research article selecting e-purse smart card technology...

Research ArticleSelecting e-Purse Smart Card Technology viaFuzzy AHP and ANP

Nurguumll DemirtaG Fenim Oumlzguumlrler Mesut Oumlzguumlrler and Ali Fuat Guumlneri

Department of Industrial Engineering Yildiz Technical University 34349 Istanbul Turkey

Correspondence should be addressed to Senim Ozgurler senimozgurlerhotmailcom

Received 19 November 2013 Accepted 9 March 2014 Published 1 June 2014

Academic Editor Huijun Gao

Copyright copy 2014 Nurgul Demirtas et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Developments in the information technologies provide important advantages to consumers and companies Nowadays smart cardtechnology starts to use e-purse applications The aim of this paper is to identify the most important decision criteria to selectthe best card technology In this study at first smart card and multiple selection techniques were explained Then the best cardtechnology was selected for an e-purse application The three types of card technologies were examined and the most importantcriteria were taken into account by the software developer while they develop card software Fuzzy analytic hierarchy process(FAHP) and analytical network process (ANP) techniques were used to compare smart card technologies

1 Introduction

Analysis with single parameter or criterion is not enoughto resolve the complex structured problem parallel to thedevelopment of science and technology The most significantinference of the single-criteria analysis is to accept the otherdimensions of the event as static and to examine only onefactor for each time However the events and the objectsoccur under the influences of a huge amount of internal andexternal factors instead of a solitary factor and demonstratecomplex formation Because of this reason events and objectsmust be defined according to the amount of agents and theircollective efficiency rather than a single agent [1] Due to thesefactors multiple parameter judgment methods are appliedin almost every field And one of these fields is smart cardtechnology

Usage and effectiveness of internet is increasing day byday One of the important results of this frequency is thatthe commerce is moving to the electronic environmentTransferring the development of the information technolo-gies to trade is effectively beneficial for both consumers andcompanies Smart card technology has been built up becauseof the necessities plus these benefits Development of thesmart card technologies has been increasing since it startedto be used in 2002 in the world and Turkey

Smart card is being used in lots of fields including healthsecurity retail telecom banking government and automo-tive field Beside this functional usage innovative approacheshave been explored while traditional boundaries were passedover in retail segment While exploring these approachestechnological improvement and especially development inthe information technologies are used as exploitable

Developing smart card application including card pay-ment system and card personalization should demonstratemulticriteria structure Multiple parameter judgment meth-ods are appropriate for smart card technology and by usingthese methods choice of smart card technologies shouldpresent more sensible decisions and suggestions

The rest of the paper is arranged as follows The smartcard technology which is appropriate for developing e-purseapplication is presented in Section 2 In Sections 3 and 4FAHP and ANP methods are explained respectively InSection 5 thesemethods are applied in order to select e-pursesmart card technology Finally the results of the paper andfuture study suggestions are considered in the last section

2 Smart Card Concept and Properties

21 Smart Card Concept Technological development is kindof ldquominimizing automation systemrdquo process Communication

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 619030 14 pageshttpdxdoiorg1011552014619030

2 Journal of Applied Mathematics

Contact smart card

Chip

(a)

Contactlesschip

Antenna

Contactlesssmart card

(b)

Antenna

Chip

Combi card

Contacts

(c)

Figure 1 (a) Contact smart card (b) contactless smart card and (c) combi card

technologies like cell phones small but functional computersystems microchips managing and controlling huge mecha-nisms and database units are the examples of thisminimizingautomation Smart cardwould be included in thisminimizingautomation process

Smart card technology has evolved a long way sincethe idea of using plastic cards to carry microelectronicchips was patented back in 1968 by Dethloff and Grotrupp[2] A smart card is a credit-card sized plastic card withan embedded computer chip that can be memory or alsoinclude a microprocessor [3ndash5] A microprocessor chip canadd delete and otherwise manipulate information in itsmemory and hence offer complex data security schemes [4]Smart card technology can enable an organization to becomemore secure efficient and interoperable while deliveringstrong authentication and security identity managementdata management customer support and communications[6] The card body is usually more than just a carrier forthe chip module It also includes information for the userand card accepters and of course security elements forprotection against forgery The card elements are printingand labeling embossing hologram signature panel tactileelements magnetic stripe chip module and antenna [7]Smart cards the result of evolution of magnetic card in useof daily life have superior memory thanmagnetic cardsThisvast memory provides ability to execute lots of functionsand present complex result As a result of this condition adatabase for customer relationship management (CRM) andstatistical data network can be provided by installing all ofthe information of any customer Also it becomes easy toreach customer almost in every branch of business like healthautomotive and retail sector

22 Smart Card Types A smart-card chip communicateswith a reader by direct physical contact or by a radiofrequency (RF) signal depending on the system design [8]Smart cards are grouped into three classes for chip-to-reader communications which are (1) contact smart cards (2)contactless smart cards and (3) combi cards

221 Contact Smart Cards Contact smart cards are the mostpopular card-connection design and are used for both cardsizes and chip types Figure 1(a) [9] depicts the contact-type

card [8] Contact cards use an eight-pin contactmicromoduleto physically connect to the card reader [8]

222 Contactless Smart Cards Contactless smart cards asshown in Figure 1(b) [9] use an antenna with approximatelya 10-centimeter (cm) range to communicate with the readerThese credit-card sized memory-chip devices derive theirpower from an RF field generated by the card reader TheRF field also transfers information to and from the cardand card reader [8] Contactless cards are better suited forenvironments where quick interaction between the card andreader is required such as high-volume physical access [10]Employee identification badges issued by large companies forbuilding access are typically contactless smart cards [8]

223 Combination Smart Cards Multipurpose combinationsmart cards are a hybrid mix of the contact and contactlessdesigns [8] These cards obtain the two systems describedabove These cards contain all the properties of the othertwo types Figure 1(c) shows the combination smart card[9] They include the eight-pin contact for communicationwith a contact-type reader and also include an antenna forcommunicationwith an RF-type reader [8]With such a cardit becomes possible to access the same chip via a contactand contactless interface with a very high level of securityIt may incorporate two noncommunicating chipsmdashone foreach interfacemdashbut preferably has a single dual interface chipproviding the many advantages of a single e-purse singleoperating architecture and so forthThemass transportationand banking industries are expected to be the first to takeadvantage of this technology [9]

23 Advantages and Usage Areas of Smart Card TechnologiesSmart card automation process has lots of function suchas to prevent time lost to reach customer to control effi-ciency Working level becomes mobilized with smart cardsThis means workplace is not bounded to an only placeand customers are reached every time Briefly customerrelationship management becomes the focus of business andproduction process and its effectiveness increases Smart cardusage provides trust and prestige to company or companyrsquosinstitutional identity Smart cards are more secure thanmagnetic cardsThe highly developed software of smart cards

Journal of Applied Mathematics 3

raises security level In case smart cards are obtained by thirdperson risk of data theft is decreased by the software thatprevents copying data and protects the cryptology algorithmInformation stored in smart cards is protected by complexsecurity mechanism Because of this reason it is hard andexpensive to copy or change data Progression to smartcard from magnetic cards decreases the card forgery causedby both fake cards and offline card process During themobilized usage the most important necessities are datasecurity and accuracy Disability to copy and change datain the card and to reach data during service is the mostsignificant property It is vital to protect both card owner andservice provider Smart cards can protect data by the help ofprivate hardware and software and can apply well developedcryptology methods

Clearly there are beneficial outcomes from the appli-cation of smart cards Realizing these benefits both forindividuals and organizations may well profoundly changethe relationship between clients or consumers and suppliersor government bodies A smart card that is your passportdriving license credit and debit card and access to your placeof work and your car ignition key will undoubtedly alterrelationships due to potential uneasiness about what datais held accessed and modified [11] Some of the potentialbenefits of smart cards are as follows Using smart cards issafer than carrying cash for an individual smart cards canimprove access to services for the disabled and elderly it is asecure means of authenticating the identity of reader deviceit is a portable and secure store of information available toall access can be made available in geographical locationswhere online communication is not possible the opportunityof fraud is reduced using smart cards social disadvantagedgroups can gain access to facilities and resources withoutfeeling stigmatized and objective selection criteria can beupheld and the risk of bias or favouritism reduced [11]

Smart cards have a wide range of potential applicationsand can be utilized for many different purposes Howeverthere are unquestionably areas that are especially suitablefor smart cards The principal characteristics of smart cardsare that they can securely store relatively small amounts ofdata and provide an environment for the secure execution ofprograms This makes them excellent candidates for use inthe entire security sector Another important characteristicis that they have a well-established format that is convenientfor manual use and handling Although the smart cardinterface does not correspond to the current PC and Internetstandards it is still easy to use which also encourages theuse of smart cards The best possible uses for smart cardsare applications that need a large number of inexpensive datastorage devices that can securely store individual or personaldata and that must perform security-related activities such asauthentication encryption andor signing [7]

In summary the range of smart card applications is grow-ing for many reasons such as security multiple applicationand portability Smart cards are used in many different appli-cations around theworld particularly for electronic paymentsecurity and authentication transportation telecommunica-tions health care loyalty programs and education Theseare some of the more popular application areas [12] It is

Payment cards

Credit cards Debit cards Electronic purse cards

Figure 2 Classification of payment cards

important to note that consumer acceptance and confidenceare crucial for the further development of smart card tech-nology as the underlying issues which demand more controlsecurity privacy flexibility and ease of use [12]

24 Smart Card as Electronic Purse The original primaryapplication of smart cards with microcontrollers was useridentification in the telecommunications sector In recentyears however smart cards have established themselves inanother market sector namely electronic payment systemsDue to the large number of cards in use the market potentialof this sector is enormous Smart cards are by nature par-ticularly suitable for payment system applications They caneasily and securely store data and their convenient size androbustness make them easy for everyone to use Since smartcards can also actively perform complicated computationswithout being influenced by external factors it is possibleto develop totally new approaches to performing paymenttransactions [3]

The idea of implementing an electronic purse in a smartcard goes back to the early days of smart card technologyHowever only since the mid-1990s has this concept beenrealized since that was when the development of large sys-tems first began Electronic payment systems and electronicpurses offer significant benefits to everyone involved Forbanks and merchants they reduce the costs associated withhandling cash Offline electronic purses largely eliminate thecosts of data telecommunications for payment transactionsThe risk of robbery and vandalism is reduced since electronicsystems contain no cash to be stolen For merchants thefact that transactions are processed more quickly is also apersuasive argument since it means that cash managementcan be optimized Classification of payment cards and e-pursesystems based on smart cards is shown in Figures 2 and 3respectively [3]

3 Fuzzy AHP Methodology

There has been growth in the number of multiple criteriadecision-making methods for assisting decision making dur-ing the past two decades These allow decision makers toevaluate various alternatives for achieving their goal [13]

Analytic hierarchy process (AHP) is a simple decision-making tool to deal with complex unstructured and mul-tiattribute problems [14] or a weight estimation techniquein many areas such as selection evaluation planning anddevelopment decision making and forecasting The tra-ditional AHP requires crisp judgments However due to


Transferrable with restrictions

Completely transferrable

Electronic purses

Electronic money Electronic cheques Prepaid cards

Figure 3 Classification of electronic purse systems based on smartcards

the complexity and uncertainty involved in real world deci-sion problems a decision maker (DM) may sometimesfeel more confident to provide fuzzy judgments than crispcomparisons [15]

The fuzzy analytic hierarchy process (FAHP) is one ofthe most popular [13] of methods that have been developedto handle fuzzy comparison matrices [15 16] The FAHPmethodology extends Saatyrsquos AHP by combining it withthe fuzzy set theory In the FAHP fuzzy ratio scales areused to indicate the relative strength of the factors in thecorresponding criteria Therefore a fuzzy judgment matrixcan be constructed The final scores of alternatives are alsorepresented by fuzzy numbers The optimum alternative isobtained by ranking the fuzzy numbers using special algebraoperators [17] There are many FAHP methods proposed byvarious authors These methods are systematic approaches tothe alternative selection and justification problem by usingthe concepts of fuzzy set theory and hierarchical structureanalysis [18 19] Decision makers usually find that it ismore confident to give interval judgments than fixed valuejudgments This is because usually heshe is unable to beexplicit about hisher preferences due to the fuzzy natureof the comparison process [16 18 19] The earliest work inFAHP appeared in van Laarhoven and Pedrycz [20] whichcompared fuzzy ratios described by triangular membershipfunctions [18 19 21] Buckley [22] determines fuzzy prioritiesof comparison ratios whose membership functions are trape-zoidal [18 19 21] Chang (1996) introduces a new approach forhandling fuzzyAHPwith the use of triangular fuzzy numbersfor pairwise comparison scale of fuzzyAHP and the use of theextent analysis method for the synthetic extent values of thepair-wise comparisons [18 19] Zhang et al (2013) proposeda multiple attribute decision making model to seek the nodeimportance of complex networks in product research anddevelopment team [23] W Zhang and Q Zhang (2013)constructed a multistage evaluation criteria system from thethree dimensions of novelty value and practicality eachof which is composed of three separate evaluation criteriasystem and the fuzzy AHP method is applied to obtain theweights of each dimension and subdimension [24]

In this study Changrsquos extent analysis [25] is used In thefollowing the steps of the extent analysis method on FAHPare given

Let 119883 = 1199091 1199092 119909

119899 be an object set and let 119880 =

1199061 1199062 119906

119898 be a goal set According to the method of

Changrsquos extent analysis each object is taken and extent anal-ysis for each goal 119892

119894 is performed respectively Therefore119898

extent analysis values for each object can be obtained withthe following signs

1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (1)

where all the 119872119895

119892119894(119895 = 1 2 119898) are triangular fuzzy

numbersThe steps of Changrsquos extent analysis can be detailed as

follows [16 18 19 21 25ndash28]

Step 1 The value of fuzzy synthetic extent with respect to the119894th object is defined as

119878119894=

119898

sum

119895=1

119872119895

119892119894otimes [

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

(2)

To obtain sum119898

119895=1119872119895

119892119894 perform the fuzzy addition operation of

119898 extent analysis values for a particular matrix such that

119898

sum

119895=1

119872119895

119892119894= (

119898

sum

119895=1

119897119895

119898

sum

119895=1

119898119895

119898

sum

119895=1

119906119895) (3)

and to obtain [sum119899

119894=1sum119898

119895=1119872119895

119892119894]minus1 perform the fuzzy addition

operation of 119872119895119892119894

(119895 = 1 2 119898) values such that

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894= (

119899

sum

119894=1

119897119894

119899

sum

119894=1

119898119894

119899

sum

119894=1

119906119894) (4)

and then compute the inverse of the vector in (4) such that

[

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (5)

Step 2 The degree of possibility of 1198722= (1198972 1198982 1199062) ge 119872

1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup119910ge119909

[min (1205831198721

(119909) 1205831198722

(119910))] (6)

and can be equivalently expressed as follows

119881 (1198722ge 1198721) = hgt (119872

1cap 1198722) = 1205831198722

(119889)

119881 (1198722ge 1198721) =

1 if1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898

1minus 1198971) otherwise

(7)

where 119889 is the ordinate of the highest intersection point 119863between 120583

1198721and 120583

1198722(see Figure 4)


M2M1

1

l2 m2 l1 u2 m1 u1d

V(M2 ge M1)

Figure 4 The intersection between1198721and119872

2

To compare1198721and119872

2 both the values of 119881(119872

1ge 1198722)

and 119881(1198722ge 1198721) are requiredThe intersection between119872

1

and1198722is shown in Figure 4 [18]

Step 3 Thedegree possibility for a convex fuzzy numberto begreater than 119896 convex fuzzy numbers119872

119894(119894 = 1 2 119896) can

be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) and (119872 ge 119872

2) and and (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 3 119896

(8)

Assume that

1198891015840(119860119894) = min119881 (119878

119894ge 119878119896) (9)

For 119896 = 1 2 119899 119896 = 119894 Then the weight vector is given by

1198821015840= (1198891015840(1198601) 1198891015840(1198601015840

2) 119889

1015840(119860119899))119879

(10)

where 119860119894(119894 = 1 2 119899) are 119899 elements

Step 4 Via normalization the normalized weight vectors are

119882 = (119889 (1198601) 119889 (119860

2) 119889 (119860

119899))119879

(11)

where119882 is a nonfuzzy number

4 Analytic Network Process (ANP)

The ANP is the most comprehensive framework for analysisof public governmental and corporate decisions It allowsdecision makers to include all the factors and tangible orintangible criteria that have a bearing on making the bestdecision [29]

Many decision problems cannot be structured hierarchi-cally because they involve the interaction and dependenceof higher-level elements on lower-level elements [30 31]Not only does the importance of the criteria determine theimportance of the alternatives as in a hierarchy but alsothe importance of the alternatives themselves determines theimportance of the criteria [30]

The ANP generalizes a widely used multicriteria decisionmaking tool the AHP by replacing hierarchies with net-works The AHP is a well-known technique that decomposesa problem into several levels in such a way that they forma hierarchy Each element in the hierarchy is supposed tobe independent and a relative ratio scale of measurement isderived from pairwise comparisons of the elements in a levelof the hierarchy with respect to an element of the precedinglevel However in many cases there is interdependenceamong criteria and alternatives The ANP can be used as aneffective tool in those cases where the interactions amongthe elements of a system form a network structure [32] Theprocess of ANP comprises four major steps [31 33 34]

Step 1 (model construction and problem structuring) Theproblem should be stated clearly and decomposed into arational system like a networkThe structure can be obtainedby the opinion of decision makers through brainstorming orother appropriate methods

Step 2 (pairwise comparison matrices and priority vectors)In ANP like AHP decision elements at each component arecompared pairwise with respect to their importance towardstheir control criterion and the components themselves arealso compared pairwise with respect to their contributionto the goal Decision makers are asked to respond to aseries of pairwise comparisons where two elements or twocomponents at a time will be compared in terms of howthey contribute to their particular upper level criterion Inaddition if there are interdependencies among elements of acomponent pairwise comparisons also needed to be createdand an eigenvector can be obtained for each element toshow the influence of other elements on it The relativeimportance values are determined on a scale of 1ndash9 (seeTable 11) where a score of 1 represents equal importancebetween the two elements and a score of 9 indicates theextreme importance of one element (row component in thematrix) compared to the other one (column component in thematrix)

A reciprocal value is assigned to the inverse comparisonthat is 119886

119894119895= 1119886119895119894 where 119886

119894119895(119886119895119894) denotes the importance of

the 119894th (119895th) element compared to the 119895th (119894th) element LikeAHP pairwise comparison in ANP is made in the frameworkof a matrix and a local priority vector can be derived asan estimate of the relative importance associated with theelements (or components) being compared by solving thefollowing formula

119860119908 = 120582max119908 (12)

where 119860 is the matrix of pairwise comparison 119908 is theeigenvector and 120582max is the largest eigenvalue of 119860 Saaty(1980) [35] proposes several algorithms for approximating119908 The following three-step procedure is used to synthesizepriorities [31 33ndash36]


(1) Sum the values in each column of the pairwise com-parison matrix

(2) Divide each element in a column by the sum of itsrespective columnThe resultant matrix is referred toas the normalized pairwise comparison matrix

(3) Sum the elements in each row of the normalizedpairwise comparison matrix and divide the sum bythe 119899 elements in the row These final numbersprovide an estimate of the relative priorities for theelements being compared with respect to its upperlevel criterion Priority vectors must be derived for allcomparison matrices

Step 3 (supermatrix formation) The supermatrix conceptis similar to the Markov chain process [34 37] To obtain

global priorities in a system with interdependent influencesthe local priority vectors are entered in the appropriatecolumns of a matrix known as a supermatrix As a resulta supermatrix is actually a partitioned matrix where eachmatrix segment represents a relationship between two nodes(components or clusters) in a system [33 34 38] Let thecomponents of a decision system be 119862

119896 119896 = 1 2 119899

which has119898119896elements denoted as 119890

1198961 1198901198962 119890

119896119898119896The local

priority vectors obtained in Step 2 are grouped and located inappropriate positions in a supermatrix based on the flow ofinfluence from a component to another component or froma component to itself as in the loop 119860 standard form of asupermatrix as follows [34 37]

119882 =

1198621

119862119870

119862119873

11989011

11989011198991

1198901198961

119890119896119899119896

1198901198731

119890119873119899119873

1198621

sdot sdot sdot 119862119870


11989011



sdot sdot sdot 119890119896119899119896


sdot sdot sdot 119890119873119899119873

[[[[[[[

[

11988211



d

1198821198701

sdot sdot sdot 119882119870119870

sdot sdot sdot 119882119870119873

d

1198821198731

sdot sdot sdot 119882119873119870

sdot sdot sdot 119882119873119873

]]]]]]]

]

(13)

As an example consider the supermatrix representation of ahierarchy with three levels [34 37]

Wℎ= [

[

119868 0 0

w21

0 0

0 W32

119868

]

]

(14)

where w21

is a vector that represents the impact of the goalon the criteria W

32is a matrix that represents the impact

of criteria on each of the alternatives and 119868 is the identitymatrix and entries of zeros corresponding to those elementsthat have no influence

For the above example if the criteria are interrelatedamong themselves a network replaces the hierarchy as shownin Figure 5 [31 33] The (2 2) entry of Wn given by W

22

would indicate the interdependency and the supermatrixwould be [34 37]

W119899= [

[

119868 0 0

w21

W22

0

0 W32

119868

]

]

(15)

Note that amatrix can replace any zero in the supermatrixif there is an interrelationship of the elements in a component

or between two components Since there usually is interde-pendence among clusters in a network the columns of asupermatrix usually sum to more than one The supermatrixmust be transformed first to make it stochastic that iseach column of the matrix sums to unity A recommendedapproach by Saaty (1996) [37] is to determine the relativeimportance of the clusters in the supermatrix with thecolumn cluster (block) as the controlling component [34 38]That is the row components with nonzero entries for theirblocks in that column block are compared according to theirimpact on the component of that column block [34 37]With pairwise comparison matrix of the row componentswith respect to the column component an eigenvector canbe obtainedThis process gives rise to an eigenvector for eachcolumn block For each column block the first entry of therespective eigenvector is multiplied by all the elements in thefirst block of that column the secondby all the elements in thesecond block of that column and so on In this way the blockin each column of the supermatrix is weighted and the resultis known as the weighted supermatrix which is stochastic[34]

Raising a matrix to powers gives the long-term relativeinfluences of the elements on each other To achieve aconvergence on the importance of weights the weighted


Goal

Criteria

Alternatives

w21

W32

(a)

Goal

Criteria

Alternatives

w21

W22

W32

(b)

Figure 5 Hierarchy and network (a) a hierarchy (b) a network

supermatrix is raised to the power of 2119896 + 1 where 119896 is anarbitrarily large number and this new matrix is called thelimit supermatrix [34 37]The limit supermatrix has the sameform as the weighted supermatrix but all the columns of thelimit supermatrix are the same By normalizing each block ofthis supermatrix the final priorities of all the elements in thematrix can be obtained [34]

Step 4 (selection of the best alternatives) If the supermatrixformed in Step 3 covers the whole network the priorityweights of alternatives can be found in the column ofalternatives in the normalized supermatrix On the otherhand if a supermatrix only comprises components that areinterrelated additional calculation must be made to obtainthe overall priorities of the alternatives The alternative withthe largest overall priority should be the one selected [34]

Asmentioned before ANP is a generalization of the AHP[33] and based on reasoning knowledge and experienceof the experts in the field ANP can act as a valuable aidfor decision making involving both tangible and intangibleattributes that are associated with the model under studyANP relies on the process of eliciting managerial inputs thusallowing for a structured communication among decisionmakers Thus it can act as a qualitative tool for strategicdecision-making problems [34]

5 Case Study

In this section selection of the best e-purse smart cardtechnology is presented for the case of a company Firstlye-purse application is explained and then FAHP and ANPmethods are applied to an e-purse smart card technologyselection problem respectively Then the results of FAHPand ANP methods are compared with each other The casestudy is about the project of the e-purse application thatis developed by Company O The proposed model includesvarious usage fields about e-purse smart card personalization

of smart card and payment with smart card Using e-pursesmart card eliminates cash interchange and provides thatprocess is made on smaller time and is enrolled stock iscontrolled at workplace and productivity is increased Inaddition to interchange in company this card also can be usedas an identity card and entrance card For the integration ofthis feature cards must be personalized Because of all thisease of use all processes are made with only one smart cardinstead of using different smart card for each process Finallyvarious characteristics that are necessary for a company arecollected in a smart card The software of these cards isprogrammed by Company O Card technologies that providerequired characteristics for programming are obtained fromcard companiesThere are three alternatives AC TU andOBcard These cards are compared according to some criteriaand the best alternative has been selected MCDM havenot been used in the case company before therefore theselections have been made heuristics

51 Determining the e-Purse Smart Card Technology SelectionCriteria In this study the selection criteria were identifiedby decision makers Thus an interview was conductedwith three decision makers who are chosen from differentareas such as developers system analysts and academiciansTwelve subcriteria were determined to select the most con-venient e-purse card technology alternative under the fourmain criteria Also three potential card technologies wereconsidered for the selection A detailed questionnaire relatedwith criteria and alternatives was prepared for the selectionprocess During the identification ofmain and subcriteria theopinions of experts from Company O were also taken intoaccount

For the three alternative card technologies criteria aresoftware technology and software performance flexibility costand service level The definitions and subcriteria of the maincriteria are summarized below and also Table 1 shows thecriteria and subcriteria

Software Technology and Software Performance (S) It is themost important criterion during the selection of smart cardfor programming of e-purse card systems It must be appro-priate for software security and reliability Performance andthe quick response to software command are other subcriteriaof the software technology and software performance ldquoSoft-ware development and easiness of software usagerdquo (SDU)ldquospeed of software workingrdquo (SS) ldquosoftware security andsoftware credibilityrdquo (SSC) and ldquotechnical sufficiencyrdquo (TS)are subcriteria of this criterion

Flexibility (F) Flexibility of smart card companies can bedefined as compatible to response customerrsquos request Thiscriterion contains ability to supply all products that customerhas requested and their urgent product demand ldquoEasilysupplying of card demandsrdquo (ESD) ldquosupplying of urgentcard demandsrdquo (SUD) and ldquosupplying of different sectordemandsrdquo (SSD) are subcriteria of this criterion

Cost (C) Software companies engaged with smart card tech-nology want to supply material with minimum price to


Table 1 Criteria and subcriteria for e-purse smart card technologyselection

Criteria Subcriteria

Softwaretechnologyand softwareperformance(S)

SDU software development and easiness ofsoftware usageSS speed of software workingSSC software security and software credibilityTS technical sufficiency

Flexibility (F)ESD easily supplying of card demandsSUD supplying of urgent card demandsSSD supplying of different sector demands

Cost (C)AB appropriate bidding according to other cardfirmsPR price reduction in comparison with othercard firms according to cards quantity

Service level(SL)

SV velocity of technical support after saleSDS sufficiency of technical support departmentafter saleOH online technical help after sale

Selection of the best e-pursesmart card technology

S

SD SS SSC TS ESD SU SSD PRAB SV SDS OH

F C SL

AC card TU card OB card

Figure 6 The hierarchy for selection of e-purse card technologyproblem

raise profitability like all production companies Providerrsquosmore appropriate price than the other providers and higherdiscount on the material according to the amount of pur-chased materials are subcriteria of cost ldquoAppropriate biddingaccording to other card firmsrdquo (AB) and ldquoprice reductionin comparison with other card firms according to cardsquantityrdquo (PR) are subcriteria of this criterion

Service Level (SL) Qualified service of provider is an impor-tant factor and it is also significant for provider to beconsidered as high performance company by the customersldquoVelocity of technical support after salerdquo (SV) ldquosufficiency oftechnical support department after salerdquo (SDS) and ldquoonlinetechnical help after salerdquo (OH) are subcriteria of this criterion

According to explanations of the aforementioned mainand subcriteria a hierarchical structure for selection ofe-purse card technology is shown in Figure 6

52 Questionnaire The three of decision makers are askedto make pairwise comparisons for main and subcriteriamentioned in Section 51 A questionnaire is provided to getthe evaluations The decision makersrsquo linguistic preferences

Table 2 Triangular fuzzy number of linguistic terms

Linguistic terms Triangular fuzzy numbersAbsolutely strong (72 4 92)Very strong (52 3 72)Fairly strong (32 2 52)Weak (23 1 32)Equal (1 1 1)Weak (23 1 32)Fairly strong (32 2 52)Very strong (52 3 72)Absolutely strong (72 4 92)

are converted into triangular fuzzy numbers by using Table 2[39] Following questions are given as a sample of questionsfrom the questionnaire for the pairwise comparisonmatrices

In the questionnaries a check mark on the right side ofthe preference ldquoequalrdquo means that the criterion on the rightside of the preference ldquoequalrdquo is more important than theone matching on the left side as it is specified by the checkmark A check mark on the left side of the preference ldquoequalrdquomeans that the criterion on the left side of the preferenceldquoequalrdquo is more important than the onematching on the rightside as it is specified by the check mark [18] A sample ofthe questionnaire forms used for comparisons of main andsubcriteria is given in Table 3

With respect to the overall goal ldquoselection of the beste-purse smart card technologyrdquo consider the following

Question 1 How important is software technology and soft-ware performance (S)when it is compared with flexibility (F)

Question 2 How important is software technology and soft-ware performance (S) when it is compared with cost (C)

Question 3 How important is software technology and soft-ware performance (S) when it is compared with service level(SL)

Question 4 How important is flexibility (F) when it iscompared with cost (C)

Question 5 How important is flexibility (F) when it iscompared with service level (SL)

Question 6 How important is cost (C) when it is comparedwith service level (SL)

Other matrices are done with same method

53 Numerical Applications of the Methods In this sec-tion the best e-purse card technology that will be usedin Company O is selected with FAHP and ANP methodsrespectively A questionnaire is made in the company toestablish importance weights (fuzzy preference numbers)that are necessary for selection of best card technology Thequestionnaires facilitate the answering of pairwise compari-son questions


Table 3 A sample from the questionnaire

Selection of thebest e-purse smartcard technology

Importance (or preference) of one criteria over another

Questions Criteria (72 4 92)Absolute

(52 3 72)Very strong

(32 2 52)Fairlystrong

(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak



(72 4 92)Absolute Criteria

Q1 S X XX FQ2 S XX X CQ3 S X XX SLQ4 F XXX CQ5 F XXX SLQ6 C XXX H

Table 4 Evaluation results of the criteria with respect to the goal

Main criteria S F C SLS (1000 1000 1000) (1500 2289 2797) (2109 2621 3130) (1778 2289 2797)

F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)

531 Applying FAHP to e-Purse Smart Card Technology Selec-tion Problem Firstly FAHP method is used for best e-pursecard technology selection In this method the factorsrsquo fuzzyweights are calculated according to each other for selectionof AC TU and OB card technologies For this criteriaand subcriteria are listed for selecting card technologiesFor determining the relative importance between main andsubcriteria three decision makers were asked to respond forpairwise comparisons According to result of questionnairegeometric mean is used to set pairwise comparisons

Geometric means are obtained as follows An examplecalculation is given

The importance of ldquosoftware technology and software per-formance (S)rdquo is (32 2 52) (32 2 52) and (52 3 72)

when it is compared with ldquoflexibility (F)rdquo Consider

(3radic(

3

2times

3

2times

5

2)3radic(2 times 2 times 3)

3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)

The evaluation results are given in Table 4Fuzzy synthetic extent values are calculated by using

evaluation results Firstly the values of fuzzy synthetic extentswith respect to themain criteria are calculated Calculation of1198781is given as an example The obtained values are shown in

Table 5 Similar calculations are done for the other matricesTable 5 depicts the results of the calculations Consider

(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)

oplus (0400 oplus 0500 oplus 0667))

+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)

Importance weights of criteria are calculated by usingfuzzy synthetic values Then probability of preference of oneto the other is calculated After this calculation the weightvectors are calculated by using the results of probability ofpreference The normalized weight vectors are values thatare used for selection of e-purse smart card technologiesTables 6 7 8 and 9 are obtained as the priority weightsof the alternatives according to each subcriterion And the


Table 5 Fuzzy synthetic extent values

1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)

Table 6 Summary combination of priority weights subcriteria of software technology and software performance

Subcriteria SDU SS SSC TSPriority weights 0199 0000 0493 0308 Alternative priority weight

AlternativesAC 0940 0764 0890 0537 0791TU 0000 0000 0000 0093 0029OB 0060 0236 0110 0370 0180

Table 7 Summary combination of priority weights subcriteria of flexibility

Subcriteria ESD SUD SSDPriority weights 0368 0338 0294 Alternative priority weight

AlternativesAC 0852 0714 0778 0784TU 0089 0089 0222 0174OB 0059 0060 0000 0042

Table 8 Summary combination of priority weights subcriteria of cost

Subcriteria AB PRPriority weights 0781 0219 Alternative priority weight

AlternativesAC 0000 0000 0000TU 1000 1000 1000OB 0000 0000 0000

Table 9 Summary combination of priority weights subcriteria of service level

Subcriteria SV SDS OHPriority weights 0359 0518 0123 Alternative priority weight



Table 10 Summary combination of priority weights main criteria of the goal

Main criteria S F C SLPriority weights 0622 0065 0000 0312 Alternative priority weights


obtained priority weights of criteria are presented in Table 10An example calculation is given

Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)

Via normalization the normalized weight vector is

119882 = (0622 0065 0000 0312) (19)

Similar calculations are done for the other matricesAccording to the final score as seen in Table 10 ldquoAC

cardrdquo technology is themost appropriated e-purse smart cardtechnology because it has the highest value with the 774priority weight

532 Applying ANP to e-Purse Smart Card Technology Selec-tion Problem In the previous section the best e-purse smart

Table 11 Nine-point intensity of importance scale and its descrip-tion

Definition Intensity of importanceEqually important 1Moderately more important 3Strongly more important 5Very strongly more important 7Extremely more important 9Intermediate values 2 4 6 8

card technology has been selected by using fuzzy AHPAccording to the result of FAHP AC card technology whichhas the highest value with the 774 priority weight is thebest e-purse smart card technology In this section the beste-purse smart card technology is selected by using ANPmethod

Both FAHP and ANP methods used the same criteria Aquestionnaire is made to establish pairwise comparisons thatare necessary for selection of best card technology Decisionmakers made individual evaluations using the nine-pointscale provided in Table 11 [40] According to the result of thequestionnaire relationship between criteria is decided andpairwise comparisons are constructed Network structure ofthe model is shown in Figure 7

After the pairwise comparisons are completed superma-trices are computed

(1) The unweighted supermatrix is created directly fromall local priorities derived from pairwise compar-isons among elements influencing each other (seeFigure 8)

(2) The weighted supermatrix is calculated by multiply-ing the values of the unweighted supermatrix withtheir affiliated cluster weights (see Figure 9)

(3) Composition of a limiting supermatrix takes placewhich is created by raising the weighted supermatrixto powers until it stabilizes (see Figure 10)

Stabilization is achieved when all the columns in thesupermatrix corresponding to any node have the same valuesAll of these steps are performed in Super Decisions whichis a software package developed for ANP applications bySaaty In Figure 11 the columnof ldquoNormalsrdquo shows the resultsAccording to the results ldquoAC cardrdquo technology which has thehighest value with the asymp066 priority weight is the mostpreferred e-purse smart card technology


Figure 7 Network structure of the ANP e-purse smart card tech-nology selection model

Figure 8 Unweighted super matrix

54 Comparison of the Results The comparison of the resultsprovided from the FAHP and the ANPmethods is illustratedin Figure 12 In FAHP method as seen in Figure 12 ldquoACcardrdquo technology is ranked in the first place with the asymp77priority weight as the most appropriated e-purse smart cardtechnology among other card technologies Besides ldquoOBcardrdquo (asymp12) and ldquoTU cardrdquo (asymp11) technologies withthe values very close to each other come as second andthird choice respectively Also in ANP method ldquoAC cardrdquotechnology is ranked in the first place with the asymp 66 priorityweight among other card technologies ldquoOB cardrdquo (asymp18) andldquoTU cardrdquo (asymp17) technologies come as second and thirdchoice respectively

When the results of FAHP and ANP methods are com-pared it is seen that the most appropriated e-purse smartcard technology is the same ldquoAC cardrdquo technology is thebest technology according to both methods But when theresults are examined it is seen that priority weights of thealternatives are not the same because of the relations betweenthe criteria and the subcriteria in the methods In the FAHPmethod the relations between the criteria and the subcriteriaare not considered but in the ANP method all of the externaland internal dependences and also feedbacks are taken intoconsideration

Figure 9 Weighted super matrix

Figure 10 Limit matrix

6 Conclusion

A field where decision making methods are applied is selec-tion of card technologies Usage of smart card technologyincreaseswith becomingwidespread of the Internet Selectingthe best technology is decided as multiple dimension forsoftware of smart card technologies that supplies customerdemands

The objective of this study is to identify important selec-tion criteria to select the best e-purse smart card technologyproviding the most customer satisfaction In the case studyat first three decision makers are chosen from different areassuch as developers system analysts and academicians Aninterview is conducted with decision makers to identify theselection criteria and potential alternatives Detailed ques-tionnaires related with criteria and alternatives are preparedfor determining the relative importance between main andsubcriteria Then FAHP and ANP methods are applied tothe e-purse smart card technology selection problem respec-tively Results of FAHP and ANPmethods are compared witheach other According to the results of methods it is seen thatthe best e-purse smart card technology is the same ldquoAC cardrdquo


Figure 11 Priorities of the alternatives obtained by ANP method

080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754

Figure 12 The results obtained from the FAHP and the ANPmethods

technology is the most appropriated card technology becauseit has the highest value than the others But when the resultsare examined it is also seen that there are some differencesbecause of the relations between the criteria and subcriteriain the methods

For further researches these methods can be applied toother smart card application areas such as security trans-portation telecommunications education and healthcareBesides for selection of smart card type smart card softwareor payment systems these methods can be used

For different smart card applications the card propertieswhich are not used in the comparison process of this studycan be taken into consideration And also the number ofevaluation criteria and alternatives can be increased Addi-tionally othermulticriteria decisionmakingmethods such asTOPSIS and ELECTRE can be used to solve e-purse smartcard technology selection problem The obtained results ofother methods can be compared with the results of this study

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

References

[1] I Dasdemir and E Gungor ldquoCok boyutlu karar verme metot-ları ve ormancılıkta uygulama alanlarırdquo ZKU Bartın OrmanFakultesi Dergisi vol 4 no 4 pp 1ndash16 2005

[2] J Domingo-Ferrer J Posegga F Sebe and V Torra ldquoAdvancesin smart cardsrdquo Computer Networks vol 51 no 9 pp 2219ndash2222 2007

[3] W Rankl and W Effing Smart Card Handbook John Wiley ampSons Chichester UK 3rd edition 2003

[4] C Liu P Yang Y Yeh and BWang ldquoThe impacts of smart cardson hospital information systemsmdashan investigation of the firstphase of the national health insurance smart card project inTaiwanrdquo International Journal ofMedical Informatics vol 75 no2 pp 173ndash181 2006

[5] K M Shelfer and J D Procaccino ldquoSmart card evolutionrdquoCommunications of the ACM vol 45 no 7 pp 83ndash88 2002

[6] Government Smart Card Handbook US General ServicesAdministration 2004 httpwwwsmartcardallianceorgreso-urcespdfsmartcardhandbookpdf

[7] W Rankl Smart Card Applications John Wiley amp Sons Chich-ester UK 2007

[8] Smart-Card Devices and Applications 2001 httpwwwnetca-ucusorgbooksegov2001pdfSmartCarpdf

[9] httpwwwmmicoinsmartcardsAboutSmartCardpdf[10] United States General Accounting Office ldquoElectronic Govern-

ment progress in promoting adoption of smart card technol-ogyrdquo Tech Rep GAO-03-144 GAO 2003

[11] S Rogerson ldquoSmart card technologyrdquo IMIS Journal vol 8 no1 1998

[12] CHM Lee YWCheng andADepickere ldquoComparing smartcard adoption in Singapore and Australian universitiesrdquo Inter-national Journal of Human Computer Studies vol 58 no 3 pp307ndash325 2003

[13] A-C Cheng C-H Chen and C-Y Chen ldquoA fuzzy mul-tiple criteria comparison of technology forecasting methodsfor predicting the new materials developmentrdquo TechnologicalForecasting and Social Change vol 75 no 1 pp 131ndash141 2008

[14] A Azadeh and H R Izadbakhsh ldquoA multi-variatemulti-attribute approach for plant layout designrdquo International Journalof Industrial Engineering Theory Applications and Practice vol15 no 2 pp 143ndash154 2008

[15] Y-M Wang Y Luo and Z Hua ldquoOn the extent analysismethod for fuzzy AHP and its applicationsrdquo European Journalof Operational Research vol 186 no 2 pp 735ndash747 2008

[16] G N Serbest and O Vayvay ldquoSelection of the most suitabledistribution channel using fuzzy analytic hierarchy process inTurkeyrdquo International Journal of Logistics Systems and Manage-ment vol 4 no 5 pp 487ndash505 2008

[17] O Duran and J Aguilo ldquoComputer-aided machine-tool selec-tion based on a Fuzzy-AHP approachrdquo Expert Systems withApplications vol 34 no 3 pp 1787ndash1794 2008

[18] C KahramanU Cebeci andD Ruan ldquoMulti-attribute compar-ison of catering service companies using fuzzy AHP the case ofTurkeyrdquo International Journal of Production Economics vol 87no 2 pp 171ndash184 2004

[19] F T Bozbura A Beskese and C Kahraman ldquoPrioritizationof human capital measurement indicators using fuzzy AHPrdquoExpert Systems with Applications vol 32 no 4 pp 1100ndash11122007


[20] P J M van Laarhoven and W Pedryca ldquoA fuzzy extension ofSaatyrsquos priority theoryrdquo Fuzzy Sets and Systems vol 11 pp 229ndash241 1983

[21] M Dagdeviren and I Yuksel ldquoDeveloping a fuzzy analytichierarchy process (AHP) model for behavior-based safetymanagementrdquo Information Sciences vol 178 no 6 pp 1717ndash1733 2008

[22] J J Buckley ldquoFuzzy hierarchical analysisrdquo Fuzzy Sets and Sys-tems vol 17 no 1 pp 233ndash247 1985

[23] W Zhang Q Zhang and H Karimi ldquoSeeking the importantnodes of complex networks in product RampD team based onfuzzy AHP and TOPSISrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 327592 9 pages 2013

[24] W Zhang and Q Zhang ldquoMulti-stage evaluation and selectionin the formation process of complex creative solutionrdquo Qualityamp Quantity 2013

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

[26] C E Bozdag C Kahraman andD Ruan ldquoFuzzy group decisionmaking for selection among computer integrated manufactur-ing systemsrdquo Computers in Industry vol 51 no 1 pp 13ndash292003

[27] C Kahraman D Ruan and I Dogan ldquoFuzzy group decision-making for facility location selectionrdquo Information Sciences vol157 no 1ndash4 pp 135ndash153 2003

[28] S Onut T Efendigil and S S Kara ldquoA combined fuzzyMCDMapproach for selecting shopping center site an example fromIstanbul Turkeyrdquo Expert Systems with Applications vol 37 no3 pp 1973ndash1980 2010

[29] S Onut U M Tuzkaya and N Saadet ldquoMultiple criteria eval-uation of current energy resources for Turkish manufacturingindustryrdquo Energy Conversion and Management vol 49 no 6pp 1480ndash1492 2008

[30] T L Saaty and L G Vargas Decision Making with the AnalyticNetwork Process Economic Political Social and Technologi-cal Applications with Benefits Opportunities Costs and RisksSpringer Pittsburgh Pa USA 2006

[31] M Dagdeviren and I Yuksel ldquoPersonnel selection using ana-lytic network processrdquo Istanbul Ticaret Universitesi Fen BilimleriDergisi vol 6 pp 99ndash118 2007

[32] E E Karsak S Sozer and S E Alptekin ldquoProduct planningin quality function deployment using a combined analyticnetwork process and goal programming approachrdquo Computersand Industrial Engineering vol 44 no 1 pp 171ndash190 2002

[33] S H Chung A H I Lee and W L Pearn ldquoAnalytic net-work process (ANP) approach for product mix planning insemiconductor fabricatorrdquo International Journal of ProductionEconomics vol 96 no 1 pp 15ndash36 2005

[34] CW Chang CWu andH C Chen ldquoAnalytic network processdecision-making to assess slicing machine in terms of precisionand control wafer qualityrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 641ndash650 2009

[35] T L Saaty The Analytic Hierarchy Process McGraw-Hill NewYork NY USA 1980

[36] L M Meade and A Presley ldquoRampD project selection using theanalytic network processrdquo IEEE Transactions on EngineeringManagement vol 49 no 1 pp 59ndash66 2002

[37] T L Saaty Decision Making with Dependence and FeedbackTheAnalytic Network Process RWSPublications Pittsburgh PaUSA 1996

[38] L M Meade and J Sarkis ldquoAnalyzing organizational projectalternatives for agile manufacturing processes an analyti-cal network approachrdquo International Journal of ProductionResearch vol 37 no 2 pp 241ndash261 1999

[39] H Baslıgil ldquoThe fuzzy analytic hierarchy process for softwareselection problemsrdquo Journal of Engineering and Natural Sci-ences vol 3 pp 24ndash33 2005

[40] M Dagdeviren S Yavuz and N Kılınc ldquoWeapon selectionusing the AHP and TOPSIS methods under fuzzy environ-mentrdquo Expert Systems with Applications vol 36 no 4 pp 8143ndash8151 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Contact smart card

Chip

(a)

Contactlesschip

Antenna

Contactlesssmart card

(b)

Antenna

Chip

Combi card

Contacts

(c)

Figure 1 (a) Contact smart card (b) contactless smart card and (c) combi card

technologies like cell phones small but functional computersystems microchips managing and controlling huge mecha-nisms and database units are the examples of thisminimizingautomation Smart cardwould be included in thisminimizingautomation process

Smart card technology has evolved a long way sincethe idea of using plastic cards to carry microelectronicchips was patented back in 1968 by Dethloff and Grotrupp[2] A smart card is a credit-card sized plastic card withan embedded computer chip that can be memory or alsoinclude a microprocessor [3ndash5] A microprocessor chip canadd delete and otherwise manipulate information in itsmemory and hence offer complex data security schemes [4]Smart card technology can enable an organization to becomemore secure efficient and interoperable while deliveringstrong authentication and security identity managementdata management customer support and communications[6] The card body is usually more than just a carrier forthe chip module It also includes information for the userand card accepters and of course security elements forprotection against forgery The card elements are printingand labeling embossing hologram signature panel tactileelements magnetic stripe chip module and antenna [7]Smart cards the result of evolution of magnetic card in useof daily life have superior memory thanmagnetic cardsThisvast memory provides ability to execute lots of functionsand present complex result As a result of this condition adatabase for customer relationship management (CRM) andstatistical data network can be provided by installing all ofthe information of any customer Also it becomes easy toreach customer almost in every branch of business like healthautomotive and retail sector

22 Smart Card Types A smart-card chip communicateswith a reader by direct physical contact or by a radiofrequency (RF) signal depending on the system design [8]Smart cards are grouped into three classes for chip-to-reader communications which are (1) contact smart cards (2)contactless smart cards and (3) combi cards

221 Contact Smart Cards Contact smart cards are the mostpopular card-connection design and are used for both cardsizes and chip types Figure 1(a) [9] depicts the contact-type

card [8] Contact cards use an eight-pin contactmicromoduleto physically connect to the card reader [8]

222 Contactless Smart Cards Contactless smart cards asshown in Figure 1(b) [9] use an antenna with approximatelya 10-centimeter (cm) range to communicate with the readerThese credit-card sized memory-chip devices derive theirpower from an RF field generated by the card reader TheRF field also transfers information to and from the cardand card reader [8] Contactless cards are better suited forenvironments where quick interaction between the card andreader is required such as high-volume physical access [10]Employee identification badges issued by large companies forbuilding access are typically contactless smart cards [8]

223 Combination Smart Cards Multipurpose combinationsmart cards are a hybrid mix of the contact and contactlessdesigns [8] These cards obtain the two systems describedabove These cards contain all the properties of the othertwo types Figure 1(c) shows the combination smart card[9] They include the eight-pin contact for communicationwith a contact-type reader and also include an antenna forcommunicationwith an RF-type reader [8]With such a cardit becomes possible to access the same chip via a contactand contactless interface with a very high level of securityIt may incorporate two noncommunicating chipsmdashone foreach interfacemdashbut preferably has a single dual interface chipproviding the many advantages of a single e-purse singleoperating architecture and so forthThemass transportationand banking industries are expected to be the first to takeadvantage of this technology [9]

23 Advantages and Usage Areas of Smart Card TechnologiesSmart card automation process has lots of function suchas to prevent time lost to reach customer to control effi-ciency Working level becomes mobilized with smart cardsThis means workplace is not bounded to an only placeand customers are reached every time Briefly customerrelationship management becomes the focus of business andproduction process and its effectiveness increases Smart cardusage provides trust and prestige to company or companyrsquosinstitutional identity Smart cards are more secure thanmagnetic cardsThe highly developed software of smart cards






Payment cards












Electronic purses






Let 119883 = 1199091 1199092 119909


1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (1)




follows [16 18 19 21 25ndash28]


119878119894=

119898

sum

119895=1

119872119895

119892119894otimes [

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

(2)

To obtain sum119898

119895=1119872119895



119898

sum

119895=1

119872119895

119892119894= (

119898

sum

119895=1

119897119895

119898

sum

119895=1

119898119895

119898

sum

119895=1

119906119895) (3)


119894=1sum119898

119895=1119872119895


operation of 119872119895119892119894

(119895 = 1 2 119898) values such that

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894= (

119899

sum

119894=1

119897119894

119899

sum

119894=1

119898119894

119899

sum

119894=1

119906119894) (4)


[

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup119910ge119909

[min (1205831198721

(119909) 1205831198722

(119910))] (6)


119881 (1198722ge 1198721) = hgt (119872

1cap 1198722) = 1205831198722

(119889)

119881 (1198722ge 1198721) =

1 if1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(7)


1198721and 120583



M2M1

1

l2 m2 l1 u2 m1 u1d

V(M2 ge M1)


2



1ge 1198722)


1



119894(119894 = 1 2 119896) can

be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) and (119872 ge 119872

2) and and (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 3 119896

(8)

Assume that

1198891015840(119860119894) = min119881 (119878

119894ge 119878119896) (9)


1198821015840= (1198891015840(1198601) 1198891015840(1198601015840

2) 119889

1015840(119860119899))119879

(10)



119882 = (119889 (1198601) 119889 (119860

2) 119889 (119860

119899))119879

(11)









119894119895= 1119886119895119894 where 119886



119860119908 = 120582max119908 (12)








119896 119896 = 1 2 119899


1198961 1198901198962 119890

119896119898119896The local


119882 =

1198621

119862119870

119862119873

11989011

11989011198991

1198901198961

119890119896119899119896

1198901198731

119890119873119899119873

1198621



11989011



sdot sdot sdot 119890119896119899119896


sdot sdot sdot 119890119873119899119873

[[[[[[[

[

11988211



d

1198821198701

sdot sdot sdot 119882119870119870

sdot sdot sdot 119882119870119873

d

1198821198731

sdot sdot sdot 119882119873119870

sdot sdot sdot 119882119873119873

]]]]]]]

]

(13)


Wℎ= [

[

119868 0 0

w21

0 0

0 W32

119868

]

]

(14)

where w21





22


W119899= [

[

119868 0 0

w21

W22

0

0 W32

119868

]

]

(15)





Goal

Criteria

Alternatives

w21

W32

(a)

Goal

Criteria

Alternatives

w21

W22

W32

(b)





5 Case Study















Service level(SL)



S


F C SL



























(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Payment cards












Electronic purses






Let 119883 = 1199091 1199092 119909


1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (1)




follows [16 18 19 21 25ndash28]


119878119894=

119898

sum

119895=1

119872119895

119892119894otimes [

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

(2)

To obtain sum119898

119895=1119872119895



119898

sum

119895=1

119872119895

119892119894= (

119898

sum

119895=1

119897119895

119898

sum

119895=1

119898119895

119898

sum

119895=1

119906119895) (3)


119894=1sum119898

119895=1119872119895


operation of 119872119895119892119894

(119895 = 1 2 119898) values such that

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894= (

119899

sum

119894=1

119897119894

119899

sum

119894=1

119898119894

119899

sum

119894=1

119906119894) (4)


[

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup119910ge119909

[min (1205831198721

(119909) 1205831198722

(119910))] (6)


119881 (1198722ge 1198721) = hgt (119872

1cap 1198722) = 1205831198722

(119889)

119881 (1198722ge 1198721) =

1 if1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(7)


1198721and 120583



M2M1

1

l2 m2 l1 u2 m1 u1d

V(M2 ge M1)


2



1ge 1198722)


1



119894(119894 = 1 2 119896) can

be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) and (119872 ge 119872

2) and and (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 3 119896

(8)

Assume that

1198891015840(119860119894) = min119881 (119878

119894ge 119878119896) (9)


1198821015840= (1198891015840(1198601) 1198891015840(1198601015840

2) 119889

1015840(119860119899))119879

(10)



119882 = (119889 (1198601) 119889 (119860

2) 119889 (119860

119899))119879

(11)









119894119895= 1119886119895119894 where 119886



119860119908 = 120582max119908 (12)








119896 119896 = 1 2 119899


1198961 1198901198962 119890

119896119898119896The local


119882 =

1198621

119862119870

119862119873

11989011

11989011198991

1198901198961

119890119896119899119896

1198901198731

119890119873119899119873

1198621



11989011



sdot sdot sdot 119890119896119899119896


sdot sdot sdot 119890119873119899119873

[[[[[[[

[

11988211



d

1198821198701

sdot sdot sdot 119882119870119870

sdot sdot sdot 119882119870119873

d

1198821198731

sdot sdot sdot 119882119873119870

sdot sdot sdot 119882119873119873

]]]]]]]

]

(13)


Wℎ= [

[

119868 0 0

w21

0 0

0 W32

119868

]

]

(14)

where w21





22


W119899= [

[

119868 0 0

w21

W22

0

0 W32

119868

]

]

(15)





Goal

Criteria

Alternatives

w21

W32

(a)

Goal

Criteria

Alternatives

w21

W22

W32

(b)





5 Case Study















Service level(SL)



S


F C SL



























(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Electronic purses






Let 119883 = 1199091 1199092 119909


1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (1)




follows [16 18 19 21 25ndash28]


119878119894=

119898

sum

119895=1

119872119895

119892119894otimes [

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

(2)

To obtain sum119898

119895=1119872119895



119898

sum

119895=1

119872119895

119892119894= (

119898

sum

119895=1

119897119895

119898

sum

119895=1

119898119895

119898

sum

119895=1

119906119895) (3)


119894=1sum119898

119895=1119872119895


operation of 119872119895119892119894

(119895 = 1 2 119898) values such that

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894= (

119899

sum

119894=1

119897119894

119899

sum

119894=1

119898119894

119899

sum

119894=1

119906119894) (4)


[

[

119899

sum

119894=1

119898

sum

119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup119910ge119909

[min (1205831198721

(119909) 1205831198722

(119910))] (6)


119881 (1198722ge 1198721) = hgt (119872

1cap 1198722) = 1205831198722

(119889)

119881 (1198722ge 1198721) =

1 if1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(7)


1198721and 120583



M2M1

1

l2 m2 l1 u2 m1 u1d

V(M2 ge M1)


2



1ge 1198722)


1



119894(119894 = 1 2 119896) can

be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) and (119872 ge 119872

2) and and (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 3 119896

(8)

Assume that

1198891015840(119860119894) = min119881 (119878

119894ge 119878119896) (9)


1198821015840= (1198891015840(1198601) 1198891015840(1198601015840

2) 119889

1015840(119860119899))119879

(10)



119882 = (119889 (1198601) 119889 (119860

2) 119889 (119860

119899))119879

(11)









119894119895= 1119886119895119894 where 119886



119860119908 = 120582max119908 (12)








119896 119896 = 1 2 119899


1198961 1198901198962 119890

119896119898119896The local


119882 =

1198621

119862119870

119862119873

11989011

11989011198991

1198901198961

119890119896119899119896

1198901198731

119890119873119899119873

1198621



11989011



sdot sdot sdot 119890119896119899119896


sdot sdot sdot 119890119873119899119873

[[[[[[[

[

11988211



d

1198821198701

sdot sdot sdot 119882119870119870

sdot sdot sdot 119882119870119873

d

1198821198731

sdot sdot sdot 119882119873119870

sdot sdot sdot 119882119873119873

]]]]]]]

]

(13)


Wℎ= [

[

119868 0 0

w21

0 0

0 W32

119868

]

]

(14)

where w21





22


W119899= [

[

119868 0 0

w21

W22

0

0 W32

119868

]

]

(15)





Goal

Criteria

Alternatives

w21

W32

(a)

Goal

Criteria

Alternatives

w21

W22

W32

(b)





5 Case Study















Service level(SL)



S


F C SL



























(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M2M1

1

l2 m2 l1 u2 m1 u1d

V(M2 ge M1)


2



1ge 1198722)


1



119894(119894 = 1 2 119896) can

be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) and (119872 ge 119872

2) and and (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 3 119896

(8)

Assume that

1198891015840(119860119894) = min119881 (119878

119894ge 119878119896) (9)


1198821015840= (1198891015840(1198601) 1198891015840(1198601015840

2) 119889

1015840(119860119899))119879

(10)



119882 = (119889 (1198601) 119889 (119860

2) 119889 (119860

119899))119879

(11)









119894119895= 1119886119895119894 where 119886



119860119908 = 120582max119908 (12)








119896 119896 = 1 2 119899


1198961 1198901198962 119890

119896119898119896The local


119882 =

1198621

119862119870

119862119873

11989011

11989011198991

1198901198961

119890119896119899119896

1198901198731

119890119873119899119873

1198621



11989011



sdot sdot sdot 119890119896119899119896


sdot sdot sdot 119890119873119899119873

[[[[[[[

[

11988211



d

1198821198701

sdot sdot sdot 119882119870119870

sdot sdot sdot 119882119870119873

d

1198821198731

sdot sdot sdot 119882119873119870

sdot sdot sdot 119882119873119873

]]]]]]]

]

(13)


Wℎ= [

[

119868 0 0

w21

0 0

0 W32

119868

]

]

(14)

where w21





22


W119899= [

[

119868 0 0

w21

W22

0

0 W32

119868

]

]

(15)





Goal

Criteria

Alternatives

w21

W32

(a)

Goal

Criteria

Alternatives

w21

W22

W32

(b)





5 Case Study















Service level(SL)



S


F C SL



























(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119896 119896 = 1 2 119899


1198961 1198901198962 119890

119896119898119896The local


119882 =

1198621

119862119870

119862119873

11989011

11989011198991

1198901198961

119890119896119899119896

1198901198731

119890119873119899119873

1198621



11989011



sdot sdot sdot 119890119896119899119896


sdot sdot sdot 119890119873119899119873

[[[[[[[

[

11988211



d

1198821198701

sdot sdot sdot 119882119870119870

sdot sdot sdot 119882119870119873

d

1198821198731

sdot sdot sdot 119882119873119870

sdot sdot sdot 119882119873119873

]]]]]]]

]

(13)


Wℎ= [

[

119868 0 0

w21

0 0

0 W32

119868

]

]

(14)

where w21





22


W119899= [

[

119868 0 0

w21

W22

0

0 W32

119868

]

]

(15)





Goal

Criteria

Alternatives

w21

W32

(a)

Goal

Criteria

Alternatives

w21

W22

W32

(b)





5 Case Study















Service level(SL)



S


F C SL



























(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Goal

Criteria

Alternatives

w21

W32

(a)

Goal

Criteria

Alternatives

w21

W22

W32

(b)





5 Case Study















Service level(SL)



S


F C SL



























(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Service level(SL)



S


F C SL



























(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















(23 1 32)Weak

(1 1 1)Equal

(23 1 32)Weak







F (0358 0437 0562) (1000 1000 1000) (1500 2000 2500) (0400 0500 0667)

C (0319 0382 0474) (0400 0500 0667) (1000 1000 1000) (0400 0500 0667)

SL (0358 0437 0562) (1500 2000 2500) (1500 2000 2500) (1000 1000 1000)





(3radic(

3

2times

3

2times

5


3radic(

5

2times

5

2times

7

2))

= (1500 2289 2797)

(16)




(1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797) = (6387 8199 9724)

((1 1 1) oplus (1500 2289 2797) oplus (2109 2621 3130)

oplus (1778 2289 2797))

+ ((0358 0437 0562) oplus (1 1 1) oplus (1500 2000 2500)


+ ((0319 0382 0474) oplus (0400 0500 0667)

oplus (1 1 1) oplus (0400 0500 0667))

+ ((0358 0437 0562) oplus (1500 2000 2500)

oplus (1500 2000 2500) oplus (1 1 1))

= (16122 19960 23823)

1198781= (6387 8199 9724) otimes (16122 19960 23823)

minus1

1198781= (

6387

238238199

199609724

16122)

1198781= (0268 0410 0603)

(17)




1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198781

1198782

1198783

1198784

(1) Matrix (0268 0410 0603) (0137 0197 0293) (0089 0119 0174) (0183 0272 0407)

(2) Matrix (0142 0228 0344) (0094 0142 0232) (0237 0371 0579) (0159 0258 0426)

(3) Matrix (0224 0360 0602) (0206 0331 0563) (0197 0309 0429)

(4) Matrix (0430 0613 0859) (0296 0387 0518)

(5) Matrix (0198 0310 0488) (0255 0413 0652) (0264 0278 0439)

(6) Matrix (0396 0545 0742) (0125 0158 0209) (0212 0297 0413)

(7) Matrix (0362 0515 0721) (0131 0167 0223) (0225 0318 0450)

(8) Matrix (0395 0545 0826) (0128 0163 0244) (0135 0292 0449)

(9) Matrix (0207 0507 0901) (0128 0164 0279) (0227 0328 0601)

(10) Matrix (0344 0519 0751) (0164 0243 0376) (0161 0238 0365)

(11) Matrix (0323 0492 0721) (0189 0279 0421) (0160 0229 0347)

(12) Matrix (0359 0513 0718) (0220 0312 0439) (0136 0176 0242)

(13) Matrix (0141 0151 0197) (0501 0575 0764) (0233 0274 0372)

(14) Matrix (0130 0171 0247) (0415 0569 0775) (0180 0261 0369)

(15) Matrix (0430 0575 0764) (0200 0274 0372) (0121 0151 0197)

(16) Matrix (0368 0545 0766) (0222 0300 0423) (0123 0155 0215)

(17) Matrix (0413 0545 0753) (0206 0293 0392) (0123 0162 0203)


















Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Consider matrix

119881 (1198781gt 1198782) = 1000

119881 (1198781gt 1198783) = 1000

119881 (1198781gt 1198784) = 1000

119881(1198781gt 1198782 1198783 1198784)

= (1000 1000 1000)

min (1000 1000 1000) = 1000

119881 (1198782gt 1198781) = 0105

119881 (1198782gt 1198783) = 1000

119881 (1198782gt 1198784) = 0595

119881(1198782gt 1198781 1198783 1198784)

= (0105 1000 0595)

min (0105 1000 0595) = 0105

119881 (1198783gt 1198781) = 0000

119881 (1198783gt 1198782) = 0323

119881 (1198783gt 1198784) = 0000

119881(1198783gt 1198781 1198782 1198784)

= (0000 0323 0000)

min (0000 0323 0000) = 0000

119881 (1198784gt 1198781) = 0502

119881 (1198784gt 1198782) = 1000

119881 (1198784gt 1198783) = 1000

119881(1198784gt 1198781 1198782 1198783)

= (0500 1000 1000)

min (0502 1000 1000) = 0502

1198821015840= (1000 0105 0000 0502)

(18)


119882 = (0622 0065 0000 0312) (19)




















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















6 Conclusion





080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











080

070

075

060

065

050

055

040

045

030

035

020

010

015

025

000

005

AC

TU

OB

Fuzzy AHP ANP07740

01110

01150

06596

01650

01754







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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