research article fuzzy approach in ranking of banks...

Research ArticleFuzzy Approach in Ranking of Banks according toFinancial Performances

Milena JakšiT1 Slaviša MoljeviT2 Aleksandar AleksiT3 Mirjana Misita4 Slavko Arsovski3

Danijela TadiT3 and Predrag MimoviT1

1Faculty of Economics University of Kragujevac Đure Pucara Starog 3 34000 Kragujevac Serbia2Faculty of Mechanical Engineering University of East Sarajevo Vuka Karadzica 30 71123 Lukavica Bosnia and Herzegovina3Faculty of Engineering University of Kragujevac Sestre Janjic 6 34000 Kragujevac Serbia4Faculty of Mechanical Engineering University of Belgrade Kraljice Marije 16 11000 Belgrade Serbia

Correspondence should be addressed to Slavko Arsovski cqmkgacrs

Received 16 March 2016 Accepted 16 May 2016

Academic Editor Kishin Sadarangani

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

Evaluating bank performance on a yearly basis and making comparison among banks in certain time intervals provide aninsight into general financial state of banks and their relative position with respect to the environment (creditors investors andstakeholders)The aimof this study is to propose a new fuzzymulticriteriamodel to evaluate banks respecting relative importance offinancial performances and their values The relative importance of each pair of financial performance groups is assessed linguisticexpressions which are modeled by triangular fuzzy numbers Fuzzy Analytic Hierarchical Process (FAHP) is applied to determinerelative weights of the financial performances In order to rank the treated banks new model based on Fuzzy Technique for OrderPerformance by Similarity to Ideal Solution (FTOPSIS) is deployed The proposed model is illustrated by an example giving reallife data from 12 banks having 80 share of the Serbian market In order to verify the proposed FTOPSIS different measures ofseparation are used The presented solution enables the ranking of banks gives an insight of bankrsquos state to stakeholders andprovides base for successful improvement in a field of strategy quality in bank business

1 Introduction

In an economic theory and practice role of banks can bedefined asmediator between the oneswith capital surplus andones in need of capital A modern competitive environmentglobalization and an unstable financial market have createda wave of important changes in banking sector of eachcountry The problems that can occur in banking sector [1]directly influence the stakeholders as well as the generaleconomy Respecting this fact it is very important to ensureand maintain performances of banks in such manner thatbanks can contribute in sustainable development of eacheconomy country and overcoming crisis Assessment ofperformances may be considered as a significant issue indifferent organizations on the level of different processesand activities [2] An assessment of bankrsquos performances as

well as final analysis needed by both internal and externalbeneficiaries of information is very significant for differentstakeholders This is a reason that considered problem hasbecome a topic of research in the last decades

Improving of bank business strategy can be achieved bymanaging bank performances financial and nonfinancialAnalysis of financial performances (FPs) is in its base ratioanalysis which uses financial reports They give informationon capital capital resources income disbursement gainedresults and cash flow Many researchers suggest that nonfi-nancial performances can be reflected on financial tables inthe long term and lead to a more realistic evaluation of abankrsquos financial standing [3] In this paper FPs and earningcapacity of banks are taken into consideration

The purpose of this study is to propose fuzzymodel basedon FPs for evaluation bank operating that is determining

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6169586 11 pageshttpdxdoiorg10115520166169586

2 Mathematical Problems in Engineering

bank performance or financial standingTheory and practiceoffer the whole set of techniques and methods of banks eval-uation As is known it is impossible to manage what cannotbe measured However measurement of bank performancesis the most important activity of bank management team

As changes in the business environment especially infinancial market are rapid and continuous relative impor-tance of FPs cannot be expressed with an exact numericalvalue It is easier for decision makers to express the sub-jectivity and imprecision of their assessments by linguisticexpressions The concept of linguistic variables is introducedby [4] it is very useful in dealing with situations which aretoo complex or not well defined to be reasonably describedin conventional quantitative expressions [5] These linguisticterms can be modeled by using the fuzzy sets theory [5 6]which allows the available information to be represented withthe granularity [7] As it is known processing of informationgranules (complex information entities) during the process ofdata abstraction and derivation of knowledge from input datais defined as granular computing [7]

The fuzzy set theory provides a strict mathematicalframework in which vague conceptual phenomena can beprecisely and rigorously studied [5] To outperform compet-ing bank institutions it requires more emphasis on internaloperational performancesThis means that it is imperative todevelop an effective way to conduct performance evaluationswhich can measure the overall organizational performanceand link it to the corporate goals

The considered problem can be stated as a multicriteriadecision making problem (MCDM) under uncertaintiesMCDM refers to finding the best opinion out of all feasiblealternatives in the presence of multiple usually conflictingdecision criteria The necessity of applying MCDM modelsin the field of financial decision making is supported by thefact that over the past decades the complexity of financialdecisions has increased rapidly [8]Themost commonly usedMCDM methods for ranking problems in various domainsare AHP [9] TOPSIS [10] and the combination of the twoMCDMmethods These two methods are used in this paper

The paper is organized as follows Section 2 presents theliterature review The basic definitions of the fuzzy set theoryand modeling of uncertainties in the relative importanceof FP groups are presented in Section 3 In Section 4 theproposed fuzzy TOPSIS approach is introduced Section 5represents the application of the recommended algorithmwith real-life data Finally Section 6 sets the conclusions

2 Literature Review

There are a large number of papers found in literature real-ized by different methods to measure of bankrsquos performancesand ranking of banksThe conventionalAnalyticHierarchicalProcess (AHP) [9] is the most widely used method Makingstrategy decisions in a bank with applying both basic andadjustedAHP applicationmodels is very significant issue [11]In that manner AHP application with specific reference tobanking could be used in the finance sector [12] In bankingsector usage of AHP application is growing especially in thesituation of global financial crisis Many authors analyzed

the bank performances using AHP with the financial andnonfinancial performance criteria [13]

The use of conventional AHPwith discrete scale is simpleand easy but it is not sufficient considering uncertaintyassociated with the mapping of onersquos perception to a number[14] Decision makers express their judgments far better byusing linguistic expressions than by representing them interms of precise numbers These linguistic expressions aremodeled by using fuzzy sets theory [5 6] In other wordsthe relative importance of criteria and the preference ofalternatives under each criterion are stated by fuzzy pairwisecomparison matrices FAHP enables mapping human per-ception to a particular number or a ratio and also consideringthe vagueness in the decision making process

There are a vast number of papers to be found in literaturewhich use fuzzy AHP (FAHP) method for evaluation andranking of banks The evaluation and rank of Turkish bankswith respect to many financial and nonfinancial performancecriteria are performed by using FAHP [15] It could beassumed that increased competition among banks and theliberalization of policies helped many institutions to take upthe banking business [16] The rank of banks can be obtainedby applying FAHP

Some researchers solve decision making problems infinancial sector especially in the field of banking by usingTOPSIS method Prioritizing different kinds of accounts inbanks objectives alternatives and criteria can be establishedfrom literature review and judgements of bankmanagersTheTOPSISfuzzy TOPSIS [17ndash19] are widely used for ranking ofentities in different research areas [15 20 21] A brief literaturereview of [22ndash27] is presented as follows

In the analyzed papers the relative importance ofattribute is assessed by decision makers that use predefinedlinguistic expressions Modeling of these linguistic expres-sions is based on the fuzzy sets theory Some authors [1718 25 28] suggest that the relative importance of attributecan be obtained by direct way In these papers determiningweight of vectors is stated as fuzzy group decision makingproblem In [17 18 25] it is assumed that decision makershave equal importance so that the aggregated weight of eachattribute can be obtained by using fuzzy averaging methodIn [22] a new aggregation procedure is developed Thisprocedure is applied for calculating of aggregated weights ofattributes in [23 28] In the rest of the analyzed papers ratingrelative importance of attributes should be based on the AHPframework It appears that the determining of the relativeimportance of attributes is more reliable when they areobtained by using pairwise comparisons than when they aredirectly obtained It is easier to make a comparison betweentwo criteria than to make an overall weight assignment Theweights of considered attributes may be calculated by usingthe different procedure In [18] the weights of treated criteriaare obtained as the average of the elements of each rowfrom the constructed fuzzy pairwise comparison matrix ofthe relative importance of attribute The geometric mean wasused in [24] for calculating relative importance of treatedattribute which is proposed in [29] In the rest of mentionedpapers the vectors weights of considered attributes are given

Mathematical Problems in Engineering 3

by using approach for handling of FAHP which is developedin [30] The obtained values of vectors weight are not fuzzynumbers [31 32]

If the values of attributes for considered alternative canbe obtained by direct measuring then these values may bedescribed by precise numbersThis assumption is introducedin [15 26] If there is a need simultaneously both crisp anduncertain criteria in the considered problem [23] may beanalyzed Under uncertain environment it is relatively dif-ficult for decision makers to provide exact precise numericalattribute values for treated alternative Because of that in therest of analyzed paper these values are described by linguisticexpressions These linguistic expressions are modeled by (a)triangular fuzzy numbers (TFNs) in [18 20 21 25 27] and(b) trapezoidal fuzzy numbers (TRFNs) in [17 23 24 28]The domains of defined TFNs and TrFNs belong to differentinterval such as [0-1] (analogy by [18 24]) [0ndash10] (analogyby [17 20 28]) and [1ndash10] (analogy by [25]) [1ndash9] (analogyby [21 23 27])

As criteria can be benefit and cost type and values ofdecision matrix can be introduced by different units itis necessary to perform their normalization By using thenormalization procedure the values of decision matrix aremapped into interval [0-1] In this way these values canbe compared Crisp values of alternatives are normalized byusing vector normalization procedure in [15 26] and linearnormalization procedure in [23] Normalization proceduresare presented in [33] In all analyzed papers the normalizeduncertain values are obtained by using linear normalizationprocedure which is proposed in [34]

Identification of the set Positive Ideal Solutions (PISs) andthe Negative Ideal Solutions (NISs) is performed in [15 26]according to the procedure which is given in conventionalTOPSIS [10] The most likely used technique for definingFuzzy Positive Ideal Solution (FPIS) and FuzzyNegative IdealSolution (FNIS) is proposed in [35] and it is used in [19 24]Authors [17] have developed a new approach for determiningFPIS and FNIS In the rest of analyzed papers [18 20 21 2325 27 28] FPIS and FNIS are determined in compliance withthe method for comparison of fuzzy numbers

Calculating separation measure of each alternative fromPIS and NIS is given by using n-dimensional Euclideandistance in [15 26] The separation measures of each alter-native from FPIS and FNIS are obtained by using (a) vertexmethod [22] applied in [19 20 25 28] (b) the proposedexpression for the distance between two fuzzy numbers [36]applied in [18 21 23 24] a new approach in [27] and (c)a novel fuzzy distance measure which has been analyzedin [37] and applied in [17] The total separate measureof each alternative from PISFPIS and NISFNIS can becalculated as sum of all obtained separate measures fromPISFPIS and NISFNIS Determining closeness to the idealsolution is based on the total separate measures The rank ofalternative corresponds to closeness coefficientThe selectionof existing or development of new approaches for calculationof separation measures have become important because ofthe importance in choosing the best rankings

Table 1 The FPs and identified KPIs

Liquidity of a bank (119895 = 1) The first liquidity ratio (119896 = 1)The second liquidity ratio (119896 = 2)

Financial structure (119895 = 2)Leverage ratio 1 (119896 = 2)Leverage ratio 2 (119896 = 4)Leverage ratio 3 (119896 = 5)

Efficiency (119895 = 3)

The first ratio of interest nonbearingcosts (119896 = 6)

The second ratio of interestnonbearing costs (119896 = 7)

The first ratio of interest nonbearingrevenue (119896 = 8)

The second ratio of interestnonbearing revenue (119896 = 9)

Profitability (119895 = 4)

Assets utilization indicator (119896 = 10)return on operating revenue (119896 = 11)

Return on assets (119896 = 12)Share multiplier (119896 = 13)Return on equity (119896 = 14)

Market position (119895 = 5)

Total book value (119896 = 15)Total capital (119896 = 16)

Total placement (119896 = 17)Bank obligations (119896 = 18)

Solvency of a bank (119895 = 6) Capital adequacy ratio (119896 = 19)

3 Modeling of the Relative Importance of FPs

Rating of the relative importance of FPs over time is basedon uncertain and imprecise knowledge by financial expertsfrom banking and economy sector Relative importance ofthe considered FPs does not depend on banks and it rarelychanges They involve a high degree of subjective judgmentsknowledge and experience of financial experts It can beassumed that it best suits the nature of human decision tomake a comparison between two FPs than to make an overallrelative importance assignmentThe financial experts expresstheir judgments far better by using linguistic expressions thanby representing them in terms of precise numbers In thispaper the fuzzy ratings of financial experts are describedby the linguistic expression modeled by fuzzy sets [5] Afuzzy set is represented by its membership function whichcan be obtained by using different approaches Howeversubjectivity in determining the membership function hasbeen considered as the weakest point in the fuzzy sets theory

In the papers which can be found in the literaturetriangular and trapezoidal functions are widely used becauseof the fact that they offer a good compromise betweendescriptive power and computational simplicity It is shownthat fuzzy sets of higher types and levels have not yet playeda significant role in the applications of the fuzzy sets theory[38]

In this paper AHP framework is used to state relativeimportance of FPs It is assumed that financial experts assessconsidering variables using predefined linguistic expressionswhich are modeled by triangular fuzzy numbers (TFNs)1198951198951015840 = (119909 1198971198951198951015840 1198981198951198951015840 1199061198951198951015840) with the lower and upper bounds1198971198951198951015840 1199061198951198951015840 and modal value1198981198951198951015840 respectively


Table 2 KPI values for treated banks

119896 = 1 119896 = 2 119896 = 3 119896 = 4 119896 = 5 119896 = 6 119896 = 7 119896 = 8 119896 = 9

119894 = 1 0099 0158 0162 0187 0266 0808 0704 0567 0127119894 = 2 0075 0120 0248 0268 0398 0763 0521 0429 0070119894 = 3 0063 0093 0420 0490 0618 0342 0051 0050 0008119894 = 4 0110 01375 0140 0227 0175 0842 0769 0660 0134119894 = 5 0135 0206 0493 0609 0753 0738 0696 0401 0050119894 = 6 0095 0155 0226 0250 0370 0712 0652 0417 0076119894 = 7 0105 0152 0280 0315 0402 0938 0876 0809 0495119894 = 8 0082 0216 0181 0197 0479 0807 0734 0650 0154119894 = 9 0081 0167 0110 0117 0220 0626 0396 0655 0158119894 = 10 0133 0214 0217 0250 0350 0895 0801 0768 0283119894 = 11 0147 0654 0221 0124 0355 0549 0456 0217 0021119894 = 12 0101 0182 0256 0285 0542 0864 0749 0636 0153

119896 = 10 119896 = 11 119896 = 12 119896 = 13 119896 = 14 119896 = 15 119896 = 16 119896 = 17 119896 = 18 119896 = 19

119894 = 1 0118 0024 8796 0211 0225 2824 496 2353 1548 19119894 = 2 0244 0040 5858 0234 0164 1816 497 1498 873 21119894 = 3 0427 0066 3607 0238 0156 942 384 825 471 38119894 = 4 0087 0016 10014 0163 0202 1928 287 1490 1232 14119894 = 5 0057 0007 5308 0037 0124 979 246 690 513 28119894 = 6 0088 0015 3515 0053 0171 658 167 494 278 35119894 = 7 0064 0033 394 0144 0611 1400 403 1157 871 16119894 = 8 0076 0019 5516 0157 0237 1291 298 1052 577 24119894 = 9 0041 0011 10126 0098 0235 731 98 588 523 18119894 = 10 0096 0032 6670 0215 0368 1010 233 773 536 17119894 = 11 0161 0014 4549 0065 0096 702 167 519 418 20119894 = 12 0123 0030 5574 0157 0240 811 210 77 339 16

Table 3

Thenormalizeddecisionmatrix

119895 = 1 119895 = 2 119895 = 3 119895 = 4 119895 = 5 119895 = 6

119894 = 1 0228 0175 0196 0279 0433 0235119894 = 2 0173 0259 0166 0277 0344 0259119894 = 3 0141 0439 0388 0311 0265 0469119894 = 4 0231 0159 0207 0275 0289 0173119894 = 5 0305 0531 0151 0233 0220 0346119894 = 6 0221 0240 0167 0176 0232 0432119894 = 7 0233 0287 0361 0266 0279 0198119894 = 8 0238 0231 0216 0228 0264 0296119894 = 9 0208 0124 0249 0270 0166 0224119894 = 10 0307 0232 0272 0275 0221 0210119894 = 11 0583 0196 0148 0203 0193 0247119894 = 12 0245 0299 0210 0229 0199 0198

If the strong relative importance of FP 1198951015840 over process 119895

holds then the pairwise comparison scale can be representedby TFN 1198951198951015840 = (1198951015840119895)

minus1= (11199061198951198951015840 11198981198951198951015840 11198971198951198951015840)

If 119895 = 1198951015840 (119895 1198951015840 = 1 119869) then the relative importance of

FP 119895 over FP 1198951015840 is represented by a single point 1 which is TFN(1 1 1)

Granularity is defined as the number of TFNs assigned tothe relative importance of FPs In this paper with respect tothe type and size of the considered problem and opinion of[39] it is assumed that the five linguistic expressions at themost were assigned to the existing linguistic variables

As it is known there are different locations in the universeof discourse In this case values in the domain of these TFNsbelong to a real set within the interval [1ndash5]

Fuzzy rating of financial experts for each pair of treatedFPs is described by linguistic expressions which can berepresented as TFNs

Very low importance 1 = (119909 1 1 2)Low importance 2 = (119909 1 2 3)Moderate importance 3 = (119909 15 3 45)High importance 4 = (119909 3 4 5)Highest importance 5 = (119909 4 5 5)

The vectors weight of FPs is calculated by approachfor handling FAHP which is developed in [29] The usedapproach has number of advantages over the traditionalFAHP [30] which is used in the above analyzed papersSome advantages are allowing more reasonable descriptionof the decision making process and reflecting the thinkingstyle of human extent analysis method cannot estimate thetrue weights from a fuzzy comparison matrix and has led


Table 4 The fuzzy decision matrix

119895 = 1 119895 = 2 119895 = 3

119894 = 1 (0036 0068 0126) (0021 0036 0069) (0024 0049 0090)119894 = 2 (0027 0052 0096) (0031 0053 0102) (0021 0041 0076)119894 = 3 (0022 0042 0078) (0052 0090 0173) (0048 0097 0178)119894 = 4 (0037 0069 0128) (0019 0033 0063) (0026 0052 0095)119894 = 5 (0048 0091 0169) (0063 0109 0210) (0019 0038 0069)119894 = 6 (0035 0066 0122) (0028 0049 0095) (0021 0042 0077)119894 = 7 (0037 0070 0129 ) (0034 0059 0113) (0045 0090 0166)119894 = 8 (0038 0071 0132) (0027 0047 0091) (0027 0054 0099)119894 = 9 (0033 0062 0115) (0015 0025 0049) (0031 0062 0114)119894 = 10 (0049 0092 0169) (0027 0048 0092) (0034 0068 0125)119894 = 11 (0093 0175 0322) (0023 0040 0077) (0018 0037 0068)119894 = 12 (0039 0073 0135) (0035 0061 0118) (0026 0052 0097)V+119895 (0093 0175 0322) (0063 0109 0210) (0048 0097 0178)Vminus119895 (0022 0042 0078) (0015 0025 0049) (0018 0037 0068)

119895 = 4 119895 = 5 119895 = 6

119894 = 1 (0010 0018 0038) (0014 0024 0032) (0013 0030 0059)119894 = 2 (0010 0017 0037) (0011 0019 0034) (0014 0033 0065)119894 = 3 (0012 0019 0042) (0009 0015 0026) (0027 0059 0118)119894 = 4 (0010 0017 0037) (0010 0016 0028) (0010 0022 0044)119894 = 5 (0009 0015 0031) (0007 0012 0022) (0020 0044 0087)119894 = 6 (0007 0011 0024) (0008 0013 0023) (0025 0054 0109)119894 = 7 (0010 0017 0036) (0009 0016 0027) (0011 0025 0050)119894 = 8 (0008 0014 0031) (0009 0015 0026) (0017 0037 0075)119894 = 9 (0010 0017 0036) (0005 0009 0016) (0013 0028 0056)119894 = 10 (0010 0017 0037) (0007 0012 0022) (0012 0026 0053)119894 = 11 (0008 0013 0027) (0006 0011 0019) (0014 0031 0062)119894 = 12 (0008 0014 0031) (0006 0011 0019) (0011 0025 0050)V+119895 (0012 0019 0042) (0014 0024 0032) (0027 0059 0118)Vminus119895 (0007 0011 0024) (0005 0009 0016) (0010 0022 0044)

to quite a number of misapplications which may lead to awrong decision to be made and fuzzy matrices cannot beconstructed [39]

4 Proposed Methodology

The issue of bank ranking is very significant for top man-agement of each bank and for enterprises that have businessrelations with analyzed banks too [40]

The banking sector priorities were improving the capitalbase time and exchange restructuring of funds with place-ment structure and efficient management of both the levelof bank exposure to possible risks and the quality of investedfunds

The conditions for harmonizing regulative with BaselII standards were obtained as a consequence of marketliberalization setting up the financial stability of the systembanking industry restructuring and its capital strengtheningadvancing supervision function and introducing interna-tional accounting standards However it is important to pointout that it requires time not only for regulative harmoniza-tion but also for harmonizing home banksrsquo practice with

qualitative and quantitative qualification standards for capitalmodels application and its dynamics

According to the real practice data it is known that themost significant business goals of enterprises may be realizedonly if there is solid cooperation between enterprises andbanks In that manner the selection of business bank isone of the most significant tasks of top management in anyenterprise This decision can be made by top managers withrespect to the rank of considered banks

In this paper a new integrated fuzzy multicriteria modelfor ranking of banks is proposed Since the rank of banksis obtained by the exact way the obtained solution is lessburdened by subjective attitudes of decisionmakers so itmaybe assumed that this assumption increases the correctness ofsolution

The considered banks are formally presented by set 119868 =

1 119894 119868 The index for a bank is denoted as 119894 and 119868 isthe total number of considered banks

In general FPs are defined according to the literature andresults of good practice Formally the FPs are presented byset of indices 119869 = 1 119895 119869 where 119895 is index for FP and119869 is the total number of treated FPs


Table 5 Comparative results of different distance measurements

Methods Banks 119888119894 Rank Methods Banks 119888119894 Rank

[22]

119894 = 1 0222 9

[43]

119894 = 1 0224 9119894 = 2 0211 10 119894 = 2 0205 10119894 = 3 0533 1 119894 = 3 0539 2119894 = 4 0184 11 119894 = 4 0186 11119894 = 5 0492 2 119894 = 5 0491 3119894 = 6 0271 8 119894 = 6 0269 8119894 = 7 0387 4 119894 = 7 0431 4119894 = 8 0278 6 119894 = 8 0282 6119894 = 9 0171 12 119894 = 9 0169 12119894 = 10 0349 5 119894 = 10 0344 5119894 = 11 0475 3 119894 = 11 0544 1119894 = 12 0272 7 119894 = 12 0271 7

Themethodbased onthehammingdistance[44]

119894 = 1 0225 9

The methodbased on theEuclideandistance[44]

119894 = 1 0194 10119894 = 2 0207 10 119894 = 2 0198 9119894 = 3 0532 1 119894 = 3 0423 3119894 = 4 0186 11 119894 = 4 0186 11119894 = 5 0491 2 119894 = 5 0496 2119894 = 6 0270 7-8 119894 = 6 0264 7119894 = 7 0387 4 119894 = 7 0362 4119894 = 8 0277 6 119894 = 8 0249 8119894 = 9 0171 12 119894 = 9 0181 12119894 = 10 0331 5 119894 = 10 0342 5119894 = 11 0476 3 119894 = 11 0576 1119894 = 12 0270 7-8 119894 = 12 0286 6

The fuzzy ratings of the relative importance of each pairof FPs are performed by financial experts It may be assumedthat managers of banking sector and main managers ofenterprises have the same opinion of FPsrsquo importance Thisassumption is based on the fact that bank managers striveto have increasing number of clients so they could put moremoney on market On the other hand enterprise managersstrive to decrease the risk of realization of business goalswhich may be achieved by selection of the most appropriatebank

The evaluation of FPs can be performed through prede-fined key performance indicators (KPIs) which are presentedby the set of indices 120581119895 = 1 119896 119870119895 The total numberof KPIs of FP 119895 119895 = 1 119869 is 119870119895 and 119896 is the index ofKPI The KPI values are determined according to financialreports The KPIs values are given from reasonable financialreport In this paper treated FIPs can be benefit type andcost type These values have different units of measures Byusing vector normalization procedure [33] the KPI values aremapped into interval [0-1] The normalized value of each FP119895 119895 = 1 119869 is calculated by using averaging operator

Calculating of the elements of fuzzy decision matrix isbased on using fuzzy algebra rules [5 6] These elements aregiven as product of vector weights of FPs and matrix of thesenormalized values

FPIS and FNIS are determined according to the weightednormalized fuzzy decision matrix by using method forcomparison of fuzzy numbers in [41 42]

The distance from FPIS and FNIS is determined byusing several different approaches presented in the literatureWe considered separation measures which are proposed in[43 44] and are based on hamming distance and Euclideandistance These separation measures are applied for TFNs inthe proposed model In this respect authors can cross-checkthe results and obtain the most representative ranking

The closeness coefficient for each bank is calculated byusing procedure of conventional TOPSIS [10] Ranking orderof all banks can be determined by using closeness coefficient

41 The Proposed Algorithm The algorithm of the proposedfuzzy TOPSIS method is carried out in the following steps

Step 1 Calculate the vectors weight of FPs by applyingmodified procedure which is developed in [15]

Order

120572119895 =[

[

119869

prod

119895=1

1198971198951198951015840]

]

1119869

120573119895 =[

[

119869

prod

119895=1

1198981198951198951015840]

]

1119869

120594119895 =[

[

119869

prod

119895=1

1199061198951198951015840]

]

1119869

119895 = 1 119869

120572 =

119869

sum

119895=1

120572119895

120573 =

119869

sum

119895=1

120573119895

120594 =

119869

sum

119895=1

120594119895

119895 = 1 119869

(1)

Then the vectors weight of FPs = [119895]1times119869 can be obtained

119895 = (120572119895 sdot 120594minus1 120573119895 sdot 120573

minus1 120594119895 sdot 120572

minus1) = (119910 119897119895 119898119895 119906119895)

119895 = 1 119869

(2)

Step 2 Transform all crisp values of KPIs 119911119894119896 into 119877119894119896 whosedomains are defined on a common scale [0 1] by applying thevector normalization method [33]

(a) For benefit type KPI 119896 119896 = 1 119870

119877119894119896 =119911119894119896

radicsum119868119894=1 1199112119894119896

(3)


Table 6 Comparative analyses of the proposed FTOPSIS

Methods Application area Attribute Alternative Separation measures Solution methods

[28] Supplier selectionin supply chains 5

5 uncertain modeled by TFNsthe linear normalization

procedure [34]Vertex method

The proposedaggregation method

FTOPSIS

[18]Decision makingmodel with fuzzy

data5


procedure [34]Procedure is proposed in [36] The fuzzy averaging

method FTOPSIS

[26] Evaluation ofhazardous waste 8 5 crisp the vector normalization

procedure [33] n-dimensional Euclidean FAHP TOPSIS

[15]Evaluation of

performances inbanking sector

13 5 crisp and uncertain vectornormalization procedure [33] n-dimensional Euclidean FAHP (handling by

[30]) TOPSIS

[17]

Assessment oftraffic police

centersperformance

430 uncertain modeled by

TrFNs the linear normalizationprocedure [34]

Vertex method The fuzzy averagingmethod FTOPSIS

[19] Illustrativeexample 2 10 uncertain modeled by TFNs Vertex method FAHP (handling by

[30]) FTOPSIS

[25] Personnel selection 114 uncertain modeled by TFNs

the linear normalizationprocedure [34]


[20] Energy planning 47 uncertain modeled by TFNs


Vertex method FAHP (handling by[30]) FTOPSIS

[27]

Organizationalstrategy

development indistributionchannel

management

45 uncertain modeled by TFNs

a proposed normalizationprocedure [27]

A proposed procedure [27] FAHP (handling by[30]) FTOPSIS

[21] Ranking of qualitygoals 8


procedure [34]Vertex method FAHP (handling by

[30]) FTOPSIS

[23] Supplier selectionof medical devices 5

7 crisp linear normalizationprocedure uncertain modeled

by TrFNs the linearnormalization procedure [25]

Procedure is proposed in [36]Proposed methodproposed in [28]

FTOPSIS

[24] Ranking ofresilience factors 6



[29]) FTOPSIS

Theproposedmodel

Ranking of banks 5 12 crisp the vectornormalization procedure [24]

Vertex method Procedure isproposed in [36] The extension

of method based on theHamming distance [34]

The extension of method basedon the Euclidean distance [34]

FAHP (handling by[29]) FTOPSIS thesensitively analyses

(b) For cost type KPI 119896 119896 = 1 119870

119877119894119896 =1119911119894119896

radicsum119868119894=1 (1119911119894119896)

2 (4)

Step 3 Calculate the normalized values of FPs 119903119894119895 119894 =

1 119868 119895 = 1 119869

119903119894119895 =

119870119895

sum

119896=1

119877119894119896 119894 = 1 119868 119896 = 1 119870 119895 = 1 119869 (5)


Step 4 Construct the weighted normalized fuzzy decisionmatrix

[V119894119895]119868times119869 (6)

where

V119894119895 = 119895 sdot 119903119894119895

V119894119895 = (119910 119897119894119895 119898119894119895 119906119894119895) 119895 = 1 119869 119894 = 1 119868

(7)

Step 5 Identify FPIS V+119895 and FNIS Vminus119895 with respect to fuzzydecision matrix as

V+119895 = (V+1 V

+119895 V

+119869 )

Vminus119895 = (Vminus1 V

minus119895 V

minus119869 )

(8)

where

V+119895 = max119894=1119868

V119894119895

V+119895 = (119910 119897+119895 119898+119895 sdot 119906+119895 )

Vminus119895 = min119894=1119868

V119894119895 119894 = 1 119868 119895 = 1 119869

Vminus119895 = (119910 119897minus119895 119898minus119895 sdot 119906minus119895 )

(9)

The values of V+119895 Vminus119895 119895 = 1 119869 are given by using method

developed in [41 42]

Step 6 Calculate separation measures the distance of eachtreated bank from V+119895 and Vminus119895 and 119889

+119894 and 119889

minus119894 respectively

Separation Measure Based on Vertex Method [22] Considerthe following

119889+119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

+119895 )2+ (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2]

119889minus119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

minus119895 )2+ (119898119894119895 minus 119898

minus119895 )2+ (119906119894119895 minus 119906

minus119895 )2]

(10)

Separation Measure Based on [43] Consider the following

119889+119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898+119895 minus 119898119894119895) sdot [(119897

+119895 minus 119897119894119895) + (119906

+119895 minus 119906119894119895)]

119889minus119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898119894119895 minus 119898

minus119895 ) sdot [(119897119894119895 minus 119897

minus119895 ) + (119906119894119895 minus 119906

minus119895 )]

(11)

Separation Measure Based on the Hamming Distance [44]Consider the following

119889+119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)]

119889minus119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)]

(12)

Separation Measure Based on the Euclidean Distance [44]Consider the following

119889+119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)2]

119889minus119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)2]

(13)


Step 7 Calculate a closeness coefficient for each bank 119888119894 119894 =1 119868

119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)

Step 8 Using closeness coefficient banks can be ranked indecreasing order

5 Numerical Example of Bank Evaluation

Regardless of theworld financial crisis and complex problemsin home economy banking system in Serbia may be consid-ered as stable A prompt recognition of business and otherchallenges of the environment and defining and maintainingcertain strategies as a successful answer to those challenges

improved the banking sector performances up to the level ofsustainable development in the last decade

Using analytical approach to consider market positionand business results in the period 1102010ndash31102015 12banks were found to show leading position as far as sectorstructure is concerned Only those banks having 80 shareof the Serbian market are taken into consideration in thispaper Their business results show a significant influence onthe overall financial results at the sector level

Common KPIs (presented in Table 1) from financialreports from National bank of Serbia have been analyzed inthe period of 1102010ndash31102015

In compliancewith the proposed algorithm the followingsteps are presented The fuzzy pairwise matrix of the relativeimportance of the considered FPs is stated

[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)

By using procedure for handling of FAHP defined in [43] thevectors weights of FP groups are

1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)

The values of identified KPIs are presented in Table 2 asmean values of KPIs in the period of 1102010ndash31102015from evidence of National bank of Serbia

Applying procedures defined in Algorithm (Step 2 andStep 3) normalized decision matrix is obtained as presentedin Table 3

Applying procedures defined in Algorithm (Step 4 andStep 5) fuzzy weighted normalized decision matrix is con-structed as presented in Table 4

The following results are obtained applying proceduresshown in Step 6 to Step 8 of the developed model and givenin Table 5

The results of the sensitivity analyses are showed inTable 5 The resolving closeness coefficient values are usedin order to verify the proposed model Considering thecalculated closeness coefficient values it is observed thatvarious resolving closeness coefficient values almost do notaffect the ranking order of the treated banks Also rankingorder of banks slightly changes with separation measureFinally when separation measures based on vertex methodare used [22] as well as separation measures based on theHamming distance [44] bank (119894 = 3) is ranked at the firstplace When other two separation measures are employedbank (119894 = 11) is ranked at the first place With respect toonly FPs there is only one most appropriate bank Accordingto obtained result it can be concluded that it is necessary toanalyze nonfinancial performances in order to determine therank of banks

With respect to bank rank main manager and financialmanager of each firm may choose bank (119894 = 3) or (119894 = 11)which presents partner of enterprise

The next step of any investigations could be the compar-ison of the proposed model to other similar FTOPSIS whichcan be found in the literature The presented FTOPSIS meth-ods are very different compared with each other researchdomains the number of attributes and alternatives deter-mining ofweights of attribute alternative values determining


separationmeasures and so forthThese main differences forcomparative analyses are presented in Table 6

Basic difference between methods which can be found inliterature and the proposed model may be stated as follows(1) FPIS and FNIS are determined by analogy conventionalTOPSIS and (2) proposed method is verified by usingdifferent separation measures

6 Conclusions

Changes that occur in the business environment in financialmarkets especially globalization increase demandon contin-uous improvement of bank effectiveness It can be achievedby applying performance measurement and managing themEmployment of analytic methods may be a good choicewhen solution of some issue needs to be found in the exactway because it is less burdened by subjective judgments ofdecisionmakersThe goal of presented paper is to introduce afuzzymodel for ranking of banksThis issue is very importantand may be illustrated by the following facts (1) appropriateimprovement strategy which has key impact on achievementand maintaining of organizational sustainability and marketposition over time and (2) stakeholders where the businessprocesses are advanced

The AHP framework is used for the rating of relativeimportance of FP It can be assumed that decision makershave specific preferences and demand prerequisites in rela-tion to profile of ideal solution It is close to human wayof thinking that decision makers used predefined linguisticexpressions which are modeled in this paper by TFNsHandling of fuzziness and uncertainties is based on using ofgeometric mean that enables making full advantage of thefuzzy sets theory In the last part of this paper the proposedFTOPSIS method is used for ranking of banks in terms oftheir FPs

The method could be very useful for management ofbanks to improve their management processes and organi-zational structure market position and so forth the mainconstraint of the proposed model is that it only deals withfinancial performances although the illustrative example hasshown that nonfinancial performances should be taken intoaccount when banks need to be ranked The flexibility of themodel is reflected by the fact that all changes such as numberand type of performances and their importance can beeasily incorporated into it The proposed fuzzy model is alsosuitable for software development Also it can be mentionedthat proposed model can be easily extended to analyze othermanagement decision problems in the different areas

Further research will be focused on (1) extending ofproposed model with nonfinancial performances (2) devel-oping software based on proposed model and (3) developingand applying fuzzy version of other MCDM methods asELECTREE and PROMETHEE and comparison of resultsobtained in this paper with the ones from other proposedfuzzy MCDMmethods

Competing Interests

The authors declare that they have no competing interests

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

[2] Y Wooluru D R Swamy and P Nagesh ldquoThe processcapability analysismdasha tool for process performance measuresand metricsmdasha case studyrdquo International Journal for QualityResearch vol 8 no 3 pp 399ndash416 2014

[3] Y Dong M Firth W Hou and W Yang ldquoEvaluating the per-formance of Chinese commercial banks a comparative analysisof different types of banksrdquo European Journal of OperationalResearch vol 252 no 1 pp 280ndash295 2016

[4] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[5] H-J Zimmermann Fuzzy Set Theory and its ApplicationsKluwer Nijhoff Boston Mass USA 2001

[6] G J Klir and T A Folger Fuzzy Sets Uncertainty andInformation Prentice Hall Upper Saddle River NJ USA 1988

[7] W Pedrycz and S M Chen Granular Computing and Decision-Making Interactive and Iterative Approaches Springer Heidel-berg Germany 2015

[8] C Zopounidis and M Doumpos ldquoMulti-criteria decision aidin financial decision making methodologies and literaturereviewrdquo Journal of Multi-Criteria Decision Analysis vol 11 no4-5 pp 167ndash186 2002

[9] T L Saaty ldquoHow to make a decision the analytic hierarchyprocessrdquo European Journal Operational Research vol 48 pp 9ndash26 1990

[10] C L Hwang and K YoonMultiple Attribute Decision Making-Methods and Applications vol 186 Springer Heidelberg Ger-many 1981

[11] C Domanski and J Kondrasiuk ldquoAnalytic Hierarchy processmdashapplications in bankingrdquo Mathematics and Statistic vol 6 pp435ndash445 2005

[12] S Bhattarai and S R Yadav ldquoAHP application in bankingunfolding utility and in a situation of financial crisisrdquo inProceedings of the 10th International Symposium on AnalyticHierarchy Process (ISAHP rsquo09) University of Pittsburgh 2009

[13] MYurdakul andY T Ic ldquoAHPapproach in the credit evaluationof the manufacturing firms in Turkeyrdquo International Journal ofProduction Economics vol 88 no 3 pp 269ndash289 2004

[14] C K Kwong and H Bai ldquoDetermining the importance weightsfor the customer requirements in QFD using a fuzzy AHF withan extent analysis approachrdquo IIE Transactions vol 35 no 7 pp619ndash626 2003

[15] N Y Secme A Bayrakdaroglu and C Kahraman ldquoFuzzyperformance evaluation in Turkish banking sector using ana-lytic hierarchy process and TOPSISrdquo Expert Systems withApplications vol 36 no 9 pp 11699ndash11709 2009

[16] D Chatterjee and B A Mukherjee ldquoStudy of the application offuzzy analytical hierarchical process (FAHP) in the ranking ofIndian banksrdquo International Journal of Engineering Science andTechnology vol 7 no 7 pp 2511ndash2520 2010

[17] S Sadi-Nezhad and K K Damghani ldquoApplication of a fuzzyTOPSIS method base on modified preference ratio and fuzzydistance measurement in assessment of traffic police centersperformancerdquoApplied Soft Computing Journal vol 10 no 4 pp1028ndash1039 2010


[18] I Mahdavi NMahdavi-Amiri A Heidarzade and R NourifarldquoDesigning a model of fuzzy TOPSIS in multiple criteriadecision makingrdquo Applied Mathematics and Computation vol206 no 2 pp 607ndash617 2008

[19] F Torfi R Z Farahani and S Rezapour ldquoFuzzy AHP todetermine the relative weights of evaluation criteria and FuzzyTOPSIS to rank the alternativesrdquo Applied Soft ComputingJournal vol 10 no 2 pp 520ndash528 2010

[20] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

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

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

[23] D Tadic M Stefanovic and A Aleksic ldquoThe evaluation andranking of medical device suppliers by using fuzzy topsismethodologyrdquo Journal of Intelligent amp Fuzzy Systems vol 27 no4 pp 2091ndash2101 2014

[24] D Tadic A AleksicM Stefanovic and S Arsovski ldquoEvaluationand ranking of organizational resilience factors by using a twostep fuzzy AHP and fuzzy TOPSISrdquo Mathematical Problems inEngineering vol 2014 Article ID 418085 13 pages 2014

[25] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[26] A T Gumus ldquoEvaluation of hazardous waste transportationfirms by using a two step fuzzy-AHP and TOPSIS methodol-ogyrdquo Expert Systems with Applications vol 36 no 2 pp 4067ndash4074 2009

[27] T Paksoy N Y Pehlivan and C Kahraman ldquoOrganizationalstrategy development in distribution channel managementusing fuzzy AHP and hierarchical fuzzy TOPSISrdquo ExpertSystems with Applications vol 39 no 3 pp 2822ndash2841 2012

[28] C-T Chen C-T Lin and S-F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

[29] X Q Wu F Pu S H Shao and J N Fang ldquoTrapezoidal fuzzyAHP for the comprehensive evaluation of highway networkprogramming shemes in Yangtze River Deltardquo in Proceedings ofthe 5th World Congress on Intelligent Control and Automation(WCICA rsquo04) vol 6 pp 5232ndash5236 Hangzhou China June2004

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

[31] C Kahraman T Ertay and G Buyukozkan ldquoA fuzzy optimiza-tion model for QFD planning process using analytic networkapproachrdquo European Journal of Operational Research vol 171no 2 pp 390ndash411 2006

[32] G Buyukozkan O Feyzioglu and E Nebol ldquoSelection of thestrategic alliance partner in logistics value chainrdquo InternationalJournal of Production Economics vol 113 no 1 pp 148ndash1582008

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

[34] H-S Shih H-J Shyur and E S Lee ldquoAn extension of TOPSISfor group decision makingrdquoMathematical and Computer Mod-elling vol 45 no 7-8 pp 801ndash813 2007

[35] G Isiklar and G Buyukozkan ldquoUsing a multi-criteria decisionmaking approach to evaluate mobile phone alternativesrdquo Com-puter Standards and Interfaces vol 29 no 2 pp 265ndash274 2007

[36] M-F Chen and G-H Tzeng ldquocrdquo Mathematical and ComputerModelling vol 40 no 13 pp 1473ndash1490 2004

[37] C Chakraborty and D Chakraborty ldquoA theoretical devel-opment on a fuzzy distance measure for fuzzy numbersrdquoMathematical andComputerModelling vol 43 no 3-4 pp 254ndash261 2006

[38] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[39] G ZhengN Zhu Z Tian Y Chen and B Sun ldquoApplication of atrapezoidal fuzzy AHP method for work safety evaluation andearly warning rating of hot and humid environmentsrdquo SafetyScience vol 50 no 2 pp 228ndash239 2012

[40] S A Malik A Mushtaq K Naseem and S A Malik ldquoExam-ining the relationship among service quality customer satis-faction and behavioral responsesmdashcomparison between publicand private sector banks of Pakistanrdquo International Journal forQuality Research vol 6 no 4 pp 365ndash380 2012

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

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

[43] B Sadeghpour-Gildeh and D Gien ldquoThe distance and thecoefficient of correlation between two random variablesrdquo inProceedings of the French Meeting on Fuzzy Logic and ItsApplications vol 1 pp 97ndash102 2001

[44] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


bank performance or financial standingTheory and practiceoffer the whole set of techniques and methods of banks eval-uation As is known it is impossible to manage what cannotbe measured However measurement of bank performancesis the most important activity of bank management team

As changes in the business environment especially infinancial market are rapid and continuous relative impor-tance of FPs cannot be expressed with an exact numericalvalue It is easier for decision makers to express the sub-jectivity and imprecision of their assessments by linguisticexpressions The concept of linguistic variables is introducedby [4] it is very useful in dealing with situations which aretoo complex or not well defined to be reasonably describedin conventional quantitative expressions [5] These linguisticterms can be modeled by using the fuzzy sets theory [5 6]which allows the available information to be represented withthe granularity [7] As it is known processing of informationgranules (complex information entities) during the process ofdata abstraction and derivation of knowledge from input datais defined as granular computing [7]

The fuzzy set theory provides a strict mathematicalframework in which vague conceptual phenomena can beprecisely and rigorously studied [5] To outperform compet-ing bank institutions it requires more emphasis on internaloperational performancesThis means that it is imperative todevelop an effective way to conduct performance evaluationswhich can measure the overall organizational performanceand link it to the corporate goals

The considered problem can be stated as a multicriteriadecision making problem (MCDM) under uncertaintiesMCDM refers to finding the best opinion out of all feasiblealternatives in the presence of multiple usually conflictingdecision criteria The necessity of applying MCDM modelsin the field of financial decision making is supported by thefact that over the past decades the complexity of financialdecisions has increased rapidly [8]Themost commonly usedMCDM methods for ranking problems in various domainsare AHP [9] TOPSIS [10] and the combination of the twoMCDMmethods These two methods are used in this paper

The paper is organized as follows Section 2 presents theliterature review The basic definitions of the fuzzy set theoryand modeling of uncertainties in the relative importanceof FP groups are presented in Section 3 In Section 4 theproposed fuzzy TOPSIS approach is introduced Section 5represents the application of the recommended algorithmwith real-life data Finally Section 6 sets the conclusions

2 Literature Review

There are a large number of papers found in literature real-ized by different methods to measure of bankrsquos performancesand ranking of banksThe conventionalAnalyticHierarchicalProcess (AHP) [9] is the most widely used method Makingstrategy decisions in a bank with applying both basic andadjustedAHP applicationmodels is very significant issue [11]In that manner AHP application with specific reference tobanking could be used in the finance sector [12] In bankingsector usage of AHP application is growing especially in thesituation of global financial crisis Many authors analyzed

the bank performances using AHP with the financial andnonfinancial performance criteria [13]

The use of conventional AHPwith discrete scale is simpleand easy but it is not sufficient considering uncertaintyassociated with the mapping of onersquos perception to a number[14] Decision makers express their judgments far better byusing linguistic expressions than by representing them interms of precise numbers These linguistic expressions aremodeled by using fuzzy sets theory [5 6] In other wordsthe relative importance of criteria and the preference ofalternatives under each criterion are stated by fuzzy pairwisecomparison matrices FAHP enables mapping human per-ception to a particular number or a ratio and also consideringthe vagueness in the decision making process

There are a vast number of papers to be found in literaturewhich use fuzzy AHP (FAHP) method for evaluation andranking of banks The evaluation and rank of Turkish bankswith respect to many financial and nonfinancial performancecriteria are performed by using FAHP [15] It could beassumed that increased competition among banks and theliberalization of policies helped many institutions to take upthe banking business [16] The rank of banks can be obtainedby applying FAHP

Some researchers solve decision making problems infinancial sector especially in the field of banking by usingTOPSIS method Prioritizing different kinds of accounts inbanks objectives alternatives and criteria can be establishedfrom literature review and judgements of bankmanagersTheTOPSISfuzzy TOPSIS [17ndash19] are widely used for ranking ofentities in different research areas [15 20 21] A brief literaturereview of [22ndash27] is presented as follows

In the analyzed papers the relative importance ofattribute is assessed by decision makers that use predefinedlinguistic expressions Modeling of these linguistic expres-sions is based on the fuzzy sets theory Some authors [1718 25 28] suggest that the relative importance of attributecan be obtained by direct way In these papers determiningweight of vectors is stated as fuzzy group decision makingproblem In [17 18 25] it is assumed that decision makershave equal importance so that the aggregated weight of eachattribute can be obtained by using fuzzy averaging methodIn [22] a new aggregation procedure is developed Thisprocedure is applied for calculating of aggregated weights ofattributes in [23 28] In the rest of the analyzed papers ratingrelative importance of attributes should be based on the AHPframework It appears that the determining of the relativeimportance of attributes is more reliable when they areobtained by using pairwise comparisons than when they aredirectly obtained It is easier to make a comparison betweentwo criteria than to make an overall weight assignment Theweights of considered attributes may be calculated by usingthe different procedure In [18] the weights of treated criteriaare obtained as the average of the elements of each rowfrom the constructed fuzzy pairwise comparison matrix ofthe relative importance of attribute The geometric mean wasused in [24] for calculating relative importance of treatedattribute which is proposed in [29] In the rest of mentionedpapers the vectors weights of considered attributes are given




























119896 = 1 119896 = 2 119896 = 3 119896 = 4 119896 = 5 119896 = 6 119896 = 7 119896 = 8 119896 = 9

119894 = 1 0099 0158 0162 0187 0266 0808 0704 0567 0127119894 = 2 0075 0120 0248 0268 0398 0763 0521 0429 0070119894 = 3 0063 0093 0420 0490 0618 0342 0051 0050 0008119894 = 4 0110 01375 0140 0227 0175 0842 0769 0660 0134119894 = 5 0135 0206 0493 0609 0753 0738 0696 0401 0050119894 = 6 0095 0155 0226 0250 0370 0712 0652 0417 0076119894 = 7 0105 0152 0280 0315 0402 0938 0876 0809 0495119894 = 8 0082 0216 0181 0197 0479 0807 0734 0650 0154119894 = 9 0081 0167 0110 0117 0220 0626 0396 0655 0158119894 = 10 0133 0214 0217 0250 0350 0895 0801 0768 0283119894 = 11 0147 0654 0221 0124 0355 0549 0456 0217 0021119894 = 12 0101 0182 0256 0285 0542 0864 0749 0636 0153

119896 = 10 119896 = 11 119896 = 12 119896 = 13 119896 = 14 119896 = 15 119896 = 16 119896 = 17 119896 = 18 119896 = 19

119894 = 1 0118 0024 8796 0211 0225 2824 496 2353 1548 19119894 = 2 0244 0040 5858 0234 0164 1816 497 1498 873 21119894 = 3 0427 0066 3607 0238 0156 942 384 825 471 38119894 = 4 0087 0016 10014 0163 0202 1928 287 1490 1232 14119894 = 5 0057 0007 5308 0037 0124 979 246 690 513 28119894 = 6 0088 0015 3515 0053 0171 658 167 494 278 35119894 = 7 0064 0033 394 0144 0611 1400 403 1157 871 16119894 = 8 0076 0019 5516 0157 0237 1291 298 1052 577 24119894 = 9 0041 0011 10126 0098 0235 731 98 588 523 18119894 = 10 0096 0032 6670 0215 0368 1010 233 773 536 17119894 = 11 0161 0014 4549 0065 0096 702 167 519 418 20119894 = 12 0123 0030 5574 0157 0240 811 210 77 339 16

Table 3


119895 = 1 119895 = 2 119895 = 3 119895 = 4 119895 = 5 119895 = 6

119894 = 1 0228 0175 0196 0279 0433 0235119894 = 2 0173 0259 0166 0277 0344 0259119894 = 3 0141 0439 0388 0311 0265 0469119894 = 4 0231 0159 0207 0275 0289 0173119894 = 5 0305 0531 0151 0233 0220 0346119894 = 6 0221 0240 0167 0176 0232 0432119894 = 7 0233 0287 0361 0266 0279 0198119894 = 8 0238 0231 0216 0228 0264 0296119894 = 9 0208 0124 0249 0270 0166 0224119894 = 10 0307 0232 0272 0275 0221 0210119894 = 11 0583 0196 0148 0203 0193 0247119894 = 12 0245 0299 0210 0229 0199 0198



minus1= (11199061198951198951015840 11198981198951198951015840 11198971198951198951015840)










119895 = 1 119895 = 2 119895 = 3

119894 = 1 (0036 0068 0126) (0021 0036 0069) (0024 0049 0090)119894 = 2 (0027 0052 0096) (0031 0053 0102) (0021 0041 0076)119894 = 3 (0022 0042 0078) (0052 0090 0173) (0048 0097 0178)119894 = 4 (0037 0069 0128) (0019 0033 0063) (0026 0052 0095)119894 = 5 (0048 0091 0169) (0063 0109 0210) (0019 0038 0069)119894 = 6 (0035 0066 0122) (0028 0049 0095) (0021 0042 0077)119894 = 7 (0037 0070 0129 ) (0034 0059 0113) (0045 0090 0166)119894 = 8 (0038 0071 0132) (0027 0047 0091) (0027 0054 0099)119894 = 9 (0033 0062 0115) (0015 0025 0049) (0031 0062 0114)119894 = 10 (0049 0092 0169) (0027 0048 0092) (0034 0068 0125)119894 = 11 (0093 0175 0322) (0023 0040 0077) (0018 0037 0068)119894 = 12 (0039 0073 0135) (0035 0061 0118) (0026 0052 0097)V+119895 (0093 0175 0322) (0063 0109 0210) (0048 0097 0178)Vminus119895 (0022 0042 0078) (0015 0025 0049) (0018 0037 0068)

119895 = 4 119895 = 5 119895 = 6

119894 = 1 (0010 0018 0038) (0014 0024 0032) (0013 0030 0059)119894 = 2 (0010 0017 0037) (0011 0019 0034) (0014 0033 0065)119894 = 3 (0012 0019 0042) (0009 0015 0026) (0027 0059 0118)119894 = 4 (0010 0017 0037) (0010 0016 0028) (0010 0022 0044)119894 = 5 (0009 0015 0031) (0007 0012 0022) (0020 0044 0087)119894 = 6 (0007 0011 0024) (0008 0013 0023) (0025 0054 0109)119894 = 7 (0010 0017 0036) (0009 0016 0027) (0011 0025 0050)119894 = 8 (0008 0014 0031) (0009 0015 0026) (0017 0037 0075)119894 = 9 (0010 0017 0036) (0005 0009 0016) (0013 0028 0056)119894 = 10 (0010 0017 0037) (0007 0012 0022) (0012 0026 0053)119894 = 11 (0008 0013 0027) (0006 0011 0019) (0014 0031 0062)119894 = 12 (0008 0014 0031) (0006 0011 0019) (0011 0025 0050)V+119895 (0012 0019 0042) (0014 0024 0032) (0027 0059 0118)Vminus119895 (0007 0011 0024) (0005 0009 0016) (0010 0022 0044)















[22]

119894 = 1 0222 9

[43]

119894 = 1 0224 9119894 = 2 0211 10 119894 = 2 0205 10119894 = 3 0533 1 119894 = 3 0539 2119894 = 4 0184 11 119894 = 4 0186 11119894 = 5 0492 2 119894 = 5 0491 3119894 = 6 0271 8 119894 = 6 0269 8119894 = 7 0387 4 119894 = 7 0431 4119894 = 8 0278 6 119894 = 8 0282 6119894 = 9 0171 12 119894 = 9 0169 12119894 = 10 0349 5 119894 = 10 0344 5119894 = 11 0475 3 119894 = 11 0544 1119894 = 12 0272 7 119894 = 12 0271 7


119894 = 1 0225 9


119894 = 1 0194 10119894 = 2 0207 10 119894 = 2 0198 9119894 = 3 0532 1 119894 = 3 0423 3119894 = 4 0186 11 119894 = 4 0186 11119894 = 5 0491 2 119894 = 5 0496 2119894 = 6 0270 7-8 119894 = 6 0264 7119894 = 7 0387 4 119894 = 7 0362 4119894 = 8 0277 6 119894 = 8 0249 8119894 = 9 0171 12 119894 = 9 0181 12119894 = 10 0331 5 119894 = 10 0342 5119894 = 11 0476 3 119894 = 11 0576 1119894 = 12 0270 7-8 119894 = 12 0286 6









Order

120572119895 =[

[

119869

prod

119895=1

1198971198951198951015840]

]

1119869

120573119895 =[

[

119869

prod

119895=1

1198981198951198951015840]

]

1119869

120594119895 =[

[

119869

prod

119895=1

1199061198951198951015840]

]

1119869

119895 = 1 119869

120572 =

119869

sum

119895=1

120572119895

120573 =

119869

sum

119895=1

120573119895

120594 =

119869

sum

119895=1

120594119895

119895 = 1 119869

(1)


119895 = (120572119895 sdot 120594minus1 120573119895 sdot 120573

minus1 120594119895 sdot 120572

minus1) = (119910 119897119895 119898119895 119906119895)

119895 = 1 119869

(2)



119877119894119896 =119911119894119896

radicsum119868119894=1 1199112119894119896

(3)








FTOPSIS


data5



method FTOPSIS



[15]Evaluation of



[30]) TOPSIS

[17]


centersperformance





[30]) FTOPSIS







[27]



management







[30]) FTOPSIS





FTOPSIS




[29]) FTOPSIS

Theproposedmodel







119877119894119896 =1119911119894119896

radicsum119868119894=1 (1119911119894119896)

2 (4)


1 119868 119895 = 1 119869

119903119894119895 =

119870119895

sum

119896=1

119877119894119896 119894 = 1 119868 119896 = 1 119870 119895 = 1 119869 (5)



[V119894119895]119868times119869 (6)

where

V119894119895 = 119895 sdot 119903119894119895

V119894119895 = (119910 119897119894119895 119898119894119895 119906119894119895) 119895 = 1 119869 119894 = 1 119868

(7)


V+119895 = (V+1 V

+119895 V

+119869 )


minus119895 V

minus119869 )

(8)

where

V+119895 = max119894=1119868

V119894119895

V+119895 = (119910 119897+119895 119898+119895 sdot 119906+119895 )

Vminus119895 = min119894=1119868

V119894119895 119894 = 1 119868 119895 = 1 119869


(9)




+119894 and 119889



119889+119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

+119895 )2+ (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2]

119889minus119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

minus119895 )2+ (119898119894119895 minus 119898

minus119895 )2+ (119906119894119895 minus 119906

minus119895 )2]

(10)


119889+119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898+119895 minus 119898119894119895) sdot [(119897

+119895 minus 119897119894119895) + (119906

+119895 minus 119906119894119895)]

119889minus119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898119894119895 minus 119898

minus119895 ) sdot [(119897119894119895 minus 119897

minus119895 ) + (119906119894119895 minus 119906

minus119895 )]

(11)


119889+119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)]

119889minus119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)]

(12)


119889+119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)2]

119889minus119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)2]

(13)



119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)








[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)


1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)











6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































119896 = 1 119896 = 2 119896 = 3 119896 = 4 119896 = 5 119896 = 6 119896 = 7 119896 = 8 119896 = 9

119894 = 1 0099 0158 0162 0187 0266 0808 0704 0567 0127119894 = 2 0075 0120 0248 0268 0398 0763 0521 0429 0070119894 = 3 0063 0093 0420 0490 0618 0342 0051 0050 0008119894 = 4 0110 01375 0140 0227 0175 0842 0769 0660 0134119894 = 5 0135 0206 0493 0609 0753 0738 0696 0401 0050119894 = 6 0095 0155 0226 0250 0370 0712 0652 0417 0076119894 = 7 0105 0152 0280 0315 0402 0938 0876 0809 0495119894 = 8 0082 0216 0181 0197 0479 0807 0734 0650 0154119894 = 9 0081 0167 0110 0117 0220 0626 0396 0655 0158119894 = 10 0133 0214 0217 0250 0350 0895 0801 0768 0283119894 = 11 0147 0654 0221 0124 0355 0549 0456 0217 0021119894 = 12 0101 0182 0256 0285 0542 0864 0749 0636 0153

119896 = 10 119896 = 11 119896 = 12 119896 = 13 119896 = 14 119896 = 15 119896 = 16 119896 = 17 119896 = 18 119896 = 19

119894 = 1 0118 0024 8796 0211 0225 2824 496 2353 1548 19119894 = 2 0244 0040 5858 0234 0164 1816 497 1498 873 21119894 = 3 0427 0066 3607 0238 0156 942 384 825 471 38119894 = 4 0087 0016 10014 0163 0202 1928 287 1490 1232 14119894 = 5 0057 0007 5308 0037 0124 979 246 690 513 28119894 = 6 0088 0015 3515 0053 0171 658 167 494 278 35119894 = 7 0064 0033 394 0144 0611 1400 403 1157 871 16119894 = 8 0076 0019 5516 0157 0237 1291 298 1052 577 24119894 = 9 0041 0011 10126 0098 0235 731 98 588 523 18119894 = 10 0096 0032 6670 0215 0368 1010 233 773 536 17119894 = 11 0161 0014 4549 0065 0096 702 167 519 418 20119894 = 12 0123 0030 5574 0157 0240 811 210 77 339 16

Table 3


119895 = 1 119895 = 2 119895 = 3 119895 = 4 119895 = 5 119895 = 6

119894 = 1 0228 0175 0196 0279 0433 0235119894 = 2 0173 0259 0166 0277 0344 0259119894 = 3 0141 0439 0388 0311 0265 0469119894 = 4 0231 0159 0207 0275 0289 0173119894 = 5 0305 0531 0151 0233 0220 0346119894 = 6 0221 0240 0167 0176 0232 0432119894 = 7 0233 0287 0361 0266 0279 0198119894 = 8 0238 0231 0216 0228 0264 0296119894 = 9 0208 0124 0249 0270 0166 0224119894 = 10 0307 0232 0272 0275 0221 0210119894 = 11 0583 0196 0148 0203 0193 0247119894 = 12 0245 0299 0210 0229 0199 0198



minus1= (11199061198951198951015840 11198981198951198951015840 11198971198951198951015840)










119895 = 1 119895 = 2 119895 = 3

119894 = 1 (0036 0068 0126) (0021 0036 0069) (0024 0049 0090)119894 = 2 (0027 0052 0096) (0031 0053 0102) (0021 0041 0076)119894 = 3 (0022 0042 0078) (0052 0090 0173) (0048 0097 0178)119894 = 4 (0037 0069 0128) (0019 0033 0063) (0026 0052 0095)119894 = 5 (0048 0091 0169) (0063 0109 0210) (0019 0038 0069)119894 = 6 (0035 0066 0122) (0028 0049 0095) (0021 0042 0077)119894 = 7 (0037 0070 0129 ) (0034 0059 0113) (0045 0090 0166)119894 = 8 (0038 0071 0132) (0027 0047 0091) (0027 0054 0099)119894 = 9 (0033 0062 0115) (0015 0025 0049) (0031 0062 0114)119894 = 10 (0049 0092 0169) (0027 0048 0092) (0034 0068 0125)119894 = 11 (0093 0175 0322) (0023 0040 0077) (0018 0037 0068)119894 = 12 (0039 0073 0135) (0035 0061 0118) (0026 0052 0097)V+119895 (0093 0175 0322) (0063 0109 0210) (0048 0097 0178)Vminus119895 (0022 0042 0078) (0015 0025 0049) (0018 0037 0068)

119895 = 4 119895 = 5 119895 = 6

119894 = 1 (0010 0018 0038) (0014 0024 0032) (0013 0030 0059)119894 = 2 (0010 0017 0037) (0011 0019 0034) (0014 0033 0065)119894 = 3 (0012 0019 0042) (0009 0015 0026) (0027 0059 0118)119894 = 4 (0010 0017 0037) (0010 0016 0028) (0010 0022 0044)119894 = 5 (0009 0015 0031) (0007 0012 0022) (0020 0044 0087)119894 = 6 (0007 0011 0024) (0008 0013 0023) (0025 0054 0109)119894 = 7 (0010 0017 0036) (0009 0016 0027) (0011 0025 0050)119894 = 8 (0008 0014 0031) (0009 0015 0026) (0017 0037 0075)119894 = 9 (0010 0017 0036) (0005 0009 0016) (0013 0028 0056)119894 = 10 (0010 0017 0037) (0007 0012 0022) (0012 0026 0053)119894 = 11 (0008 0013 0027) (0006 0011 0019) (0014 0031 0062)119894 = 12 (0008 0014 0031) (0006 0011 0019) (0011 0025 0050)V+119895 (0012 0019 0042) (0014 0024 0032) (0027 0059 0118)Vminus119895 (0007 0011 0024) (0005 0009 0016) (0010 0022 0044)















[22]

119894 = 1 0222 9

[43]

119894 = 1 0224 9119894 = 2 0211 10 119894 = 2 0205 10119894 = 3 0533 1 119894 = 3 0539 2119894 = 4 0184 11 119894 = 4 0186 11119894 = 5 0492 2 119894 = 5 0491 3119894 = 6 0271 8 119894 = 6 0269 8119894 = 7 0387 4 119894 = 7 0431 4119894 = 8 0278 6 119894 = 8 0282 6119894 = 9 0171 12 119894 = 9 0169 12119894 = 10 0349 5 119894 = 10 0344 5119894 = 11 0475 3 119894 = 11 0544 1119894 = 12 0272 7 119894 = 12 0271 7


119894 = 1 0225 9


119894 = 1 0194 10119894 = 2 0207 10 119894 = 2 0198 9119894 = 3 0532 1 119894 = 3 0423 3119894 = 4 0186 11 119894 = 4 0186 11119894 = 5 0491 2 119894 = 5 0496 2119894 = 6 0270 7-8 119894 = 6 0264 7119894 = 7 0387 4 119894 = 7 0362 4119894 = 8 0277 6 119894 = 8 0249 8119894 = 9 0171 12 119894 = 9 0181 12119894 = 10 0331 5 119894 = 10 0342 5119894 = 11 0476 3 119894 = 11 0576 1119894 = 12 0270 7-8 119894 = 12 0286 6









Order

120572119895 =[

[

119869

prod

119895=1

1198971198951198951015840]

]

1119869

120573119895 =[

[

119869

prod

119895=1

1198981198951198951015840]

]

1119869

120594119895 =[

[

119869

prod

119895=1

1199061198951198951015840]

]

1119869

119895 = 1 119869

120572 =

119869

sum

119895=1

120572119895

120573 =

119869

sum

119895=1

120573119895

120594 =

119869

sum

119895=1

120594119895

119895 = 1 119869

(1)


119895 = (120572119895 sdot 120594minus1 120573119895 sdot 120573

minus1 120594119895 sdot 120572

minus1) = (119910 119897119895 119898119895 119906119895)

119895 = 1 119869

(2)



119877119894119896 =119911119894119896

radicsum119868119894=1 1199112119894119896

(3)








FTOPSIS


data5



method FTOPSIS



[15]Evaluation of



[30]) TOPSIS

[17]


centersperformance





[30]) FTOPSIS







[27]



management







[30]) FTOPSIS





FTOPSIS




[29]) FTOPSIS

Theproposedmodel







119877119894119896 =1119911119894119896

radicsum119868119894=1 (1119911119894119896)

2 (4)


1 119868 119895 = 1 119869

119903119894119895 =

119870119895

sum

119896=1

119877119894119896 119894 = 1 119868 119896 = 1 119870 119895 = 1 119869 (5)



[V119894119895]119868times119869 (6)

where

V119894119895 = 119895 sdot 119903119894119895

V119894119895 = (119910 119897119894119895 119898119894119895 119906119894119895) 119895 = 1 119869 119894 = 1 119868

(7)


V+119895 = (V+1 V

+119895 V

+119869 )


minus119895 V

minus119869 )

(8)

where

V+119895 = max119894=1119868

V119894119895

V+119895 = (119910 119897+119895 119898+119895 sdot 119906+119895 )

Vminus119895 = min119894=1119868

V119894119895 119894 = 1 119868 119895 = 1 119869


(9)




+119894 and 119889



119889+119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

+119895 )2+ (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2]

119889minus119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

minus119895 )2+ (119898119894119895 minus 119898

minus119895 )2+ (119906119894119895 minus 119906

minus119895 )2]

(10)


119889+119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898+119895 minus 119898119894119895) sdot [(119897

+119895 minus 119897119894119895) + (119906

+119895 minus 119906119894119895)]

119889minus119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898119894119895 minus 119898

minus119895 ) sdot [(119897119894119895 minus 119897

minus119895 ) + (119906119894119895 minus 119906

minus119895 )]

(11)


119889+119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)]

119889minus119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)]

(12)


119889+119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)2]

119889minus119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)2]

(13)



119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)








[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)


1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)











6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895 = 1 119895 = 2 119895 = 3

119894 = 1 (0036 0068 0126) (0021 0036 0069) (0024 0049 0090)119894 = 2 (0027 0052 0096) (0031 0053 0102) (0021 0041 0076)119894 = 3 (0022 0042 0078) (0052 0090 0173) (0048 0097 0178)119894 = 4 (0037 0069 0128) (0019 0033 0063) (0026 0052 0095)119894 = 5 (0048 0091 0169) (0063 0109 0210) (0019 0038 0069)119894 = 6 (0035 0066 0122) (0028 0049 0095) (0021 0042 0077)119894 = 7 (0037 0070 0129 ) (0034 0059 0113) (0045 0090 0166)119894 = 8 (0038 0071 0132) (0027 0047 0091) (0027 0054 0099)119894 = 9 (0033 0062 0115) (0015 0025 0049) (0031 0062 0114)119894 = 10 (0049 0092 0169) (0027 0048 0092) (0034 0068 0125)119894 = 11 (0093 0175 0322) (0023 0040 0077) (0018 0037 0068)119894 = 12 (0039 0073 0135) (0035 0061 0118) (0026 0052 0097)V+119895 (0093 0175 0322) (0063 0109 0210) (0048 0097 0178)Vminus119895 (0022 0042 0078) (0015 0025 0049) (0018 0037 0068)

119895 = 4 119895 = 5 119895 = 6

119894 = 1 (0010 0018 0038) (0014 0024 0032) (0013 0030 0059)119894 = 2 (0010 0017 0037) (0011 0019 0034) (0014 0033 0065)119894 = 3 (0012 0019 0042) (0009 0015 0026) (0027 0059 0118)119894 = 4 (0010 0017 0037) (0010 0016 0028) (0010 0022 0044)119894 = 5 (0009 0015 0031) (0007 0012 0022) (0020 0044 0087)119894 = 6 (0007 0011 0024) (0008 0013 0023) (0025 0054 0109)119894 = 7 (0010 0017 0036) (0009 0016 0027) (0011 0025 0050)119894 = 8 (0008 0014 0031) (0009 0015 0026) (0017 0037 0075)119894 = 9 (0010 0017 0036) (0005 0009 0016) (0013 0028 0056)119894 = 10 (0010 0017 0037) (0007 0012 0022) (0012 0026 0053)119894 = 11 (0008 0013 0027) (0006 0011 0019) (0014 0031 0062)119894 = 12 (0008 0014 0031) (0006 0011 0019) (0011 0025 0050)V+119895 (0012 0019 0042) (0014 0024 0032) (0027 0059 0118)Vminus119895 (0007 0011 0024) (0005 0009 0016) (0010 0022 0044)















[22]

119894 = 1 0222 9

[43]

119894 = 1 0224 9119894 = 2 0211 10 119894 = 2 0205 10119894 = 3 0533 1 119894 = 3 0539 2119894 = 4 0184 11 119894 = 4 0186 11119894 = 5 0492 2 119894 = 5 0491 3119894 = 6 0271 8 119894 = 6 0269 8119894 = 7 0387 4 119894 = 7 0431 4119894 = 8 0278 6 119894 = 8 0282 6119894 = 9 0171 12 119894 = 9 0169 12119894 = 10 0349 5 119894 = 10 0344 5119894 = 11 0475 3 119894 = 11 0544 1119894 = 12 0272 7 119894 = 12 0271 7


119894 = 1 0225 9


119894 = 1 0194 10119894 = 2 0207 10 119894 = 2 0198 9119894 = 3 0532 1 119894 = 3 0423 3119894 = 4 0186 11 119894 = 4 0186 11119894 = 5 0491 2 119894 = 5 0496 2119894 = 6 0270 7-8 119894 = 6 0264 7119894 = 7 0387 4 119894 = 7 0362 4119894 = 8 0277 6 119894 = 8 0249 8119894 = 9 0171 12 119894 = 9 0181 12119894 = 10 0331 5 119894 = 10 0342 5119894 = 11 0476 3 119894 = 11 0576 1119894 = 12 0270 7-8 119894 = 12 0286 6









Order

120572119895 =[

[

119869

prod

119895=1

1198971198951198951015840]

]

1119869

120573119895 =[

[

119869

prod

119895=1

1198981198951198951015840]

]

1119869

120594119895 =[

[

119869

prod

119895=1

1199061198951198951015840]

]

1119869

119895 = 1 119869

120572 =

119869

sum

119895=1

120572119895

120573 =

119869

sum

119895=1

120573119895

120594 =

119869

sum

119895=1

120594119895

119895 = 1 119869

(1)


119895 = (120572119895 sdot 120594minus1 120573119895 sdot 120573

minus1 120594119895 sdot 120572

minus1) = (119910 119897119895 119898119895 119906119895)

119895 = 1 119869

(2)



119877119894119896 =119911119894119896

radicsum119868119894=1 1199112119894119896

(3)








FTOPSIS


data5



method FTOPSIS



[15]Evaluation of



[30]) TOPSIS

[17]


centersperformance





[30]) FTOPSIS







[27]



management







[30]) FTOPSIS





FTOPSIS




[29]) FTOPSIS

Theproposedmodel







119877119894119896 =1119911119894119896

radicsum119868119894=1 (1119911119894119896)

2 (4)


1 119868 119895 = 1 119869

119903119894119895 =

119870119895

sum

119896=1

119877119894119896 119894 = 1 119868 119896 = 1 119870 119895 = 1 119869 (5)



[V119894119895]119868times119869 (6)

where

V119894119895 = 119895 sdot 119903119894119895

V119894119895 = (119910 119897119894119895 119898119894119895 119906119894119895) 119895 = 1 119869 119894 = 1 119868

(7)


V+119895 = (V+1 V

+119895 V

+119869 )


minus119895 V

minus119869 )

(8)

where

V+119895 = max119894=1119868

V119894119895

V+119895 = (119910 119897+119895 119898+119895 sdot 119906+119895 )

Vminus119895 = min119894=1119868

V119894119895 119894 = 1 119868 119895 = 1 119869


(9)




+119894 and 119889



119889+119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

+119895 )2+ (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2]

119889minus119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

minus119895 )2+ (119898119894119895 minus 119898

minus119895 )2+ (119906119894119895 minus 119906

minus119895 )2]

(10)


119889+119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898+119895 minus 119898119894119895) sdot [(119897

+119895 minus 119897119894119895) + (119906

+119895 minus 119906119894119895)]

119889minus119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898119894119895 minus 119898

minus119895 ) sdot [(119897119894119895 minus 119897

minus119895 ) + (119906119894119895 minus 119906

minus119895 )]

(11)


119889+119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)]

119889minus119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)]

(12)


119889+119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)2]

119889minus119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)2]

(13)



119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)








[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)


1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)











6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[22]

119894 = 1 0222 9

[43]

119894 = 1 0224 9119894 = 2 0211 10 119894 = 2 0205 10119894 = 3 0533 1 119894 = 3 0539 2119894 = 4 0184 11 119894 = 4 0186 11119894 = 5 0492 2 119894 = 5 0491 3119894 = 6 0271 8 119894 = 6 0269 8119894 = 7 0387 4 119894 = 7 0431 4119894 = 8 0278 6 119894 = 8 0282 6119894 = 9 0171 12 119894 = 9 0169 12119894 = 10 0349 5 119894 = 10 0344 5119894 = 11 0475 3 119894 = 11 0544 1119894 = 12 0272 7 119894 = 12 0271 7


119894 = 1 0225 9


119894 = 1 0194 10119894 = 2 0207 10 119894 = 2 0198 9119894 = 3 0532 1 119894 = 3 0423 3119894 = 4 0186 11 119894 = 4 0186 11119894 = 5 0491 2 119894 = 5 0496 2119894 = 6 0270 7-8 119894 = 6 0264 7119894 = 7 0387 4 119894 = 7 0362 4119894 = 8 0277 6 119894 = 8 0249 8119894 = 9 0171 12 119894 = 9 0181 12119894 = 10 0331 5 119894 = 10 0342 5119894 = 11 0476 3 119894 = 11 0576 1119894 = 12 0270 7-8 119894 = 12 0286 6









Order

120572119895 =[

[

119869

prod

119895=1

1198971198951198951015840]

]

1119869

120573119895 =[

[

119869

prod

119895=1

1198981198951198951015840]

]

1119869

120594119895 =[

[

119869

prod

119895=1

1199061198951198951015840]

]

1119869

119895 = 1 119869

120572 =

119869

sum

119895=1

120572119895

120573 =

119869

sum

119895=1

120573119895

120594 =

119869

sum

119895=1

120594119895

119895 = 1 119869

(1)


119895 = (120572119895 sdot 120594minus1 120573119895 sdot 120573

minus1 120594119895 sdot 120572

minus1) = (119910 119897119895 119898119895 119906119895)

119895 = 1 119869

(2)



119877119894119896 =119911119894119896

radicsum119868119894=1 1199112119894119896

(3)








FTOPSIS


data5



method FTOPSIS



[15]Evaluation of



[30]) TOPSIS

[17]


centersperformance





[30]) FTOPSIS







[27]



management







[30]) FTOPSIS





FTOPSIS




[29]) FTOPSIS

Theproposedmodel







119877119894119896 =1119911119894119896

radicsum119868119894=1 (1119911119894119896)

2 (4)


1 119868 119895 = 1 119869

119903119894119895 =

119870119895

sum

119896=1

119877119894119896 119894 = 1 119868 119896 = 1 119870 119895 = 1 119869 (5)



[V119894119895]119868times119869 (6)

where

V119894119895 = 119895 sdot 119903119894119895

V119894119895 = (119910 119897119894119895 119898119894119895 119906119894119895) 119895 = 1 119869 119894 = 1 119868

(7)


V+119895 = (V+1 V

+119895 V

+119869 )


minus119895 V

minus119869 )

(8)

where

V+119895 = max119894=1119868

V119894119895

V+119895 = (119910 119897+119895 119898+119895 sdot 119906+119895 )

Vminus119895 = min119894=1119868

V119894119895 119894 = 1 119868 119895 = 1 119869


(9)




+119894 and 119889



119889+119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

+119895 )2+ (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2]

119889minus119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

minus119895 )2+ (119898119894119895 minus 119898

minus119895 )2+ (119906119894119895 minus 119906

minus119895 )2]

(10)


119889+119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898+119895 minus 119898119894119895) sdot [(119897

+119895 minus 119897119894119895) + (119906

+119895 minus 119906119894119895)]

119889minus119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898119894119895 minus 119898

minus119895 ) sdot [(119897119894119895 minus 119897

minus119895 ) + (119906119894119895 minus 119906

minus119895 )]

(11)


119889+119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)]

119889minus119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)]

(12)


119889+119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)2]

119889minus119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)2]

(13)



119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)








[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)


1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)











6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















FTOPSIS


data5



method FTOPSIS



[15]Evaluation of



[30]) TOPSIS

[17]


centersperformance





[30]) FTOPSIS







[27]



management







[30]) FTOPSIS





FTOPSIS




[29]) FTOPSIS

Theproposedmodel







119877119894119896 =1119911119894119896

radicsum119868119894=1 (1119911119894119896)

2 (4)


1 119868 119895 = 1 119869

119903119894119895 =

119870119895

sum

119896=1

119877119894119896 119894 = 1 119868 119896 = 1 119870 119895 = 1 119869 (5)



[V119894119895]119868times119869 (6)

where

V119894119895 = 119895 sdot 119903119894119895

V119894119895 = (119910 119897119894119895 119898119894119895 119906119894119895) 119895 = 1 119869 119894 = 1 119868

(7)


V+119895 = (V+1 V

+119895 V

+119869 )


minus119895 V

minus119869 )

(8)

where

V+119895 = max119894=1119868

V119894119895

V+119895 = (119910 119897+119895 119898+119895 sdot 119906+119895 )

Vminus119895 = min119894=1119868

V119894119895 119894 = 1 119868 119895 = 1 119869


(9)




+119894 and 119889



119889+119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

+119895 )2+ (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2]

119889minus119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

minus119895 )2+ (119898119894119895 minus 119898

minus119895 )2+ (119906119894119895 minus 119906

minus119895 )2]

(10)


119889+119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898+119895 minus 119898119894119895) sdot [(119897

+119895 minus 119897119894119895) + (119906

+119895 minus 119906119894119895)]

119889minus119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898119894119895 minus 119898

minus119895 ) sdot [(119897119894119895 minus 119897

minus119895 ) + (119906119894119895 minus 119906

minus119895 )]

(11)


119889+119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)]

119889minus119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)]

(12)


119889+119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)2]

119889minus119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)2]

(13)



119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)








[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)


1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)











6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[V119894119895]119868times119869 (6)

where

V119894119895 = 119895 sdot 119903119894119895

V119894119895 = (119910 119897119894119895 119898119894119895 119906119894119895) 119895 = 1 119869 119894 = 1 119868

(7)


V+119895 = (V+1 V

+119895 V

+119869 )


minus119895 V

minus119869 )

(8)

where

V+119895 = max119894=1119868

V119894119895

V+119895 = (119910 119897+119895 119898+119895 sdot 119906+119895 )

Vminus119895 = min119894=1119868

V119894119895 119894 = 1 119868 119895 = 1 119869


(9)




+119894 and 119889



119889+119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

+119895 )2+ (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2]

119889minus119894

=

119869

sum

119895=1

radic1

3sdot [(119897119894119895 minus 119897

minus119895 )2+ (119898119894119895 minus 119898

minus119895 )2+ (119906119894119895 minus 119906

minus119895 )2]

(10)


119889+119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898+119895 minus 119898119894119895) sdot [(119897

+119895 minus 119897119894119895) + (119906

+119895 minus 119906119894119895)]

119889minus119894 =

119869

sum

119895=1

radic1

6sdot (119897119894119895 minus 119897

+119895 )2+ 2 sdot (119898119894119895 minus 119898

+119895 )2+ (119906119894119895 minus 119906

+119895 )2+ (119898119894119895 minus 119898

minus119895 ) sdot [(119897119894119895 minus 119897

minus119895 ) + (119906119894119895 minus 119906

minus119895 )]

(11)


119889+119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)]

119889minus119894 =

1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)

+max (10038161003816100381610038161003816119898119894119895 minus 119898minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)]

(12)


119889+119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897+119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

+119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

+119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

+119895

10038161003816100381610038161003816)2]

119889minus119894 =

radic1

2sdot

119869

sum

119895=1

[max (10038161003816100381610038161003816119897119894119895 minus 119897minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119898119894119895 minus 119898

minus119895

10038161003816100381610038161003816)2+max (10038161003816100381610038161003816119898119894119895 minus 119898

minus119895

1003816100381610038161003816100381610038161003816100381610038161003816119906119894119895 minus 119906

minus119895

10038161003816100381610038161003816)2]

(13)



119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)








[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)


1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)











6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888119894 =119889minus119894

119889minus119894 + 119889+119894

(14)








[[[[[[[[[[[[[[[[[[

[

(119909 1 1 1) 2 1 4 5 3

1

2

(119909 1 1 1)1

2

3 4 1

1

1

2 (119909 1 1 1) 4 5 2

1

4

1

3

1

4

(119909 1 1 1) 1

1

21

5

1

4

1

5

1

1

(119909 1 1 1)1

21

3

1

1

1

2

2 2 (119909 1 1 1)

]]]]]]]]]]]]]]]]]]

]

(15)


1 = (0321 0300 0553)

2 = (0118 0205 0395)

3 = (0124 0250 0460)

4 = (0037 0063 0135)

5 = (0033 0056 0098)

6 = (0057 0126 0252)

(16)











6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












6 Conclusions





Competing Interests


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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