Guo X Huang Y Li and J Guo ldquoA Pareto supplierselection algorithm for minimum the life cycle cost of complexproduct systemrdquo Expert Systems with Applications vol 42 no9 pp 4253ndash4264 2015

[7] T L Saaty ldquoHow to make a decision the analytic hierarchyprocessrdquo European Journal Operation-al Research vol 48 no1 pp 9ndash26 1990

[8] C-L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications vol 186 of Lecture Notes in Economicsand Mathematical Systems Springer Heidelberg Germany1981

[9] B Roy ldquoClassement et choix en presence de points de vue mul-tiples (Lamethode ELECTRE)rdquoRevue Francaise drsquoInformatiqueet de Recherche Operationnelle vol 2 no 8 pp 57ndash75 1968

[10] D Tadic D D Milanovic M Misita and B Tadic ldquoNewintegrated approach to the problem of ranking and supplierselection under uncertaintiesrdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 225 no 9 pp 1713ndash1724 2011

[11] HAGuvenir andE Erel ldquoMulticriteria inventory classificationusing a genetic algorithmrdquo European Journal of OperationalResearch vol 105 no 1 pp 29ndash37 1998


[12] A Aleksic M Stefanovic D Tadic and S Arsovski ldquoA fuzzymodel for assessment of organization vulnerabilityrdquo Measure-ment vol 51 no 1 pp 214ndash223 2014

[13] G J Klir and T A Folger Fuzzy Sets Uncertainty and Informa-tion Prentice Hall Upper Saddle River NJ USA 1988

[14] W Pedrycz and F Gomide An Introduction to Fuzzy SetsAnalysis and Design MIT Press Cambridge Mass USA 1997

[15] H-J Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwer Academic Boston Mass USA 4th edition 2001

[16] P Kaur and S Chakrabortyb ldquoA new approach to vendorselection problem with impact factor as an indirect measure ofqualityrdquo Journal ofModernMathematics and Statistics vol 1 no1 pp 8ndash14 2007

[17] H M Fazel Zarandi B I Turksen and S Saghiri ldquoSupplychain crisp and fuzzy aspectsrdquo International Journal of AppliedMathematics and Computer Science vol 12 no 3 pp 423ndash4352002

[18] B E Flores andDCWhybark ldquoImplementingmultiple criteriaABC analysisrdquo Journal of OperationsManagement vol 7 no 1-2pp 79ndash85 1987

[19] M-C Yu ldquoMulti-criteria ABC analysis using artificial-intelligence-based classification techniquesrdquo Expert Systemswith Applications vol 38 no 4 pp 3416ndash3421 2011

[20] A Hadi-Vencheh and A Mohamadghasemi ldquoA fuzzy AHP-DEA approach for multiple criteria ABC inventory classifica-tionrdquo Expert Systems with Applications vol 38 no 4 pp 3346ndash3352 2011

[21] J Puente D de la Fuente P Priore and R Pino ldquoAbc classi-fication with uncertain data A fuzzy model vs a probabilisticmodelrdquoApplied Artificial Intelligence vol 16 no 6 pp 443ndash4562002

[22] C-W Chu G-S Liang and C-T Liao ldquoControlling inventoryby combiningABC analysis and fuzzy classificationrdquoComputersamp Industrial Engineering vol 55 no 4 pp 841ndash851 2008

[23] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[24] K Govindan and M B Jepsen ldquoELECTRE a comprehensiveliterature review onmethodologies and applicationsrdquo EuropeanJournal of Operational Research vol 250 no 1 pp 1ndash29 2016

[25] L H Alencar A T de Almeida and D C Morais ldquoAmulticriteria group decision model aggregating the preferencesof decision-makers based on electre methodsrdquo Pesquisa Opera-cional vol 30 no 3 pp 687ndash702 2010

[26] G AMontazer H Q Saremi andM Ramezani ldquoDesign a newmixed expert decision aiding system using fuzzy ELECTRE IIImethod for vendor selectionrdquo Expert Systems with Applicationsvol 36 no 8 pp 10837ndash10847 2009

[27] A H Marbini and M Tavana ldquoAn extension of the ElectreI method for group decision-making under a fuzzy environ-mentrdquo Omega vol 39 no 4 pp 373ndash386 2011

[28] D Tadic A T Gumus S Arsovski A Aleksic and MStefanovic ldquoAn evaluation of quality goals by using fuzzy AHPand fuzzy TOPSIS methodologyrdquo Journal of Intelligent amp FuzzySystems vol 25 no 3 pp 547ndash556 2013

[29] D-Y Chang ldquoApplications of the extent analysis method onfuzzy AHPrdquo European Journal of Operational Research vol 95no 3 pp 649ndash655 1996

[30] J MMerigo andM Casanovas ldquoUsing fuzzy numbers in heavyaggregation operatorsrdquo International Journal of InformationTechnology vol 4 no 4 pp 267ndash272 2008

[31] SM Baas andHKwakernaak ldquoRating and ranking ofmultiple-aspect alternatives using fuzzy setsrdquo Automatica vol 13 no 1pp 47ndash58 1977

[32] D Dubois and H Prade ldquoDecision-making under fuzzinessrdquoin Advances in Fuzzy Set Theory and Applications pp 279ndash302North-Holland Amsterdam Netherlands 1979

[33] J C Pomerol and S Barba-RomeoMulticriteria Decision Man-agement Principles and Practice Kluwer Nijhoff PublishingBoston Mass USA 2000

[34] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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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


of analysed inventory items ranked at first place belong togroup A next 15 correspond to group B and the rest ofinventory items correspond to group C Selection of theclassification criterion depends on the kind of problem beingconsidered The issue of selection of suppliers may be statedas a multicriteria optimization task In literature there isa lot of papers that employ different methods for rankingand assessment of suppliers The most used methods areAnalytical Hierarchy Process (AHP) [7] Technique forOrderof Preference by Similarity to Ideal Solution (TOPSIS) [8]ELimination Et Choix Traduisant la REalite (ELECTRE) [9]or combination of two or more methods [10] In compliancewith the rank of suppliers decision makers may choose themost suitable suppliers work building partnership relations

The conventional ABC methodology may sometimes notbe appropriate to provide a good classification of inventoryitems in practice [11] If there is a need to make the clas-sification more sophisticated more than one classificationcriterion and imprecise data about items have to be usedThe classification task becomes a multicriteria classificationproblem in the presence of uncertainty

In this paper the authors focus on the supplier classifica-tion in global supply chain where number of suppliers is largedue to demands of global market The number and types ofthe selection criteria are not clearly defined in the literatureand there are no specific guidelines that treat this issue Thespecificity of construction industry products their variabledemand over time and in particular high cost indicatethe importance of suppliersrsquo classification problem A newfuzzy multicriteria ABC model for classification of suppliersin construction industry is proposed Three eliminatorycriteria are selected to make a base of potential suppliers (1)customer care (2) quality (ratio between price and specificperformances of products) and (3) delivery method (ratiobetween costs and delivery rate)

The objective of paper is to define the appropriate pur-chasing strategy in the global supply chain through two steps(1) classification of a large number of potential suppliers byapplying proposed fuzzymulticriteriaABC (2) the ranking ofclassifiedA group of suppliers (optimal portfolio of suppliers)by using the proposed model consisted from fuzzy AHP andfuzzy ELECTREmethods For the classification problem theproposed fuzzy multicriteria ABC methodology is suitablesince it treats uncertain criteria in appropriate way andit is easy and suitable for usage leading to determinationof optimal portfolio of suppliers This optimal portfolio ofsuppliers consisted of those who are suitable for furtherselection of the group that will be used for building partner-shipThemethods of multicriteria decisionmakingmethodsfuzzy AHP and fuzzy ELECTRE are used for ranking ofselected suppliers In this manner resources such as time andcosts of defining appropriate supply strategy may be reducedsignificantly and in the same time effectiveness of supplyprocess is increased In the final consequence this should leadto increase effectiveness of all business processes in globalsupply chain

Each of three considered criteria is described by using lin-guistic expressions specified by enterprisemanagement teamThese linguistic expressions are modelled using triangular

fuzzy numbers by analogy to Aleksic et al [12] Uncertaintyin criteria values used in supplier classification are mod-elled by fuzzy sets [13ndash15] The motivation for using thismethodology came from its suitability for handling impreciseand ambiguous data and it supports usage of decisionmaking methods which operate with such data It may besaid that fuzzy sets theory resembles the human reasoningin its use of approximate information and uncertainty togenerate decisions [16] In thismanner fuzzy sets have severaladvantages over theories that treat similar problems (a) theyare based on a natural language (b) they are conceptually easyto understand (c) they could be combined with conventionalmethods and techniques for dealing and reasoning withuncertain data and (d) they can capture most nonlinearrelations in problems of arbitrary complexity [17]

The paper is organized in the followingwayThe literaturereview is given in Section 2 Section 3 presents evaluationframework and modelling of uncertainties In Section 4the proposed algorithm is presented Section 5 is used forthe verification of the model using an illustrative exampleConclusions are presented in Section 6

2 Literature Review

A number of papers presented in the literature were dealingwith the problem of items classification in uncertain envi-ronments for inventory control purposes and proposed fuzzymulticriteria ABC classification approaches after realizingthe importance of considering multiple criteria in the ABCanalysis [18] In addition to the overall cost some othercriteria such as lead time inventory holding cost limitationof the warehouse space and order cost were recognized asbeing important for items classification

In the literature as well as in the practice determinationof the optimal portfolio of suppliers is based onmathematicalmodels Possibly there may be a lot of potential suppliers soin the first step the eliminatory criteria should be appliedBuilding partnersrsquo relations with suppliers should be per-formed after the portfolio of suitable suppliers is generatedHowever there is no wide literature which considers supplierselection when there is large number of potential suppliers

Techniques of artificial intelligence such as backprop-agation networks support vector machines and 119896-nearestneighbors may be used for ABC inventory classificationtaking into account several criteria [19] Also ABC classifi-cation may be supported by using different methods such asAnalytic Hierarchy Process and Data Envelopment Analysis[20] In some cases existing models may be analysed andimproved such as extension of theNg-model formulticriteriainventory ABC classification [20] In this way classificationmay be performed by respect to multicriteria and theirweights simultaneouslyThe criteria weightsmay be obtainedby using DEA [20] If criteria are uncertain such as thefact that demand frequency and costs are in focus fuzzyABC model for classification of items according to theirvalue may be applied [21] As the values of classificationcriteria are uncertain they may be described by triangularfuzzy numbers [15] and may be aggregated into one bymultiplying two corresponding criteria values In this way





















1198961198961015840 = (119909 119897

119890

1198961198961015840 119898119890

1198961198961015840 119906119890

1198961198961015840)





1198961198961015840 = (

119890

1198961015840119896)minus1

=

(1119906119890

1198961015840119896 1119898119890

1198961015840119896 1119897119890

1198961015840119896)


119905

119894119896119872119905

119894119896 119880119905









[119890

1198961198961015840]119870times119870

(1)


1198961198961015840 =

119864

sum

119890=1

120596119890sdot 119890

1198961198961015840 (2)


[1198961198961015840]119870times119870

(3)


119896 with respect


119896= (

119870

sum

1198961015840=1

1198971198961198961015840

119870

sum

1198961015840=1

1198981198961198961015840

119870

sum

1198961015840=1

1198971198961198961015840)

sdot (

119870

sum

119896=1

119870

sum

1198961015840=1

1198971198961198961015840

119870

sum

119896=1

119870

sum

1198961015840=1

1198981198961198961015840

119870

sum

119896=1

119870

sum

1198961015840=1

1199061198961198961015840)

(4)


119882119901= ((Bel (

1)) (Bel (

119896)) (Bel (

119870))) (5)



119896) This value




119894119896



method

V119894119896=1

119879sdot

119879

sum

119905=1

V119905119894119896 (6)



as follows

119894119896= 119908119896sdot V119894119896 (7)




(1199082 1) and (119908



1198941 1198942 1198943) from the



radic

3

sum

119896=1

(119908119896minus 119894119896)2

(8)



119896=1119870(120583119894119896







radic

3

sum

119896=1

(0 minus 119894119896)2

(9)






1198941015840119896ge

119894119896 (119894 1198941015840



1 1198681le 119868 If

1198941015840119896lt 119894119896(119894 1198941015840

= 1 1198681) then






1198941198941015840]119868times119868

is performed where

1198881198941198941015840 =

0 1198941015840119896lt 119894119896if 1198781198941198941015840 = 0

sum

119896isin1198781198941198941015840

1199081198961198941015840119896ge 119894119896if 1198781198941198941015840 = 0

(10)

1198991198941198941015840 =

0 1198941015840119896gt 119894119896

max119896isin119873119878

1198941198941015840(119889 (1198941015840119896 119894119896))

max119896=1119870

119889 (1198941015840119896 119894119896)

otherwise(11)



119889 (119894119896 1198941015840119896)

= radic1

3sdot [(119897119894119896minus 1198971198941015840119896)2

+ (119898119894119896minus 1198981198941015840119896)2

+ (119906119894119896minus 1199061198941015840119896)2

]

(12)



expressions

119888119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198881198941198941015840

119899119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840

(13)


[1198981198941198941015840]119868times119868




1198941198941015840 lt

119888119904119903or 1198991198941198941015840 gt 119899119904119903


1198941198941015840 ge

119888119904119903and 1198991198941198941015840 le 119899119904119903






[[[[[

[

1 1 1 11

1198721

1198671

1198721

119871

1

1198711

1198721

1198721

119867

1 1 1 1 1119872 1 119871

1 1 1 1

]]]]]

]

(14)


23=

4

sum

119890=1

120596119890sdot 119890

1198961198961015840

= 03 sdot (1 1 1) + 03 sdot (1 3 5) + 02 sdot (1 1 1)

+ 02 sdot (1 1 5) = (1 16 3)

(15)


[[

[

(1 1 1) (015 0425 1) (02 0505 1)

(1 235 6667) (1 1 1) (1 16 3)

(1 1980 5) (0333 0625 1) (1 1 1)

]]

]

(16)










11987821= 1198961 1198962

11987311987821= 1198963

119878241

= 1198962

119873119878241

= 1198961 1198963

119878236

= 1198962

119873119878236

= 1198961 1198963

11987812= 1198963

11987311987821= 1198961 1198962

119878141

= 1198962

119873119878141

= 1198961 1198963

119878136

= 1198962

119873119878136

= 1198961 1198963

119878412

= 1198961 1198963

119873119878412

= 1198962

119878411

= 1198961 1198963

119873119878411

= 1198962

1198784136

= 1198962

1198731198784136

= 1198961 1198963

119878362

= 1198961 1198963

119873119878362

= 1198962

119878361

= 1198961 1198963

119873119878361

= 1198962

1198783641

= 1198961 1198963

1198731198783641

= 1198962

(17)


119862 =

[[[[[

[

mdash 069 047 047

031 mdash 047 047

053 053 mdash 047

053 053 053 mdash

]]]]]

]

(18)


119896 = 1 2 3 are

119889 (21 411)

= radic1

3sdot [(0151 minus 0118)

2

+ (0192 minus 0148)2

+ (0201 minus 0179)2

]

= 00342

119889 (22 412)

= radic1

3sdot [(0294 minus 030)

2

+ (0352 minus 0376)2

+ (0411 minus 0423)2

]

= 00159





119889 (23 413)

= radic1

3sdot [(0209 minus 0163)

2

+ (0256 minus 0209)2

+ (0283 minus 0236)2

]

= 00467

(19)


119899241

max119896=13

(00342 00467)

max119896=123

(00342 00159 00467)=00467

00467

= 1

(20)




radic

3

sum

119896=1

(119908119896minus 119894119896)2


radic

3

sum

119896=1

(119908119896minus 119894119896)2




radic

3

sum

119896=1

(0 minus 119894119896)2


radic

3

sum

119896=1

(0 minus 119894119896)2




119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)


Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2


119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)



(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)


119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























1198961198961015840 = (119909 119897

119890

1198961198961015840 119898119890

1198961198961015840 119906119890

1198961198961015840)





1198961198961015840 = (

119890

1198961015840119896)minus1

=

(1119906119890

1198961015840119896 1119898119890

1198961015840119896 1119897119890

1198961015840119896)


119905

119894119896119872119905

119894119896 119880119905









[119890

1198961198961015840]119870times119870

(1)


1198961198961015840 =

119864

sum

119890=1

120596119890sdot 119890

1198961198961015840 (2)


[1198961198961015840]119870times119870

(3)


119896 with respect


119896= (

119870

sum

1198961015840=1

1198971198961198961015840

119870

sum

1198961015840=1

1198981198961198961015840

119870

sum

1198961015840=1

1198971198961198961015840)

sdot (

119870

sum

119896=1

119870

sum

1198961015840=1

1198971198961198961015840

119870

sum

119896=1

119870

sum

1198961015840=1

1198981198961198961015840

119870

sum

119896=1

119870

sum

1198961015840=1

1199061198961198961015840)

(4)


119882119901= ((Bel (

1)) (Bel (

119896)) (Bel (

119870))) (5)



119896) This value




119894119896



method

V119894119896=1

119879sdot

119879

sum

119905=1

V119905119894119896 (6)



as follows

119894119896= 119908119896sdot V119894119896 (7)




(1199082 1) and (119908



1198941 1198942 1198943) from the



radic

3

sum

119896=1

(119908119896minus 119894119896)2

(8)



119896=1119870(120583119894119896







radic

3

sum

119896=1

(0 minus 119894119896)2

(9)






1198941015840119896ge

119894119896 (119894 1198941015840



1 1198681le 119868 If

1198941015840119896lt 119894119896(119894 1198941015840

= 1 1198681) then






1198941198941015840]119868times119868

is performed where

1198881198941198941015840 =

0 1198941015840119896lt 119894119896if 1198781198941198941015840 = 0

sum

119896isin1198781198941198941015840

1199081198961198941015840119896ge 119894119896if 1198781198941198941015840 = 0

(10)

1198991198941198941015840 =

0 1198941015840119896gt 119894119896

max119896isin119873119878

1198941198941015840(119889 (1198941015840119896 119894119896))

max119896=1119870

119889 (1198941015840119896 119894119896)

otherwise(11)



119889 (119894119896 1198941015840119896)

= radic1

3sdot [(119897119894119896minus 1198971198941015840119896)2

+ (119898119894119896minus 1198981198941015840119896)2

+ (119906119894119896minus 1199061198941015840119896)2

]

(12)



expressions

119888119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198881198941198941015840

119899119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840

(13)


[1198981198941198941015840]119868times119868




1198941198941015840 lt

119888119904119903or 1198991198941198941015840 gt 119899119904119903


1198941198941015840 ge

119888119904119903and 1198991198941198941015840 le 119899119904119903






[[[[[

[

1 1 1 11

1198721

1198671

1198721

119871

1

1198711

1198721

1198721

119867

1 1 1 1 1119872 1 119871

1 1 1 1

]]]]]

]

(14)


23=

4

sum

119890=1

120596119890sdot 119890

1198961198961015840

= 03 sdot (1 1 1) + 03 sdot (1 3 5) + 02 sdot (1 1 1)

+ 02 sdot (1 1 5) = (1 16 3)

(15)


[[

[

(1 1 1) (015 0425 1) (02 0505 1)

(1 235 6667) (1 1 1) (1 16 3)

(1 1980 5) (0333 0625 1) (1 1 1)

]]

]

(16)










11987821= 1198961 1198962

11987311987821= 1198963

119878241

= 1198962

119873119878241

= 1198961 1198963

119878236

= 1198962

119873119878236

= 1198961 1198963

11987812= 1198963

11987311987821= 1198961 1198962

119878141

= 1198962

119873119878141

= 1198961 1198963

119878136

= 1198962

119873119878136

= 1198961 1198963

119878412

= 1198961 1198963

119873119878412

= 1198962

119878411

= 1198961 1198963

119873119878411

= 1198962

1198784136

= 1198962

1198731198784136

= 1198961 1198963

119878362

= 1198961 1198963

119873119878362

= 1198962

119878361

= 1198961 1198963

119873119878361

= 1198962

1198783641

= 1198961 1198963

1198731198783641

= 1198962

(17)


119862 =

[[[[[

[

mdash 069 047 047

031 mdash 047 047

053 053 mdash 047

053 053 053 mdash

]]]]]

]

(18)


119896 = 1 2 3 are

119889 (21 411)

= radic1

3sdot [(0151 minus 0118)

2

+ (0192 minus 0148)2

+ (0201 minus 0179)2

]

= 00342

119889 (22 412)

= radic1

3sdot [(0294 minus 030)

2

+ (0352 minus 0376)2

+ (0411 minus 0423)2

]

= 00159





119889 (23 413)

= radic1

3sdot [(0209 minus 0163)

2

+ (0256 minus 0209)2

+ (0283 minus 0236)2

]

= 00467

(19)


119899241

max119896=13

(00342 00467)

max119896=123

(00342 00159 00467)=00467

00467

= 1

(20)




radic

3

sum

119896=1

(119908119896minus 119894119896)2


radic

3

sum

119896=1

(119908119896minus 119894119896)2




radic

3

sum

119896=1

(0 minus 119894119896)2


radic

3

sum

119896=1

(0 minus 119894119896)2




119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)


Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2


119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)



(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)


119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











as follows

119894119896= 119908119896sdot V119894119896 (7)




(1199082 1) and (119908



1198941 1198942 1198943) from the



radic

3

sum

119896=1

(119908119896minus 119894119896)2

(8)



119896=1119870(120583119894119896







radic

3

sum

119896=1

(0 minus 119894119896)2

(9)






1198941015840119896ge

119894119896 (119894 1198941015840



1 1198681le 119868 If

1198941015840119896lt 119894119896(119894 1198941015840

= 1 1198681) then






1198941198941015840]119868times119868

is performed where

1198881198941198941015840 =

0 1198941015840119896lt 119894119896if 1198781198941198941015840 = 0

sum

119896isin1198781198941198941015840

1199081198961198941015840119896ge 119894119896if 1198781198941198941015840 = 0

(10)

1198991198941198941015840 =

0 1198941015840119896gt 119894119896

max119896isin119873119878

1198941198941015840(119889 (1198941015840119896 119894119896))

max119896=1119870

119889 (1198941015840119896 119894119896)

otherwise(11)



119889 (119894119896 1198941015840119896)

= radic1

3sdot [(119897119894119896minus 1198971198941015840119896)2

+ (119898119894119896minus 1198981198941015840119896)2

+ (119906119894119896minus 1199061198941015840119896)2

]

(12)



expressions

119888119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198881198941198941015840

119899119904119903=

1

119868 sdot (119868 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840

(13)


[1198981198941198941015840]119868times119868




1198941198941015840 lt

119888119904119903or 1198991198941198941015840 gt 119899119904119903


1198941198941015840 ge

119888119904119903and 1198991198941198941015840 le 119899119904119903






[[[[[

[

1 1 1 11

1198721

1198671

1198721

119871

1

1198711

1198721

1198721

119867

1 1 1 1 1119872 1 119871

1 1 1 1

]]]]]

]

(14)


23=

4

sum

119890=1

120596119890sdot 119890

1198961198961015840

= 03 sdot (1 1 1) + 03 sdot (1 3 5) + 02 sdot (1 1 1)

+ 02 sdot (1 1 5) = (1 16 3)

(15)


[[

[

(1 1 1) (015 0425 1) (02 0505 1)

(1 235 6667) (1 1 1) (1 16 3)

(1 1980 5) (0333 0625 1) (1 1 1)

]]

]

(16)










11987821= 1198961 1198962

11987311987821= 1198963

119878241

= 1198962

119873119878241

= 1198961 1198963

119878236

= 1198962

119873119878236

= 1198961 1198963

11987812= 1198963

11987311987821= 1198961 1198962

119878141

= 1198962

119873119878141

= 1198961 1198963

119878136

= 1198962

119873119878136

= 1198961 1198963

119878412

= 1198961 1198963

119873119878412

= 1198962

119878411

= 1198961 1198963

119873119878411

= 1198962

1198784136

= 1198962

1198731198784136

= 1198961 1198963

119878362

= 1198961 1198963

119873119878362

= 1198962

119878361

= 1198961 1198963

119873119878361

= 1198962

1198783641

= 1198961 1198963

1198731198783641

= 1198962

(17)


119862 =

[[[[[

[

mdash 069 047 047

031 mdash 047 047

053 053 mdash 047

053 053 053 mdash

]]]]]

]

(18)


119896 = 1 2 3 are

119889 (21 411)

= radic1

3sdot [(0151 minus 0118)

2

+ (0192 minus 0148)2

+ (0201 minus 0179)2

]

= 00342

119889 (22 412)

= radic1

3sdot [(0294 minus 030)

2

+ (0352 minus 0376)2

+ (0411 minus 0423)2

]

= 00159





119889 (23 413)

= radic1

3sdot [(0209 minus 0163)

2

+ (0256 minus 0209)2

+ (0283 minus 0236)2

]

= 00467

(19)


119899241

max119896=13

(00342 00467)

max119896=123

(00342 00159 00467)=00467

00467

= 1

(20)




radic

3

sum

119896=1

(119908119896minus 119894119896)2


radic

3

sum

119896=1

(119908119896minus 119894119896)2




radic

3

sum

119896=1

(0 minus 119894119896)2


radic

3

sum

119896=1

(0 minus 119894119896)2




119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)


Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2


119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)



(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)


119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[[[[[

[

1 1 1 11

1198721

1198671

1198721

119871

1

1198711

1198721

1198721

119867

1 1 1 1 1119872 1 119871

1 1 1 1

]]]]]

]

(14)


23=

4

sum

119890=1

120596119890sdot 119890

1198961198961015840

= 03 sdot (1 1 1) + 03 sdot (1 3 5) + 02 sdot (1 1 1)

+ 02 sdot (1 1 5) = (1 16 3)

(15)


[[

[

(1 1 1) (015 0425 1) (02 0505 1)

(1 235 6667) (1 1 1) (1 16 3)

(1 1980 5) (0333 0625 1) (1 1 1)

]]

]

(16)










11987821= 1198961 1198962

11987311987821= 1198963

119878241

= 1198962

119873119878241

= 1198961 1198963

119878236

= 1198962

119873119878236

= 1198961 1198963

11987812= 1198963

11987311987821= 1198961 1198962

119878141

= 1198962

119873119878141

= 1198961 1198963

119878136

= 1198962

119873119878136

= 1198961 1198963

119878412

= 1198961 1198963

119873119878412

= 1198962

119878411

= 1198961 1198963

119873119878411

= 1198962

1198784136

= 1198962

1198731198784136

= 1198961 1198963

119878362

= 1198961 1198963

119873119878362

= 1198962

119878361

= 1198961 1198963

119873119878361

= 1198962

1198783641

= 1198961 1198963

1198731198783641

= 1198962

(17)


119862 =

[[[[[

[

mdash 069 047 047

031 mdash 047 047

053 053 mdash 047

053 053 053 mdash

]]]]]

]

(18)


119896 = 1 2 3 are

119889 (21 411)

= radic1

3sdot [(0151 minus 0118)

2

+ (0192 minus 0148)2

+ (0201 minus 0179)2

]

= 00342

119889 (22 412)

= radic1

3sdot [(0294 minus 030)

2

+ (0352 minus 0376)2

+ (0411 minus 0423)2

]

= 00159





119889 (23 413)

= radic1

3sdot [(0209 minus 0163)

2

+ (0256 minus 0209)2

+ (0283 minus 0236)2

]

= 00467

(19)


119899241

max119896=13

(00342 00467)

max119896=123

(00342 00159 00467)=00467

00467

= 1

(20)




radic

3

sum

119896=1

(119908119896minus 119894119896)2


radic

3

sum

119896=1

(119908119896minus 119894119896)2




radic

3

sum

119896=1

(0 minus 119894119896)2


radic

3

sum

119896=1

(0 minus 119894119896)2




119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)


Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2


119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)



(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)


119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119889 (23 413)

= radic1

3sdot [(0209 minus 0163)

2

+ (0256 minus 0209)2

+ (0283 minus 0236)2

]

= 00467

(19)


119899241

max119896=13

(00342 00467)

max119896=123

(00342 00159 00467)=00467

00467

= 1

(20)




radic

3

sum

119896=1

(119908119896minus 119894119896)2


radic

3

sum

119896=1

(119908119896minus 119894119896)2




radic

3

sum

119896=1

(0 minus 119894119896)2


radic

3

sum

119896=1

(0 minus 119894119896)2




119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)


Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2


119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)



(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)


119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












radic

3

sum

119896=1

(119908119896minus 119894119896)2


radic

3

sum

119896=1

(119908119896minus 119894119896)2




radic

3

sum

119896=1

(0 minus 119894119896)2


radic

3

sum

119896=1

(0 minus 119894119896)2




119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)


Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2


119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)



(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)


119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896 = 1 119896 = 2 119896 = 3

119894 = 2 (0151 0192 0201) (0294 0352 0411) (0209 0256 0283)119894 = 1 (0162 0209 0214) (0276 0352 0411) (0182 0232 0263)119894 = 41 (0118 0148 0179) (030 0376 0423) (0163 0209 0236)119894 = 36 (0085 0116 0146) (0322 0411 0440) (0163 0202 0240)


Rank119894 = 2 4119894 = 1 3119894 = 41 1-2119894 = 36 1-2


119873 =

[[[[[

[

mdash 1 1 1

0583 mdash 0759 1

0340 0433 mdash 1

0623 0579 0790 mdash

]]]]]

]

(21)



(13))

119888119904119903=

1

4 sdot (4 minus 1)

4

sum

119894=1

4

sum

119894=1

1198881198941198941015840 =

1

12sdot 6 = 05

119899119904119903=

1

4 sdot (4 minus 1)

119868

sum

119894=1

119868

sum

119894=1

1198991198941198941015840 =

1

12sdot 9107 = 0759

(22)


119872 =

[[[[[

[

mdash 0 0 0

0 mdash 1 0

1 1 mdash 0

1 1 0 mdash

]]]]]

]

(23)











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











6 Conclusion









Competing Interests


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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